{"id": "adap-org/9710003", "meta": {"categories": ["adap-org", "nlin.AO"], "created": "1997-10-21", "extraction": {"body_chars": 56931, "cleaning": {"detected_repeated_margin_lines": ["505-665-6598", "FAX 505-665-2083", "Los Alamos National Laboratory", "0", "PREPRINT October 26, 2018 0"], "page_count": 48, "removed_boilerplate_lines": 395}, "method": "pypdf_no_ocr", "source_pdf_bytes": 857522, "text_chars": 57517}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9710003", "primary_category": "adap-org", "source": "arxiv", "title": "TRANSIMS traffic flow characteristics", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9710003"}, "text": "TRANSIMS traffic flow characteristics\n\nAbstract\nKnowledge of fundamental traffic flow characteristics of traffic simulation models is an essential requirement when using these models for the planning, design, and operation of transportation systems. In this paper we discuss the following: a description of how features relevant to traffic flow are currently under implementation in the TRANSIMS microsimulation, a proposition for standardized traffic flow tests for traffic simulation models, and the results of these tests for two different versions of the TRANSIMS microsimulation.\n\narXiv:adap-org/9710003v1 21 Oct 1997\nTRANSIMS traffic flow characteristics\nKai Nagel †, (505) 665-0921 phone, (505) 982-0565 fax, kai@lanl.gov email\nPaula Stretz †, (505) 665-6598 phone, (505) 665-7464 fax, stretz@lanl.gov em ail\nMartin Pieck †, (505) 665-0086 phone, (505) 665-7464 fax, martin@tsasa.lanl.g ov email\nShannon Leckey†, (505) 665-3733 phone, (505) 665-7464 fax, shannon@tsasa.la nl.gov email\nRick Donnelly †,∗, (505) 665-3733 phone, (505) 665-7464 fax,\nRick.Donnelly@worldnet.att.net email\nChristopher L. Barrett † (505) 665-0430 phone, (505) 665-7464 fax, barrett@tsasa.lan l.gov\nemail\n† Los Alamos National Laboratory, MS 997, Los Alamos NM 87545, USA\n∗ Parsons Brinckerhoff, Inc., 5801 Osuna Road NE # 220, Albuquerqu e NM 87109, USA\nPREPRINT October 26, 2018\nAbstract\nKnowledge of fundamental traffic flow characteristics of traffi c simulation\nmodels is an essential requirement when using these models f or the planning,\ndesign, and operation of transportation systems. In this pa per we discuss the\nfollowing: a description of how features relevant to traffic fl ow are currently\nunder implementation in the TRANSIMS microsimulation, a pr oposition for\nstandardized traffic flow tests for traffic simulation models, a nd the results of\nthese tests for two different versions of the TRANSIMS microsi mulation.\nkeywords: traffic simulation, traffic flow, intersections\n\nI. INTRODUCTION\nOne could probably reach agreement that the traffic flow behavior o f traffic simulation mod-\nels should be well documented. Yet, in practice, this turns out to be somewhat difficult.\nMany traffic simulation models are under continuous development, an d the traffic flow dy-\nnamics documented in a certain publication has probably been refined and extended until\nthe paper gets actually published.\nIt makes thus sense to agree on a certain set of tests for traffic fl ow dynamics which should\nalways be run and documented together with any “real” results. In this paper, we propose\nsuch a suite of traffic flow measurements. We are well aware of the f act that some of\nthe results in this paper are arguably unrefined with respect to rea lity. Yet, as we stated\nabove, we are continuously working on improvements, and this public ation represents both\na snapshot of where we currently stand and an argument for a sta ndardized traffic flow\ntest suite for simulation models. We hope that this publication will both open the way\nfor a constructive dialogue on which standardized traffic flow tests should be run for traffic\nsimulation models, and which of the features of our traffic simulation m odels may need\nimprovement.\nWhen designing a traffic microsimulation model, the first idea might be to measure all as-\npects of human driving and put them in algorithmic form into the compu ter. Unfortunately,\nsuch attempts cause many problems. The first is a data collection pr oblem, because one\ncan certainly not measure “all” aspects of human driving and is thus f aced with the double\nsided problem that the necessary data collection process is extrem ely costly and still selec-\ntive. Second, what if the macroscopic flow properties of such a mod el are clearly wrong,\nfor example producing an hourly flow rate that is much to high or to low ? Since, in such a\nmodeling approach, one does not know the connection between the many parameters of the\nmodel and the emergent properties (such as flow), one is left to ra ndom trial and error.\nFor that reason, the TRANSIMS (TRansportation ANalysis and SIM ulation System [1])\nmicrosimulation starts with a minimal approach. A minimal set of driving rules is used to\nsimulate traffic, and this set of rules is only extended when it becomes clear that a certain\nimportant aspect of traffic flow behavior cannot be included with the current rule set. Besides\nthe conceptual clarity, this also has the advantage that it is usually computationally fast –\nminimal models have few rules and thus run fast on computers. This a rgument also makes\nit clear that one wants to remain flexible with respect to refinements of the model: If certain\nrefinements are unnecessary with respect to a certain question, one would want to switch\nthem off both for conceptual clarity and for computational speed .\nThe questions that TRANSIMS is currently designed for are transp ortation planning ques-\ntions. The most important zeroth order result of a transportatio n microsimulation should\nbe the delays, since, once they occur, they dominate travel times, and also hind er discharge\nof the transportation system, thus leading to grid-lock. Delays ar e caused by congestion,\nand congestion is caused by demand being higher than capacity. This implies that the first\nthing the TRANSIMS traffic microsimulation has to get right are capac ity constraints (and\npossibly their variance). Capacity constraints are caused by a var iety of effects:\n• Undisturbed roadways such as freeways have capacity constrain ts given by the maxi-\n\nmum of the flow-density diagram.\n• Typical arterials have their capacity constraints given by traffic ligh ts.\n• In the case of unprotected turning movements (yield, stop, ramp s, unprotected left,\netc.), the capacity constraints are given as a function of the traffi c on the “interfering\nlanes”. For example, the number of vehicles making an unprotected left turn depends\non the oncoming traffic.\nBuilding a simulation which can be adjusted against all these diagrams s eems a hopeless task\ngiven the enormous amount of degrees of freedom. The TRANSIMS approach for that reason\nhas been to generate the correct behavior from a few much more basic parameters. The\ncorrect behavior with respect to the above criteria can essentially be obtained by adjusting\ntwo parameters: (i) The value of a certain asymmetric noise parame ter in the acceleration\ndetermines maximum flow on freeways and through traffic lights; (ii) t he value of the gap\nacceptance determines flow for unprotected movements.\nThe remainder of this paper will first describe the algorithms TRANSI MS uses for the most\nimportant traffic movements, and then describe the resulting flow c haracteristics.\nII. RULES\nA. Single lane uni-directional traffic\nOur traffic simulation is based on a cellular automata technique, i.e., a ro ad is composed\nof cells, and each cell can either be empty, or occupied by exactly on e vehicle [2,3], see\nFig. 1 (a). Since movement has to be from one cell to another cell, ve locities have to be\ninteger numbers between 0 and vmax, where the unit of velocity is [cells per time-step]. It\nturns out that reasonable values are [3,4]:\n• length of a box = 1 /ρ jam = 7.5 m ( ρjam = density of vehicles in a jam).\n• time step = 1 sec\n• maximum velocity = 5 boxes per time step = 5 ·7. 5 m / sec = 135km/h ≈ 85mph\nFor other conditions, such as higher or lower speed limits, this can be adapted.\nNote that this approach implies a coarse graining of the spatial and temporal resolution and\ntherefore of the velocities. A vehicle which has a speed of, say, 4 in t his model stands for a\nvehicle which has a speed anywhere between 3 . 5 ·7. 5 meters/sec ≈ 95 km/h (59 mph) and\n4. 49999 ·7. 5 meters/sec ≈ 121 km/h (75 mph).\nVehicles move only in one direction. For an arbitrary configuration (v elocity and position),\none update of the traffic system consists of two steps: a velocity u pdate step consisting of\nthree consecutive rules, and a movement step according to the re sult of the velocity update.\nThe whole update is performed simultaneously for all vehicles. The co mplete configuration\nat time step t is stored and the configuration at time step t + 1 is computed from that “old”\n\ninformation. Computationally we calculate in time step t (with the three rules) the new\nvelocity of each car and write this newly calculated velocity in the same site without moving\nthe car (velocity update). After that we move all cars according t o their newly calculated\nvelocity (movement update).\n1. (velocity update)\nFor all particles i simultaneously, do the following:\nIF ( vi ≥ gapi )\nvi :=\n{ gapi − 1 with probability pnoise if possible 1\ngapi else (close following/braking)\nELSE IF ( vi < v max )\nvi :=\n{ vi with probability pnoise\nvi + 1 else (acceleration)\nELSE (i.e. ( vi = vmax AND vi < gap i )\nvi :=\n{ vmax − 1 with probability pnoise\nvmax else (free driving)\nENDIF\n2. (movement update)\nMove all particles i to xi(t + 1) = xi(t) + vi.\nThe index i denotes the position (an integer number) of a vehicle, v (i ) its current velocity,\nv max its maximum speed, gap(i ) the number of empty cells ahead, and pnoise is a random-\nization parameter.\nThe first velocity rule represents noisy car following or braking. If t he vehicle ahead is\ntoo close, the vehicle itself attempts to adjusts its velocity such th at it would, in the next\ntime-step, reach a position just behind where the vehicle ahead is at the moment. Yet, with\nprobability pnoise, the vehicle is a bit slower than this.\nThe second velocity rule represents noisy acceleration. Essentially , the acceleration is linear\n(i.e. independent from current speed), but with probability pnoise, no acceleration happens in\nthe current time step (maybe as a result of switching gears etc.). I nstead of an acceleration\nsequence of 0 → 1 → 2 → 3 → . . . , a possible acceleration sequence can now be 0 → 0 →\n1 → 2 → 2 → 2 → 3 → . . . .\nThe last velocity rule represents free driving. Instead of remaining always at the same\nspeed, such vehicles fluctuate between vmax (with probability 1 − pnoise) and vmax − 1 (with\nprobability pnoise). Note that a vehicle which is set to vmax− 1 will go through the acceleration\nstep next time, thus in the next time step either staying at vmax − 1 with probability pnoise\nor getting back to vmax. Note that the resulting average speed of a freely driving vehicle is\nthus vmax − pnoise.\nThis microsimulation is also fairly well understood from a theoretical p erspective; see [5,6]\nfor more information.\n\nB. Lane changing for passing\nFor multi-lane traffic, the model consists of parallel single lane models with additional rules\nfor lane changing. Here we describe the two lane model which can be m odified to any kind\nof multi lane model. Lane changing is modeled by an additional update s tep, which is added\nbefore the velocity update. The new sequence of steps is present ed below. Steps two and\nthree are the same in the single lane model and they are executed se parately for each lane.\n1. Lane changing decision\n2. Velocity update\n3. Vehicle movement\nAccording to this lane changing rule set the vehicles are only moving sid eways during the\nlane changing step; forwards movement is done in the vehicle moveme nt step. One should,\nthough, look at the combined effect of the lane changing and the mov ement, and then\nvehicles will usually have moved sideways and forwards. The decision to change lane is\nimplemented as strictly parallel update, i.e. each vehicle is making its de cision based upon\nthe configuration at the beginning of the update.\n• Lane changing decision for passing\n– IF neighboring position xo(i) in other lane is vacant\n∗ THEN Calculate:\n· gap(i ) Gap Forward in Current Lane,\n· gapo(i ) Gap Forward in Other Lane,\n· gapb(i ) Gap Backward in Other Lane,\n· IF ( gap(i ) < v (i ) AND gapo(i ) > gap (i) )\n− THEN weight1 = 1\n− ELSE weight1 = 0\n· weight2 = v(i) − gapf (i)\n· weight3 = vmax(i) − gapb(i).\n∗ IF ( weight1 > weight 2 ) AND ( weight1 > weight 3 )2\n· THEN mark vehicle for lane change 3\n2Weights are used because of extensibility towards “lane cha nging for plan following”. See below.\n3In the current version, the lane change is actually still rej ected with a probability of 0.01 even\nwhen all the rules are fulfilled. This is in order to break the f ollowing artifact or variations of\nit: Assume one lane is completely occupied and one is complet ely empty. The above rule set will\nresult in these vehicles just changing back and forth betwee n the lanes—the vehicles will never get\nsmeared out across the lanes. See Ref. [7] for more details.\n\nThe rules are working in the following way (see Fig. 1 (b)): First we look at the neighboring\nposition in the target lane. If this cell is vacant, we calculate the gap forward in the current\nlane (gap), the gap forward in the target lane ( gapo), and the gap backward in the target lane\n(gapb). With these results we calculate the weight1 to weight3 described above. Finally if\nthe weight comparisons render true the car will change to the new la ne. After executing the\nlane changing decision we calculate the new velocity for all cars and mo ve them according\nto this velocity.\nFor three or more lanes, a simultaneous implementation of the lane ch anging decision can\nlead to collisions. For example, in a three-lane road two vehicles on the left and right lane\ncould decide to go to the same spot in the middle lane. From an algorithm ic point of view,\nthis is possible because the lane changing decision is based on the confi guration on time t;\nbut it is also an entirely realistic situation. 4 To avoid collision we only allow lane changes\nin a certain direction in each time step:\n• IF the time step is even\nTHEN start procedure lane changing decision to the left for cars on the middle and\nthen on the right lane\n• IF the time step is odd\nTHEN start procedure lane changing decision to the right side for cars on the middle\nand then on the left lane\nThus, left lane changes occur only on even time steps, right lane cha nges occur only on odd\ntime steps. This behavior is collision free.\nC. Lane changing for plan following\nVehicles in TRANSIMS follow route plans, i.e. they know ahead of time th e sequence of\nlinks they intend to follow. This means that, when they approach an in tersection, they need\nto get into the correct lanes in order to make the intended turn. Fo r example, a vehicle\nwhich intends, according to its route plan, to make a left turn at the next intersection needs\nto get into one of the lanes which actually allow a left turn.\nThis is achieved in TRANSIMS by supplementing the basic lane changing r ules with a bias\ntowards the intended lanes. This bias increases with increasing urge ncy, i.e. with decreasing\ndistance to the intersection. Technically, this is achieved by adding a nother weight to the\nacceptance conditions for lane changing:\n• IF (weight1 + weight4 > weight 2) AND ( weight1 + weight4 > weight 3)\nTHEN change lane\n4In a deeper sense, the problem is caused by the fact that the un derlying decision making dynamics\nhas a time scale which is smaller than the time resolution of t he simulation. The simulation thus\nmust resolve the conflict by other means.\n\nweight4 is calculated according to\nweight4 = max\n[\nd∗ − d\nvmax\n, 0\n]\nfor lane changes in the desired direction as long as the vehicle is not in o ne of the correct\nlanes, cf. Fig. 1 (c). d is the remaining distance to the intersection, d∗ is a parameter;\nboth are given in the unit of “cells”. d∗ is currently set to 70 cells, i.e. approx. 500 m\nor 1/3 of a mile, throughout the simulation. In consequence, weight4 increases from zero\nto d∗/v max = 14 during the approach to the intersection. If weight4 = 0, then it does\nnot influence lane changing decision. weight4 = 1 has the same effect as a slower vehicle\nahead on the same lane. Further increases of weight4 more and more override the security\ncriterions that the forward and the backward gap on the destinat ion lane need to be large\nenough. weight4 > v max lets the vehicle make the lane change even if only the neighboring\ncell on the destination lane is free.\nD. Unprotected turning movements\nA necessary element of traffic simulations are unprotected turning movements. By this we\nmean that that for the movement the driver intends to make, some other lanes have priority.\nExamples are stop signs, yield signs, on-ramps, unprotected left t urns.\nThe general modeling principle for this in TRANSIMS is based on a gap ac ceptance in the\ninterfering lanes, see Fig. 1 (d). Interfering lanes are the lanes wh ich have priority; for\nexample, for a stop-controlled left turn onto a major road this wou ld be all lanes coming\nfrom the left plus the leftmost lane coming from the right. In order t o accept the turn, there\nhas to be a sufficient gap on each of these lanes.\nNote that “gap divided by the velocity of the oncoming vehicle” is the o ncoming vehicle’s\ntime headway, which is the typical measure used in the Highway Capac ity Manual [8]. If\none wants a time headway on an interfering lane of at least 3 seconds , then a vehicle with\na velocity of 4 cells/second would have to be at least 12 cells away from the intersection.\nThe current TRANSIMS microsimulation uses a gap acceptance (gap between intersection\nand nearest car to the intersection which is approaching) of 3 times the oncoming vehicle’s\nvelocity, i.e. when the gap on each interfering lane is larger than or eq ual to the first vehicle\non that lane, the move is accepted. For example, if the oncoming veh icle has a speed of 3, at\nleast 9 empty cells have to be between the oncoming vehicle and the int ersection. A special\ncase is if the oncoming vehicle has the velocity zero, in which case no ga p is necessary. 5\n5The condition for the “case study” microsimulation of TRANS IMS [9,10] was that a movement\nwas accepted if, for all interfering lanes, the gap was large r than vmax. That means that for fast\noncoming traffic the acceptance was higher than in the newer ve rsion, but for low speed oncoming\ntraffic the acceptance rate was lower—with the extreme case th at no turns were possible against\noncoming traffic of speed zero.\n\nE. Signalized intersections\nIn TRANSIMS, we distinguish between signalized intersections and un signalized intersec-\ntions because they are modeled differently in TRANSIMS. In signalized intersections, the\npriorities are changing in time and regulated by signals. In unsignalized intersections, the\npriorities are fixed.\nWhen a simulated vehicle approaches a signalized intersection, the algorithm first decides if,\naccording to its current speed, it potentially wants to leave the link, i.e. its current speed (in\ncells per update) is larger than or equal to the remaining number of c ells on the link. 6 If a\nvehicle wants to leave the link, the algorithm checks the “traffic cont rol”, which determines\nif the vehicle can leave the link. If it encounters a red light, it can not leave the link and no\nfurther action is taken. If it encounters a protected (green arr ow) or caution (yellow) signal,\nthe vehicle is allowed to enter the intersection. If it encounters a pe rmitted signal (green, for\nexample permitted left turn against oncoming traffic), the vehicle ch ecks all interfering lanes\nfor the gap that is larger or equal to 3 times the oncoming vehicle’s ve locity (see Subsec. II D\nabove).\nIf the movement into the intersection is accepted, the vehicle is mov ed into an “intersection\nqueue”; there is one queue for each incoming lane. This queue models vehicle behavior\ninside an intersection. The vehicle gets a “time stamp”, before which it is not allowed to\nleave the intersection; this time stamp is representative for the du ration of the movement\nthrough the intersection. The intersection queues have finite cap acity; once they are full, no\nmore vehicles are accepted and the vehicles start to queue up on th e link. This models the\nfinite vehicle storing capacity of an intersection.\nOnce a vehicle is ready to leave the intersection, it moves to the first cell on the destination\nlink if available.7 The speed of the vehicle is not changed when it is in the intersection qu eue\nso it exits on the destination link in the first cell with the same velocity t hat it had when it\nentered the queue.\nNote that vehicles turning against interfering traffic make their dec ision to accept the turn\nwhen they enter the intersection queue, not when they leave it. This can have the eff ect\nthat a vehicle enters the intersection queue when there is no oncom ing traffic, but, because\nof other vehicles ahead of it in the same queue, cannot make its turn immediately. Yet, since\nthe turn was already accepted, it will be executed as soon as all veh icles ahead in the same\nqueue have cleared the queue and a cell on the destination link is availa ble. The turn can\noccur during oncoming traffic. So in some sense vehicles will go “throu gh” each other. Yet,\nnote that on average the result is still correct. The approach des cribed above will not let\nmore vehicles through the intersection than a gap acceptance calc ulated when leaving the\nintersection queue. The above logic was chosen for simplification pur poses since unsignalized\nintersections (see below) do not have queues and thus need to make their acceptance decisions\nwhen entering the intersection.\n6Vehicles may accelerate or slow down before they actually re ach the intersection. See below.\n7Algorithmically, it only “reserves” a cell. See below.\n\nF. Unsignalized intersections\nUnsignalized intersections in TRANSIMS have no internal queues, i.e. vehicles go right\nthrough them. 8 Also, vehicles leaving an unsignalized intersection go down the destina tion\nlink as far as prescribed by their velocity, not just into the first cell as in the signalized\nintersections. Apart from these two differences, unsignalized inte rsections are similar to\nsignalized ones.\nWhen a simulated vehicle approaches an unsignalized intersection, th e algorithm first decides\nif, according to its current speed, it potentially wants to leave the lin k, i.e. its current speed\n(in cells per update) is larger than or equal to the remaining number o f sites on the link. If a\nvehicle wants to leave the link, the algorithm checks the “traffic cont rol”, which determines\nif the vehicle can leave the link. Currently occuring traffic controls ar e: no control, yield,\nand stop.\nIf a “no control” is encountered, the vehicle is moved to its destinat ion cell without any\nfurther checks. For example, if a vehicle has a velocity of 5 cells per u pdate and 2 more\ncells to go on its link, then it attempts to go 3 cells into the destination lin k. If that cell is\nalready reserved (either by another “reservation” or by a real v ehicle), then the next closer\ncell is attempted, etc., until the algorithm either finds an empty cell or returns that the\ndestination lane is full. “No control” is usually used for the major direc tions, i.e. for the\nlanes which have priority.\nIf a “yield” is encountered, the vehicle checks the gap on all interfe ring lanes. According to\nthe same rules as above, on all interfering lanes the gap needs to be larger or equal three\ntimes the first vehicle’s speed on that lane. If the movement is accep ted, the destination cell\nis selected according to the same rules as with the “no control” case .\nIf it encounters a “stop”, the vehicle is brought to a stop. Only whe n the vehicle has a\nvelocity of zero for at least one time step on the last cell of the link, is it allowed to continue.\nIf the result of the regular velocity update indeed accelerates the vehicle,9 then it attempts\nto go through the intersection. On all interfering lanes the gap, ac cording to the same rules\nas above, needs to be larger or equal to three times the first vehic le’s speed on that lane. If\nthe movement is accepted, a vehicle coming from a stop sign will always go to the first cell\non the destination link (if empty) and will have a velocity of one.\nG. Parking locations\nIn the current TRANSIMS microsimulation, vehicular trips start and end at parking loca-\ntions. Each link in the microsimulation, except for freeway ramps, fr eeway links, and some\n8Again, technically the vehicles only reserve cells on the de stination links. The actual move\nthrough the intersection happens later and can also be postp oned if after the velocity update the\nvehicle actually does not make it to the intersection.\n9I.e. there is a probability of 1 − pnoise that the vehicle will not accelerate in the given time step.\n\n“virtual” links such as centroid connectors, has at least one parkin g location. Parking lo-\ncations thus represent the aggregated parking options on that lin k. Parking locations have\nrules about how vehicles enter and exit the simulation:\n• Each vehicle in TRANSIMS has a complete route plan, together with a s tarting time.\nAt the starting time, the vehicle is added to a queue of vehicles that w ant to leave the\nsame parking location. When the vehicle is the first one in the queue, it attempts to\nenter the link. The acceptance logic is in spirit similar to the logic of the u nsignalized\nintersections, i.e. vehicles check the available gap and make their dec ision based on\nthat. Parking accessory logic is not the focus of the current pape r, and since that logic\nmay change in TRANSIMS in the near future and we also expect no influ ence on the\nresults presented here, we omit further technical details.\n• A vehicle that has reached its destination parking location according to its plan will\nleave the microsimulation. It is simply removed from the traffic.\nH. Parallel logic\nTRANSIMS is designed to run on parallel computers, such as coupled workstations, desktop\nmulti-processors, or supercomputers. The parallelization approa ch used for the microsimu-\nlation is a geographical distribution, i.e. different geographical part s of the simulated area\nare computed on different CPUs.\nThe current TRANSIMS microsimulation has these boundaries always in the middle of links.\nThis is done in order to keep the complexity of the parallel computing lo gic as far away as\npossible from the complexity of the intersection logic.\nInformation needs to be exchanged at the boundaries several tim es per update in order to\nkeep the dynamics consistent. For example, if a vehicle changes lane s and end up close in\nfront of another one, that other one is probably forced to brake . Now, if the lane changing\nvehicle is on one CPU and the following one on another, one needs to co mmunicate the lane\nchange. This will be called “Update boundaries” in the following section .\nI. Complete scheduling\nFor a complete transportation microsimulation, we need to specify w hen movements are ac-\ncepted, and also how conflicts are resolved. For example, vehicles s imultaneously attempting\nto change lanes into the middle lane represent such a conflict. Anoth er conflict is two vehicles\nfrom two different links competing for the same site on the destinatio n link.\nThe complete update of the current TRANSIMS microsimulation is as f ollows. Assume that\nthe state at time t is the result of the last update. Let t1, t2, etc. be intermediate partial\ntime steps.\n1. Vehicles which are ready to leave intersection queues from signaliz ed intersections\nreserve cells on outgoing lanes. They only attempt to reserve the fi rst cell on the link;\n\ntheir velocity is the same as it was when they entered the intersectio n. When the cell\nis occupied (either by another “reservation” or by a vehicle), then the vehicle cannot\nleave the intersection. Note that there can be a conflict between d ifferent queues for\nthe same destination cell. The current solution in TRANSIMS is that qu eues are\nserved on a first come first served basis in some arbitrarily defined w ay, i.e. a queue\nwhich happens to be treated earlier in the microsimulation has a slightly higher chance\nof unloading its vehicles. — Result: t1 information.\n2. Vehicles change Lanes. Use information from time t1 to calculate situation at time t2.\n3. Exit from Parking. Results in t3 information.\n4. Exchange boundary information for parallel computing.\n5. Non-signalized intersections reserve sites on target lanes. Not e that there can be a\nconflict of two incoming links competing for the same destination cell. T he current\nsolution in TRANSIMS is that links are served on a first come first serv ed basis,\ni.e. a link which happens to be treated earlier in the microsimulation has a slightly\nhigher chance of unloading its vehicles. Note that this conflict only ha ppens between\nminor links. Major links never compete for the same outgoing link exce pt when there\nis a network coding error; and for the competition between major a nd minor links,\nthe major link always wins because of the interfering lanes conditions .10 Result: t4\ninformation.\n6. Calculate speeds and do movements. If a vehicle scheduled for an intersection does\nnot go through the intersection as a result of the velocity update, the reservation is\ncancelled. Vehicles which go through unsignalized intersections have p set to zero,\ni.e. if it turns out that the result of the velocity update indeed brings them into the\nintersection, they need to go to the site on the destination lane whic h was reserved\nearlier. Result: t5 = t + 1 information.\n7. Exchange boundary information and migrate vehicles for parallel computing.\nIII. ST ANDARDIZED FLOW TEST SUITE FOR SIMULA TION MODELS\nIn order to control the effect of driving rules, TRANSIMS provides controlled tests for traffic\nflow behavior. These tests are simplified situations where elements o f the microsimulation\ncan be tested in isolation. This test suite uses the standard microsim ulation code in the same\nway it is used for full-scale regional simulations, and it also uses the sa me input and output\nfacilities: The test network is currently defined via a table in an oracle data base, in the same\nformat as the Dallas/Fort Worth network is kept. Input of vehicles is, following individual\nvehicle’s plans, via parking locations, the same way vehicles enter reg ional simulations.\n10Note that the situation slightly different when the speed of th e vehicle on the major link is\nzero—see below.\n\nOutput is collected on certain parts of the network on a second-by -second basis, the same\nway it can be collected for regional microsimulations. The collected ou tput is then post-\nprocessed to obtain the aggregated results presented in this pap er.\nThe test cases we look at in this paper are the following (see also Fig. 1 (e)):\n• One-lane traffic, in order to see if car following behavior generates r easonable funda-\nmental diagrams.\n• Three-lane traffic, in order to see if the addition of passing lane chan ging behavior still\ngenerates reasonable fundamental diagrams, and in order to look at lane usage.\n• Stop sign, yield sign, and left turns against oncoming traffic, in order to see it the logic\nfor non-signalized intersections generates acceptable flow rates .\n• A signalized intersection, in order to see of we obtain reasonable flow rates, and in\norder to check lane changing behavior for plan following purposes.\nA. Measured quantities\nWe look at three minute averages of the following quantities:\n• Local Flow. Flow q is defined as usual by:\nq = N\nT [vehicles/hour ]\nN is the number of cars which pass a certain site at a time period T .\n• Local Density. Density is in principle easily defined, ρ = N/L , where N is the\nnumber of vehicles on a piece of roadway of length L. Yet, given current sensoring\ntechnology, this is not easy to achieve since one would need a sensor which counts,\nsay once a second, cars on a predefined stretch of length L of the roadway. For that\nreason, empirical papers sometimes resort to occupancy, which is the fraction of time\na given sensor has been occupied by a vehicle. Current TRANSIMS me asures density\naccording to its original definition, i.e., once a time step, we count the number of\nvehicles on a stretch of roadway of L = 5 sites = 5 × 7. 5 m = 37 . 5 m. 11 We add these\ncounts for k = 180 measurement events and then divide the resulting number by L\nand by k:\nρ = N\nk ∗ L\n11The “magical” number of L = 5 sites is equal to the maximum velocity of vmax = 5 sites/update.\nThis ensures that each vehicle is counted at least once.\n\nThe result can be scaled to convenient units, for example “vehicles p er km”.\nNote that this way of computing density averages the counts over a length of 37.5 m,\nwhich is longer than the usual sensor extensions. The effect of this should be system-\natically studied.\n• Local velocity. It is well known that one can measure velocity either analogous\nto our flow definition (local velocity) or analogous to our density defi nition (space-\naveraged velocity). Under non-stationary conditions, the measu rements give different\nresults, since, for example, the first definition never counts vehic les with velocity zero.\nLocal velocity is easier to measure in practice; the space-average d velocity is easier\nto interpret since it is equal to the travel velocity and it is also the ve locity which\nneeds to be used in the fundamental relationship between flow, den sity, and velocity,\nq = ρ ·v. Since in a simulation model, both are similarly easy to measure, we meas ure\nthe more useful travel velocity. Once a time step, we sum up the ind ividual velocities\nof all vehicles on a stretch of roadway of L = 5 sites = 5 × 7. 5 m = 37 . 5 m. We add\nthese sums for k = 180 measurement events and then divide the resulting number by\nN and by k, where N is the same number as obtained during the density measurement\nabove:\nv =\n∑ v\nk ∗ N\n• Lane usage. Lane usage of a particular lane is the number of cars on this lane divide d\nby the number of cars on all lanes. It can be computed as:\nfi = ρi\n∑ n\nj=1 ρj ·n ,\nwhere i is the lane we look at and n is the number of lanes.\nB. T est networks\nEssentially two test networks are used: a circle of 1 000 sites = 0.75 k m in various configura-\ntions, and a simple signalized intersection. Most of the test are run o n the circle networks.\nThe circle can have one or two or three lanes. In all tests, the circle is slowly loaded with\ntraffic via a parking location at x = 1 sites. Velocity, flow, and density are measured on\n486 ≤ x ≤ 490, thus generating the fundamental diagrams for one-lane, tw o-lane, and\nthree-lane traffic. Since the circle gets slowly loaded, the complete f undamental diagram is\ngenerated during one run.\nFor testing yield signs and stop signs, an incoming lane is added on the r ight side of traffic at\nx = 501 (i.e. the first cell for the incoming traffic is 501). The characte ristics of the incoming\ntraffic is measured on a measurement box on the last 5 sites of the inc oming lane. The\nincoming lane is operated at maximum flow, i.e. with as many vehicles as po ssible entering\nat its beginning. The incoming vehicles are removed at x = 900 via a parking accessory.\n\nThe result of this measurement is typically a diagram showing the flow o f incoming vehicles\non the y-axis versus the flow on the circle on the x-axis.\nFor testing left turns against oncoming traffic, an opposing lane is ad ded so that it ends at\nx = 500. The traffic control here is again a “yield” logic; the difference f rom before is that\nvehicles only traverse the interfering traffic, they do not join it.\nLast, a three-lane intersection approach is used. The left lane mak es a left turn, the middle\nlane goes straight, the right lane makes a right turn. Incoming vehic les have plans about\ntheir intended movement at the intersection and attempt to reach the corresponding lane.\nThe intersection has signals with 1 minute green phase and 1 minute re d phase. The typical\noutput from this run is the flow of vehicles which go through the inter section, and the\nnumber of vehicles which cannot make their intended turn because t hey did not reach their\nlane.\nThe results are shown in Figs. 2 to 4. Note that the HCM has the same curves for stop sign\nand for yield sign, whereas we obtain higher flows through yield signs, as should be the case.\nIV. SOME STUDY RESUL TS\nMost of the results presented here were generated with an exper imental code. The disad-\nvantage of an experimental code is that actual implementation in th e production version\nmay still introduce changes in the results due to small discrepancies . The advantage is that\nturnover (compile times, complexity of code, etc.) is much higher tha n with a production\nversion. We used that advantage to test many different rules. In t he following, we want to\npresent a small subsection of tests.\nAll results presented in this section refer to the situation of a 1-lan e minor street merging\ninto a 1-lane major street, with the intersection control being a yie ld sign. Fig. 5 (a)\nrepeats the result from above for convenience. Figs. 5 (b)–(c) s how the result of different\naverage free speeds (e.g. result of speed limits) in the simulation (sa me speed limit for\nboth streets). A high average free speed of approx. 130 km/h ( ≈ 80 mph, generated by\nvmax = 5), maybe a freeway merge, generates a flow of approx. 2000 ve h/hour/lane in the\nincoming lane when there is no traffic on the major road. From there, maximum incoming\nflow decreases continuously. Lower average free speeds of appr ox. 75 km/h (50 mph) and\n50 km/h (30 mph) generate lower maximum incoming flows and are gene rally closer to the\nHighway Capacity Manual curve. Yet, it should also be clear from the se curves that the flow\non the minor road as a function of flow on the major road is also a func tion of the speed\nlimit and not only of the gap acceptance, which is constant in all three simulations.\nAnother series of experiments shows the effect of different accep tance logics. Fig. 5 (d)\nshows, when compared to Fig. 5 (a), the difference between “acce pt when gap ≥ 3vback” vs.\n“accept when gap > 3vback”. This seems like a negligible difference in the rules; yet, the\nresults are quite different in the congested regime. Whereas in the fi rst, quite many vehicles\nare able to get into the congested major road, in the second, only f ew of them make it. The\ndifference is easiest explained by looking at a vehicle of speed zero on t he major road just in\nfront of the merge point, with space for a vehicle downstream of th e merge point. With the\nfirst rule, a vehicle at the yield sign will accept the move and move in fro nt of the vehicle\n\non the major road, in the second case, it will not. Both scenarios se em to be plausible to\nus; only systematic measurements can probably resolve which one is better for a simulation\nmodel.\nA last series of experiments shows the effect of different values for the gap acceptance.\nFigs. 5 (e) and (f) show “accept when gap > v back and gap > v max. Clearly, more vehicles\nare accepted, leading to a higher flow of turning vehicles as a functio n of the flow on the\nmajor road. Note that the flow via the yield sign is never higher than 1 800 minus the flow\non the major road. This reflects the fact that the major road can not have a higher flow\nthan 1800 veh/h/lane (free speed approx 50 mph); traffic throug h the yield sign can thus at\nmost fill the major road to capacity. This explains why the much weak er gap acceptances to\nnot produce even more difference in the regime where the major roa d is uncongested. The\nsituation is clearly different for unprotected turns across instead of into traffic, as can be\nseen for the left turns in the next section.\nV. COMP ARISON TO CASE STUDY LOGIC\nThe gap acceptance logic presented here and used in the current T RANSIMS microsimula-\ntion is different from the logic used in the “Dallas/Fort Worth Case Stu dy” [9,10]. The logic\nduring that case study was: “Accept an unprotected movement if in all interfering lanes the\ngap is larger than vmax = 5.” This means that at low flow rates on the major road, more\nturns were accepted, whereas at high flow rates on the major roa d, less turns were accepted.\nFig. 6 compares the results for the current gap-acceptance logic and the one used in the case\nstudy for the case where the major road is a 3-lane road. Note tha t the results for the turns\ninto other traffic are not that much different whereas the result for th e turns across other\ntraffic yields dramatically higher flows with the case study logic. This is d ue to the fact that\nfor turns into other traffic, there is a capacity constraint of the form that the j oint flows\nfrom the major and the incoming road cannot exceed capacity of th e major road. Such a\nconstraint obviously does not exist for turns across the major road.\nVI. SUMMAR Y AND CONCLUSION\nIn transportation simulation models for larger scale questions such as planning, the flow\ncharacteristics of the traffic dynamics are in some sense more impor tant than the microscopic\ndriving dynamics of the vehicles itself. This becomes especially true sin ce a “complete”\nrepresentation of human driving is impossible anyway, both due to kn owledge constraints\nand due to computational constraints. Yet, calibrating a traffic sim ulation model against\nall types of desired behavior (for example against all HCM curves an d values mentioned in\nthis paper) seems a hopeless task given the high degrees of freedo m.\nTRANSIMS thus attempts to generate plausible macroscopic behav ior from simplified mi-\ncroscopic rules. This paper described the more important aspects of these rules as currently\nimplemented or under implementation in TRANSIMS. Before we implemen t rules in the\nTRANSIMS production version, we usually try to run systematic stu dies with more exper-\n\nimental versions. The results of the traffic flow behavior from that study were presented.\nAlso, we showed the effects of some changes in the rules for the exa mple of a yield sign.\nFinally, some comparisons were made between the logic currently und er implementation and\nthe logic used for the Dallas/Fort Worth case study.\nOne problem with microscopic approaches is that, in spite of all possib le diligence, subtle\ndifferences between design and actual implementation can make a sig nificant difference in\nthe macroscopic outcome. For that reason, this paper should also be seen as an argument\nfor a standardized traffic flow test suite for simulation models. We pr opose that simulation\nmodels, when used for studies, should first run these tests to see how the macroscopic flow\ndynamics actually is. We think that the combination of results presen ted in Figs. 2 to 4\nare a good test set, although extensions may be necessary in the f uture (e.g. merge lanes,\nweaving, etc.). We will attempt to provide future TRANSIMS results also with updated\nversions of the results of the traffic flow tests.\n\nREFERENCES\n[1] TRANSIMS, TRansportation ANalysis and SIMulation System, Los Alamos National\nLaboratory, Los Alamos, U.S.A. See http://www-transims.tsasa.lan l.gov.\n[2] Kai Nagel. Freeway traffic, cellular automata, and some (self-or ganizing) criticality.\nIn R.A. de Groot and J. Nadrchal, editors, Physics Computing ’92 , page 419. World\nScientific, 1993.\n[3] K. Nagel and M. Schreckenberg. A cellular automaton model for freeway traffic. J. Phys.\nI France, 2:2221, 1992.\n[4] C.L. Barrett, S. Eubank, K. Nagel, J. Riordan, and M. Wolinsky. I ssues in the repre-\nsentation of traffic using multi-resolution cellular automata. Los Alam os Unclassified\nReport 95-2658, Los Alamos National Laboratory, 1995.\n[5] K. Nagel. Particle hopping models and traffic flow theory. Phys. Rev. E , 53(5):4655,\n1996.\n[6] K. Nagel. Fluid-dynamical vs. particle hopping models for traffic flo w. In D.E.Wolf,\nM.Schreckenberg, and A.Bachem, editors, Traffic and granular flow , page 41. World\nScientific, Singapore, 1996.\n[7] M. Rickert, K. Nagel, M. Schreckenberg, and A. Latour. Two lan e traffic simulations\nusing cellular automata. Physica A, 231:534, 1996.\n[8] Transportation Research Board. Highway Capacity Manual . Special Report No. 209.\nNational Research Council, Washington, D.C., 3rd edition, 1994.\n[9] R.J. Beckman et al. TRANSIMS Dallas/Fort Worth case study repo rt. Technical report,\nLos Alamos National Laboratory, TSA-Division, Los Alamos, NM 8754 5, 1997. To be\nreleased.\n[10] K. Nagel and C.L.Barrett. Using microsimulation feedback for tr ip adaptation for real-\nistic traffic in Dallas. International Journal of Modern Physics C , 8(3):505–526, 1997.\n\nFIGURES\n1 5 2\ngap\n1 vehicle with velocity 1 cell per time-step (a)\n3 2\nforward gapbackward gap\ngap\nforward gap\ngap\nSituation I\nSituation II\nbackward gap\n(b)\nWrong Lane\nTurn Lane\nWeight 4\n12 12120 1 1 1 1 1 2 131312 13 13 13 14\nNo Weight 4 added for lane changing\nWeight 4 added for lane changing (c)\ngap = 3 * velocity(oncoming vehicle) (d)\nLocationx=1\nFlow\nParking\nLane\nBox\nOpposing Lane\nMeasurement\nIncoming\nParking\nAccessory\nx=501\nx=490\nx=486\nx=900\n(e)\n\nFIG. 1. (a) Definition of gap and examples for one-lane update rules. Traffic is moving to th e\nright. The leftmost vehicle accelerates to velocity 2 with p robability 0.8 and stays at velocity 1 with\nprobability 0.2. The middle vehicle slows down to velocity 1 with probability 0.8 and to velocity 0\nwith probability 0.2. The right most vehicle accelerates to velocity 3 with probability 0.8 and\nstays at velocity 2 with probability 0.2. Velocities are in “ cells per time step”. All vehicles are\nmoved according to their velocities at a later phase of the up date. (b) Illustration of lane changing\nrules. Traffic is moving to the right; only lane changes to the l eft are considered. Situation I: The\nleftmost vehicle on the bottom lane will change to the left be cause (i) the forward gap on its own\nlane, 1, is smaller than its velocity, 3; (ii) the forward gap in the other lane, 10, is larger than\nthe gap on its own lane, 1; (iii) the forward gap is large enoug h compared to its own velocity:\nweight2 = v − gapf = 3 − 10 = − 7 < 1 = weight1; (iv) the backward gap is large enough:\nweight3 = vmax − gapb = 5 − 6 = − 1 < 1 = weight1. Situation II: The second vehicle from the\nright on the right lane will not accept a lane change because t he gap backwards on the target lane\nis not sufficient. (c) Value of weight4 when in wrong lane during the approach to the intersection.\n(d) Example of a left turn against oncoming traffic. The turn is acc epted because on all three\noncoming lanes, the gap is larger or equal to three times the fi rst oncoming vehicle’s velocity. (e)\nTest networks.\n\n0 10 20 30 40 50 60 70 80 90 100\nDensity [v/km/lane]\nFlow [v/hr/lane]\nFundamental Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 5\n0 500 1000 1500 2000 2500\nFlow [veh/hr/lane]\nVelocity [km/hr]\nFlow − Velocity Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 5\n0 10 20 30 40 50 60 70 80 90 100\nDensity [veh/km/lane]\nVelocity [km/hr]\nDensity − Velocity Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 5\n(a)\n0 5 10 15 20 25\n1200Flow T−Intersection [veh/hr/lane]\nTime [min]\nTime − Flow Diagram for traffic light controlled T−intersection\n(b)\nFIG. 2. (a) One-lane traffic: Flow vs. density, travel velocity vs. flow, a nd travel velocity vs.\ndensity. (b) Number of vehicles going through the intersection and numbe r of vehicles “off plan”\nper green phase, re-scaled to hourly flow rates per lane.\n\n0 10 20 30 40 50 60 70 80 90 100\nDensity [v/km/lane]\nFlow [v/hr/lane]\nFundamental Diagram for 3−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 5\n0 500 1000 1500 2000 2500\nFlow [veh/hr/lane]\nVelocity [km/hr]\nFlow − Velocity Diagram for 3−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 5\n0 10 20 30 40 50 60 70 80 90 100\nDensity [veh/km/lane]\nVelocity [km/hr]\nDensity − Velocity Diagram for 3−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 5\n0 500 1000 1500 2000 2500\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\nFlow [veh/hr/lane]\nPer Lane Usage [density in lane / density all lanes]\nLane Usage − Flow Diagram for 3−lane Circle simulation\nRight and Middle Lanes\n0 10 20 30 40 50 60 70 80 90 100\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\nDensity [veh/km/lane]\nPer Lane Usage [density in lane / density all lanes]\nLane Usage − Density Diagram for 3−lane Circle simulation\nRight and Middle Lanes\nFIG. 3. Three-lane circle: Flow vs. density, travel velocit y vs. flow, travel velocity vs. density,\nlane usage vs. flow, and land usage vs. density. The asymmetry in the lane usage at low densities\nis due to the fact that the parking locations start filling in v ehicles on the right lane, and they only\nmove to the left when traffic on the right lane becomes dense.\n\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Stop Sign [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 3\ngap_back = 3 * Vel\n−−− HCM\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Stop Sign [veh/hr/lane]\nFlow − Flow Diagram for 2−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 3\ngap_back = 3 * Vel\n−−− HCM\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 3\ngap_back = 3 * Vel\n−−− HCM\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 2−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 3\ngap_back = 3 * Vel\n−−− HCM\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Left Turn [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 3\ngap_back = 3 * Vel\n−− HCM\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Left Turn [veh/hr/lane]\nFlow − Flow Diagram for 2−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 3\ngap_back = 3 * Vel\n−− HCM\nFIG. 4. Flow through stop sign, yield sign, and unprotected l eft turn. Left column: one-lane\ntraffic on major road (circle). Right column: two-lane traffic o n major road (circle). Solid line:\nHighway Capacity Manual [8]. Note that for “left turn across two lanes” (bottom right) the\ninterfering volume is the sum of both lanes, i.e. twice the va lue show on the x-axis.\n\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 3\ngap_back > 3 * Vel\n−−− HCM\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 5\ngap_back > 3 * Vel\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 2\ngap_back > 3 * Vel\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 3\ngap_back >= 3 * Vel\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 3\ngap_back > Vel\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 3\ngap_back > V_MAX\nFIG. 5. Comparison between different rules for the case of a 1-l ane minor road controlled by a\nyield sign merging into a 1-lane major road. (a) Figure as shown earlier, i.e. “accept if gap > 3vback”\nand vmax = 3. (b) – (c) Effect of different maximum velocities vmax = 5 and vmax = 2. (d) Effect\nof a slightly different acceptance rule “accept if gap ≥ 3vback” ( vmax = 3). (e) – (f ) Effect of\nweaker gap acceptances “accept if gap > v back” and “accept if gap > v max” ( vmax = 3).\n\n1000 1200 1400 1600 1800 2000 2200 2400\nFlow Circle [veh/hr/lane]\nFlow Stop Sign [veh/hr/lane]\nFlow − Flow Diagram for 3−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 5\ngap_back = 3 * Vel\n1000 1200 1400 1600 1800 2000 2200 2400\nFlow Circle [veh/hr/lane]\nFlow Stop Sign [veh/hr/lane]\nFlow − Flow Diagram (TRANSIMS) for 3−lane Circle simulation\n1000 1200 1400 1600 1800 2000 2200 2400\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 3−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 5\ngap_back = 3 * Vel\n1000 1200 1400 1600 1800 2000 2200 2400\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram (TRANSIMS) for 3−lane Circle simulation\n1000 1200 1400 1600 1800 2000 2200 2400\nFlow Circle [veh/hr/lane]\nFlow Left Turn [veh/hr/lane]\nFlow − Flow Diagram for 3−lane Circle simulation\nCircle = 1000\np_noise = 0.2\nV_MAX = 5\ngap_back = 3 * Vel\n1000 1200 1400 1600 1800 2000 2200 2400\nFlow Circle [veh/hr/lane]\nFlow Left Turn [veh/hr/lane]\nFlow − Flow Diagram (TRANSIMS) for 3−lane Circle simulation\nFIG. 6. Comparison between current TRANSIMS microsimulati on gap acceptance logic and\nthe one used in the case study where the major road has three la nes. Flow through stop sign, yield\nsign, and unprotected left turn into/across one-lane traffic on major road. Left column: current\nTRANSIMS microsimulation. Right column: case study TRANSI MS microsimulation. Note that\nthe results for the turns into other traffic are not that much different whereas the result for t he\nturns across other traffic yields much higher flows with the case study logic .\n\n14 of 14\nMovement\nv=4v=2 v=3\nBefore\nAfter v=2 v=2 v=5\nRules:\nIf gap > speed, speed = speed + 1.\nIf gap < speed, speed = gap. (No collisions)\nSometimes slow down for no reason.\n\n13 of 14\nLeft Lane Change\nv=3\nGap = 3\nGap Forward = 5\nGap Backward = 5\nDesired Speed = 4\nCurrent Speed = 3\nWeight1 = 4 > 3 AND 5 > 3 = 1\nWeight2 = 3 - 5 = -2\nWeight3 = 5 - 5 = 0\nLane Change = (1 > -2) AND (1 > 0) = 1\n\n12 of 14\nStop-After Movement\nv=1\nLink 3 Link 2\nLink 1\nv=2\nv=5v=4\nv=3 v=5 v=2\n\n11 of 14\nStop--Before Movement\nv=0\nLink 3 Link 2\nLink 1\nv=3\nv=5v=5\nv=3 v=5 v=3\n\n10 of 14\nLink 4\nLink 1\nLink 2\nLink 3\n\n9 of 14\nSignal - Phase 2\nLink 3 Link 2\nLink 1\n\n8 of 14\nSignal - Phase 1\nLink 3 Link 2\nLink 1\n\n7 of 14\nRules for Traffic Signal\nProtected, Caution (green, yellow):\nProceed if intersection buffer not full.\nWait (red):\nMove as far as possible on current link,\n(gap > 0), then wait.\nPermitted:\nProceed if gap on interfering lanes >= maxV,\nintersection buffer not full.\n\n6 of 14\nRules for Yield\nProceed if:\nGap on interfering lanes >= maxV.\nDestination cell on destination link vacant.\n\n5 of 14\nRules for Stop\nProceed if:\nStopped for at least 1 time step.\nGap on interfering lanes >= maxV.\nDestination cell on destination link vacant.\n\n4 of 14\nPlan Following Rules\nLane Changes to follow plan ignored until vehicle is within the\nconsideration distance (70 cells from intersection).\nIs current lane an acceptable lane for plan following?\nYes - bias vehicle to stay in current lane (Weight 4 = -1).\nNo - bias vehicle to change lanes based on distance from\nintersection. (Weight 4 = MaxSpeed - (Distance From\nIntersection - MaxSpeed) / 13)\nExecute lane change rules modified to include plan following\nweight (Weight 4).\nWeight 1 = (Gap in current lane < desired speed AND Gap\nForward in new lane > Gap in current lane) + Weight4.\n\n3 of 14\nLane Change Rules\nProbability of 0.5 - skip lane change.\nCell at Current Position in New Lane Must be vacant.\nCalculate Gap in Current Lane, Gap Forward in New Lane, Gap\nBackward in New Lane.\nWeight 1 = Gap in current lane < desired speed AND Gap\nForward in new lane > Gap in current lane. [1,0]\nWeight 2 = Current speed - Gap Forward in new lane.\nWeight 3 = Max Speed - Gap Backward.\nChange Lanes if:\n(Weight 1 > Weight 2) AND (Weight 1 > Weight 3).\n\n2 of 14\nStop-After Movement\nLink 2Link 3\nLink 1\nv=3\nv=5\nv=1v=3\n\n1 of 14\nStop-Before Movement\nLink 2Link 3\nLink 1\nv=0\nv=4\nv=5v=3\n\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nflow blue cars [v/hr]\nflow red cars[v/hr]\nFlow blue cars unprotected left turn vs flow red cars one_lane\n\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Stop Sign [veh/hr/lane]\nFlow − Flow Diagram for 3−lane Circle simulation\nCircle = 1000\nP_Brake = 0.2\nV_MAX = 5\ngap_back = 3 * Vel\n\n0 1000 2000 3000 4000 5000 6000 7000\nflow blue cars [v/hr]\nflow red cars[v/hr]\nFlow blue cars unprotected left turn vs flow red cars three_lane\n\n/0/1\n/0/0/0/0\n/0/0/0/0\n/0/0/0/0\n/0/0/0/0\n/1/1/1/1\n/1/1/1/1\n/1/1/1/1\n/1/1/1/1\nParking\nAccessory\nCPN I CPN II\n0 10 20 30 40 50 60 70 80 90 100\nDensity [v/km/lane]\nFlow [v/hr/lane]\nFundamental Diagram for 3−lane Circle simulation\nCircle = 1000\nP_Brake = 0.2\nV_MAX = 5\n\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Stop Sign [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\nP_Brake = 0.2\nV_MAX = 3\ngap_back = 3 * Vel\n\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 3−lane Circle simulation\nCircle = 1000\nP_Brake = 0.2\nV_MAX = 5\ngap_back = 3 * Vel\n\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\nP_Brake = 0.2\nV_MAX = 3\ngap_back = 3 * Vel\n\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Yield Sign [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\nP_Brake = 0.2\nV_MAX = 3\ngap_back = 2 * Vel\n\n0 200 400 600 800 1000 1200 1400 1600 1800 2000\nFlow Circle [veh/hr/lane]\nFlow Left Turn [veh/hr/lane]\nFlow − Flow Diagram for 1−lane Circle simulation\nCircle = 1000\nP_Brake = 0.2\nV_MAX = 3\ngap_back = 3 * Vel"}
{"id": "adap-org/9711001", "meta": {"categories": ["adap-org", "nlin.AO", "q-bio"], "created": "1997-11-04", "extraction": {"body_chars": 31869, "cleaning": {"detected_repeated_margin_lines": [], "page_count": 8, "removed_boilerplate_lines": 35}, "method": "pypdf_no_ocr", "source_pdf_bytes": 350531, "text_chars": 32569}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9711001", "primary_category": "adap-org", "source": "arxiv", "title": "A condition for the genotype-phenotype mapping: Causality", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9711001"}, "text": "A condition for the genotype-phenotype mapping: Causality\n\nAbstract\nThe appropriate choice of the genotype-phenotype mapping in combination with the mutation operator is important for a successful evolutionary search process. We suggest a measure to quantify the quality of this combination by addressing the question whether the relation among distances is carried over from one space to the other. Search processes which do not destroy the neighbourhood structure are termed strongly causal. We apply the proposed measure to parameter and structure optimisation problems in order to assess the combination (mapping, mutation operator) and at the same time to be able to propose improved settings.\n\narXiv:adap-org/9711001v1 4 Nov 1997\nA condition for the genotype–phenotype mapping: Causality\nBernhard Sendhoff ∗ Martin Kreutz\nInstitut f¨ ur Neuroinformatik, Ruhr-Universit¨ at Bochum\n44780 Bochum, Germany\nWerner von Seelen\npublished in:\nT. B¨ ack (Ed.) (1997) Proceedings of the Seventh International\nConference on Genetic Algorithms (ICGA’97) , Morgan Kauff-\nman, 73-80.\nAbstract\nThe appropriate choice of the genotype →\nphenotype mapping in combination with the\nmutation operator is important for a success-\nful evolutionary search process. We suggest a\nmeasure to quantify the quality of this com-\nbination by addressing the question whether\nthe relation among distances is carried over\nfrom one space to the other. Search pro-\ncesses which do not destroy the neighbour-\nhood structure are termed strongly causal .\nWe apply the proposed measure to parameter\nand structure optimisation problems in order\nto assess the combination (mapping, muta-\ntion operator) and at the same time to be\nable to propose improved settings.\n1 Introduction\nThe optimisation process in evolutionary algorithms is\nlargely influenced by the mapping from the genotype\nspace to the phenotype space. Especially for structure\noptimisation problems a measure of the quality of the\ncombination (mapping, mutation, crossover) would be\ndesirable. In this paper we propose such a measure\nbased upon the observation that Darwinian evolution\ntakes gradual changes to the optimum, although in\nbiological evolution other phenomena like punctuated\nequilibria are also observed.\nWe demand that the search process is locally strongly\ncausal with respect to the mutation operator, that is:\nsmall variations on the genotype space due to mutation\nimply small variations in the phenotype space . This\nway the neighbourhood structure under the mapping\nG → P is conserved, see Figure 1. The distance on\n∗{bs,kreutz}@neuroinformatik.ruhr-uni-bochum.de\nthe genotype space is defined via the mutation prob-\nability. The need for a strong causal exploration of\nthe search space has been expressed before (Rechen-\nberg 1994; Lohmann 1993). However, in the following\nwe want to quantify the degree to which the setting\n(mapping, mutation operator) satisfies the causality\ncondition.\nThe distance measure and therefore the causality con-\ndition in section 2 only depends on the mutation and\nnot on the crossover operator. This does not represent\nany opinion whether one or the other is the driving\nforce in evolutionary algorithms. However, we believe\nthat the mutation operator usually is responsible for\nsmall steps in the phenotype space, hence for gradual\nchanges which we want to analyse. Furthermore, we\nassume a locally smooth fitness function and define\nconditions for the genotype → phenotype mapping for\nthis problem domain. Thus, unlike in correlation based\nanalysis, (Jones et al. 1995; Manderick et al. 1991),\nwe do not explicitly refer to a fitness landscape, in-\nstead we focus on the conservation of neighbourhood\nstructures.\nIn the next section we will propose a condition for a\nstrongly causal search process and quantify it by intro-\nducing a probabilistic interpretation of the condition.\nSection 3 presents a first application in the domain\nof parameter optimisation problems and the following\nsection is concerned with the structure optimisation of\nneural networks, where complicated genotype → phe-\nnotype mappings are commonly used.\n2 A Condition for Causality\nIn section 1 we claim that for the successful intro-\nduction of new information by mutation the mutation\noperator should preserve the neighbourhood structure\nin the corresponding evolutionary spaces. We believe\nthat strong causality is necessary in evolutionary algo-\nrithms\n\ng kg i\ng j\npi\npi\nstrong causal\nweak causal\nnon causal\nphenotype space\nmapping\ngenotype space\np\nk\np\nk\npj\npj\npj\npi\np\nkpi\np\nk\npj\nFigure 1: Examples for strongly, weakly and non\ncausal genotype–phenotype mappings under the influ-\nence of mutation. Circles denote genotypes ( gi, gj, gk),\ngj and gk are results of mutations from gi. The corre-\nsponding phenotypes ( pi, pj, pk) are shown as squares.\nThe first strongly causal mapping does not destroy the\nneighbourhood structure in genotype space, the sec-\nond weakly causal mapping, maps small mutations in\ngenotype space to large distances in phenotype space\nand vice versa. The last example shows a non-causal\nmapping.\n• to allow for controlled small steps in the pheno-\ntype space which are provoked by small steps in\nthe genotype space. Especially in the vicinity of\nan optimum we need small steps to gradually ap-\nproach the optimum.\n• for the ability of self-adaptation of any strategy\nparameters, since with the lack of strong causality\nthe information about the past is meaningless and\nadaptation is impossible.\nIn order to formulate the causality condition we have\nto define the term small variation in a mathematical\nsense. Therefore, we introduce a measure of distance\nin the genotype and phenotype space. For the mathe-\nmatical correctness we have to show that the measure\nin the respective spaces endows these spaces with a\nmetric. For distances in the genotype space we pro-\npose a “universal” measure which is based on the prob-\nability of reaching genotype gj from genotype gi. In\nthis respect it resembles definitions of distance used in\nevolutionary biology, (Schuster 1995a): “. . . the notion\nof distance in genotype space is given by the smallest\nnumber of individual mutations required for the inter-\nconversion of two genotypes . . . ” . Furthermore, this\nmeasure is general enough to be applicable to a wide\nrange of evolutionary algorithms.\nWe introduce the following notations: Genotype space\nG = {gi} and phenotype space P = {pi}. Both G and\nP can also be continuous spaces. The mapping be-\ntween the spaces is f : G ↦→ P, thus pi = f (gi). The\noperators mutation and crossover act upon the space\nG, the selection operator acts upon the fitness space\nF and therefore on P. We parameterise the mutation\noperator by a real valued vector ⃗ σ∈ I Rl.\nNow, we will introduce the definition of distance on G,\nbased on the mutation probability P (gi\n⃗ σ\n→ gj) of reach-\ning gj from gi in G via mutation which is characterised\nby ⃗ σ.\nd(gi, gj) = − log\n( 1\nPid\nP (gi\n⃗ σ\n→ gj)\n)\n(1)\nPid = P (g\n⃗ σ\n→ g) (2)\nThis definition is only sensible if we claim that P (gi\n⃗ σ\n→\ngj) < P id and that the probability not to mutate is in-\ndependent of g, which is satisfied by most evolutionary\nalgorithms1. The logarithm in eq. (1) is introduced in\norder to make the distance measure additive instead\nof multiplicative. The properties of this measure are\ndiscussed in (Sendhoff et al. 1997).\nEq. (1) allows for the comparison between different\nEAs independent of any particular metric on the geno-\ntype space, like Hamming distance or Euclidian dis-\ntance.\nNow, we can proceed with the definition of causality.\nCondition: Strong causality\n∀gi, gj, gk ∃ ⃗σ′, ε with ⃗ σ∈ Uε( ⃗σ′)\n||f (gi) − f (gj)|| < ||f (gi) − f (gk)||\n⇐ ⇒ − log\n(\nP (gi\n⃗ σ\n→gj)\nPid\n)\n< − log\n(\nP (gi\n⃗ σ\n→gk)\nPid\n)\n⇐ ⇒ P (gi\n⃗ σ\n→ gj) > P (gi\n⃗ σ\n→ gk) (3)\nThe additional condition that ⃗ σcan be drawn from\nanywhere inside a sphere with radius ε (ε can be suf-\nficiently small) around ⃗σ′ guarantees that the effect of\nmutation continuously varies with ⃗ σ. That is, besides\nthe existence of an appropriate ⃗ σ, we have to guarantee\nthat it is possible to locate. Mathematically, the space\n1 In GAs P (gi\n⃗ σ\n→ gj) < P id corresponds to a mutation\nrate p < 0.5 ( p = 0.5 leads to random initialisation) and in\nES to normally distributed mutations with zero mean.\n\nof all mutation parameters which satisfy the causal-\nity condition is not empty and additionally not of\nmeasure zero.\nWe have indicated, that our analysis is concerned with\nthe local behaviour of evolutionary search. Therefore,\ncondition (3) should not be seen as a global condition.\nThe term local is difficult to define. However, an ab-\nsolute measure of locality is not necessary since we are\ninterested in the relative performance of EAs.\nCondition (3) defines strong causality in both direc-\ntions. Small distances and variations on the phenotype\nspace imply small distances and variations in the geno-\ntype space with respect to the probability of jumping\nthis distance via mutation and vice versa. However,\nin most EAs the second direction is more important.\nThat is, small variations in the genome provoke small\nvariation in the phenotype.\nSo far we have only set up a qualitative condition for\nstrong causality. In order to compare between EAs,\nwe have to find a quantitative version of condition (3).\nWe will rephrase it in the light of a probabilistic inter-\npretation.\nAssuming the gi, gj, gk to be random variables with\nuniform distribution, both sides of condition (3) be-\ncome boolean random variables. As a shortcut, we\nintroduce the symbols A and B:\nA := ||f (gi) − f (gj)|| < ||f (gi) − f (gk)|| (4)\nB := − log\n(\nP (gi\n⃗ σ\n→gj)\nPid\n)\n< − log\n(\nP (gi\n⃗ σ\n→gk)\nPid\n)\n⇐ ⇒ B := P (gi\n⃗ σ\n→ gj) > P (gi\n⃗ σ\n→ gk) (5)\nSince we assume the distribution of gi, gj, gk to be\nknown, we can derive the probabilities P (A), P (B),\nand P (A, B). We can now, with the help of Bayes’\nlaw, recast the two directions (genotype ↔ phenotype)\nin the following way:\nProbabilistic condition : Strong causality\n∀gi, gj, gk ∃ ⃗σ′, ⃗ εwith ⃗ σ∈ Uε( ⃗σ′)\nG ⇒ P : P (A|B) = P (A, B)\nP (B) = 1 (6)\nP ⇒ G : P (B|A) = P (A, B)\nP (A) = 1 (7)\nThe value of P (A|B) serves as a quantitative measure\nfor the causality in EAs. If the neighbourhood rela-\ntions in both spaces are uncorrelated for every point,\nthen the system is weakly but not strongly causal\nP (A, B) = P (A)·P (B) and therefore P (A|B) = P (A),\nP (B|A) = P (B), thus distance relations in the pheno-\ntype space are statistically independent from distance\nrelations in the genotype space and vice versa 2. One\nexample for such systems is the class of Monte Carlo\nalgorithms where the transition probability between\nany pair of genotypes is constant. For constant tran-\nsition probabilities, B in equation (5) is constant for\nall genotype combinations and does therefore not pro-\nvide any information about the distance relation in the\nphenotype space.\nIn evolutionary molecular biology measures similar to\nthis probabilistic formulation of the causality condi-\ntion are employed in the context of the analysis of the\n“sequence–structure” mapping (Schuster 1995b).\n3 Parameter Optimisation\nFirstly, we employ one of the mainstream paradigms\nof EAs – the evolution strategy (ES) and show that\nthe ES is strongly causal in terms of our proposed\ncondition. As an example for an EA, which violates\nthe causality condition, we analyse the canonical ge-\nnetic algorithm (GA) applied to parameter optimisa-\ntion. We propose a new mutation operator for the GA\nwhich observes strong causality to a greater extent and\nshow that this also increases the performance.\n3.1 Evolution Strategy\nWe firstly focus on the transition probability. In the\ncanonical ES G = P = I Rn and the genotype → phe-\nnotype mapping is the identity f : G → P = id I Rn . It\nuses normally distributed mutation steps which are in-\ndependent of the genotype gi ∈ G . That is, the transi-\ntion gi\n⃗ σ\n→ gk is defined by adding a normally distributed\nnumber z = ( z1, . . . , zn) with zi ∼ N (0, σ2). Hence,\nthe pdf of this transition can be expressed in terms of\nz\ngi\n⃗ σ\n→gk : gj = gi + z (8)\np(gi\n⃗ σ\n→gk) = p(z = gj − gi)\n= 1\n√\n2π\nn\nσn exp\n(\n− ∥gk−gi∥2\n2σ2\n)\n(9)\npid = p(z = 0) = 1√\n2π\nn\nσn\n(10)\nInserting the transition pdf in the causality condition\n(3) with f : G → P = id I Rn results in\n∥f (gi) − f (gj)∥ < ∥f (gi) − f (gk)∥\n2Whether the system is non-causal in the sense of being\nnon-deterministic is not determined by eqs. (3,6,7), since\nwe do not observe whether the mapping from genotype to\nphenotype space itself is probabilistic or not.\n\n⇐ ⇒ ∥ gi − gj∥ < ∥gi − gk∥\n⇐ ⇒ exp\n(\n−∥gi − gj∥2)\n> exp\n(\n−∥gi − gk∥2)\n⇐ ⇒ p(gi\n⃗ σ\n→gj) > p(gi\n⃗ σ\n→gk) (11)\nwhich holds for all combinations of gi, gj, gk, σ.\nThe examination of the metric conditions of the dis-\ntance measure of an ES and some notes on the self-\nadaptation of σ are presented in (Sendhoff et al. 1997).\n3.2 Genetic Algorithms\nIn the case of genetic algorithms (GA) the genotype\nspace consists of binary strings of length L, therefore\nG = {0, 1}L. Canonical GAs mutate by changing each\nbit position from 0 → 1 and 1 → 0, respectively, with\nthe probability pm. Thus, pm corresponds to the mu-\ntation parameter σ. Let hij denote the Hamming dis-\ntance between gi and gj.\nIn order to examine the causality condition we use\nthe Euclidian metric on the phenotype space P and\nchoose the standard binary coding for the genotype–\nphenotype mapping f : G → P . Using the following\nnotations\ngi = {xi(n) | xi(n) ∈ {0, 1}} (12)\nhij =\nL−1∑\nn=0\n|xi(n) − xj (n)| (13)\nf (gi) =\nL−1∑\ni=0\nxi(n) 2n (14)\nP (gi\n⃗ σ\n→ gj) = phij\nm (1 − pm)L−hij (15)\nthe causality condition is expressed as\n⏐\n⏐\n⏐\n⏐\n⏐\nL−1∑\nn=0\n(xi(n)−xj(n)) 2n\n⏐\n⏐\n⏐\n⏐\n⏐ <\n⏐\n⏐\n⏐\n⏐\n⏐\nL−1∑\nn=0\n(xi(n)−xk(n)) 2n\n⏐\n⏐\n⏐\n⏐\n⏐\n⇐ ⇒ phij\nm (1−pm)L−hij > p hik\nm (1−pm)L−hik (16)\nAssuming pm < 0.5 the right hand side of (16) can\nbe expressed as hij < h ik. Therefore the causality\ncondition reads\n⏐\n⏐\n⏐\n⏐\n⏐\nL−1∑\nn=0\n(xi(n)−xj(n)) 2n\n⏐\n⏐\n⏐\n⏐\n⏐ <\n⏐\n⏐\n⏐\n⏐\n⏐\nL−1∑\nn=0\n(xi(n)−xk(n)) 2n\n⏐\n⏐\n⏐\n⏐\n⏐\n⇐ ⇒\nL−1∑\nn=0\n|xi(n)−xj(n)| <\nL−1∑\nn=0\n|xi(n)−xk(n)| (17)\nwhich obviously does not hold in general, not even\nlocally.\nAs a measure of the extent to which the GA satis-\nfies the causality condition we employ the probabilistic\nversion of the condition. After some extensive calcula-\ntions which are presented in (Sendhoff et al. 1997) we\nget P (A|B) ≈ 0.51 and P (B|A) ≈ 0.62. That is, the\nchance of a small mutation of a genotype resulting in a\nsmall change of the corresponding phenotype is about\n51%. The probability that a small change of a pheno-\ntype is caused by a small mutation of the genotype is\nsomewhat higher, about 62%. Thus, in the case of a\ncanonical GA, the mapping from the genotype to the\nphenotype is not strongly causal. In our opinion the\ncombination of binary coding and point mutation is\nnot well suited for continuous parameter optimisation\ncombined with locally smooth fitness functions and is\nthe reason why ES, which observes strong causality,\noutperforms the GA in most cases in this problem do-\nmain.\n3.3 A New Mutation Operator\nWe have seen that in GA the standard mutation oper-\nator together with the binary encoding does not satisfy\nthe causality condition in general. Possible solutions\nto this problem are to use a different encoding scheme,\ne.g. the Gray code 3, to change the mutation operator\nand keep the encoding scheme, and to change both.\nIn the remainder of this section we will partly out-\nline an approach, presented in detail in (Sendhoff\net al. 1996), which sticks to the concept of point mu-\ntation, but uses a position dependent mutation rate\npm = pm(i). This will provide us with an interesting\nexample of an EA, where a modification of the muta-\ntion operator enhances the causality and, as we will\nsee, also the performance.\n2 4 6 80\n0.1\n0.2\n0.3\n0.4\n0.5\ni\nP(i) σ=10\nσ=0.5\nFigure 2: Top: pm(i) for σ = 10 .0 (dashed curve\n- numerical approximation), bottom: pm(i) (rescaled\nwith a factor 8) for σ = 0.5 (dashed curve - numerical\napproximation).\nAs we have seen above, the ES is a strongly causal\noptimisation procedure. Therefore, we translate the\nconcept of mutation by adding normally distributed\n3Although we show in (Sendhoff et al. 1997) that the\nGray code does not increase the causality substantially.\n\nnumbers in ES to point mutation in GAs. Thus, we\ncalculate a probability distribution which will on aver-\nage resemble the summation of a normally distributed\nnumber. Depending on the standard deviation σ of the\nunderlying normal distribution we get different distri-\nbutions of the mutation rates pm(i), see Figure 2. For\nthe efficient use of the new mutation operator we de-\nrived a numerical approximation of pm(i; σ) which is\npresented in (Sendhoff et al. 1996).\n(a)\nSphere model\n0 5.0·105 1.0·106 1.5·106 2.0·106 2.5·106\nfunction evaluations\n10-15\n10-10\n10-5\nbest function value\n(b)\nAckley's function\n0 5.0·105 1.0·106 1.5·106 2.0·106 2.5·106\nfunction evaluations\n10-6\n10-4\n10-2\nbest function value\nFigure 3: Convergence plots of optimisation runs.\nThe solid line shows the results obtained by position-\ndependent mutations. Dashed lines the ones from the\ncanonical GA. (population size: 50; dimension: n =\n30; encoding length: 32 bits using Gray-code)\nThe numerical estimates of the causality measure are\nP (A|B) ≈ 0.73 and P (B|A) ≈ 0.74. In order to sup-\nport our hypothesis that increasing the strong causal-\nity in an optimisation process leads to an improved\nperformance we apply the modified GA to two stan-\ndard optimisation problems. Results are given for the\nsphere model and Ackley’s function, see figure 3. The\nnew GA converges faster to a better value than the\ncanonical GA. Figure 3 shows that in case of the sphere\nmodel the increase of convergence speed is of order 10 5\nand for the Ackley function of order 10 1.5.\n4 Causality in Structure Optimisation\nThe problem to choose the right genotype → pheno-\ntype mapping is of particular importance in the do-\nmain of structure optimisation. We will here concen-\ntrate on the structure optimisation of neural networks.\nWe will regard the set of all possible connection ma-\ntrices as the phenotype space, and allow the matrix to\nhave entries from {0, 1} or from {0, ..., Nsym}. Since\nthere is no measure on the space of these matrices\nwhich relates directly to the performance of the neu-\nral network without evaluating the network, we will\nuse the standard Euclidian distance measure for the\nphenotype space, ( yMj\nnk denotes the entry at (row n,\ncolumn k) of the matrix Mj)\ndE(Mi, Mj) = 1\nN 2\nN∑\nn=1\nN∑\nk=1\n|yMi\nnk − yMj\nnk | (18)\nThere is no a priori structure assumed for the network,\nhence the matrix is not constrained to any layered net-\nwork structure. If ynk > 0 there exists a connection\nbetween neuron n and neuron k. We are not restricted\nto the upper triangular part of the matrix, thus in prin-\nciple feedback connections can be specified. If the ynk\nare restricted to the values {0, 1}, we only specify the\nconnection between the neurons. If we extend the al-\nlowed values to all integers in the set {0, ..., Nsym}, it is\npossible to further define initial values for the weights\nand the thresholds. In connection with gradient de-\nscent algorithms for the fine tuning of the weights,\nthis approach has been successful, see (Sendhoff et al.\n1997), and we will therefore include it in the following\nexaminations.\nThere have been several proposals on how to organise\nthe genotype space and the mapping f : G → P for the\noptimisation of the structure of neural networks, see\nalso (Whitley 1995). Most of them can be categorised\ninto three principal approaches, the direct encoding,\nthe recursive or grammar encoding and the cellular\nencoding. The first attempts to use evolutionary (ge-\nnetic) algorithms for structure optimisation employed\nthe direct encoding method, (Miller et al. 1989), and\nit probably still is the most frequently used method.\nThe recursive encoding has been introduced by Kitano\n(1990), in order to overcome the bad scaling behaviour\nof the direct encoding method for large networks and to\nfavour a modular structure of the network. The third\napproach, the cellular encoding , proposed by Gruau\n(1993), uses a tree representation of operators which\nconstruct the network. The structure of the tree and\ntherefore of the network is optimised by genetic pro-\ngramming. In the following we will examine the direct\nencoding and the recursive encoding with respect to\nthe proposed measure, eqs. (6, 7). Therefore, we will\nexamine whether the neighbourhood structure on G\nis carried over to P; whether the system is strongly\ncausal. We will restrict ourselves to the direction,\n\nG → P and we will not sample uniformly in G. The\nreason is, that for the mutation operator p± (see eq.\n(19)) the probability to reach the genotype gj and gk\nfrom gi is zero for almost all uniformly sampled triples.\nThus, if we want to examine the system (mapping,\np±), we sample gi uniformly and obtain gj and gk via\nmutation from gi with the probability pinit. By tun-\ning pinit, we are at the same time able to determine\nhow local the three chromosomes are. We then derive\nthe probability to reach gj and gk from gi by “normal\nmutation”.\n4.1 The direct encoding method\nIn the direct encoding method the chromosome con-\nsists of the whole connection matrix. Usually all ma-\ntrix rows are concatenated to form the chromosome,\nwhose elements we want to denote by xn. The range\nof allowed values for x and y can be {0, 1} or from\nthe integer set {0, ..., Nsym}. The following operators\n(χ ∈ [0, 1[ is a uniformly sampled number) have been\nused\np±x =\n{\nx + 1 ; χ < 0.5\nx − 1 ; χ ≥ 0.5 (19)\npux = ⌊χ · (Nsym + 1)⌋ (20)\npu replaces x by a new integer with equal probability\nfrom the set {0, ..., Nsym}. We used the Euclidian mea-\nsure of distance for matrices on the phenotype space\nand a distance measure which only counts structural\ndifferences dI (Mi, Mj)\ndI (Mi, Mj) = 1\nN 2\nN∑\nn=1\nN∑\nk=1\n|Θ( yMi\nnk ) − Θ( yMj\nnk )| (21)\nΘ( x) =\n{ 0 ; x ≤ 0\n1 ; x > 0\nThe results for the probability P (A|B), that is the\nprobability that ( dE,I denotes dE or dI )\nA := dE,I (Mi, Mj) > dE,I (Mi, Mk) (22)\nholds in phenotype space, given that\nB := − log (P (gi → gj)) > − log (P (gi → gk)) (23)\nis true in genotype space, are presented in Table 1.\nThe standard setting, ( x ∈ { 0, 1}, dI, pu) is strongly\ncausal in the G → P direction. However, if the al-\nlowed values are extended to an interval of integers,\nall settings have problems at least for the structural\ndistance measure. Thus, we conclude that even direct\nencoding methods are not strongly causal straightfor-\nwardly if we depart from the basic setting.\ndE/p± dI /p± dE/pu dI /pu\nx ∈ {0, 1} – – – 0.0\nx ∈ {0, .., Nsym} 0.0 0.614 0.662 0.564\nTable 1: Numerical estimation of the probabilities\nP (A|B), using combinations of the two different dis-\ntance measures and mutation operators. The probabil-\nities have been estimated from 10 5 trials ( Nsym = 10\nand pinit = 0.25).\na 1 = 7\n2 5 7 9 1 . . .\n5 4 8 7 3 6 9 1 . . . . . .\n( )\n9 1\nS c\nL c\nFigure 4: One element is replaced by four elements\nin the recursion step via the small chromosome SC →\nlarge chromosome LC mapping.\n4.2 The recursive encoding method\nIn all encoding methods apart from the one discussed\nabove a more or less intrinsic mapping is introduced\nfrom the genotype to the phenotype space. We al-\nready argued why this is sensible and we now want to\nexamine to what extent a recursive encoding method\nis strongly causal. The coding, described in (Sendhoff\net al. 1997), consists of four chromosomes, where only\nthe first two are important for the building process\nof the connection matrix. In each iteration step ev-\nery element of the connection matrix is replaced by a\n2 × 2 matrix of new elements. The new elements are\nspecified by a mapping from the small chromosome\nSC to the large chromosome LC. The length of the\nsmall chromosome NSC is variable, the length of the\nlarge one is fixed by the condition NLC = 4 · NSC . At\neach step i the first place N (yi\nnk) of each connection\nmatrix element yi\nnk in SC is determined; for example\nposition N (y1 = 7) = 3 in Figure 4. The element is\nthen replaced by the four elements at the positions\n(\n4 · (N (yi\nnk) − 1) + 1, 4 · (N (yi\nnk) − 1) + 2,\n4 · (N (yi\nnk) − 1) + 3, 4 · (N (yi\nnk) − 1) + 4\n)\n(24)\nin the large chromosome LC . Figure 4 shows the re-\nplacement of an element y1 = 7 by the four elements\n(3, 6, 9, 1). In case yi\nnk is not in SC , it is replaced by\nfour so called terminal symbols (in the notation of in-\nteger strings, the most convenient choice is zero). A\nterminal symbol is in turn always replaced by another\nfour terminal symbols in an recursion step. Figure 5\nshows the evolution of a 8 × 8 connection matrix Mcon\n\na 1\n(\na 11\na 22\n2 . . . ( (\na 11\na 44\n3 (\na 12\na 21\n3 a 22\n. . .\n.\n.\n.\nS\nL\nc\nc\n(\na 11\n4 a 12\n4 a 18\na 88\n4 a 81\na 71\n(\n.\n.\n.\n.\n.\n.\n. . .\n. . .\n.\n.\n.\nS c L c\n. . .\nFigure 5: Scheme of the recursive development of the\nconnection matrix up to a size of 8 × 8.\nfollowing the introduced rules. This network connec-\ntion matrix is a function of the mutation and crossover\nprobabilities, the chromosome length dSC , the number\nof iteration steps Nsteps and of the size of the set of in-\ntegers {1, ..., Nsym} of allowed values for both strings.\nWe restrict ourselves to mutations on SC and exam-\n(a)\n50 5\n0.3\n0.35\n0.4\n0.451 - P(A|B)\nNsym\ndSc\n(b)\n50 10\n0.3\n0.35\n0.4\n0.451 - P(A|B)\ndSc\nNsym\nFigure 6: Since it is easier to visualise, the probabil-\nity to violate the causality condition (1 − P (A|B)) is\nshown for the (a) Euclidian distance measure dE and\n(b) structure distance measure dI . The values have\nbeen estimated from 10 5 trials ( pinit = 0.25).\nine the probability (1 − P (A|B)) as a function of dSC\nand Nsym, the results are shown in Figure 6 (a) for\nthe Euclidian distance measure dE(Mi, Mj) and in (b)\nfor the structure distance measure dI (Mi, Mj). Only\nthe results for the p± mutation operator are shown,\nbecause the values for pu are only slightly lower and\nshow a similar behaviour as in Figure 6. We note that\nthe system is generally not strongly causal, especially\nfor specific combinations ( dSC , Nsym). Furthermore,\nthe best (lowest, since causality violations are shown)\nvalues on average are reached if the encoding param-\neters dSC and Nsym differ only slightly, thus we con-\nclude dSC ≈ Nsym. We also note that the differences\nbetween dE and dI are only marginal both qualita-\ntively and quantitatively. Thus, from the point of view\nof causality the combined optimisation of the structure\nand the initial weight values seems to be sensible.\nWe will now, similar to section 3 in the domain of pa-\nrameter optimisation, try to lower the probability of\ncausality violations with the help of a new position de-\npendent mutation operator. Firstly, we have to iden-\ntify typical settings which are responsible for causality\nviolations. One is a direct consequence of the redun-\ndant nature which from the viewpoint of accumulated\nmutation is also advantageous. Hence, we do not try\nto change the encoding to be less redundant, but in-\nstead we change the mutation operator, so that the\nmutation probability rises for redundant chromosome\nentries. If pm is the probability to mutate and Nxk(SC )\ndenotes the number of occurrences of symbol xk(SC )\nin SC before the position of xk(SC ), we write\npm(xk(SC )) = pm · (Nxk(SC ) + 1) (25)\nSecondly, we observe that all elements from SC which\noccur in the first four elements in LC have a large im-\npact on the connection matrix, since the first element\nin SC is always mapped onto this first block of elements\nin LC. Therefore, we suggest a second modification to\nthe mutation operator:\npm(xk(SC )) = p2\nm\n∀ xk(SC ) ∈ { x1(LC), . . . , x4(LC)} (26)\nFigure 7 (a) and (b) show the results for the probabil-\nity of causality violating steps for the mutation oper-\nator with the modifications eqs. (25, 26) compared to\nthe fixed mutation (dashed curve) rate pm. In Figure\n7 (a) we kept Nsym constant and changed the length\nof SC , since we expect that in this case modification\n(25) will have the largest impact because the amount\nof redundant elements in SC rises with dSC . Indeed,\nwe observe that (1 − P (A|B)) is considerably reduced\nand that the effect is more pronounced for larger val-\nues of dSC . Figure 7 (b) shows experiments carried\nout for the combinations Nsym = dSC which, as we\npointed out earlier, are the best choices for the coding\nparameters. The new mutation operator reduces the\nprobability of causality violations also in these cases,\n\n(a)\n10 20 30 40 50\n0.3\n0.32\n0.34\n0.36\n0.38\n0.4\n0.42\n1 - P(A|B)\ndSc\nNsym = 10\n(b)\n(10, 10) (20,20) (30,30) (40,40) (50,50)\n0.32\n0.34\n0.36\n0.38\n0.4\n1 - P(A|B)\nNsym dSc( , )\nFigure 7: The probability to violate the causality con-\ndition (1 − P (A|B)) (see also Fig. 6) estimated from\n105 trials. (a) Nsym = 10 is kept constant and (b) the\nrelation Nsym = dSC is fixed. The interlaced curve\nshows the values for the mutation operator with the\nmodifications (25, 26) and the dashed curve for the\nstandard mutation rate pm (pinit = 0.25).\nhowever the difference to the fixed mutation rate is\nsmaller than in Figure 7 (a). Thus, we conclude that\nminor modifications of the mutation operator can al-\nready have an causality enhancing effect on the search\nprocess and that it is worthwhile to analyse the (geno-\ntype → phenotype, mutation) system with respect to\nthe question why and for which specific settings prob-\nlems can occur.\n5 Conclusion\nIn this paper we suggested a condition which the\nsetting (genotype → phenotype mapping, mutation)\nshould fulfill in order to allow gradual changes for a\nlocal search on the phenotype space which can be con-\ntrolled via the mutation parameter on the genotype\nspace. We applied the probabilistic causality condi-\ntion to problems in the domain of parameter optimisa-\ntion and structure optimisation both analytically and\nconstructively. Thus, besides examining the search\nprocess, we also suggested variations in the mutation\noperator to improve the setting with respect to our\ncondition. Especially in the later domain, where com-\nplicated mappings are commonly used, we believe the\nmeasure could be a useful tool for constructing evolu-\ntionary algorithms. In the case of parameter optimisa-\ntion, Figures 3 show that the setting which enhances\nthe causal behaviour at the same time improves the\nperformance.\nAcknowledgements\nThis research work is part of the BMBF SONN project\nunder Grant No. 01IB401A9.\nReferences\nGruau, F. (1993). Genetic synthesis of modular neural\nnetworks. In S. Forrest (Ed.), Proc. 5th Int. Conf. Genetic\nAlgorithms, pp. 318–325. Morgan Kaufmann.\nJones, T. and S. Forrest (1995). Fitness distance cor-\nrelation as a measure of problem difficulty for genetic\nalgorithms. In L. Eshelman (Ed.), Proc. 6th Int. Conf.\nGenetic Algorithms , pp. 184–192. Morgan Kaufmann.\nKitano, H. (1990). Designing neural networks using ge-\nnetic algorithms with graph generation system. Complex\nSystems 4 , 461–476.\nLohmann, R. (1993). Structure evolution and incomplete\ninduction. Biol. Cybern. 69 (4), 319–326.\nManderick, B., M. de Weger, and P. Spiessens (1991). The\ngenetic algorithm and the structure of the fitness land-\nscape. In R. Belew and L. Booker (Eds.), Proc. 4th Int.\nConf. Genetic Algorithms , pp. 143–150. Morgan Kauf-\nmann.\nMiller, G. and P. Todd (1989). Designing neural net-\nworks using genetic algorithms. In D. Schaffer (Ed.), Proc.\n3rd Int. Conf. Genetic Algorithms , pp. 379–384. Morgan\nKaufmann.\nRechenberg, I. (1994). Evolutionsstrategie ’94. Friedrich\nFrommann Holzboog.\nSchuster, P. (1995a). Artificial life and molecular evolu-\ntionary biology. In F. Moran et al. (Eds.), Advances in\nArtificial Life , pp. 3–19. Springer.\nSchuster, P. (1995b). How to search for RNA structures\n- theoretical concepts in evolutionary biotechnology. J.\nBiotechnology 41 , 239–257.\nSendhoff, B. and M. Kreutz (1996). Analysis of possi-\nble genome–dependence of mutation rates in genetic al-\ngorithms. In T. Fogarty (Ed.), Evolutionary Computing ,\nVolume 1143 of LNCS, pp. 257–269. Springer.\nSendhoff, B. and M. Kreutz (1997). Evolutionary opti-\nmization of the structure of neural networks using a re-\ncursive mapping as encoding. In Proc. 3rd Int. Conf. Arti-\nficial Neural Networks and Genetic Algorithms . Springer.\nSendhoff, B., M. Kreutz, and W. von Seelen (1997).\nCausality and the analysis of evolutionary algorithms.\nSubmitted to IEEE Trans. Evolutionary Computation .\nWhitley, L. (1995). Genetic algorithms and neural net-\nworks. In J. Periaux and G. Winter (Eds.), Genetic Algo-\nrithms in Engineering and Computer Science , Chapter 11.\nJohn Wiley."}
{"id": "adap-org/9712001", "meta": {"categories": ["adap-org", "nlin.AO"], "created": "1997-12-10", "extraction": {"body_chars": 31135, "cleaning": {"detected_repeated_margin_lines": ["1"], "page_count": 16, "removed_boilerplate_lines": 27}, "method": "pypdf_no_ocr", "source_pdf_bytes": 370477, "text_chars": 31691}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9712001", "primary_category": "adap-org", "source": "arxiv", "title": "Experiences with iterated traffic microsimulations in Dallas", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9712001"}, "text": "Experiences with iterated traffic microsimulations in Dallas\n\nAbstract\nThis paper reports experiences with iterated traffic microsimulations in the context of a Dallas study. ``Iterated microsimulations'' here means that the information generated by a microsimulation is fed back into the route planner so that the simulated individuals can adjust their routes to circumvent congestion. This paper gives an overview over what has been done in the Dallas context to better understand the relaxation process, and how to judge the robustness of the results.\n\narXiv:adap-org/9712001v1 10 Dec 1997\nExperiences with iterated traffic microsimulations\nin Dallas\nKai Nagel\nLos Alamos National Laboratory, TSA-DO/SA Mail Stop M997,\nLos Alamos NM 87544, USA, kai@lanl.gov\nMay 21, 2018\nAbstract\nThis paper reports experiences with iterated traffic microsi mulations\nin the context of a Dallas study. “Iterated microsimulation s” here means\nthat the information generated by a microsimulation is fed b ack into the\nroute planner so that the simulated individuals can adjust t heir routes to\ncircumvent congestion. This paper gives an overview over wh at has been\ndone in the Dallas context to better understand the relaxati on process,\nand how to judge the robustness of the results.\n1 Introduction\nThe advent of ever more powerful workstation computers makes large scale\ntransportation microsimulation projects feasible. At the same time , traditional\ntools are having a hard time treating all the complexities of modern co ngested\ntransportation systems. As a result, there has been considerab le progress in\nthe area of transportation microsimulation in recent years, both in terms of\ntheoretical understanding of micro-models [1, 2, 3, 4] and in terms of practi-\ncal implementations [5, 6, 7, 8, 9, 10, 11]. Yet, having good transpo rtation\nmicrosimulations is only part of the problem. When running such a micro simu-\nlation, one finds oneself confronted with the question of how to “dr ive” it: when\na driver in the microsimulation approaches an intersection, how does she know\nwhich way to proceed?\nThe traditional answer to this question are turning percentages: for exam-\nple 50% of the vehicles go straight, 20% left, and 30% right. One is, th ough,\nimmediately faced with a data collection problem: It is improbable that o ne\nknows the turn counts for all intersections of a regional area; an d if one does\nnot, one needs methods to “generate” the missing information [10]. Also, for\ntransportation planning purposes one recognizes quickly that these numbers are\nnot very useful because they change easily with infrastructure c hanges.\n\nA somewhat better way is to use origin-destination matrices. Yet, t he prob-\nlems are the same: The information is not available, especially for non- work\ntrips; and these matrices change with major infrastructure chan ges.\nAn answer to this is to start from demographics: It is improbable tha t peo-\nple’s income changes or that they move in a matter of weeks in respon se to trans-\nportation infrastructure changes. From there, for a microsimulation project, the\nfirst step is to generate “synthetic” populations from the demogr aphic data [12].\nThe next task is to derive activities (sleep, work, shop, . . .) for each member of\na synthetic population, and then to derive the transportation dem and from this.\nSo far no project has succeeded in completely executing this progr am, i.e. to use\nactivities based on demographics to “drive” a transportation micro simulation.\nThere are considerable challenges “on the way”. For example, even when\ngiven a time-dependent origin-destination matrix and a traffic netwo rk, how does\none allocate the trips to the network? Traditional assignment meth ods can be\nshown to be dynamically inconsistent under heavily congested condit ions. One\nof the major problems is that for links where demand is higher than ca pacity,\nthe link travel time depends on for how long the congested condition has been\nin place and not just on demand only, and thus the traditional link tra vel time\nfunctions are invalid.\nA way out is to replace the traditional cost functions by a microsimula tions.\nIn short, one allocates traffic streams to the road network, runs the microsim-\nulation and collects link travel times, re-allocates some of the traffic streams,\nre-runs the microsimulation, etc. This mimics “day-to-day” dynamic s, i.e. “over\nnight” some drivers decide to try a new route the next day. Obvious ly, one faces\nconsiderable challenges, such as: Which fraction of the population s hould be re-\nplanned? How do we reach fast convergence of the process? Does the process\nconverge at all? Can we measure convergence? Is a real traffic sys tem con-\nverged? This paper will report results related to these questions in the context\nof a Dallas study. After providing the context (Sec. 2) and showing the first re-\nsults (Sec. 3), the paper will summarize systematic feedback stud ies (Sec. 4) and\npossible “structural” convergence criterions (Sec. 5). After th at, this paper will\nlook at different aspects of the “robustness” of results; first un der the change\nof a random seed (Sec. 6), then under the change of the complete microsimu-\nlation (Sec. 7), then in comparison to reality (Sec. 8). Section 9 disc usses the\nrobustness question in somewhat more general, pointing out that o ne first needs\nto define the question that the simulation is supposed to answer. Th e paper is\nconcluded by a short summary (Sec. 10).\n2 The context\nThe context of the results reported here is the Dallas/Fort Worth case study of\nthe TRANSIMS project. The goal of the case study was to demons trate that\nthe approach has the general capability of generating output tha t is useful for\nstake-holder analysis and scenario evaluation [13]. The setting of th e study was\na 5 miles times 5 miles area (“study area”) around the busy freeway in tersection\n\nbetween the LBJ freeway and the Dallas North Tollway north of Dallas down-\ntown. The problem was approached by starting out with a “focused network”\nand a production-attraction (PA) matrix. The focused road netw ork contained\nall streets inside the study area, but got considerably thinner with inc reasing\ndistance from the study area. A production-attraction matrix is e ssentially a 24-\nhour origin-destination matrix with some land-use information include d (trips\nfrom work to home are entered as trips from home to work, with the result that\nthe zones where trips originate in these matrices need to be residen tial zones).\nThe PA matrix for Dallas/Fort Worth contained approximately 10 million trips\nduring a 24-h period.\nSince TRANSIMS is a microscopic approach, the first thing that was d one\nwas to decompose the PA matrix into individual trips. This was done us ing a 24-\nhour time use function, i.e. each trip had a starting time, and the dist ribution\nof starting times reflected rush period traffic. Out of these trips, only the\ntrips starting between 5am and 10am were considered. These trips were routed\nalong the road network, and only the trips which went through the s tudy area\nwere retained (about 300 000 trips). This set of trips, sometimes c alled “initial\nplanset”, forms the basis of all studies presented here. See [13, 1 4] for further\ndetails.\n3 First results\nJust using the routes from the initial planset and sending them thro ugh a traffic\nsimulation1 typically generates a result as in Fig. 1, where many streets are\noccupied by jammed traffic. The problem behind this is, obviously, tha t the\nroute generation phase for the initial planset did not take into acco unt what\nother people were going to do. Yet, predicting what everybody else will do in\na straightforward way is impossible; imagine yourself in a situation whe re you\nwake up one morning in a city where you need to go to work, and the on ly two\npieces of information you have is a map of the transportation syste m and the\ninformation that everybody else is in the same situation as you.\nThe way we solve this is by iterations between microsimulation and plann er.\nA possible method is the following: the microsimulation is run, link travel times\nare extracted from the microsimulation, a certain percentage of t he travelers\ncomputes new routes based on these link travel times, the microsim ulation is run\nagain with these new plans, etc. New routes are computed using time -dependent\nfastest path based on the starting time, starting location, destin ation, and link\ntravel times from the microsimulation. Other iteration schemes are possible,\nsee, e.g., [15].\nAn open question here is which percentage of the travelers should b e re-\nplanned. First, we started out with a fairly high percentage, 10%, a nd used\n1Note that this remark implies that we are using microsimulat ions based on route plans ,\ni.e. each driver knows before her trip starts exactly the seq uence of links she wishes to take\nthrough the network.\n\nFigure 1: Microsimulation run based on initial planset, snapshot at 9:3 0 am.\nthat until the results started oscillating, which was after the 7th it eration.2 In\niterations number 8–12, 5% of the trips were re-routed; in iteratio ns number\n13–14, 2% of the trips. A typical result after 14 iterations can be s een in Figs. 2\nand 3. Clearly, the traffic jams have cleared, and traffic is not only mo re evenly\ndistributed, but because the system is more efficient, traffic does n ot back up.\nFig. 2 looks “plausible”, whereas Fig. 1 does not. Further informatio n can be\nfound in [14].\n2The reason for the oscillation is easy to understand: Assume you have two route alterna-\ntives, and one is slightly faster. If you re-route a certain p ercentage of people, than the other\nalternative will be faster. Without additional measures, t his will be an undamped oscillation.\n\nFigure 2: Microsimulation run based on the planset after 14 iteration s, snapshot\nat 9:30 am. Compare to Fig. 1.\n4 Systematic feedback studies\nUsing a relaxation scheme such as the above is not very practical (b ecause\nit needs human supervision) and scientifically not very convincing: Qu estions\nsuch as “Does the process converge? If so, can we quantify ‘dista nce’ from\nthe converged state? Is real traffic converged?” come up. One n ext step is\nthus to more systematically test relaxation schemes and relaxation properties.\n(Since we do not know if the process “converges” in the mathematic al sense, we\nwill talk about “relaxation” instead.) Rickert [5, 16] has run such tes ts in the\nsame Dallas/Fort Worth context, but with a different microsimulation [6]. His\nmicrosimulation is somewhat less realistic (most important points: no s ignal\nplans, no turn pockets, no lane changing for turning behavior), bu t it currently\n\nFigure 3: Microsimulation run based on the planset after 14 iteration s, snapshot\nat 9:30 am, detail. Traffic lights are shown by their colors at the end of links.\nCars can be color coded according to specified criteria – in this specifi c example,\ncars that were not able to follow their intended plans because they c ould not\nget into the desired lane are shown red.\nruns more than 20 times faster than the microsimulation used in the la st section.\nSome comparisons between the results obtained by both microsimula tions will\nbe shown below; we believe using a different microsimulation will not have any\nsignificant impact on the general results that will be discussed in this section.\nFig. 4 shows the sum of all travel times as a function of the iteration number\nfor different iteration schemes. For example, the full line (also mark ed by +)\nis for the same relaxation scheme as described in the last section. Th e data\nmarked by the × symbol is for an iteration series where in each iteration only\n1% of the travelers was re-planned. The dotted lines are for iterat ion series\nwhere 5% of the travelers were re-planned; in one case these 5% we re selected\nrandomly, in the other case they were selected according to “age” : A traveler\n\n1e+08\n1.5e+08\n2e+08\n2.5e+08\n3e+08\n3.5e+08\n4e+08\n4.5e+08\n20 40 60 80 100\nsum travel time at 10:00 am [sec]\niteration\nrandom, 3*20%, 3*10%, 3*5%, 2*2% (run 1)\nrandom, 110*1%, RP2 (run 4)\nlinear age, 5% (run 11)\nrandom, 5% (run 12)\nFigure 4: Sum of all travel times as a function of the iteration. Clear ly, the sum\nof all travel times decreases with the iterations because congest ion clears and\nthe efficiency of the whole system increases. Different iteration sch emes result\nin different relaxation speeds. From [16].\nwho had not tried a new route for a long time was more prone to re-pla nning.\nSeveral observations can be made from this plot:\n• The simulations relax to a value of approx. 1 .1 ·108 seconds for the sum of\nall travel times. Clearly, the iteration described by the full line and “ +”\nis not yet there.\n• The iterations marked by the × symbol are relaxing very slowly. The\nreason for this is that 1% re-planning means that the probability of not\nhaving been re-planned after 110 iterations is 0 .99110 ≈ 0.33, i.e. about one\nthird of all trips still follow their initial route which has been computed\nwithout any feedback information.\n• As the line for “age-dependent” re-planning shows, much faster r elaxation\nschemes are possible even when they are non supervised.\nFor further information, see [5, 16].\n5 Characteristics of fastest paths in relaxed vs.\nunrelaxed traffic situations\nThe above quantity, sum of all travel times, has the disadvantage that as a\nrelaxation criterion it is only meaningful in the context of a relaxation series:\nOne can follow its behavior, and from that one can decide that the se ries is now\n“relaxed” or not. This approach is not useful for answering the qu estion where\nthe “real” system is. Is it somewhere along such a relaxation line, but not quite\nat the bottom? Is it totally different? The sum of all trip times is a numb er\nthat cannot be compared across different networks: That numbe r would, for\nexample, depend on the size of the region.\n\nFor answering such a question, one needs a more “structural” app roach, one\nthat looks “directly” at the properties that supposedly relax. The property\nthat relaxes in the above approach is the “incentive to deviate”. Ou r behav-\nioral assumption is that people, given the starting time and location a nd the\ndestination, will switch routes until they find a reasonably fast rou te. To a\ncertain extent, this follows the traditional assumption both for th e Wardropian\nequilibrium in transportation and for a Nash equilibrium in game theory in that\nrational players choose their best strategy (= fastest route), based on the as-\nsumption that everybody else is rational. Yet, in reality people do not search for\nthe fastest route at all costs; also, the above relaxation scheme when inspected\ncloser reveals that in it people do not select the route with the faste st expected\ntravel time, even not in the average.\nSo the question becomes if we can measure a quantity which reflects the\nincentive to deviate; and if so, if this quantity decreased with the ite ration\nnumbers. A different way to formulate the question is: How much add itional\ntime would you need for your second-best route? One of the proble ms here is\nthat “second-best” route is not well-defined; simply computing k–shortest paths\nin an algorithmic way will generate many solutions that most people will n ot\nconsider alternatives; such as leaving a freeway at an off-ramp, cr ossing the\nintersection, and getting on again at the other side of the intersec tion. Yet,\nother approaches based on “reasonableness” cannot be conside red satisfying for\nthe present question [17].\nIn order to have a quantitatively sound criterion, we used k–fastest paths but\nthen calculated the “difference to the fastest path”. By this, one can distinguish\nbetween routes which differ only little from the fastest path and rou tes which\ndiffer a lot. We then plot that quantity as a function of, say, addition al travel\ntime. The information provided thus is (Fig. 5a): Given I accept x seconds\nmore travel time, how different are my possible routes from my fast est route?\nOne can now calculate this information for many different origin-dest ination\npairs. Averaging the resulting information for given amounts of add itional travel\ntime results in Fig. 5b, i.e. the figure provides an answer to the quest ion: How\ndifferent will my route be in the average from the fastest route if I accept\nx seconds of additional travel time?\nNow, note that the two curves in Fig. 5b have been computed with tw o\ndifferent link travel time sets: (i) “unrelaxed congested” uses link t ravel times\nobtained from an “unrelaxed” simulation (i.e. based on the “initial plan set”)\nat a congested time of the day; (ii) “relaxed congested” uses link tr avel times\nobtained from a “relaxed” simulation (i.e. after many iterations) at t he same\ncongested time of the day. Clearly, when accepting a certain amoun t of ad-\nditional travel time, the options one has under relaxed conditions a re in the\naverage much more diverse than under unrelaxed conditions. In ot her words,\nmany different routes will provide near-optimum performance for t he individual\ndriver; the optimum becomes “flat”. This means that the system ar ranges itself\nin a way that the “incentive to deviate” is fairly small, because even if y ou do\nnot use the optimal route, the gains from switching are fairly small. F or further\ninformation, see [18].\n\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n0 20 40 60 80 100 120\nsimilarity to fastest path\nadditional travel time [sec]\nOD pair #\n0.45\n0.5\n0.55\n0.6\n0.65\n0.7\n0.75\n0.8\n0.85\n0 20 40 60 80 100 120\nsimilarity to fastest path\nadditional travel time [sec]\naverage similarity to fastest path for given additional travel time\nunrelaxed congested\nrelaxed congested\nFigure 5: Plotting “similarity to fastest path” as a function of “addit ional travel\ntime”. Top: Example. One observes that, if one allows more and more addi-\ntional travel time, one obtains more and more options that are ver y different\nfrom the fastest option. Bottom: Average over 955 origin destination pairs. One\nsees that, for given additional travel time, one has more diverse o ptions in the\nrelaxed congested case. From [18].\n6 Random fluctuations\nOur microsimulations are stochastic, i.e. they use random numbers d uring the\ncomputation of their dynamics. That means that a simple change of a random\nseed can change the entire dynamic trajectory of the simulation. F ig. 6 is an\nexample how dramatically the situationcan change with only the chang e of a\nrandom seed shown (compare to Fig. 2). Clearly, using stochastic d ynamics\nmakes formal definitions, analytical proofs, and computational p roofs of conver-\ngence much harder. Yet, we believe that it is a necessary feature o f reality; also,\nit makes the computational search process more robust against g etting “stuck”\nin implausible situations.\nA result of stochasticity is that transportation simulation project s can, for\nany given question, at best return a “distribution” of possible answ ers, i.e.\ndeterministic answers are not possible. Note that, if the distributio n is not\n\nFigure 6: Demonstration of random fluctuations. Microsimulation ru n based on\nplanset after 14 iterations, snapshot at 9:30 am. Compare to Fig. 2 , the only\ndifference between the runs is a changed random seed.\nGaussian, then using the arithmetic mean as a replacement for the d eterministic\nanswer can be misleading, for example in the case of a bi-modal distrib ution.\nThis indicates that more systematic studies of the effects of stoch astic dy-\nnamics need to be done.\n7 Different microsimulations\nAt the current stage, it is unclear in how far transportation simulat ion projects\nwill be able to provide “robust” answers to questions that are relev ant to society.\nAs a minimum requirement, one should be able to extract “similar” answ ers from\ndifferent codes. Fig. 7 shows an example of using a different microsimu lation\n\nFigure 7: Using a different microsimulation in the same study context. Compare\nto Figs. 2 and 6. Unfortunately, a plot showing individual vehicles as in the other\nplots was not available; but to a certian extent, a comparison is poss ible. Gray\nmeans link density (occupancy) below 10%, green means below 30% (w hich\ncorresponds roughly to capacity), yellow means below 50%, red mea ns below\n70%, and purple means above 70%. From [16].\nin the same context; in fact, the microsimulation is the same as the on e used\nfor the results described in Sec. 4. The figure should be compared w ith Figs. 2\nand 6. Clearly, there are similarities. Traffic on the eastern part of t he east-west\nfreeway is in general heavy whereas most of the other simulation ar ea turns out\nto be outside the congested regime. In some sense, Fig. 7 is in betwe en Figs. 2\nand 6, which only differ by the initial random seed. Beyond these some what\ngeneral observations, comparison between different microsimulat ions is not very\nwell defined problem; see Sec. 9 for a further discussion of this sub ject and [19]\nfor more information.\n\n8 Comparison to reality\nIn the context of the Dallas study, we had some turn counts from s elected\nintersections available. Before turning to the results, some details about the\ncomparison have to be noted, which are a consequence of the fact that the\nnorth-south freeway in the study area did not exist in its full length before\n1990:\n• The main inputs for our microsimulations are the network and the trip\ntables (PA matrices).\n• The trip table that we use is from before 1990, i.e. before the northern\npart of the north-south freeway existed .\n• The network that we use is from after 1990, i.e. after the northern part of\nthe north-south freeway was opened.\n• The reality counts are from 1996, i.e. from a time where the network was\nsimilar to the one in the simulation, but the trips had adjusted to the\nexistence of a much faster way to travel north-south in the nort hern part\nof the study area.\nIn consequence, we would expect that our studies underestimate north-south\ntraffic in our study area compared to the 1996 counts.\nWhen looking at Fig. 8, this is clearly the case. But this is not the only\nfeature. One also notices that in fairly general our simulation over- estimates\ntraffic on minor roads and through turns; this is most probably a con sequence\nof the way the re-planner works because it only looks at travel times and not,\nfor example, at inconveniences caused by sharp turns or by stop s igns. These\nresults reflect work in progress; further results will be published e lsewhere [19].\n9 Robustness of (micro-)simulation results\nThe previous two sections ask the question of how to compare differ ent mi-\ncrosimulations or of how to compare microsimulations with reality. In p rinciple,\nthis should be easy: microsimulations generate microscopic observa bles, and so\nwe simply extract the same information from the microsimulations and from\nreality and we compare them. Yet, it is unclear which information to exactly\nto extract: Which is the most meaningful information? For example, does the\nfact that microsimulation XYZ has 20% too many right turns on a cert ain in-\ntersection really matter? Or what is more important: 100 vehicles mo re at the\nend of an already large traffic jam, or 100 vehicles more on unconges ted roads?\nOne needs to relate this problem to the question one attempts to an swer\nwith a specific simulation project. The level of fidelity of a simulation pr oject\nwill always depend on the level of effort spent; and it seems reasona ble to us to\nadjust the level of effort to what is really necessary for a given que stion. Also,\nwith a given question at hand, the problem of comparing simulations be comes\n\nFigure 8: Comparison with reality between 8:30am and 8:45am. At most in-\ntersections, there are three principal bars, one for right turns , one for traffic\ngoing straight, and one for left turns. For each of these principal bars, the\nmiddle (black/white) bar shows the observed value, the outside (gr ay) bar the\nvalue from the simulation. For example, for the intersection in the rig ht up-\nper corner, the simulation is underestimating both southbound and northbound\nthrough traffic. – When interpreting this figure, one needs to know that the\nrightmost north-south road and the topmost east-west roads a re major arteri-\nals, the dark grey roads are freeways, roads immediately parallel t o the freeways\nare frontage roads, and all other roads are minor. Only then it bec omes clear\nthat we are overestimating traffic that goes through “inconvenien t” routes, i.e.\nthrough sharp turns and over minor roads. Also, we underestimat e north-south\ntraffic in general due to the data inconsistency described in the tex t. Note\nthat we have made no attempt to “calibrate” the simulation to these values.\nFrom [19].\n\n5 6 7 8 9 10 11 12\nmedian speed [m/sec]\nsimulation time [h]\nPAMINA (run 11, it 58/59/60)\nTRANSIMS\nFigure 9: Comparison of median “geographical” speeds between two simula-\ntions. What we mean by geographical speed is “geographical distan ce” divided\nby “travel time”, i.e. not the average driving speed. This measures accessability\nof an area; in this case accessabiliby of the interior of the simulation a rea when\ncoming from the outside. From [19].\nbetter defined: One does not need any more to decide if two simulatio ns are\n“equal”, but only if two different simulations give the same answer to t he spec-\nified questions. This latter approach is much better accessible to th e statistical\ntools at hand; admittedly, it is not very satisfying because it means t hat for any\nnew question one needs, in some sense, to start over.\nFig. 9 presents one such possible result. Plotted is the average spe ed for\nreaching the center of the simulation area when coming from the out side. The\nspeed here is calculated by “geographical distance divided by trip tim e”, i.e. it\ndoes not reflect driving speed but is a measure of “accessibility”: Low speeds\nreflect that the destination is difficult to reach. Our plot is a simplified v ersion\nof a type of questions that are really important in the context of “s take-holder\nanalysis”. For example, does the introduction of a light rail make tra vel to the\ndowntown area faster for people who do not own a car? (Probably y es.) Does it\nmake the same travel faster for people who own a car? (Probably n ot, at least\nnot during uncongested conditions.)\nNow note that Fig. 9 only gives part of the full answer. We certainly s ee\nthat in both simulations, accessibility of the center of the simulated a rea drops\nduring the morning rush period. We also see that the drop in accessib ility is\ndifferent between both microsimulations. Yet, if the question were f or example\nto find the time of worst accessibility during the morning rush period, both\nsimulations would give the same answer.\nLast, but not least, note that this paper deliberately side-steps t he ques-\ntion of the trustworthiness of the microsimulation itself. In this pap er, we have\nconcentrated on “macroscopic” aspects, such as large scale fluc tuations and the\ninterplay between route planner and microsimulation. Yet, it is also us eful to\ntest and document microscopic aspects of traffic models. The discu ssion of what\nto measure here and how is ongoing; but there seems to be some slow ly growing\n\nconsensus that certain building blocks of the flow characteristics, such as flow\nthrough a stop sign as function of the traffic on the major road, un realistic as\nthese examples may be, should be part of these systematic tests. The devel-\nopment of more complex test suites would certainly be desirable. For further\ninformation on this in the context of TRANSIMS, see [20].\n10 Summary\nIterated transportation microsimulations provide a very powerfu l addition to\nthe tools of transportation planning. Their power lies both in the cap ability\nto represent time-dependent scenarios in their “true” time-depe ndent dynam-\nics and in the possibility to access individual, microscopic quantities dire ctly.\nYet, one needs to note that research in this area is at its beginning; no con-\nsistent theory is available to help and even the building of intuition is in its\ninitial stages. This paper provides a summary of what has been done in the\ncontext of a TRANSIMS study of the Dallas area in order to provide e xactly\nsome of this intuition. These results can be summarized as follows: (i) Iter-\nations between router and microsimulation adjust routes in a way th at traffic\nbecomes “plausible”. (ii) The number of iterations that are necessa ry until the\nprocess is plausibly “relaxed” can be significantly reduced by using “in telligent”\nrelaxation schemes. (iii) It seems possible to distinguish “unrelaxed” from “re-\nlaxed” conditions by looking at fastest paths in both situations. The result is\ninterpretable in the sense that in relaxed transportation systems , there is not\nmuch difference in travel time between the strictly fastest path an d very dif-\nferent routes. (iv) In stochastic microsimulations, simply changing the random\nseed can generate large fluctuations. (v) Using a different microsim ulation in\nthe same scenario produces results that look similar, but that are d ifficult to\ncompare in general. (vi) Comparing to reality has the same caveats, plus the\nproblem of data consistency. Nevertheless, the comparison prov ides useful in-\nsights. (vii) In general, comparison between microsimulations need t o be geared\nto specific questions. An example of such a question is provided.\nAcknowledgments\nLos Alamos National Laboratory is operated by the University of Ca lifornia for\nthe U.S. Department of Energy under contract W-7405-ENG-36. This article is\nwork performed under the auspices of the U.S. Department of Ene rgy.\nReferences\n[1] K. Nagel. Particle hopping models and traffic flow theory. Phys. Rev. E ,\n53(5):4655, 1996.\n[2] S. Krauss et al, this volume.\n\n[3] A. Schadschneider et al, this volume.\n[4] Y. Sugiyama et al, this volume.\n[5] M. Rickert, in preparation.\n[6] M. Rickert and K. Nagel. Experiences with a simplified microsimulation\nfor the Dallas/Fort Worth area. International Journal of Modern Physics\nC, 8(3):483–504, 1997.\n[7] M. Van Aerde, B. Hellinga, M. Baker, and H. Rakha. INTEGRATION : An\noverview of traffic simulation features. Transportation Research Records, in\npress.\n[8] Forschungsverbund f¨ ur Verkehr und Umwelt (FVU) NR W. See\nhttp://www.zpr.uni-koeln.de/Forschungsverbund-Verkehr-NR W/.\n[9] G.D.B. Cameron and C.I.D. Duncan. PARAMICS–Parallel microscopic\nsimulation of road traffic. J. Supercomputing, 10(1):25, 1996.\n[10] J. Esser et al, this volume.\n[11] B. Chopard, this volume.\n[12] R.J. Beckman, K.A. Baggerly, and M.D. McKay. Creating synthet ic\nbaseline populations. Transportation Research A, Policy and Practice ,\n30A(6):415–429, 1996.\n[13] R.J. Beckman et al. TRANSIMS Dallas/Fort Worth case study rep ort. Los\nAlamos Unclassified Report LA-UR to be released, Los Alamos Nationa l\nLaboratory, TSA-Division, Los Alamos NM 87545, USA, 1997.\n[14] K. Nagel and C.L.Barrett. Using microsimulation feedback for tr ip adapta-\ntion for realistic traffic in Dallas. International Journal of Modern Physics\nC, 8(3):505–526, 1997.\n[15] K. Nagel. Individual adaption in a path-based simulation of the fr eeway\nnetwork of Northrhine-Westfalia. International Journal of Modern Physics\nC, 7(6):883, 1996.\n[16] M. Rickert et al, in preparation.\n[17] D. Park and L.R. Rilett. Identifying multiple and reasonable paths in\ntransportation networks: A heuristic approach. Transportation Research\nRecord, In press.\n[18] T. Kelly and K. Nagel, submitted. Also LA-UR 97-4453.\n[19] K. Nagel et al, in preparation.\n[20] K. Nagel, P. Stretz, M. Pieck, S. Leckey, R. Donnelly, and C.L. B arrett.\nTRANSIMS traffic flow characteristics, submitted. Also LA-UR 97-3 530."}
{"id": "adap-org/9712004", "meta": {"categories": ["adap-org", "nlin.AO", "q-bio"], "created": "1997-12-13", "extraction": {"body_chars": 38969, "cleaning": {"detected_repeated_margin_lines": [], "page_count": 14, "removed_boilerplate_lines": 7}, "method": "pypdf_no_ocr", "source_pdf_bytes": 64547, "text_chars": 39565}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9712004", "primary_category": "adap-org", "source": "arxiv", "title": "Sentient Networks", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9712004"}, "text": "Sentient Networks\n\nAbstract\nIn this paper we consider the question whether a distributed network of sensors and data processors can form \"perceptions\" based on the sensory data. Because sensory data can have exponentially many explanations, the use of a central data processor to analyze the outputs from a large ensemble of sensors will in general introduce unacceptable latencies for responding to dangerous situations. A better idea is to use a distributed \"Helmholtz machine\" architecture in which the collective state of the network as a whole provides an explanation for the sensory data.\n\nadap-org/9712004 13 Dec 1997\nSentient Networks\nGeorge Chapline\nLawrence Livermore National Laboratory\nchapline1@llnl.gov\nAbstract\nThe engineering problems of constructing autonomous networks of\nsensors and data processors that can provide alerts for dangerous situations\nprovide a new context for debating the question whether man-made systems\ncan emulate the cognitive capabilities of the mammalian brain. In this paper we\nconsider the question whether a distributed network of sensors and data\nprocessors can form “perceptions” based on sensory data. Because sensory\ndata can have exponentially many explanations, the use of a central data\nprocessor to analyze the outputs from a large ensemble of sensors will in\ngeneral introduce unacceptable latencies for responding to dangerous\nsituations. A better idea is to use a distributed “Helmholtz machine” architecture\nin which the sensors are connected to a network of simple processors, and the\ncollective state of the network as a whole provides an explanation for the\nsensory data. In general communication within such a network will require time\ndivision multiplexing, which opens the door to the possibility that with certain\nrefinements to the Helmholtz machine architecture it may be possible to build\nsensor networks that exhibit a form of artificial consciousness.\n1. Introduction\nDuring the next few decades autonomous networks of sensors and data\nprocessors are going to come into widespread use for industrial process\nmonitoring and detection of threats to populations and infrastructures. If the\nindividual sensors have sufficient sensitivity and the environmental data are\nunambiguous, then these sensor networks can be operated as robots; which\nmeans the sensory data can be used to directly activate responses. Obviously\nautomatic warning systems that can quickly respond to malfunctions in an\nindustrial process or hostile acts which threaten populations or essential\ninfrastructures would have many practical applications. Unfortunately, in\npractice it may be very difficult on the basis of outputs from sensors to decide in\na timely way that something untoward is happening. For example, it might be\nthat state of the art sensors do not have sufficient sensitivity or discrimination to\nprovide unambiguos signals regarding malfunctions or hostile acts. Even in\nnetworks where the individual detectors are very sensitive it may be necessary\nto correlate the outputs from sensors in different locations and also possibly\nfrom different kinds of sensors in order to determine whether a dangerous\n\nsituation has arisen. Another frequent problem that occurs with the\ninterpretation of real world sensor data is that signal to noise ratios for the\nsignatures of interest are not sufficiently large to provide unambiguous signals.\nThese problems are similar to those faced by many species of animals in the\nnatural world. Actually almost all animals can respond to dangers. However, in\nall invertebrates (with the possible exception of cephalopods) this response is\nalways a pre-conditioned automatic response. For example in order to navigate\naround obstacles the brains of flying insects must almost instantaneously\nrecognize how these obstacles are moving relative to the insect, and then\nimmediately send appropriate signals to the muscles controling its flight. It is the\ngoal of most current robotics research projects to duplicate this kind of\nautomatic response in various kinds of man-made electromechanical devices.\nIn vertebrates, on the other hand, there is a new anatomical feature - the\ncerebral cortex - which provides an additional channel for responding to\nsensory inputs. As a consequence vertebrates can respond to dangers either by\nmaking use of a rapid midbrain response system similar to that in invertebrates\nor in a somewhat slower fashion by making use of their cerebral cortex. In the\ncerebral cortex of vertebrates sensory data is processed in a much more\nsophisticated way than in the brains of invertebrates. Not only is the extraction\nof features from the sensory data more elaborate, but studies in both humans\nand animals of the functions of the cerebral cortex suggest that the processing\nof sensory data in the cerebral cortex is primarily concerned with the formation\nof explanations for the sensory data; i.e. \"perceptions\" [1].\nOne of the obvious evolutionary advantages of being able to take into account\nvarious possible explanations for sensory data is that this greatly enhances an\nanimal's capability to cope with stealthy predators. A vivid example of the\nnecessity of having such a capability is provided by the dilemma faced by a\nzebra on the African savanah. A lion sitting in plain view is very likely not to be\nan immediate threat, whereas recognizing that one is being stalked by a lion\nwill often require making that determination on the basis of manifestly\nambiguous data - for example, a slight movement of the grass or the state of\nagitation of birds or other animals. It is clear that faced with ambiguous\ninformation an animal may or may not want to respond. Indeed it is tempting to\nspeculate that vertebrates employ some kind of statistical inference engine in\norder to determine whether in ambiguius situations there is a danger that\nrequires flight or some other prompt respone. In fact there is strong\ncircumstancial evidence [2] that humans in effect make use of a statistical\ninference system in order to make decisions. Although initially it might seem\nimplausible that humans - not to mention animals with no obvious capabilities\nfor understanding abstract concepts - could utilize probabilistic reasoning, it is\nperhaps logically inescapable [3] that the vertebrate cerebral cortex must be\ncapable of inferring and somehow encoding estimates of the posterior\nprobabilities for the occurrence of various situations or events that are important\nfor survival. Historically it was suggested more than a century ago by Helmholtz\nthat the main function of the human perceptual system is to infer probable\ncauses for sensory inputs.\n\nIn the following section 2 we briefly review the well known statistical basis for\nthe assertion that it is the availability of posterior probabilities that makes it\npossible for a network of sensors and processors to provide explanations for\nsensory data. Unfortunately it is unknown exactly how the vertebrate cerebral\ncortex evaluates and records posterior probabilites. Nevertheless some parallel\ncomputation architectures have appeared in the literature which, at least in\nprinciple, should make it possible to use posterior probabilities to provide\nexplanations for sensory data. Although parallel computation architectures for\nthe interpretation of sensor outputs have not yet come into widespread use in\nengineering systems, there are reasons for expecting that this type of\narchitecture will find widespread use in the future when real time responses are\ndesired. For example, even in the case of robotic feedback control systems\nwhere the use of parallel computation architectures may not be mandatory the\nsignificant speed advantages of parallel analog circuits will make the use of\nVLSI parallel analog circuits [4] increasingly attractive for the real time\ninterpretation of sensor data [5]. In the case of distributed networks of sensors\nrapid interpretation of the sensory data will almost certainly require new\ncomputational paradigms. In particular, when the sensory data are ambiguous,\nthe use of parallel computation is almost manditory because in general there\ncan be exponentially many explanations for a particular set of sensor outputs.\nThe rationale for parallel computation in those cases where one is faced with an\nexponential proliferation of possible explanations for the sensor outputs is that\nthe required computational task of finding the most likely explanation is\nintractible for a single sequential data processor. This will be especially true\nwhen finding an explanation for sensor data involves using Markov chain Monte\nCarlo methods to “invert” a stochastic model for the world. On the other hand, as\nnoted by Hopfield and Tank in a celebrated paper [6], parallel computation\nusing a “neural network” architecture provides a remarkable capability for\nfinding good solutions to computationally intractible problems. Consequently\nalthough various kinds of computational schemes, e.g. rule based expert\nsystems, might be employed in special cases to rapidly explain sensory data,\nartificial neural networks do provide a natural universal computational\nframework to use in an autonomous system seeking an explanation for sensor\ndata precisely because of their ability to provide good solutions to\ncomputationally difficult problems. Furthermore although there are various\nparallel computation architectures that might be adopted to the problem of\nexplaining sensor data we would like to focus in the following on the fact that\nphysical systems of interacting 2-state units, viz. “spins”, in thermal equilibrium\nwith a heat bath can duplicate the behaivor of artificial neural networks like\nthose introduced by Hopfield, and in fact have certain advantages over\ncompletely deterministic neural networks (this is the basic idea behind\nsimulated annealing [7]).\nThe main point we would like to make in this paper is that in a distributed\nnetwork of autonomous sensors and data processors alternative interpretations\nof sensor inputs can be represented by the collective state of physically\nseparated binary units, each of which has activation level 0 or 1. In other words\n\nan autonomous network of sensors and data processors can be conceived of as\nthe hardware embodiment of an abstract statistical system of interacting spins\nfunctioning as a neural network. In section 4 we describe one particular way,\nwhich we call the Helmholtz machine after the corresponding abstract neural\nnetwork architecture, in which a hardware network of sensors and data\nprocessors might be made to emulate a spin system functioning as a statistical\ninference engine . In section 5 we briefly discuss how a hardware Helmholtz\nmachine might be built using existing data processing and communication\ntechnologies. In particular we compare and contrast the intelligent agents\nrequired for a Helmholtz machine with the TCP/IP interfaces used in Ethernet\nnetworks. In section 6 we address the question whether a distributed network of\nsensors and simple data processors could behave as though it were\n“conscious”.\n2. Statistical theory of pattern recognition\nIt has been understood for some time that pattern recognition systems\nare in essence machines that utilize either preconceived probability\ndistributions or empirically determined posterior probabilities to classify\npatterns. In the ideal case where the a priori probability distribution p(α ) for the\noccurence of various classes α of feature vectors and probability densities p(x|\nα ) for the distribution of data sets x within each class are known, then the best\npossible classification procedure would be to simply choose the class α for\nwhich the posterior probability\nP x p p x\np p x( | ) ( ) ( | )\n( ) ( | )α α α\nβ β\nβ\n=\n∑\n(1)\nis largest. Unfortunately in the real world one is typically faced with the situation\nthat neither the class probabilities p(α ) nor class densities p(x|α) are precisely\nknown, so that one must rely on empirical information to estimate the\nconditional probabilities P(α | x) needed to classify data sets. In practice this\nmeans that one must adopt a parametric model for the class probabilities and\ndensities, and then use empirical data to fix the parameters θ of the probability\nmodel. Once values for the model parameters have been fixed, then sensory\ndata can be classified by simply substituting values for the model probabilities\np(α; θ ) and p(x|α; θ ) into equation (1).\nUnfortunately determining values for the model parameters from\nempirical data is itself a computationally intractible problem. This means that in\npractice one is usually limited to using models of relatively modest complexity,\nand consequently one is always faced with the issue of choosing the best\npossible values for the model patameters. One popular way of measuring how\ngood a paticular set of model parameters is at reproducing the observed data,\n\nknown as the maximum likelihood (ML) estimator, can be motivated by noting\nthat the formula for the posterior probability given in equation (1) can be\nformally interpreted as the canonical Boltzmann distribution for the population\nof energy levels of a physical system in equilibrium with a heat bath. In\nparticular if one defines the \"energy\" of a classification α to be\nE p p xα α α= − log ( ) ( | ) , (2)\nthen the posterior probability introduced in (1) can be formally expressed in the\nform\nP x e\ne\nE\nE( | )α\nα\nα\nα\n=\n−\n−\n∑\n, (3)\nwhere the energies are those defined in equation (2). If we suppose that the\nenergy levels Eα defined in equation (2) define a physical system whose energy\nlevels are populated according to a canonical distribution of the form (3), then\nthe thermodynamic free energy of this system will be given by\nF (x) = { ( ) ( ( )log ( ))}E P P Pα\nα\nα α α∑ − − , (4)\nwhere we have used P( )α as shorthand notation for the canonical distribution\n(3). If instead of the true probability distributions p(α ) and p(x|α) one uses\nmodel probabilities p(α; θ ) and p(x|α; θ) to calculate a probability\ndistribution Pθ α( ) for different classifications of a data set x, then equation (3) will\nno longer necessarily be satisfied and the free energy calculated from eqation\n(4) will in general differ from the true free energy. In particular we would have\nthat\nF (x) = F (x, θ) − P P Pθ θ\nα\nα α α( )log[ ( ) / ( )]∑ (5)\nwhere F (x, θ ) is the free energy calculated using the distribution Pθ α( ) .The\nquantity P P Pθ θ\nα\nα α α( )log[ ( ) / ( )]∑ in the second term in equation (5) is always\npositive and measures of the difference in bits between the model\ndistribution Pθ α( ) and the true distribution P( )α . This distance measure, known\nas the Kullback- Leibler divergence, is the basis for the ML estimator that is\nwidely used by statisticians to measure how well a given set of model\nprobabilities reproduces the empirical data [8 ]. Minimization of the Kullback-\nLeibler distribution with respect to the model parameters θ is one way of\nchoosing the best possible values for these parameters. It should also be noted\n\nthat the estimated free energy F (x, θ ) is always greater than the equilibrium\nfree energy F (x) , so that the best possible probability distribution to use for\nclassifications is that which minmizes the estimated free energy. This analogy\nbetween pattern recognition and statistical physics opens the door to using\ninsights from theoretical physics to solve pattern recognition problems.\nAn ingenious physics model in which the posterior probabilities P( α| x; θ)\nare naturally represented in the canonical Boltzmann distribution form (3) was\nintroduced in 1985 by Ackley, Hinton, and Sejnowski [9 ]. In this model, known\nas the Boltzmann machine, data sets and their “explanations” are represented\nby configurations of binary units with activation levels ai= 0 or 1.The energy\nfunction for the assembly of binary units is assumed to have a form similar to\nthat used by physicists to describe a system of interacting spins in a magnet:\nE(a) = −\n≠\n∑\n2 w a aij\ni j\ni j\n+ θi\ni\nia∑ , (6)\nwhere a ={ ai} denotes the set of activation levels, theθi are biases, and the\nweights wij describes the interaction strength between binary units i and j In the\nwork of Ackley et. al. these interactions are assumed to be symmetric; i.e. wij =\nwji. If the system of binary units is assumed to be in contact with a heat bath at\nsome fxed temperature the probabilty distribution for the activation levels will\napproach a stationary equilibrium whose form is just the Boltzmann distribution\n(3) corresponding to the energy function (6). In the case of the Boltzmann\nmachine the probability distribution Pθ α( ) will be the probability distribution for\nthe activation levels in a certain subset, referred to as the hidden units, of all\nbinary units. The remaining binary units, referred to as the visible units,\nrepresent the environmental data x. The model parameters θ for the Boltzmann\nmachine are the connection strengths wij and biases θi for the binary units.\nThese parameters are determined by minimizing the Kullbach-Leibler\ndivergence between the probability distribution Pθ α( ) with the visible unit\nactivation levels fixed and the probability distribution for classifications with the\nactivation levels of the visible units allowed to vary freely. Used as a pattern\nrecognition device the Boltzmann machine has the virtue that non-trivial\ncorrelations between different instances of environmental data are\nautomatically represented, and used in the classification of data sets. This\nmeans that the classifications provided by the Boltzmann machine take into\naccount more information than just the relationship between a class and its\nfeature vectors. Unfortunately Boltzmann machines have not found many\npractical applications because determination of the connection strengths and\nbiases for realistic data sets is very slow using even the fastest supercomputers.\nMoreover even when the connection strengths and biases are known the\nclassification of data sets is slow because of the necessity for repetitive\n\nsampling of a joint probability distribution for the activation levels of the hidden\nunits.\n3. Belief networks\nAs noted in the introduction vertebrates are often faced with the problem\nof finding an explanation for sensory data whose interpretation is not\nimmediately obvious. This is a rather more subtle problem than the more\nfamiliar problem of classifying static patterns because finding an explanation for\nsensory data will in general involve taking into account structural relationships\nthat are inherent in the entire ensemble of environmental input data. One of the\nmain goals of artificial intelligence research is the development of algorithms\nwhich can recognize and manipulate the structural relationships inherent in\ndata sets. Almost effortlessly the human brain is able to capture the essence of\nthese structural relationships by the use of hierarchical knowledge trees or\n“ontologies”. At the present time there is a great deal of interest within the\nartificial intelligence community in developing algorithmic tools for contructing\nand utilizing ontologies for applications such as medical diagnosis and\nbattlefield management [10]. It remains unclear though whether these purely\nalgorithmic approaches to artificial intelligence will lead to systems that can\ndeal with the ambiguities and complexities of real world sensor data.\nIn existing computerized decision systems which use probabilistic\nreasoning the knowledge base is represented by a description of the\ndependencies between the variables of the system [11]. In a belief network\nthese dependencies are represented by a probability distribution for the states\nof the nodes in a layered network which is a product of conditional probabilities\nfor each unit given the values of the units which proceed it in some ordering.\nMany types of belief networks are possible, but typically belief networks have a\ntree-like structure and the conditional probabilities have the Markov property;\ni.e. the random variables at nodes not connected by a branch are conditionally\nindependent given those variables which are so connected. In addition belief\nnetworks often incorporate unobserved latent variables known as hidden\nvariables. Software implementations of Markov belief networks with hidden\nvariables have been successfully used for some quite difficult real world pattern\nrecognition problems such as speech recognition [12], and represent a\npromising general approach to the interpretation of complex data sets. Indeed\nthe success of hidden variable Markov models for speech recognition should\ncertainly be kept in mind when designing networks of sensors and data\nprocessors to interpret environmental data, and is also perhaps a significant\nclue as to how the cerebral cortex of vertebrates forms perceptions .\nModels for the cerebral cortex which might function as belief networks\nwere first introduced by Little, Shaw, and Vasudevan [13], and indeed these\nmodels were among the first artificial neural networks. The nodes of these\nnetworks can be represented by binary units with activation level\nai =0 or 1, and\n\nit has been pointed out [13 ] i that if the Markov transition probability for the\nactivation level of a single node in these models have the sigmoidal form\np(ai(n+1) | a (n)) = σ[β(1-2 ai(n+1)) w aij\nj\nj∑ ], (7)\nwhere σ(x) = 1/[1+exp(-x)] then these models resemble Boltzmann machines.\nThe vector a(n) = { ai(n)} in equation (7) denotes the set of activation levels at\nlayer n of the network. In the original work of Little, Shaw, and Vasudevan the\nway activation levels vary from layer to layer was thought of as describing the\ntime evolution of the whole network; whereas in their use as belief networks the\nnumber n will generally be taken to represent actual physical layers of the\nnetwork. A little algebra shows that the Markov transition probability for the\nensemble of activation levels in a layer can be written in the form\nP[a (n+1) | a(n)] = e\ne\na n X n\na n X n\na n\nβ\nβ\n( ) ( )\n( ) ( )\n( )\n+ •\n+ •\n+\n∑\n, (8)\nwhere Xi(n) = w a nij\nj\nj∑ ( ) -θi ia and we have used vector multiplication notation in\nthe exponents. If the interactions between units are symmetric, then the\ntransition matrix (8) will eventually become independent of n, and the\nprobability distribution for activation levels will have the canonical Boltzmann\nform; if activation levels in the first layer are fixed we recover equation (3). If the\nconnection strengths between units are asymmetric, i.e. wij ≠ wji, then the\nactivation levels will vary from layer to layer and depend on the activities in the\nfirst layer. The activation levels in the initial layer represent the environmental\ndata, while the activation levels in later layers can be thought of as representing\nthe explanation of the the environmental data [15]. It should be noted in this\nconnection that 2-dimensional arrays of binary units within each layer provide a\nconvenien way of representing and classifying feature vectors using redundant\npopulation codes [16]. Thus Boltzmann machines with asymmetric weights\nprovide an attractive way to realize fault tolerent belief networks.\nOne of the main practical problems with using conventional belief\nnetworks to explain sensor data is that given a set of conditional probabilities\nand associated decision tree which constitutes a model for the world, finding\nexplanations for input data will in typically involve using Markov chain Monte\nCarlo methods to invert the world model. Because of the need for repetitive\nsampling, this cannot in general be done in real time. Another practical problem\nwith belief networks is that determining the network parameters from the\nenvironmental data is very difficult if there are too many parameters. Fortunately\na descendent of the the Boltzmann machine form of belief networks, the\nHelmholtz machine, has recently emerged that offers the promise of alleviating\nthese problems.\n\n4. Helmholtz machine networks\nThe Helmholtz machine architecture introduced by Hinton et.al. [17] is a\nstochastic neural network that resembles the Boltzmann machine in that the\nnodes of the network consist of units whose activation levels are quantized to\nbe either 0 or 1. Another similarity between the Helmholtz and Boltzmann\nmachines is that some of the binary units represent environmental input data,\nwhile the remaining \"hidden units\" represent possible explanations for the input\ndata. All information concerning conditional probabilities is contained in the\nvalues of the connection strengths wij between nodes; indeed it is possible that\nthese weights play much the same role in the Helmholtz machine as the\nsynaptic connection in the cerebral cortex. The activation of the ith hidden unit\nin the network is chosen stochastically in accordance with the probability p xj( ) =\np( ai(n+1) | a (n)) that with a particular set x of sensory inputs the ith hidden unit\nin the nth layer has activation 1. The probabilities p xj( ) are calculated from the\nweights for the connection between the ith unit and the activities of the units in\nthe previous layer using the sigmoidal formula (7). Thus the activities of units\nare determined in a quasi-deterministic way similar to that in a feed-forward\nneural network. If one assumes that the activities of the binary units within a\ngiven layer are independent, then the probability of a particular explanation α\n={a(n), n>1} will be given by the product :\nQ(α) =\nn>\n∏\n[ ( ] [ ( )]p a p aj\nj\na\nj\naj j\n∏ −\n−\n; (9)\nso that the binary units that are turned on contribute with weight p xj( ) while the\nunits that are turned off contribute with weight 1- p xj( ) .\nAll of the information concerning the structure of the external world that is\nneeded to explain a given ensemble of sensory data is encoded into the\nconnection weights wij used in eq. 6 , and the main obstacle in utilizing a\nHelmholtz machine is determining the values of these parameters.\nUnfortunately when the set of input data is complex it is a computationally\nintractible problem to precisely determine these parameters. However, one can\nobtain approximate values for the connection weights of a Helmholtz machine\nbased on the observation (cf. eq’s 3 and 5) that the negative free energy -F (x,\nθ, Q ) calculated using the non-equilibrium distribution Q(α) provides an lower\nbound for the logarithm of the probability of generating a particular set x of\nsensory inputs. In the Helmholtz machine scheme of Hinton et. al. the\nparameters θ correspond to the connection weights and biases of a separate\nfeed forward network which is used offline to generate the “true” posterior\nprobability distribution Pθ α( ) . The connection strengths wij which are used in the\nrecognition network to explain input data are determined simultaneously with\nthe parameters θ by using gradient descent methods to minimize the free\n\nenergy function F(x, θ, Q ). Because the conditioning of almost any kind of\nnetwork that used to interpret observed data will be computationally tedious, it is\nperhaps to be expected that artificial neural networks will increasingly be used\nfor the purpose of determining the parameters of belief networks; and the use of\na separate generative network for this purpose by Hinton et. al. may be a first\nstep in this direction.\nOnce the connection strengths of the Helmholtz machine have been\nfixed, then the probabilities for likely explanations of given sensory data can be\nrapidly determined using Eq’s 7 and 8. Thus the Helmholtz machine\narchitecture avoids the problem with the Boltzmann machine of having to\nrepetitively sample a probability distribution in order to recognize input data.\nFurthermore with the use of the independence ansatz, eq. 8, this architecture\nallows one to sidestep the combinatorial problem that arises when there are\nexponentially many explanations. In summary, in addition to making the\nlearning of the structure inherent in a set of input patterns tractable, the separate\ngenerative and recognition models used in a Helmholtz machine allow one to\narrive at definitive interpretations of instances of environmental data in real time.\nOf course from the perspective of actually building sensor networks the\ncrucial feature of the Helmholtz machine concept is that it can apparently be\nrealized in a straightforward way in hardware as a distributed system of sensors\nand simple data processors. In addition it would be natural in such a distributed\nsystem to use population codes to represent both feature vectors and possible\nexplanations of the environmental data, thus decentralizing the cognitive\nprocess of the machine. Once the connection strengths for each individual\nhidden unit are determined they can be stored at the location of the hidden unit\nand used in a simple data processor to stochastically determine the activation\nlevel of that particular hidden unit in response to sensory inputs. It should be\nemphasized that a very nice feature of this way of realizing the Helmholtz\nmachine architecture is that it is very robust against failures of individual\ncomponents; i.e. a significant number of individual sensors and/or hidden units\nwould have to fail before the system as a whole completely loses its ability to\ninterpret sensory data.\nThe actual way in which the sensors and hidden units of a Helmholtz\nmachine are deployed for the detection of threats or undesirable situations will\ndepend on the particular application. For example, in the case of a biological\nwarfare warning system, one might be interested in detecting expanding clouds\nof small particles. For this application the particle size detectors need to be\ncoupled to the hidden units in such a way as to detect the motion and\nexpansion of a cloud of spore sized particles. Traditional methods for confirming\nthe presence of biological pathogens such as immunoassays would be too slow\nto use in an emergency response system. However, other techniques, e.g.\nspectral analysis of backscattered UV radiation, might provide sufficient hints\nwhich together with the cloud tracking units could be used to infer whether\nmicrobiological warfare agents have been released. This example is in fact a\ngood illustration of the kind of situation where a Helmholtz machine could to be\nused to fuse multi-modality sensor data and compute a probability that a\n\ndangerous siuation had arisen even though the data provided by each sensor\nmodality is highly ambiguous. It should be noted that nothing about the physics\nof how clouds of microbiological agents spread or the physics of remote\nsensing needs to be understood a priori by the Helmholtz machine. A Helmholtz\nmachine is capable of self-learning, and all that is required to recognize that a\nbiological warfare attack is underway is that the Helmholtz machine be trained\nto recognize the regularities and structure of the sensory inputs expected for\nsuch an attack.\n5. Intelligent agents for sentient networks\nIn a Helmholtz machine network both the processed data and\ncomputational processes are distributed throughout the network. If the sensor\nand data processor nodes are physically separated then some means must be\nprovided for these nodes to communicate with each other. This communication\nsupport must be capable of relaying the activation levels of the binary units in\none level of the network to the units in the next level of the network within a\nrelevancy time interval. Since each node of the network maintains a list of the\nother nodes it must communicate with, it may be convenient to make use of\nEthernet networking technology to implement the Helmholtz machine\narchitecture. Indeed one advantage of configuring the Helmholtz machine as an\nEthernet is that one could take advantage of the IEEE standards (1451.1 and\n1451.2) for connecting sensors to microprocessors and the sensor plus\nmicroprocessor to a network [18]. A significant caveat here though is that in\npractice it may only be possible to configure the Helmholtz machine as an\nEthernet if the nodes can be connected by wires or cables.\nIn cases where the nodes of the network are geographically separated it\nmay be necessary to use wireless communications; and in these cases it may\nbe more desirable to use some other interface between the sensor plus\nmicroprocessor and the network than the TCP/IP agent used in Ethernets. In\ngeneral though one will want to use time division multiplexing for the\ncommunications in order to minimize the total energy expended in network\ncommunications. In particular if one uses an internal clock for the network as a\nwhole in which each node is assigned a talk and listen time slot, then all\ncommunication devices can be powered down except during those few time\nslots in which they participate.\nThe total bandwidth required for the network communications will be\ndetermined by the response time desired and the number of nodes that must\ncommunicate with each other during a clock cycle. Generally, each hidden\nprocessor unit needs to communicate with every binary unit in the previous\nlayer of the network. However, only a small number of bits needs to be\ntransmitted between any two nodes at any given time, and this will reduce local\nbandwidth requirements. Furtherrmore in many practical cases the connection\nstrengths between distant units may be negligible so that communication need\nbe carried out only between nearby neighbors. Thus it appears that for the\nnetwork as a whole an Internet-like model for the communications where local\nnetworks feed data to hubs which can communicate with each other using much\n\nhigher bandwidth transmission channels might be appropriate. In any case\nbecause of the practical requirement to conserve energy resources\nasynchronous sampling techniques like those used by pager receivers would\nprobably not be acceptable for the nodes of a distributed Helmholtz machine\nnetwork. In the next section we suggest that this practical requirement may have\nsome profound consequences for our understanding of the cerebral cortex.\n6. Artificial consciousness?\nIf a network of sensors and simple processors configured as a Helmholtz\nmachine might be considered to be sentient , an obvious question is under what\nconditions would such networks become “conscious”? The neurophysiological\nmeaning of consciousness is still a matter of considerable controversy.\nNevertheless following the suggestion by Crick and Koch [19 ] that\nconsciousness is intimately related to the presence in the cerebral cortex of 40\nHz oscillations, there is increasing evidence [20] that coordination of the\nactivities within the cerebral cortex by an internal clock is somehow responsible\nfor the unique awareness of the external world associated with consciousness.\nIn addition an intriguing mathematical interpretation for a correlation between\nthe use of an internal clock in a neural network and conscious awareness has\nrecently emerged [21]; namely construction of a continuous 3-dimensional\nmanifold from a foliation of 2-dimensional representations of feature vectors will\nnot be topologically possible without using an internal clock if the 2-dimensional\nrepresentations lie on a topologically non-trivial surface.\nAs noted previously layering in a belief network is essential for\naccomodating hierarchical explanations of the input data. Furthermore in the\ncase of a Helmholtz machine it is natural to represent these hierarchical\nfeatures or “interpretations” using 2-dimensional population codes. An\ninteresting question is under what circumstances can a foliation of 2-\ndimensional population code representations be fused together to form a 3-\ndimensional manifold? There is compelling evidence that the human brain\nmakes use of continuous 3-dimensional representations of the external world\n[22], and indeed construction of such representations may be a very nice way of\nfusing different data modalities in artificial threat detection systems. The\ninteresting topological fact is that this will only be possible if there is an internal\nclock and each node is assigned a time slot. In fact 2-dimensional data\nrepresentations will necessarily lie on a topologically non-trivial surface if they\nare part of a Hopfield-like network. Further this circumstance will occur naturally\nif the feature vectors are self-organized [21], and in addition Hopfield-like\nnetworks could serve the purpose of error correction for feature vectors [23]. An\nadditional benefit [21] of fusing topologically non-trivial representations of\nfeature vectors is that the only stable excited states for the network are those\nthat correspond to excitations on the boundary; i.e. variations in input data. This\nof course is very reminescent of the phenomenology of the conscious cerebral\ncortex. Thus perhaps our most exciting observation is that with some slight\nmodifications the Helmholtz machine architecture might provide a framework for\ncreating artificially consciousness in a network of sensors and data processors.\n\nAcknowledgements\nI am very grateful to John Woodworth and Alan Spero for encouraging me to\nthink about the collective properties of sensor networks. I would also like to\nthank Jim Barbieri, Bill Buchanan and Dave Fuess for helpful discusssions. This\nwork was supported under DOE contract W-7405-ENG-48.\nReferences\n1. A. S. Bregman, “Perceptual Interpretation and the Neurobiology of\nPerceptions” in The Mind-Brain Continuum ed. R. Llinas and P. S.\nChurchland (MIT Press 1996).\n2. D. Kahneman and A. Tversky, Cognitive Psychology\n3 , 430 (1972).\n3. H. Reichenbach, “Predictive Knowledge” in The Rise of Scientific Philosophy\n(University of California Press 1951).\n4. G. Chapline, C. Y. Fu, and B. Law, “Neural Networks on Chips” in Energy and\nTechnology Review Oct.-Dec. 1992 , ed. J. Sefcik (Lawrence Livermore\nNational Laboratory 1992).\n5. N. Franceschini, “Engineering Applications of Small Brains” in Future\nElectron Devices\n7 , 38 (1996).\n6. J. J. Hopfield and D. W. Tank, Biol. Cybern. 52 , 141 (1986).\n7. S. Kirkpatrick, C. D. Gelatt, and M. P. Veccchi, Science 220 , 671 (1983).\n8. G. E. Hinton and T. J. Sejnowski, “Learning and Unlearning in Boltzmann\nMachines” in Parallel Distributed Processing , ed. D. E. Rumelhart et. al.\n(MIT Press, 1986).\n9. S. Kullback, Information Theory and Statistics (Wiley, 1959).\n10. For an introduction to current research in this area see the Web site for\nStanford University’s Knowledge Systems Laboratory.\n11. J. Pearl, Probabilistic Inference in Intellegent Systems (Morgan Kaufmann\n1988).\n\n12. L. R. Rabiner and B. H. Juang, Fundamentals of Speech Recognition\n(Prentice Hall 1993).\n13. W. A. Little, Math. Biosci.\n19 , 101 (1974); G. L. Shaw and R. Vasudevan,\nMath. Biosci. 21 , 207 (1974).\n14. B. Apolloni and D. de Falco, Neural Comp. 3 , 402 (1991).\n15. R. M. Neal, Neural Comp . 4 , 832 (1992).\n16. R. S. Zemel and G. E. Hinton, Neural Comp. 7 , 549 (1995).\n17. P. Dayan, G. Hinton, and R. Neal, Neural Comp. 7 , 889 (1995).\n18. J. Warrior, “Smart Sensor Networks of the Future” in Sensors Mar. 1997, 40.\n19. F. Crick and C. Koch, Seminars in the Neurosciences 2, 263 (1990).\n20. R. Llinas and D. Pare, “The Brain as a Closed System Modulated by the\nSenses” in The Mind-Brain Continuum ed. R. Llinas and P. S. Churchland\n(MIT Press 1996).\n21. G. Chapline, Network: Comp. Neural Syst.\n8 , 185 (1997).\n22. R. N. Shepard, Psychonomic Bulletin & Review 1 , 2 (1994).\n23. G. H. Hinton and T. Shallice, Psychol. Rev. 98 , 74 (1991).\ni “Cooperative Monitoring Program, Business Plan”, US Department of Energy,\nOffice of Nonproliferation and National Security, Office of Research and\nDevelopment, 1996."}
{"id": "adap-org/9712005", "meta": {"categories": ["adap-org", "nlin.AO", "q-bio"], "created": "1997-12-15", "extraction": {"body_chars": 35187, "cleaning": {"detected_repeated_margin_lines": ["2"], "page_count": 16, "removed_boilerplate_lines": 41}, "method": "pypdf_no_ocr", "source_pdf_bytes": 161589, "text_chars": 36301}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9712005", "primary_category": "adap-org", "source": "arxiv", "title": "Effects of neutral selection on the evolution of molecular species", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9712005"}, "text": "Effects of neutral selection on the evolution of molecular species\n\nAbstract\nWe introduce a new model of evolution on a fitness landscape possessing a tunable degree of neutrality. The model allows us to study the general properties of molecular species undergoing neutral evolution. We find that a number of phenomena seen in RNA sequence-structure maps are present also in our general model. Examples are the occurrence of \"common\" structures which occupy a fraction of the genotype space which tends to unity as the length of the genotype increases, and the formation of percolating neutral networks which cover the genotype space in such a way that a member of such a network can be found within a small radius of any point in the space. We also describe a number of new phenomena which appear to be general properties of neutrally evolving systems. In particular, we show that the maximum fitness attained during the adaptive walk of a population evolving on such a fitness landscape increases with increasing degree of neutrality, and is directly related to the fitness of the most fit percolating network.\n\narXiv:adap-org/9712005v1 15 Dec 1997\nEffects of neutral selection on\nthe evolution of molecular species\nM. E. J. Newman\nSanta Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501. U. S.A.\nRobin Engelhardt\nCenter for Chaos And Turbulence Studies, Dept. of Chemistry,\nUniversity of Copenhagen, Universitetsparken 5, Copenhagen Ø, Denmark\nAbstract\nWe introduce a new model of evolution on a fitness landscape po ssessing a tunable\ndegree of neutrality. The model allows us to study the genera l properties of molecular\nspecies undergoing neutral evolution. We find that a number o f phenomena seen in\nRNA sequence-structure maps are present also in our general model. Examples\nare the occurrence of “common” structures which occupy a fra ction of the genotype\nspace which tends to unity as the length of the genotype incre ases, and the formation\nof percolating neutral networks which cover the genotype sp ace in such a way that\na member of such a network can be found within a small radius of any point in the\nspace. We also describe a number of new phenomena which appea r to be general\nproperties of neutrally evolving systems. In particular, w e show that the maximum\nfitness attained during the adaptive walk of a population evo lving on such a fitness\nlandscape increases with increasing degree of neutrality, and is directly related to\nthe fitness of the most fit percolating network.\n1 Introduction\nBiological molecules such as proteins and RNAs undergo evolution jus t as\norganisms do, selected for their ability to perform certain function s by the\nreproductive success which that ability imparts on their hosts. It is believed\nthat many mutations of a molecule are evolutionarily neutral in the se nse\nthat they do not change the fitness of the molecule to perform the function\nfor which it has been selected. We have many examples of proteins wh ich\nPreprint submitted to Proc. R. Soc. London B 20 November 2018\n\nappear to possess approximately the same conformation and to pe rform the\nsame function in different species, but which have different sequenc es. Such\nproteins may differ only by a single amino acid or may have whole regions\nwhich have been substituted or inserted, or they may even be so diff erent as\nto appear completely unrelated. A mutation is said to be neutral if it c hanges\na molecule into one of these functional equivalents, leaving the viabilit y of its\nhost unchanged. This idea was first explored in detail by Kimura [1,2].\nIn fact, as Ohta has pointed out [3], it is not necessary that the fitn ess of a\nmolecule remain precisely the same under a given mutation for that mu tation\nto be considered neutral. A mutation which produces a change in fitn ess will\ncause the population sizes of the original and mutant strains to dive rge expo-\nnentially from one another over time. However, the time constant o f this ex-\nponential is inversely proportional to the change in fitness. Thus if the change\nin fitness is small, its effects will be felt only on very long time-scales. If these\ntime-scales are significantly longer than the time-scale on which muta tions\noccur (the inverse of the mutation rate), then the change in fitne ss will never\nbe felt. In effect, the mutation rate places a limit on the resolution wit h which\nselection can detect changes of fitness, so that small fitness cha nges are effec-\ntively, if not precisely, neutral.\nIt is possible that the concept of neutral selection can also be applie d to\nthe evolution of entire organisms. Certainly there are changes pos sible in an\norganism’s genome which have no immediate effect on its reproductive success,\nor which produce an effect sufficiently small that selection cannot de tect it on\nthe available time-scales. In this paper we will primarily use the languag e\nof molecular evolution, but the reader should bear in mind that the ide as\ndescribed may have wider applicability.\nDespite the long history of the idea, many aspects of neutral evolu tion are\nstill not well understood. In particular, we have very little idea of th e general\nbehaviours that can be expected of systems (molecules or organis ms) with a\nsignificant degree of evolutionary neutrality. The primary reason f or this gap\nin our understanding is that, despite many decades of hard work, w e still have\na rather poor idea of the way in which genomic sequences map onto mo lecular\nstructures and hence onto a fitness measure. In the case of ent ire organisms\nthe equivalent problem is that of calculating the genotype-phenoty pe map-\nping, which is even less well understood. One simple case in which neutr al\nevolution has been investigated in some detail is that of RNA structu re [4–7],\nalthough calculations so far are limited to secondary structures, a nd even these\ncannot be calculated with any reliability, so that these studies should be taken\nmore as a qualitative guide to the behaviour of systems undergoing n eutral\n\nselection than an accurate representation of the real world. The trouble with\nthis approach however is that RNAs are not a sufficiently general mo del that\nthe results gained from their study can be applied to other systems , such as\nprotein evolution or the evolution of whole organisms.\nAt the other extreme, studies have been performed of extremely simple mathe-\nmatical models of neutral evolution in the context of genetic algorit hms [8,9].\nAn example is the “Royal Road” genetic algorithm studied by van Nimwe -\ngen et al. [10,11]. These models possess highly artificial fitness functions cho-\nsen specifically to show a high degree of neutrality whilst at the same t ime\nbeing simple enough to yield to analytic methods. Like RNA secondary s truc-\ntures, these models have given us some insight into the type of effec ts we may\nexpect neutral evolution to produce, but, like RNAs, they are not sufficiently\ngeneral to be sure that these insights apply to other systems as w ell.\nIn this paper, therefore, we propose a new mathematical model o f neutral\nevolution. This model is an abstract model of a genotype to fitness map in the\nspirit of the Royal Road model. This approach allows us to sidestep th e prob-\nlems of incorporating the chemistry of real molecules in our calculatio ns and\nto investigate the properties of the system more quickly and in grea ter detail\nthan is possible with, for example, RNA structure calculations. In ad dition,\nthe model is more general than either the Royal Road fitness func tion or the\nRNA sequence-structure maps of Refs. [5,6]. In fact, it possesse s regimes in\nwhich it mimics the behaviour of both of these systems, as well as pro tein- and\norganism-like regimes. Because the behaviours of our model cover such a wide\nrange of possibilities, it seems reasonable to conjecture that gene ric features\nof the model which span all of these regimes may be common to most s ystems\nundergoing neutral selection. This is the power of our model, and th ese general\nresults are the results that we will concentrate on in this paper; we believe\nthat the generic behaviours of our model should be visible in the evolu tion of\nreal systems such as proteins which are, as yet, beyond our abilitie s to study\ndirectly.\nIn Section 2 we introduce our landscape model of neutral evolution . In Sec-\ntion 3 we discuss its properties and compare these with previous res ults for\nother systems undergoing neutral evolution. In Section 4 we discu ss the im-\nplications of our results for evolving molecular species. In Section 5 w e give\nour conclusions.\n\n2 The model\nSelective neutrality arises as a result of the many-to-one nature o f the sequence-\nstructure or genotype-phenotype maps found in biological syste ms. Many pro-\ntein sequences, for example, map onto the same tertiary structu re, and since\nthe fitness is primarily a function of the structure, such sequence s possess (at\nleast approximately) the same fitness. We wish to construct a mode l of this\nphenomenon without resorting to actual calculations of the struc ture of any\nparticular class of molecules. Only in this way can we hope to create a m odel\nwhich is general enough to represent the behaviours of many differ ent such\nclasses. Our approach is to employ a “fitness landscape” model of t he type\nfirst proposed by Wright [12,13] which maps sequence (or genotype ) directly\nto fitness. Structures (or phenotypes) appear in our model as c ontiguous sets\nor “neutral networks” of sequences possessing the same fitnes s.\nOur model is a generalization of the N K model proposed by Kauffman [14,15],\nwhich is itself a generalization of the spin glass models of statistical ph ysics [16].\nConsider a sequence of N loci, which correspond to the nucleotides in an RNA\nor to amino acids in the case of a protein. At each locus i we have a value\nxi drawn from an appropriate alphabet, such as {A,C,G,U} for RNAs, or the\nset of 20 amino acids in the case of proteins. We denote the size of th e al-\nphabet by A. Each locus interacts with a number K of other “neighbour”\nloci, which may be chosen at random or in any other way we wish. (Kauff man\nrefers to these interactions as epistatic interactions, though th is nomenclature\nis strictly only appropriate to the case where we are modelling the fitn ess of\nwhole organisms.) In the case of RNAs, bases most often interact w ith one\nother base to form either a Watson-Crick or a G-U pair. Some bases have\nboth pairing and tertiary interactions. Some, in the single stranded regions,\nhave very little interaction with any others. Thus a value of K = 1 might\nbe approximately correct for RNA. 1 For proteins, which have more complex\ntypes of interactions, a higher value of K may be appropriate.\nEach locus i makes a contribution wi to the fitness of the sequence, whose\nmagnitude depends on the value xi at that locus and also on the values at\neach of the K neighbouring loci. There are AK+1 possible sets of values for\nthe K + 1 loci in this neighbourhood, and hence AK+1 possible values of wi.\n1 It is possible to generalize the model to allow different loci t o interact with dif-\nferent numbers of neighbours, which gives a behaviour more r epresentative of true\nRNAs and proteins. In this paper however, we will confine ours elves to the case of\nconstant K for simplicity. As we will see, even with this constraint the model is still\nable to duplicate the behaviours seen in real systems.\n\nFollowing Kauffman and Johnsen [14] we choose this set of values at ra ndom.\nHowever, Kauffman and Johnsen chose the values to be random rea l numbers\nin the interval 0 ≤ wi < 1. We by contrast choose them to be integers in\nthe range 0 ≤ wi < F . Thus if F = 2 for example, each contribution wi is\neither zero or one. Now we define the fitness W of the entire sequence to be\nproportional to the sum of the contributions at each locus:\nW = 1\nN(F − 1)\n∑\ni\nwi. (1)\nThe fitness of all sequences thus falls in the range from zero to one , and there\nare N F − N + 1 possible fitness values in this range.\nIn the limit in which F → ∞ the probability that two sequences will possess\nthe same fitness becomes vanishingly small, and our model therefor e possesses\nno neutrality and is in fact exactly equivalent to the N K model. However,\nwhen F is finite the probability of two sequences possessing the same fitnes s\nis finite, so that the model possesses neutrality to a degree which in creases\nas F decreases. Neutrality is greatest when F takes the smallest possible\nvalue of two. Two sequences with the same fitness may be equivalent either\nto molecules which fold into the same conformation and perform the s ame\nfunction, or they may be equivalent to molecules with different confo rmations\nbut approximately the same contribution to the reproductive succ ess of the\nhost organism. The ruggedness of the landscape is controlled by th e parameter\nK, and is largest when K takes the maximal value of N − 1 [14,17]. In the\nnext section, we investigate the properties of the landscapes gen erated by our\nmodel, and show that with the right choice of parameters they can b e used to\nmimic real biological systems, such as RNAs.\n3 Evolution on neutral landscapes\nThe topology of a fitness landscape depends on the types of mutat ion al-\nlowed to molecules evolving on it. In biological evolution, point mutation s—\nmutations of the value at a single locus—are the most common. In this case,\na neutral network is defined to be a set of sequences which all poss ess the\nsame fitness and which are connected together via such point muta tions. In\nthe molecular case, we assume that closely similar sequences have th e same\nfitness because they fold into the same conformation, so that the se neutral\nnetworks correspond to (tertiary) structures. In the organis mal case they cor-\n\nrespond to phenotypes. 2\nThe model described in the last section possesses neutral networ ks of exactly\nthis type. The total fitness W in the model ranges from zero to one, but the\ngreatest number of sequences have fitness close to W = 0 .5. (In the extreme\ncase where K = N − 1 the distribution of W is binomial. When K < N − 1 it is\napproximately but not exactly so. Examples of these distributions a re shown\nin Figure 1 (a).) We would therefore expect the largest neutral net works to be\nthose with fitness close to W = 0.5 and this is indeed what we find in practice.\nTypically there are a large number of small neutral networks and a s mall\nnumber of large ones. In Figure 1 we show histograms of the sizes of the\nneutral networks for N = 20 and various values of K. For the RNA-like case\nK = 1, the histogram appears to be convex, indicating a distribution wh ich\nfalls off faster than a power law. The same behaviour has been in seen in RNA\nstudies by Gr¨ uneret al. [6]. As K increases the distribution flattens, and by\nthe time we reach K = 5 it is markedly concave. Thus the behaviour seen in\nRNAs is not in this case generic. For some intermediate value of K close to\nK = 2, the distribution appears to be power-law in form, perhaps indica ting\nthe divergence of some scale parameter governing the distribution , in a manner\nfamiliar from the study of critical phenomena [18].\nWe find that the total number of neutral networks SN grows exponentially as\naN with sequence length. In Figure 1 (b) we show the number of networ ks in\nour model for K = 1, both for two-letter {G,C} alphabets, and for a four-\nletter {A,C,G,U} alphabet. We find that a ≈ 1.5 for the A = 2 case and\na ≈ 2.3 for the A = 4 case. Interestingly, Stadler and co-workers [6,7,20] have\nperformed the same calculations for RNA sequences using the full s econdary-\nstructure calculation, and they also find an exponential increase in the number\nof structures with sequence length with values of a = 1.6 and a = 2.35 for the\ntwo- and four-letter cases respectively. This suggests to us tha t this behaviour\nis more general than the specific secondary-structure map emplo yed in the\nStadler calculations.\nThe largest neutral networks on our landscapes percolate, which is to say, they\nfill the sequence space roughly uniformly, in such a way that no sequ ence is\n2 One might argue that the individual fitness levels should cor respond to structures,\nnot the neutral networks. However, there are presumably man y structures which\nare not similar enough to be easily accessible from one anoth er by point mutations,\nand yet which possess similar fitnesses, at least to the mutat ion-limited accuracy\nwith which selection can distinguish. Thus it seems more app ropriate to draw a\ncorrespondence between networks and structures in the case of this model.\n\n1 10 100 1000 10000 100000\nsize of network\n−4\n−3\n−2\n−1\nfrequency of occurrence\n5 10 15\nfitness W\nfrequency\n5 10 15 20\nlength N\nneutral networks\n(a)\n(b)\nFig. 1. Histogram of the frequency of occurrence of neutral n etworks as a function\nof size for a particular realization of a landscape with N = 20, A = 2, F = 2, and\n(circles) K = 1, (squares) K = 2, (diamonds) K = 5. Inset (a): the frequency of\noccurrence of sequences as a function of fitness for N = 20, A = 2, F = 2, and\n(solid line) K = 19, (dotted line) K = 0. Inset (b): the number of neutral networks\nas a function of N for K = 1, F = 2, and (circles) A = 2, (squares) A = 4.\nmore than a certain distance away from a member of the percolating network.\nDetermining which networks are percolating is not an easy task. We h ave\ndeveloped two different measures to help identify percolating netwo rks, and\nwe describe them further in Ref. [19]. Gr¨ uner et al. [6] introduced instead the\nidea of a “common” network, which is one which contains greater tha n the\naverage number of sequences. We can employ this definition with our model\ntoo. We find that the common networks in the model form a small fra ction\nof the total number of networks, that fraction decreasing expo nentially as\nN increases, as shown in the inset to Figure 2. The same result is found in\nRNAs [6].\nIn the main frame of Figure 2 we show the fraction of sequences whic h fall in\nthe common networks as a function of N. As the figure shows, our numerical\nstudies indicate that this fraction increases with sequence length, tending to\none in the limit of large N. Even though the common networks form a smaller\nand smaller fraction of all networks as N becomes large, they nonetheless cover\nmore and more of the sequence space. These results have interes ting evolu-\n\n8 10 12 14 16 18 20\nlength of sequence N\n0.6\n0.7\n0.8\n0.9\n1.0\nfraction of sequences in common networks\n8 12 16 20\nlength N\n−2\n−1\ncommon networks\nFig. 2. The fraction of all sequences which fall in common net works as a function of\nN for a particular realization of the model with A = 2, F = 2, and (circles) K = 2,\n(squares) K = 4, (diamonds) K = 6. Inset: the number of common networks as a\nfraction of the total number of networks for the same landsca pes.\ntionary implications: they imply that as sequences become longer, a la rger\nand larger majority of structures (the small networks) are vanis hingly un-\nlikely to occur through natural selection. Evolution can only find the smaller\nand smaller fraction of “common” structures. The same conclusion s have been\nreached in the case of RNAs. However, the results presented her e indicate that\nthese conclusions are not specific to RNAs, and probably apply to mo st sys-\ntems undergoing neutral evolution.\nNext we have examined the dynamics of populations evolving on our lan d-\nscapes. These studies have yielded some of the most interesting re sults of this\nwork. In their studies with N K landscapes, Kauffman and Johnsen [14] made\na useful approximation in representing evolving populations by their single\ndominant sequence. This approximation is only valid in the case in which t he\ntime-scale for mutation is much longer than the time-scale on which se lection\nacts. For the moment we will assume this to be the case. A “random h ill-\nclimber” is a population of this type, represented by a single dominant strain,\nwhich tries mutations—point mutations in the present case—until it fi nds one\nwith higher fitness than the current strain. In this way the hill-climbe r per-\n\n0 4 8 12 16\nnumber of fitness levels F\n0.6\n0.7\n0.8\n0.9\n1.0mean fitness\n0 20 40 60 80\ntime t\n0.0\n0.2\n0.4\n0.6\n0.8\nfitness W\nFig. 3. The maximum fitness attained by a random hill-climber averaged over ten\nsimulations with N = 20, K = 4 and A = 2, as a function of the neutrality param-\neter F (circles). The lower curve (squares) is the fitness of the mos t fit percolating\nneutral network averaged over the same ten runs. Inset: the fi tness of one of the\nhill-climbers in the simulation as a function of time.\nforms an adaptive walk through sequences of ever-increasing fitn ess until it\nreaches a local fitness optimum. To study neutral landscapes we m odify this\nstrategy so that the hill-climber samples adjacent sequences at ra ndom until it\nfinds one of fitness greater than or the same as itself. Such a climber will move\nat random on a neutral network until it finds a mutation which takes it to a\nnetwork of higher fitness. In the upper curve of Figure 3 we show t he average\nfitness attained by such a walker over ten simulations on our landsca pes as a\nfunction of the neutrality parameter F . Recall that neutrality increases with\ndecreasing F . As the figure shows, the hill-climber on average finds higher fit-\nness maxima for higher degrees of neutrality. In other words, neu trality helps\nthe population to attain a greater fitness. This is certainly an idea wh ich has\nbeen entertained before in the literature, but it is lent a new convict ion when\nwe see it emerge in the behaviour of a general model such as this.\nThe lower curve on Figure 3 shows the fitness of the most fit percola ting net-\nwork averaged over the same ten landscapes. The curve follows qu ite closely\nthe form of the curve for the fitness of the local maxima found by t he hill-\n\nclimber. Our explanation of this result is as follows. The climber moves d iffu-\nsively on a neutral network until it finds a one-mutant neighbour wh ich belongs\nto a network of greater fitness, at which point it shifts to that net work. This\nprocess continues until it reaches a non-percolating network, at which point it\nis confined to the region occupied by the network and can only get as high as\nthe local maximum within that region. Thus the highest fitness attain able on\na landscape with neutrality depends directly on the highest fitness a t which\nthere are percolating networks. Since the landscapes with the gre atest degree\nof neutrality also have more and fitter percolating networks, this e xplains why\nhigher fitnesses are attained on landscapes with lower values of F .\nThe inset to Figure 3 shows the fitness of one of our hill-climbers as a f unction\nof time, and we can clearly see the steps in this function where the clim ber finds\nits way onto a neutral network of higher fitness. Similar steps have been seen,\nfor example, in laboratory experiments on the evolution of bacteria [21,22].\nAlthough it appears in the figure that nothing happens in the periods between\nthese jumps, it is at these times that the climber diffuses around its n etwork,\ntesting new mutations to find one of higher fitness. It is this diffusive motion\nwhich allows us to find higher fitness sequences on landscapes with hig her\ndegrees of neutrality. Van Nimwegen et al. [10] have dubbed these periods of\napparent stasis “epochs”. They also bear some similarity to the pala eontologi-\ncal “punctuated equilibria” described by Eldredge and Gould [23,24], a lthough\nthere are many other possible explanations for the periods of stas is seen in\nfossil evolution.\nTo investigate the epochs in more detail, we have performed simulatio ns of\ntrue populations evolving on our landscape. In these simulations we t ake a\npopulation of M sequences which at each generation reproduce with proba-\nbility proportional to their fitness, in such a way that the total pop ulation\nsize remains constant. Reproduction is also subject to mutation at some rate\nq per locus: with probability q the value at any locus in a sequence changes\nto a new randomly chosen one when the sequence is reproduced.\nIn Figure 4 we show the results of such a simulation for a population of\nM = 1000 sequences with N = 20 and a mutation rate of q = 5 × 10− 6. For this\nvalue of q the mutation rate is low enough that the hill-climber approximation\nused above is reasonable and, as the figure shows, the epochs visib le in the\ncase of the hill-climber appear also in the average fitness measured in the\npopulation simulations (solid line). For the simple case of the Royal Roa d\ngenetic algorithm, van Nimwegen et al. [11] have studied the epochs extensively\nand given a number of analytic and numerical results which may gener alize to\n\n0 400 800 1200 1600\ntime t\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nmean population fitness\n0 400 800 1200 1600\ntime t\n0.0\n1.0\n2.0\n3.0\n4.0\n5.0\npopulation entropy\nFig. 4. Solid line: the average fitness of a population of M = 1000 sequences evolving\non a landscape with N = 20, K = 4, A = 2 and F = 2. The mutation rate is\nq = 5 × 10− 6. Dotted line: the entropy of the population, which is a measu re of the\ndiversity of sequences present.\npresent case. 3\nThe length of the epochs in Figures 3 and 4 increase, on average, wit h in-\ncreasing fitness. This behaviour was also seen in the Royal Road, an d occurs\nbecause as the fitness increases the number of structures with h igher fitness\nstill dwindles. 4 The length of the epochs in fact depends on the rate of diffu-\nsion of the population across the neutral network, which in turn de pends on q,\nand on the density of “points of contact” between the network an d other net-\nworks of higher fitness [2]. Another interesting feature of the epo chs, also seen\nin the Royal Road, is that their average fitness does not correspo nd exactly\nto the fitness of any of the networks. Typically the average fitnes s is a little\nlower than the fitness of the dominant structure in the population b ecause\n3 In fact, the Royal Road model can be reproduced as a special ca se of the present\nmodel in which the epistatic interactions between loci take a particular structure,\nbeing collected in blocks, rather than placed randomly betw een pairs of loci.\n4 It would do this under any circumstances, but the problem is m ade particularly\nsevere by the binomial distribution of fitness values mentio ned above, which has an\nexponentially decreasing tail for high fitness.\n\ndeleterious mutants are constantly arising. Even though these mu tants are\nselected against, there are at any time enough of them in the popula tion to\nmake a noticeable difference to the average fitness. Since the numb er of pos-\nsible mutants with lower fitness than the dominant sequence increas es with\nincreasing fitness, it is also possible to get error threshold effects w ith increas-\ning fitness [25,26]. As the fitness increases, there may come a point w here the\nrate at which deleterious mutants arise in the population exceeds th e rate at\nwhich they are suppressed by selection, and at this point further im provement\nin fitness becomes impossible. Thus there may be a dynamical limit on th e\nfitness of populations, independent of the limit imposed by the struc ture of\nthe landscape discussed above. (This is true of landscapes without neutral\nevolution too, though the effect is much more prominent in the neutr al case.)\nIn Figure 4 we also show the entropy of the population as a function o f time\n(dotted line). The entropy is defined as\nS = −\n∑\ni\npi log pi, (2)\nwhere pi is the average probability of finding a particular member of the\npopulation with sequence i. 5 In this case, we see that the entropy is low\nduring the epochs, indicating that the population is compact, and he nce that\nthe strong-selection approximation made in the case of the random hill-climber\napplies. Only during the selection events themselves, when the popu lation\nmoves to a new neutral network does the entropy increase.\nSimulations similar in spirit to ours have been performed for population s of\ntRNAs by Fontana and co-workers [5,27]. In these simulations the au thors\nchose a “target” structure which was artificially selected for, and they also\nobserved epochs in the evolution as the population passed through a succession\nof increasingly fit structures on its way to the target.\n4 Discussion\nThe aim of this work is to study a model of neutral evolution which is ge neral\nenough to encompass behaviours typical of other more specific mo dels which\nhave been employed in the past. In this way we can reproduce in a gen eral\n5 In fact, calculating this entropy for a population of finite s ize in a large sequence\nspace is not a trivial task. The technical details of the calc ulation will be addressed\nin more detail in another paper [19].\n\ncontext the results which have been observed as special cases an d hence in-\nvestigate the extent to which these results are general propert ies of fitness\nlandscapes possessing neutrality, or particular to the systems in w hich they\nwere first observed. In this spirit, we put forward the following con jectures\nabout the fitness landscapes on which biological molecules evolve, ba sed on\nthe results of the investigations outlined in this paper.\n(1) The total number of possible structures increases exponent ially with se-\nquence length. The exponential constant of this increase appear s to be\napproximately numerically equal in both the general model and the o nly\nspecific case in which it has been studied, that of RNA secondary str uc-\nture.\n(2) There are a large number of structures which correspond to a small num-\nber of sequences, and a small number of structures which corres pond to a\nlarge number of sequences. The exact form of the histogram of st ructure\nfrequency, shown in Figure 1, varies depending on the parameters of our\nmodel. However, for certain values of the parameters it has a form similar\nto that seen in RNA studies, whilst for others it appears to follow a po wer\nlaw.\n(3) The “common” structures—ones which correspond to a large n umber of\nsequences—constitute an exponentially decreasing fraction of th e total\nnumber as sequence length increases. Conversely, however, the y cover a\nfraction of the sequence space which tends to unity for long seque nces.\n(4) The evolution of populations, at least on short time-scales, is do minated\nby the presence of neutrality. Neutrality helps the population to fin d\nstructures of high fitness without having to cross fitness barrier s. The\nhighest fitness which can be found in this way is closely related to the\nfitness of the highest percolating neutral network, which itself de pends on\nthe amount of neutrality. For landscapes with a higher degree of neutrality\ntherefore, the population typically reaches a higher fitness.\n(5) The fitness may be limited by error threshold effects, which are p artic-\nularly severe for landscapes of this type, because the size of the n eutral\nnetworks (and hence the ratio of numbers of beneficial and harmf ul mu-\ntants) falls exponentially with increasing fitness.\n(6) Evolution proceeds in jumps separated by “epochs” in which the fitness\nappears to change very little. In fact, the population uses these e pochs to\ndiffuse across the current neutral network, allowing it to search a larger\nportion of sequence space for beneficial mutations.\n\n5 Conclusions\nTo conclude, we believe that by studying a simple and general model o f a neu-\ntral landscape, we should be able to distinguish properties of specifi c systems\nundergoing neutral selection from properties common to all such s ystems. We\nhave found a number of potential candidates for inclusion in a list of s uch\ncommon properties. There are many interesting lines of investigatio n which\nwe have not been able to pursue in this short work, including details of the\nstructure and size of the neutral networks such as percolation m easures and\ncovering radii, details of population dynamics on these networks inclu ding\nentropy and other statistical measures of the structure of suc h populations,\ncalculations of the length of epochs, of the maximum fitness obtaina ble on\nthese landscapes, of the effects of error threshold effects on ma ximum fitness,\nand many effects of the variation of the parameters of the model, p articu-\nlarly the effects of the variation of the level of epistasis K and the neutrality\nparameter F . Some of these questions will be addressed in a forthcoming work.\nAcknowledgements\nThe authors would like to thank James Crutchfield, Melanie Mitchell, Er ik\nvan Nimwegen and Paolo Sibani for interesting discussions. This work was\nsupported in part by the Santa Fe Institute and DARPA under gran t number\nONR N00014–95–1–0975.\nReferences\n[1] Kimura, M. 1955 Solution of a process of random genetic dr ift with a continuous\nmodel. Proc. Natl. Acad. Sci. USA 41, 144.\n[2] Kimura, M. 1983 The Neutral Theory of Molecular Evolution. Cambridge\nUniversity Press.\n[3] Ohta, T. 1972 Population size and rate of evolution. J. Mol. Evol. 1, 305.\n[4] Schuster, P., Fontana, W., Stadler, P. F. and Hofacker, I . L. 1994 From sequences\nto shapes and back: A case study in RNA secondary structures. Proc. R. Soc.\nLondon B 255, 279.\n\n[5] Huynen, M., Stadler, P. F. and Fontana, W. 1996 Smoothnes s within\nruggedness: The role of neutrality in adaptation. Proc. Nat. Acad. Sci. 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P. 1982 Self-replication wit h error: A model for\npolynucleotide replication. Biophys. Chem. 16, 329.\n[27] Fontana, W. and Schuster, P. 1987 A computer model of evo lutionary\noptimization. Biophys. Chem. 26, 123."}
{"id": "adap-org/9712006", "meta": {"categories": ["adap-org", "nlin.AO"], "created": "1997-12-23", "extraction": {"body_chars": 39818, "cleaning": {"detected_repeated_margin_lines": ["2"], "page_count": 22, "removed_boilerplate_lines": 83}, "method": "pypdf_no_ocr", "source_pdf_bytes": 93961, "text_chars": 41573}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9712006", "primary_category": "adap-org", "source": "arxiv", "title": "Adaptive Competition, Market Efficiency, Phase Transitions and Spin-Glasses", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9712006"}, "text": "Adaptive Competition, Market Efficiency, Phase Transitions and Spin-Glasses\n\nAbstract\nWe analyze a simple model of adaptive competition which captures essential features of a variety of adaptive competitive systems in the social and biological sciences. Each of N agents, at each time step of a game, joins one of two groups. The agents in the minority group are awarded a point, while the agents in the majority group get nothing. Each agent has a fixed set of strategies drawn at the beginning of the game from a common pool, and chooses his current best-performing strategy to determine which group to join. For a fixed N, the system exhibits a phase change as a function of the size of the common strategy pool from which the agents initially draw their strategies. For small pool sizes, the system is in an efficient market phase. All information that can be used by the agents' strategies is traded away, no agent can accumulate more points than would an agent making random guesses, and thus the commons suffer,since relatively few points are awarded to the agents in total. For large initial strategy pool sizes, the system is in an inefficient market phase, in which there is predictive information available to the agents' strategies, and some agents can do better than random at accumulating points. In this phase, the total number of points awarded to the agents is greater than in a game in which all agents guess randomly, and so the commons do relatively well. At a critical size of the strategy pool marking the cross-over from the efficient market to the inefficient market phases, the commons do best. This critical size of the pool grows monotonically with N. The behavior of this system has some features reminiscent of a spin-glass.\n\nFigure 1\nPast Prediction\n0 0 0 1\n0 0 1 1\n0 1 0 0\n0 1 1 1\n1 0 0 0\n1 0 1 0\n1 1 0 0\n1 1 1 1\n\nAdaptive Competition, Market Efficiency, Phase Transitions and\nSpin-Glasses\nRobert Savit, Radu Manuca and Rick Riolo\nProgram for the Study of Complex Systems and Physics Department\nUniversity of Michigan\nAnn Arbor, MI 48109\nAbstract\nIn this paper we analyze a simple model of adaptive competition which captures essential\nfeatures of a variety of adaptive competitive systems in the social and biological sciences.\nIn this model, each of N agents, at each time step of a game, joins one of two groups.\nThe agents in the minority group are awarded a point, while the agents in the majority\ngroup get nothing. Each agent has a fixed set of strategies drawn at the beginning of the\ngame from a common pool, and chooses his current best-performing strategy to\ndetermine which group to join. We find that for a fixed N, the system exhibits a phase\nchange as a function of the size of the common strategy pool from which the agents\ninitially draw their strategies. For small pool sizes, the system is in an efficient market\nphase in which all information that can be used by the agents' strategies is traded away,\nand no agent can accumulate more points than would an agent making random guesses.\nIn addition, in this phase the commons suffer, and relatively few points are awarded to\nthe agents in total. For large initial strategy pool sizes, the system is in an inefficient\nmarket phase, in which there is predictive information available to the agents' strategies,\nand some agents can do better than random at accumulating points. In addition, in this\nphase, the total number of points awarded to the agents is greater than in a game in which\nall agents guess randomly, and so the commons do relatively well. At a critical size of\nthe strategy pool marking the cross-over from the efficient market to the inefficient\nmarket phases, the commons do best. This critical size of the pool grows monotonically,\nand in a very simple way with N. The behavior of this system has features reminiscent\nof a spin-glass in statistical physics, with the small pool size phase being, in a certain\nsense, more glassy than the large pool phase. We argue that the structure we have\nelucidated has important implications for a range of phenomena in the social and\nbiological sciences, as well as for the general study of adaptive, competitive systems.\n12/3/97 a\n\nIntroduction\nMost systems in the biological and social sciences involve a number of interacting\nagents, each making behavioral choices in the context of an environment that is formed,\nin large part, by the collective action of the agents themselves, and with no centralized\ncontroller acting to coordinate agent behavior. In the most interesting and difficult to\nanalyze cases, the agents have heterogeneous strategies, expectations and beliefs [Arthur,\n1994]. In some cases the agents' strategies may be self-validating, at least for a limited\ntime. For example, in the financial markets a wide-spread belief that a commodity will\nrise in price may perforce result in a price rise for that commodity [Arthur et al, 1996].\nBut unless there are fundamental reasons for the price rise, such bubbles eventually tend\nto burst, so that widely-shared strategies are often self-defeating in the long run. Thus, in\nmany systems, and most clearly in those in which agents compete for scarce resources,\nsuccessful agents will employ strategies that differentiate them from their competitors, so\nthat the agents will place themselves in groups in which target resources are not over-\nutilized. Examples of such systems include animals foraging for food, firms searching\nfor profitable technological innovations, packets looking for paths through the internet,\nor people trying to go to (overly) popular events or places. Furthermore, from the point\nof view of overall system performance, the best strategy sets are those that result in\ncoordinated resource utilization so that average agent experience is relatively good, and\nthe scarce resource is consumed near its limiting rate. Examples of systems of competing\nagents in which such coordinated allocation of resources is critical include ecological\ncommunities [Cody and Diamond, 1975], routers trying to send packets over the Internet\n[Kahin and Keller, 1997], and humans trying to decide on which night to go to a popular\nbar [Arthur, 1994].\nThese systems are enormously complicated and their detailed dynamics may depend on\nparticular characteristics of the agents and their interactions. Nevertheless, there also are\nfundamental properties which are shared by all these systems. If we have any hope of\never understanding these kinds of collective adaptive systems, or even of understanding\nthe terms in which we should analyze them, we must first understand the dynamics\nimposed by their most basic shared properties.\nIn this paper we analyze a simple model that incorporates the basic adaptive and\nfeedback features of systems of agents competing for a scarce resource [Challet and\nZhang, 1997]. In this model, each agent chooses to be in one of two groups at each time\n\nstep, and those agents in the minority group are awarded a point. Each agent has his own\nset of strategies, chosen initially from a pool of available strategies, from which he uses\nthe currently best-rated strategy (based on past performance) to select a group to join.\nEach strategy uses information about which group was in the minority during (a few)\nprevious time steps to predict which will be the minority group during the current time\nstep. We find that as a function of the size of the strategy pool available to the agents,\nthere is a transition which separates an efficient market phase from an inefficient market\nphase. If the number of available strategies is not sufficiently large, then the system is in\na phase, in which the agents are \"frustrated\" in their attempt to find minority positions,\nand many features of the aggregate behavior of the system in this phase are analogous to\nthose of the glassy phase of a spin-glass [Mezard, Parisi and Virasoro, 1987; Fischer and\nHertz, 1991]. In this phase the market is efficient, and it can be shown that all the\nrelevant information accessible to the agents' strategies has been traded away by their\ncompetition [Malkiel, 1985; Fama, 1970; Fama, 1991]. Thus, no agent can ever exceed a\n50% rate of success. (Remarkably, however, there is still information in the record of\nwhich group was the minority group as a function of time, but that information cannot be\nused by the agents' strategies.) In addition, the collective behavior of the agents is\ngenerally substantially worse than in a random market in which the agents join one of the\ntwo groups independently and randomly with equal probability. Above a critical size of\nthe strategy pool, the system is in an inefficient market phase. Here there is predictive\ninformation available to the agents' strategies and some agents can achieve better than a\n50% success rate. In addition, the collective experience of the agents is much improved\nand is substantially better than in a random market, indicating an emergent coordination\namong the agents. As a function of the size of the strategy pool, the best collective\nperformance occurs near the critical size of the strategy pool which marks the cross-over\nfrom the efficient market phase to the inefficient market phase. The critical size of the\nstrategy pool scales with N, the number of agents playing the game, in a very simple\nway. These unexpected emergent properties have profound implications for the study\nand epistemology of competitive markets in the biological and social sciences.\nWe first describe the model, then present the results of our analysis and finally discuss\nthe implications of our work for the study of adaptive, competitive social and biological\nsystems.\n\nThe Model\nThe simple model of competition we discuss here consists of N agents playing a game\n[Challet and Zhang, 1997]. The rules of the game are as follows: At each time step of\nthe game, each of the N agents playing the game joins one of two groups, labeled 0 or 1.\nEach agent that is in the minority group at that time step is awarded a point, while each\nagent belonging to the majority group gets nothing. An agent chooses which group to\njoin at a given time step based on the prediction of a strategy. The strategy uses\ninformation from the historical record of which group was the minority group as a\nfunction of time. A strategy of memory m is a table of 2 columns and 2m rows. An\nexample of an m=3 strategy is shown in Fig. 1. The left column contains all the eight\npossible combinations of three 0's and 1's. To use this strategy, an agent observes which\ngroup was the minority group during the last three time steps, and finds that string of 0's\nand 1's in the left hand column of the table. The corresponding entry in the right hand\ncolumn contains that strategy's determination of which group (0 or 1) the agent should\njoin during the current time step.\nIn each of the games discussed here, all strategies used by all the agents have the same\nvalue of m. At the beginning of the game each agent is randomly assigned s (>1) of the\n22m possible strategies, with replacement.1 For his current play the agent chooses his\nstrategy that would have had the best performance over the history of the game up to\nthat time. Ties between strategies are decided by a coin toss. Following each round of\ndecisions, the cumulative performance of each of the agent's strategies is updated by\ncomparing each strategy's latest prediction with the current minority group. Because the\nagents each have more than one strategy, the game is adaptive in that agents can choose\nto play different strategies at different moments of the game in response to changes in\ntheir environment; that is, in response to new entries in the time series of minority groups\nas the game proceeds. Because the environment (i.e. the time series of minority groups)\nis created by the collective action of the agents themselves, this system has very strong\nfeedback, reminiscent of, for example, the financial markets.\n\nResults\nIn what follows we will report and interpret the results of this game for a range of values\nof N (odd), m and s=2. The qualitative results also hold for other values of s that are not\nextremely large.2 To achieve stable results, the game must be run for a long enough\ntime. What is long enough depends on N and m. For our runs, in which N ranged\nbetween 11 and 1001, we found that 10,000 time steps were generally sufficient, except\nfor the largest values of N, which required runs of 100,000 time steps. To start the game,\nwe also create a short (of order m) random history of 0's and 1's, so that the strategies can\nbe initially evaluated. The asymptotic statistical results of any run do not materially\ndepend on what this random string is.\nTo begin to understand the behavior of this system, consider the time series of the\nnumber of agents belonging to group 1 (≡L1). (This information is not available to the\nagents but it is available to the researchers.) The mean of this series is generally close to\n50% for all values of N, m and s (we shall return to this point below), and so the standard\ndeviation, σ, of this time series is a measure of how well the commons do: The smaller\nσ, the more total points are awarded to all agents combined. That is, if there are typically\nmany fewer than 50% of the agents in the minority, then σ will be large and there will be\nfew total points awarded, while if σ is small, then most of the time the minority group\nwill consist of only slightly fewer than half of the agents, and more total points will be\nawarded.\nThe behavior of σ is quite remarkable. In Fig. 2, we plot σ for these time series as a\nfunction of m for N=101 and s=2. For each value of m, 32 independent runs were\nperformed. The horizontal dashed line in this graph is at the value of σ for the random\ngame, i.e. for the game in which the agents randomly choose 0 or 1, independently and\nwith equal probability at each time step.\nNote the following features:\n1. For small m, the average value of σ is very large (much larger than in the random\ncase). In addition, for m<6 there is a large spread in the σ 's for different runs\nwith different (random) initial distributions of strategies to the agents, but with\nthe same m.\n\n2. There is a minimum in σ at m=6 at which σ is less than the standard deviation of\nthe random game. We shall refer to the value of m at which the σ vs. m curve\n(for fixed N) has its minimum as mc.3 Thus, in Fig. 2, mc=6. Also, for m≥mc,\nthe spread in the σ's appears to be small relative to the spread for m<mc.\n3. As m increases beyond 6, σ slowly increases, and for large m approaches the\nvalue for the random game.\nThe system clearly behaves in a qualitatively different way for small and large m. To\nfurther study the dynamics in these two regions, we consider the time series of minority\ngroups, (≡G) the data publicly available to the agents. We want to study the information\ncontent of strings of consecutive elements of this time series of various lengths (including\nstrings of length m) for different values of m and N. To do this, we consider the\nconditional probability P(1|uk). This is the conditional probability to have a 1\nimmediately following some specific string, uk, of k elements of G. For example,\nP(1|0100) is the probability that 1 will be the minority group at some time, given that\nminority groups for the four previous times were 0,1,0 and 0, in that order. Recall that in\na game played with memory m, the strategies use only the information encoded in strings\nof length m to make their choices. In Fig. 3, we plot P(1|uk) for G, the time series of\nminority groups generated by a game with m=4, N=101 and s=2. Fig. 3a shows the\nhistogram for k=m=4 and Fig. 3b shows the histogram for k=5. Note that the histogram\nis flat at 0.5 in Fig. 3a, but, remarkably, is not flat in Fig. 3b. Thus, any agent using\nstrategies with memory (less than or) equal to 4, will find that those strings of minority\ngroups contain no predictive information about which group will be the minority at the\nnext time step. But recall that this time-series was itself generated by players playing\nstrategies with m=4. Therefore, in this sense, the market is efficient and no strategy\nplaying with memory (less than or) equal to 4 can, over the long run, accumulate more\npoints than would be accumulated randomly. (In principle, an agent switching strategies\nadaptively can do better than random, but we have explicitly verified that this does not\nhappen in this phase. The expicit reason for this is described in the results section and in\nmore detail in [Manuca, Riolo and Savit, 1998].) But note also that the time series of\nminority groups is not a random (IID)4 string. There is information in this string, as\nindicated by the fact that the histogram in Fig. 3b is not flat. However, that information\nis not available to the agents playing the m=4 game who collectively generated that string\nin the first place.\n\nWe can repeat this analysis for m≥6 (N=101, s=2). For this range of m, the\ncorresponding histogram for k=m is not flat, as we see in Fig. 4 for the m=6 game. In\nthis case, there is significant information available to agents playing the game with\nmemory m and the market is not efficient. Indeed, some individual agents can\naccumulate more points than they would by simply making random guesses. We have\nverified these statements by explicitly calculating the points won by individual agents in\ngames with different values of m. For m<6 (N=101, s=2), no agent ever achieves results\nstatistically greater than 50%, while for m≥6 some agents do win more than 50% of the\ntime (statistically significantly). It is noteworthy that even for m=5, the histogram of\nconditional probabilities with k=m=5 is flat, even though σ is less than in the random\ngame.\nHow does the system behavior change as we change the number of agents? One can\nrepeat the calculation of σ for different N. One finds, plotting σ vs. m for fixed N, that\nin all cases one obtains a graph similar to that in Fig. 2, but in which the position of the\nminimum, mc, is proportional to lnN. In addition, σ and the spread in σ behave in very\nsimple ways with changes in N which differ depending on whether m is greater than or\nless than mc. In Fig. 5 we study the behavior of σ as a function of N for m=3 and m=16.\nFor the range of values of N used in these figures, m=3 is to the left of the minimum in\nthe curve of σ vs. m (i.e. 3<mc for all these values of N) and m=16 is to the right of the\nminimum (16>mc). In Fig. 5a we plot σ vs. N on a log-log scale. We see that for m=3 σ\nis proportional to N, while for m=16, σ is proportional to N1/2. This is typical: for fixed\nm, and m<mc, σ is proportional to N, while for fixed m and m>mc σ is proportional to\nN1/2. In Fig. 5b, we plot, again for m=3 and 16, the spread in σ, i.e., the standard\ndeviation of the σ's (≡∆σ ) as a function of N, on a log-log scale. Here we also see power\nlaw behavior: for m=3 ∆σ is proportional to N, while for m=16, ∆σ is proportional to\nN1/2. As before, this behavior is representative of the two behaviors seen for values of\nm<mc and values of m>mc, respectively.\nThe transition between these very different behaviors is at mc~lnN. We have found,\nusing scaling arguments that, to a first approximation, σ2/N is a function only of 2m\nN\n≡z.\nTo see this explicitly, we plot in Fig. 6 σ2/N as a function of z on a log-log scale for\nvarious N and m (with s=2). We see first that all the data fall on a nearly universal\ncurve. The minimum of this curve is near 2mc/N=zc≈0.5, and separates the two different\nphases. The slope for z<zc approaches -1 for small z, while the slope for z>zc\n\napproaches zero for large z, consistent with the results of Fig. 5.5 Because σ2/N depends\nonly on z, it is clear that for fixed z σ is proportional to N1/2 for any fixed z, both above\nand below zc. In addition, it can be shown that, for fixed z ∆σ is approximately\nindependent of N, approaching a z-dependent constant as N→∞ . The N→∞ limit of ∆σ\nis large for small values of z and decreases monotonically with increasing z. It is unclear\nwhether or not ∆σ is non analytic at zc.\nDiscussion\nIn this section, we will first present some qualitative arguments that explain the different\nbehaviors for small and large m, and the scaling results with N. More detailed\nexplanations along with results of corroborating simulations will appear in a forthcoming\npublication [Manuca, Riolo, and Savit]. Following that, we will discuss some\nimplications of this study for a wide variety of social and biological systems.\nConsider first the small m region (m<mc). In this region, the time series of the number\nof agents belonging to group 1 (≡L1) has a very unusual structure. Consider the time\nseries of minority groups, and focus on one particular binary string, χ, of length m.\nNow, from L1 form the time series of the number of agents which choose group 1 in\nresponse to the first, third, fifth and other odd occurrences of χ. Now form the time\nseries of the number of agents which choose group 1 in response to all the even\noccurrences of χ. It turns out that the standard deviation of the series of responses to the\nodd occurrences of χ is of order N1/2, while the standard deviation of the time series of\nresponses to the even occurrences of χ is of order N. This is true for all possible strings\nof length m in the time series of minority groups. Thus, in the whole time series of the\nnumber of agents belonging to group 1, there is a bursty structure with large (order N)\nexcursions from the mean separated by smaller excursions of order N1/2. For large N,\nthe contributions of the large, order N excursions to the standard deviation dominate, and\nso σ ∝ N in this region.\nThis behavior can be understood as follows: Consider the pool of all 22m strategies. For\ns=2, each agent chooses two strategies randomly from the pool. For very large N, the\nstrategies played by the agents are a good approximation to the population of the pool.\nNow, divide the strategies into two groups according to their response to some string, say\nχ. The first time χ appears in the time series of minority groups, the agents will choose\nto join group 0 or 1, and which strategies they play will be a more or less random sample\nof the strategies in the pool. Consequently, we expect that the population of group 1 will\n\ndeviate from 50% by a number of order N1/2. Suppose that the minority group turns out\nto be group 1. Then, in the pool of strategies, all those whose response to χ is 1 will get\nan additional point. The next time the string χ appears in the series of minority groups,\nthose strategies whose response to χ is 1 will be preferentially chosen. Now, if N is\nlarge, then approximately 75% of the agents will join group 1. (i.e., on average, only\nthose agents both of whose strategies respond to χ with a 0 will join group 0.)\nConsequently, the minority group at this time step will be group 0, and all those\nstrategies which respond to χ with a 0 will get a point. Thus, after the second occurrence\nof χ, all strategies will have gained one point due to their response to χ, some following\nthe first occurrence, and the others following the second occurrence. There will therefore\nbe no strong preference for the agents' responses to the third occurrence of χ (but see the\ndiscussion in [Manuca, Riolo and Savit, 1998]), and deviations in the membership of\ngroup 1 at the third occurrence of χ will again be of order N1/2. This period two process\nrepeats, for all strings and gives rise to the structure described above. A more detailed\ndescription of this dynamics will appear in [Manuca, Riolo and Savit, 1998].\nThe precise distribution of strategies to the agents at the beginning of the game differs\nfrom run to run. The existence of the period two oscillations described above does not,\nfor large N, depend on the details of that distribution. However, the coefficient in front\nof N in the expression for the standard deviation in the responses to even occurrences of\na given string does depend on the distribution of initial strategies. Consequently, the size\nof σ will vary from run to run, but, in leading order, will still be proportional to N.\nThus, the spread in the σ's is also proportional to N in leading order in the low m phase\n(m<mc), as we saw in Fig. 5.\nThe existence of different runs with different σ's is reminiscent of the glass phase of a\nspin-glass system. The analogy goes deeper: In the spin-glass case different samples of\nthe glassy material behave differently as a result of frozen in randomly distributed\nimpurities. From one sample to the next, the precise way in which the impurities are\ndistributed affects the thermodynamic behavior of that sample. A spin-glass order\nparameter, which is non-zero in the glassy phase just expresses the fact that the\nthermodynamics of different samples is different. In our case, the strategies are\ndistributed to the agents randomly, but the precise way in which that randomness is\nrealized in one game affects the size and distribution of the membership of groups 0 and\n1, thus leading to different time-averaged behavior which is most pronounced in the low\nm phase. Mathematically, ∆σ , the spread in the σ's, bears some resemblance to a spin-\n\nglass order parameter. Furthermore, while the expectation value of the number of agents\nbelonging to group 1 is nearly 50% in the high m phase, it deviates more strongly from\n50% in the low m phase, and differs from run to run. This is also analogous to the\nbehavior of a spin-glass, in that the magnetization in the glassy phase is non-zero and\ndiffers from sample to sample. Although there is some sense in which the low m phase is\nglassy, we do not mean to imply that the transition at mc is a transition from a glassy to a\nnormal phase. Both the high and low m phases partake of glassy behavior and have\nelements of broken ergodicity due to the initial distribution of strategies, although the\nglassiness is more pronounced in the low m phase. This analogy will be more fully\ntreated in a forthcoming publication [Manuca, Riolo and Savit].\nIn contrast to the behavior in the low m phase, for fixed m\n≥mc σ, the standard deviation\nof L1, scales with N1/2. In this high m phase, for a given N, the strategy space is\nsufficiently large so that the strategies assigned to the N agents are not a representative\nsample of the entire strategy space. Consequently, the arguments used in describing the\nlow m phase do not hold, and there are not the same very strong temporal correlations in\nthe response of the system to successive occurrences of a given string. The scaling of σ\nwith N1/2 just represents the variations in the membership of N agents whose decisions\nare not very tightly coupled. However, there are still strong dependencies among the\ndecisions of the agents. In fact, it is the remarkable emergent coordination among the\nagents' decisions that accounts for the fact that σ is below the random result in this phase.\nThe precise way in which this comes about will be discussed in detail elsewhere\n[Manuca, Riolo and Savit, 1998], but the issue can be thought of in the following way:\nStrategies that would have been most effective in the past at predicting minority groups\nare more likely to be used by the agents. Those strategies have specific predictions for\neach string of length m. Consequently, there is an induced dependence among the\ndecisions of the agents which can be expressed as nontrivial conditional probabilities.\nThat is, the probability that at some time agent i chooses group 1, changes depending on\nthe choices of the other agents at that time.6 It can be shown that, for fixed m, such\nnontrivial conditional probabilities cause deviations in σ from that of the random game.\nIn the high m phase, this term is negative, giving rise to a lower σ.\nNote that the region of greatest coordination (smallest σ) is when z=2m/N is of order\none. We can understand this in the following way: Dependencies among the strategies\nare induced by the agent's selection of the best strategies, as described above. But each\nchosen strategy dictates a response to 2m different strings of length m. Thus, as m\n\nincreases, for fixed N, it becomes increasingly difficult for strategies to coordinate all of\ntheir entries. (I.e., at any moment there are N strategies in play, and they must minimize\nfluctuations over 2m choices.) Consequently, for fixed N, maximal coordination will be\nachieved for some finite m (=mc), whose value is a monotonically increasing function of\nN. As m increases for fixed N, coordination becomes less effective, and σ approaches\nthe result of the random game from below. It can be shown that the correction term is\nproportional to 1/2m [Manuca, Riolo and Savit].7\nIt is quite remarkable that σ2/N lies on a universal curve as a function of the scaling\nvariable z=2m/N, as shown in Fig. 6. One immediate consequence of this is that the\ncritical value of m, mc is proportional to lnN as we have found. An intriguing result here\nis that for maximum coordination N should be roughly the same size as the dimension of\nthe strategy space. Thus, it is the dimension of the strategy space that seems to be the\nrelevant measure of the diversity of strategies available to the agents.\nImplications\nThis work points up a number of very important and general features that must be\nconfronted in any analysis of adaptive competition among N players. First, it is clear that\nthe size of the strategy space available to the agents and the number of agents playing the\ngame are crucial parameters in determining the qualitative behavior of the system.\nMoreover, their ratio, z, seems to be the predominant parameter for many of the most\nimportant features of the game. In particular, if, for a given number of agents, the\nstrategy space is not large enough, then the game will be efficient in a well-defined\nsense, but the commons will suffer. There appears to be a critical size of the strategy\nspace that gives rise to the greatest common good, at least as measured by σ, and\neverywhere in the inefficient market phase the common good is enhanced over a strictly\nrandom game by an implicit emergent coordination. Note that in this game there is no\ncentral controlling authority. The improvement of the common good in the inefficient\nphase is a purely emergent effect. This general structure, an efficient, but poorly\nperforming market when the strategy space is small, a market whose average\nperformance is close to random when the strategy space is large (but which, nonetheless,\nmay produce wealthy agents [Manuca, Riolo and Savit, 1998]), and good performance,\nboth for the commons and for individuals when the dimension of the strategy space is\nmatched to the number of players is likely to be a characteristic that transcends the\nspecific model discussed here. These observations have clear relevance to a number of\nimportant issues. For example, there may be significant public policy implications for\n\nthe design of markets and other regulatory issues, as well as implications for the study of\na wide range of competitive systems including ecological systems.\nThe phase structure of this model as a function of 2m/N is quite intriguing. In particular,\nit is quite remarkable that there is such a clear transition between the efficient and\ninefficient phases. Although some of the results of this system may be specific to the\nmodel, the observation of the general phase structure and the features it shares with a\nspin-glass has clear implications for the epistemology of adaptive systems. It suggests, at\nleast as a first step, a set of concepts and tools which are likely to be fruitful in thinking\nabout, and analyzing adaptive competitive systems.\nOne of our most noteworthy results is that the agents so efficiently and selectively trade\naway the information accessible to them in the efficient phase (z<zc). The overly\ncompetitive dynamics in this phase is remarkably successful at removing information\nfrom the reach of the players, and thus removing any possibility that any agent could\nperform well. What is even more remarkable is the selective way in which this\ninformation is removed. There is still significant information contained in the history of\nminority groups, as we see in Fig. 3b. But it is simply not available to the players whose\ndecisions produced that series in the first place. The precise way in which this\ninformation is removed from the system will be discussed elsewhere [Manuca, Riolo and\nSavit, 1998].\nIt is clear that the cost of over competition both to the individual and to the collective\nmay be much higher than one would a priori have thought. Not only can no agent\nindividually perform well, but the over-competition eliminates the possibility that the\ncommon good could be well-served. This may be an important insight for a variety of\ndecision makers. For example, such insights could affect the cost-benefit analysis\nassociated with a company's decision about whether to invest in technological innovation\nin an already crowded market. There is also an important methodological and conceptual\nlesson from these observations: The way in which information is removed from access\nto agents in an efficient or overcrowded market may be considerably more subtle than\none might have thought. This suggests a reevaluation of the concept of an efficient\nmarket, and the tools and vocabulary used to describe it.\nAlthough the behavior we have elucidated is very intriguing, it is important to remember\nthat there are many effects that may play a major role in specific systems and which\n\ncould alter the emergent structure, fundamentally. For example, while this model is\nadaptive, it is not evolutionary. There is no discovery of new strategies by the agents, no\nmutation, no recombination, no sex. Evolutionary dynamics may drastically alter the\nphase structure of the system. In fact, we believe that some kinds of evolutionary\ndynamics may remove the efficient, over competitive phase and drive the system to an\neffective strategy space corresponding to the region around mc. Nevertheless, any\nanalysis of specific systems which share the competitive dynamics we have discussed\nhere, must take account of the structure we have described, for it is the ground upon\nwhich the description of more specific systems, possibly with more complex dynamics, is\nbased.\nFinally, and perhaps most importantly, our work raises the question of what really are the\nfundamental terms in which we ought to think about N-agent adaptive competitive\nsystems. For example, the fact that in the efficient phase (m<mc), σ is so strongly\ndependent on the initial distribution of strategies and, is proportional to N for fixed m,\nraises the question of how such simple quantities should be interpreted. Clearly, the\nvariance of this time series does not carry the information one might suppose, and at the\nvery least must be supplemented with a specification of the spread in the variance. As\nanother example, the fact that the market is efficient in this phase, but that the time series\nof minority groups is not random suggests the need for a more sophisticated approach to\ncharacterizing publicly available information. The fact is that there is no well developed\nepistemology for complex adaptive systems, and we are still quite unsure of what the\nimportant issues are or what are the most robust ways of characterizing the dynamics of\nsuch systems. But the study of simple models, and the elucidation of the variety of\nbehaviors which they manifest can lead us toward a deeper understanding of how to\nproperly frame the questions that we can sensibly ask, and sensibly answer, for complex\nsystems.\n\nFootnotes\n1. As we shall see, the dynamics of adaptivity are crucial to our results. It is, therefore,\nessential that s>1 so that the agents have more than one strategy with which to play. For\ns=1 the game devolves into a game with a trivial periodic structure. It is also worth\nnoting, parenthetically, that the majority game, in which each agent in the majority group\ngets a point has trivial periodic structure.\n2. The dependence of the results of the game on s are interesting and will be discussed in\ndetail elsewhere. However, the qualitative picture we present here obtains for s<<22m\n3. As we shall see below, the value of m at the minimum of the curve, mc, increases\nwith increasing N.\n4. IID stands for independent and identically distributed, and indicates a sequence whose\nentries are chosen independently, from some probability distribution that does not\nchange. In our case, IID would mean that the 0's and 1's were all chosen independently,\nand that the probability to choose 1 did not change over time. This is the simplest, most\nintuitive meaning of \"random\".\n5. For different values of s there are systematic changes in the shape of this scaling\ncurve. These will be discussed elsewhere [Manuca, Riolo and Savit, 1998].\n6. Another way to say this is that, as a result of the competition among each agent's\nstrategies, the joint probability for any set of agents to make specific choices is not equal\nto the product of the individual probabilities for each agent's choice.\n7. There is another finite-size effect that contributes to a σ that is lower than would be\nobtained in the random game. Independent of induced coordination, for a given initial\ndistribution of strategies, the decisions of the N agents following a given string of m 0's\nand 1's in the time series of minority groups will be constrained. Therefore, for finite N,\nthe probability distribution of the number of agents choosing group 1 following a specific\nm-string will not, in general, be symmetric. It can be shown that skewed distributions\nlower the standard deviation of a random process. However, as m increases, the agents\nwill, over time, respond to a larger number of different m-strings, each with a different\nprobability distribution of, say 1's. Consequently, as m increases, the effect of the\nskewness in each of the responses to a specific string will be averaged away.\n\nFigure captions\nFig. 1. An example of an m=3 strategy.\nFig. 2. σ as a function of m for N=101 and s=2. 32 independent runs of 10,000 time\nsteps were performed for each value of m. σ for each run is indicated by a dot. The\nhorizontal dashed line is at the value of σ for the random game described in the text.\nNote the broad spread in the values of σ for m<6.\nFig 3a. A histogram of the conditional probability P(1|uk) with k=4 for the game played\nwith m=4. There are 16 bins corresponding to the 16 possible combinations of 4 0's and\n1's. The bin numbers, when written in binary form yield the strings, u k.\nFig. 3b A histogram of the conditional probability P(1|uk) with k=5 for the game played\nwith m=4. There are 32 bins corresponding to the 32 possible combinations of 5 0's and\n1's. The bin numbers, when written in binary form yield the strings, u k.\nFig. 4 A histogram of the conditional probability P(1|uk) with k=6 for the game played\nwith m=6. There are 64 bins corresponding to the 64 possible combinations of 6 0's and\n1's. The bin numbers, when written in binary form yield the strings, u k.\nfig. 5a. σ as a function of N for fixed m, for two values of m (3 and 16), in the two\nphases of the system, on a log-log scale. Note that for m=3, σ∝Ν, while for m=16,\nσ∝Ν 1/2.\nFig. 5b. The spread in σ, ∆σ , as a function of N for fixed m, for two values of m (3 and\n16), in the two phases of the system, on a log-log scale. Note that for m=3, ∆σ∝Ν, while\nfor m=16, ∆σ∝Ν 1/2.\nFig 6. σ2/N as a function of z≡2m\nN\nfor various values of N, on a log-log scale.\n\nReferences\n[Arthur, 1994] W. Brian Arthur, \"Complexity in Economic Theory: Inductive Reasoning\nand Bounded Rationality\", Amer. Econ. Assoc. Papers and Proc 84: 406-411.\n[Arthur et al, 1996] W. B. Arthur, J. Holland, B. LeBaron, R. Palmer, and P. Tayler,\n\"Asset Pricing Under Endogenous Expectations in an Artificial Stock Market\", Santa Fe\nInstitute Working Paper 96-12-093.\n[Challet and Zhang, 1997] D. Challet and Y.-C. Zhang, \"Emergence of Cooperation and\nOrganization in an Evolutionary Game\", preprint, University of Fribourg.\n[Cody and Diamond, 1975] Ecology and Evolution of Communities, M. L. Cody and J.\nM. Diamond (eds.), (Harvard Univ Press, Cambridge MA).\n[Fama, 1970] E. F. Fama, \"Efficient Capital Markets: A Review of Theory and\nEmpirical Work\", Journal of Finance; 25(2), 383-417.\n[Fama, 1991] E. F. Fama, \"Efficient Capital Markets: II\", Journal of Finance, 46(5),\n1575-1617.\n[Fischer and Hertz, 1991] Spin Glasses, K. H. Fischer and J. A. Hertz (Cambridge\nUniversity Press, Cambridge)\n[Kahin and Keller, 1997] Coordination of the Internet, B. Kahin and J. Keller (eds),\n(MIT Press, Cambridge).\n[Malkiel, 1985], A Random Walk Down Wall Street, 4th Edition, B. G. Malkiel, (W. W.\nNorton, New York)\n[Manuca, Riolo and Savit, 1998] R. Manuca, R. Riolo and R. Savit, in preparation.\n[Mezard, Parisi and Virasoro, 1987] Spin Glass Theory and Beyond, M. Mezard, G.\nParisi and M. A. Virasoro (World Scientific, Singapore).\n\n0 2 4 6 8 10 12 14 16\nFigure 2: N=101 S=2\nm\nSt. dev.\n\n0 2 4 6 8 10 12 14 16\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\nm−bit past\nP(1|m−bit past)\nFigure 3a: m = 4\n0 5 10 15 20 25 30\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n(m+1)−bit past\nP(1|(m+1)−bit past)\nFigure 3b: m = 4\n\n0 10 20 30 40 50 60\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\nm−bit past\nP(1|m−bit past)\nFigure 4: m = 6\n\nm=3\nslope=1\nslope=1/2\nFigure 5a: s=2\nN\nSt. Dev.\nm=3\nslope=1\nslope=1/2\n−2\n−1\nFigure 5b: s=2\nN\nSt. Dev. Spread\n\nN=11\nslope=−1\n−3\n−2\n−1\n−2\n−1\nFigure 6: s=2\nz\nStdev^2/N"}
{"id": "adap-org/9801001", "meta": {"categories": ["adap-org", "nlin.AO"], "created": "1997-12-31", "extraction": {"body_chars": 21747, "cleaning": {"detected_repeated_margin_lines": ["1"], "page_count": 7, "removed_boilerplate_lines": 21}, "method": "pypdf_no_ocr", "source_pdf_bytes": 131815, "text_chars": 21978}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9801001", "primary_category": "adap-org", "source": "arxiv", "title": "Geometric statistical inference", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9801001"}, "text": "Geometric statistical inference\n\nAbstract\nFinite sample size corrections to the reparametrization-invariant solution of the inverse problem of probability are computed, and shown to converge uniformly to the correct distribution.\n\narXiv:adap-org/9801001v1 31 Dec 1997\nGeometric statistical inference\nVipul Periwal\nDepartment of Physics, Princeton University, Princeton, N ew Jersey 08544\nFinite sample size corrections to the reparametrization-i nvariant solution of the inverse problem\nof probability are computed, and shown to converge uniforml y to the correct distribution.\nI. INTRODUCTION\nA basic problem in statistics, with applications in divers fields, is the det ermination of the probability distribution\nthat underlies a finite set of experimental results involving continuo us variables. From the practical point of view,\none usually has a definite finite-parameter model for the probability distribution on theoretical grounds, and the\nparameters are fixed by fitting to the data set. As is obvious, this p arametric inference introduces a theoretical bias,\nsince the true distribution may not even lie in the parameter set chos en. It is therefore of interest to attempt a direct\ndetermination of the probability distribution without resorting to fin ite-dimensional approximations. Of course, for\nany finite data set, one obtains only a probabilistic description of the probability distribution, but the spread of\npossible probability distributions decreases as the size of the data s et is increased [1].\nThis set-up is fundamental in pattern theory, a context in which re parametrization invariance is of great importance.\nIn visual information processing algorithms, for example, if the infe rred pattern depends on, say, the orientation of the\nobject, then the object recognition capabilities of the algorithm ar e not likely to be efficient. In speech recognition,\nvariations in the tempo of speech should, again, be treated as repa rametrizations of the same underlying pattern, as\nshould frequency variations.\nIn the Bayesian approach to the problem, one starts by assuming a space of probability distributions, {w}, within\nwhich the true distribution lies, and given a set of observations, f , one obtains a probabilistic description of the true\ndistribution by using\nprob(w|f ) = prob(f |w) · prob(w)\nprob(f ) ; (1)\nin words, the probability of the distribution w being the true distribution, given f, is the probability of f given\nthe distribution w, multiplied by the probability of w occuring in the space of all distributions, normalized by the\nprobability of f occuring in any of the distributions in the set {w}. The maximum likelihood (ML) estimate of w is\nthen that distribution w that maximizes prob( f |w) · prob(w). To go further in this direction, one needs to figure out\nwhat the a priori probability prob( w) should be, and how to compute the ML estimate. Clearly, to avoid bia s, one\nwants the space of distributions to be as general as possible, cons istent with computability.\nIn one dimension, Bialek, Callan and Strong [2] have recently given an e legant formulation of this problem for\ncontinuous distributions in one dimension. They used Bayes’ rule to w rite the probability of the probability distribution\nQ, given the data {xi}, as\nP [Q(x)|x1, x2, ..., xN ] = P [x1, x2, ..., xN |Q(x)]P [Q(x)]\nP (x1, x2, ..., xN ) = Q(x1)Q(x2) · · ·Q(xN )P [Q(x)]∫\n[dQ(x)]Q(x1)Q(x2) · · · Q(xN )P [Q(x)] , (2)\nwhere the factors Q(xi) arise because each xi is chosen independently from the distribution Q(x), and P [Q] encodes\nthe a priori hypotheses about the form of Q. The optimal least-square estimate of Q, Q est(x, {xi}), is then\nQest(x; {xi}) = ⟨Q(x)Q(x1)Q(x2) · · ·Q(xN )⟩(0)\n⟨Q(x1)Q(x2) · · ·Q(xN )⟩(0) , (3)\nwhere ⟨· · ·⟩(0) denotes an expectation value with respect to the a priori distribution P [Q(x)].\nIn this field-theoretic setting, Ref. [2] assumed that the prior dist ribution P [Q] should penalize large gradients, so\nwritten in terms of an unconstrained variable φ ≡ ln(ℓQ) ∈ (−∞, +∞), they assumed\nPℓ[φ(x)] = 1\nZ exp\n[\n− ℓ\n∫\ndx(∂xφ)2\n]\n× δ\n[\n1 − 1\nℓ\n∫\ndxeφ(x)\n]\n, (4)\nwith ℓ a global length parameter that they later fixed by means of other consider ations. This form for the prior\ndistribution is very simple, and quite minimal in terms of underlying assu mptions, so is almost an ideal example of\nthe Bayesian set-up.\n\nThere is, however, an important aspect in which this prior distributio n does not measure up—a probability distri-\nbution is a density, and hence transforms in a specific manner under reparametrizations of the data, e.g. given a set\nof N observations xi, we want to infer a distribution wx that is related in a very simple way to the distribution wexp\ninferred from exp( xi) :\nwx(z) = exp( z) · wexp(exp(z)). (5)\nThis reparametrization covariance of the estimated distribution tu rns out to be a strong constraint on the distribution\nprob(w), and on the actual computation of the estimated distribution. In re cent work, I found a geometric modification\nof the approach of Ref. [2], which satisfies this desired reparametr ization covariance [4]. It is the details of this\nsolution, especially the corrections for finite sample size (which are o bviously of paramount importance for practical\napplications), that are explained in the present paper.\nThis paper is organized as follows: Section 2 contains a short review o f Ref. [4], to keep the presentation here\nrelatively self-contained. In Section 3, the one-dimensional solutio n for finite sample sizes is given, along with a\ndiscussion of corrections arising from fluctuations about the sadd lepoint distribution. In section 4, I present some\nconclusions, with comments on the solution of the inference problem in dimensions larger than 3.\nII. REVIEW\nInstead of the form (eq. 4) used in Ref. [2], I write Q(x) ≡\n√\nh(x) exp(φ(x)), with h(x) a metric in one dimension.\nHence\n√\nh(x) is a scalar density, and so is Q, which is crucial for a reparametrization covariant solution. An intuit ion\nfor the rˆ ole thath plays is that it is a local analogue of ℓ, determining binning intervals locally. Set\nP [φ, h] ≡ 1\nZ exp\n[\n− ℓ\n∫\ndx\n√\nh(x)\n−1\n(∂xφ)2\n]\nδ\n[\n1 − 1\nℓ\n∫\ndx\n√\nh(x)eφ(x)\n]\n, (6)\nThe inverse of the metric is just 1 /h(x) and the reparametrization invariant volume element is\n√\nh(x)dx, so P [φ, h]\nis reparametrization invariant. Now, we want to evaluate\n⟨Q(x1)Q(x2) · · · Q(xN )⟩(0)\n=\n∫\nDφ Dh\nDif f+\nP [φ, h]\nN∏\ni=1\n√\nh exp[φ(xi)] (7)\n= 1\nZ\n∫ dλ\n2π\n∫\nDφ Dh\nDif f+\nexp [−S(φ, h; λ)] , (8)\nwhere\nS(φ, h; λ) = 1\n∫\ndx\n√\nh(x)\n−1\n(∂xφ)2 + iλ\n∫\ndx\n√\nh(x)eφ(x) −\nN∑\ni=1\n[φ(xi) + 1\n2 ln h(xi)] − iλ. (9)\nNotice that the integral over all metrics has been divided by the volu me of the group of orientation preserving\ndiffeomorphisms. In one dimension, this division eliminates all but one glo bal degree of freedom from the metric.\nThere is no operational way to distinguish between the factor\n√\nh(xi) and exp( φ(xi))—these must occur together in\nQ(xi). Taking the local symmetry into account, the number of local degre es of freedom is the same as in the approach\nof Bialek et al. [2].\nThe equations of motion that follow from varying S are\n− 1\n2 (φ′)2 1\nh + iλ exp(φ) −\n∑\ni\n1√\nh(xi)\nδ(x − xi) = 0 , (10)\n−\n( 1√\nh\nφ′\n) ′\n+ iλ\n√\nh exp(φ) −\n∑\ni\nδ(x − xi) = 0 , (11)\n∫\ndx\n√\nh exp(φ) = 1 , (12)\nwhere I have used primes to denote d/dx. δ (x) denotes the scalar density such that\n∫\ndxδ(x) = 1 . Introduce a variable\n\ny(x) =\n∫ x√\nh(s)ds, (13)\nthen it follows from eq. 10 and eq. 11 that\nd2φ\ndy2 = 1\n( dφ\ndy\n) 2\n. (14)\nNote that the use of eq. 13 is limited by the non-singularity of\n√\nh.\nIt is now necessary to be careful about the limits of integration. Th is care in the boundary terms is unnecessary at\nN = ∞, in accord with the fact that any finite data set will not indicate the tr ue limits of the probability distribution.\nSuppose that x ranges from x− to x+, then integrating eq. 11 I find\nN − iλ +\n∫ x+\nx−\ndx( 1√\nh\nφ′)′ = 0 , (15)\nso, using eq. 12, it follows that\nexp(−φ) = ( y+ − y−) (y − c)2\n(y− − c)(y+ − c) (16)\nwhere c is an arbitrary constant of integration. c is restricted only by c > y + > y −, or y+ > y − > c, since exp( φ)\nmust be positive. This ‘cyclic’ constraint on y+, y−, c arises because the algebra of infinitesimal reparametrizations of\nthe line has a subalgebra isomorphic to sl(2, R).\nIn order to determine y(x), which satisfies\n(y+ − c)(y − y−)\n(y+ − y−)(y − c) =\n∫ x\nx−\n[\nN\n∑\ni\nδ(x′ − xi)\n]\ndx′ , (17)\nobserve that the cross ratio on the left is projectively invariant, i.e., invariant under transformations of the form\nz ↦→αz + β\nγz + δ , (18)\nwith α, β, γ, δ real, αδ − βγ = 1 . This amounts to a three-parameter family of equivalent solutions y(x) determined\nby the data. This projective invariance can be fixed by setting c = ∞, y+ = 1 and y− = 0, which implies φ = 0.\nOperationally, this means that the next measured data point will be o bserved with equal probability at any value\nof y in the interval [0 , 1]. An important point to notice about this solution is that it is only valid at N = ∞. At finite\nN, the metric determined by this solution is not smooth, being a sum over δ functions, hence we must take more care\nin solving eq.’s 10,11,12 for finite N. Indeed, the action evaluated at this solution is infinite for finite N.\nIII. FINITE N CORRECTIONS\nObviously, for any practical application of this theory, one needs a systematic scheme for computing finite sample\nsize corrections. To understand the theory at finite N, and to make contact with the higher-dimensional theory\ncommented on in Section 4, we rewrite eq. 9 as\nS = 1\n∫\ndy(∂yφ)2 + iλ\n( ∫\ndyeφ − 1\n)\n+ N\n[\n−\n∫\ndy ˆPy(y)φ(y) −\n∫\ndx ˆPx ln\n√\nh(x)\n]\n, (19)\nwhere y(x) is as in eq. 13, ˆPx(x) ≡ 1\nN\n∑ δ(x − xi), and ˆPy(y)y′ ≡ ˆPx(x). It is clear in this form that there are two\nindependent functional variations of S, one with respect to φ(y), and the other with respect to y(x) :\n−∂2\ny φ + iλ exp(φ) − N ˆPy(y) = 0 , (20)\n∂x\n[ ˆPx(x)\ny′(x)\n]\n= 0 . (21)\n\nFor N large, suppose ˆPx(x) = P (x)+ 1√\nN ρ(x), with P (x) a continuous density and ρ(x) : ⟨ρ(x)ρ(x′)⟩ = P (x)δ(x−x′),\nfollowing Ref. [2]. Now, a solution to eq.’s 20 and 21 at leading order in N is φ0 = 0 , and\n√\nh(x) ≡ ∂xy0(x) = P (x),\nas reviewed in Sect. 2. Indeed, it would seem from eq. 17 that this so lution is exact, and no modification for finite N\nis necessary. This is not true, since for finite sample sizes eq. 17 defines a singular metric, an d hence is inconsistent\nwith our calculations which have assumed that the metric is non-singu lar implicitly.\nWe can, however, solve eq. 11 to get φ = φ\n[ √\nh, ˆPx\n]\n, and then use this solution in eq. 10 to determine h. We shall\nfind that the leading correction at finite N is a correction to φ at O(N −1/2), with no correction to\n√\nh = P (x). This\nordering of the computational steps parallels the calculation in [2], wit h the determination of h here the analogue of\nthe determination of the length scale in [2]. What is different in the pres ent approach, is that the formulæ derived\nhere are reparametrization-covariant, e.g., square roots of densities do not appear. To leading order, we find iλ = N,\nand\nexp(φ0) = P (x)√\nh\n. (22)\nNotice that eq. 22 has the correct reparametrization covariance by construction, with a function equated to a ratio of\ndensities.\nAt the next order, expanding eq. 11 about φ0, iλ = N, we find\n[\n− 1√\nh(x)\n∂x\n1√\nh(x)\n∂x + N eφ0\n]\nφ1 ≡\n(\n−∆ h + N eφ0\n)\nφ1 =\n[ √\nN ρ(x)√\nh(x)\n+ ∆ hφ0\n]\n. (23)\nEq. 23 is easily solved to obtain\nφ1(x) =\n∫\ndx′√\nh(x′)\n[ √\nN ρ(x′)√\nh(x′)\n+ ∆ hφ0(x′)\n]\nK(x, x′), (24)\nwith the leading term in K given by\nK(x, x′) ≈ 1\n√\nN\n[\neφ0(x)eφ0 (x′)\n] −1/4\nexp\n(\n−\n√\nN\n∫ max(x,x′)\nmin(x,x′)\n√\nheφ0/2dx′′\n)\n. (25)\nNote the reparametrization invariance of eq. 24.\nWe can now insert eq.’s 22,24 in eq. 9, and vary with respect to h to obtain h = h( ˆPx). This order of solving\nthe saddlepoint equations is somewhat simpler than solving the equat ions obtained by independent variations—the\nresults are, obviously, independent of this order of solution. We fin d\nSred[h] = −N\n∫\ndx\n[\nP + 1√\nN\nρ\n]\nln P + 1\n∫ (\nN P φ1\n[\nln\n√\nh − ln P − φ1\n]\n−\n√\nN ρ\n[\nln\n√\nh + φ1 − ln P\n])\n, (26)\nwith φ1 = φ1(h). Now, it is important to note that while Sred appears to include terms with powers of N that should\nbe ignored at first glance, the variation of φ1 with respect to h is O(1) even though φ1 = O(N −1/2). Thus, varying\neq. 26 with respect to h, we find\n2 N P δφ1\nδ\n√\nh\n(\nln\n√\nh − ln P\n)\n+ O(\n√\nN ) = 0\nso to leading order in N, √h0 = P, φ0 = 0 so our old result is recovered. However, at the next order, we fi nd no\ncorrection to h of O(N −1/2). Thus the leading finite size correction is entirely encoded in eq. 24, wit h\n√\nh = P. At\nhigher orders in N −1/2, there are, of course, further corrections to φ, and to h.\nConsider now the computation of the functional integral expande d about the saddlepoint solution found above.\nThe division by the volume of the group of diffeomorphisms eliminates th e integration over fluctuations in the metric,\nso we are left with computing the integral over the fluctuations in φ. This integration gives the operator determinant\ndet−1/2(−∆ h + N eφ1). This determinant can be computed in the large N limit by the standard van Vleck technique,\nexplained in Ref. [5], for example. Thus\ndet\n(\n−∆ P 2 + N eφ1\n) −1/2\n∝ exp\n(\n−\n√\nN\n∫\ndxP (x)eφ1/2\n)\n, (27)\n\nwhich differs from the corresponding expression in Ref. [2]. Eq. 27 is r eparametrization invariant. Putting everything\ntogether, eq. 8 is given by\nN∏\ni=1\nP (xi) exp\n(\n−\n√\nN\n∫\ndxP (x)eφ1/2 − 1\n∫\ndxP (x)−1(∂xφ1)2\n)\n. (28)\nEq. 28 can be used to evaluate the least-square estimate of the inf erred distribution.\nFinally, we compute the magnitude of the correction φ1 :\n⟨φ1(x)2⟩connected ∼ 1\n√\nN\n, (29)\nindependent of x, implying uniform convergence to the solution reviewed in Sect. II. This uniform conv ergence is\nof great practical importance, since it implies that our estimated dis tribution will make sense even in regions where\nP is small. The uniformity can be directly traced to the reparametrizat ion invariance of our computations, which\nenables the binning size to be adjusted locally enabling a more accurat e determination of the distribution. As this\nuniformity is of great importance for practical applications, it is all t he more gratifying that a re-analysis motivated\nby conceptual concerns regarding reparametrization covarianc e leads to a computational improvement.\nIV. CONCLUSIONS\nThe statistical inference problem is of great interest in biophysics, for example, when x is a vector variable. Bialek,\nCallan and Strong [2] gave an account of a higher dimensional variant of their theory, but I want to add a few remarks\nhere regarding a geometric version sketched in [4], taking into accou nt reparametrization invariance. In Sect.’s II\nand III, we saw that a reparametrization-covariant Bayesian set -up could be obtained by introducing a metric, and a\nscalar field, dividing out by the group of diffeomorphisms. Since a metr ic in one dimension has only one global degree\nof freedom, this left only one local scalar degree of freedom, which is what we expect to need to fit the data, since the\ndata determines a scalar density on the interval [ x−, x+].\nIn higher dimensions, the aim is still to determine a scalar density, but the number of local degrees of freedom\nin the metric in d dimensions is d(d + 1)/2 and reparametrizations form a d parameter local symmetry group. It is\nclear then that a geometric formulation will require a reparametriza tion invariant constraint to reduce the number of\ndegrees of freedom in the metric, to the required d + 1 local degrees of freedom.\nReparametrization invariant constraints must be given by the vanis hing of tensor quantities, so in this case we have\nto construct appropriate tensors from the metric. Investigatin g the constraints provided by the vanishing of various\ncurvature tensors, since we expect only a scalar local degree of f reedom to survive after imposing the constraint and\ndividing out by reparametrizations, we are naturally led to consider c onformally Euclidean metrics, which can indeed\nbe characterized by the vanishing of a tensor constructed from t he curvature, the Weyl curvature, W δ\nαβγ . W vanishes\nidentically in d ≤ 3, so the geometric theory will only be valid for d > 3.\nThe proposed generalization of eq. 6 to higher dimensions is now given by\nP [hαβ] ∝ exp\n(\n−\n∫\nddx\n√\nhL\n)\nδ\n(\n1 −\n∫\nddx\n√\nh(x)\n) ∏\nx\nδ\n(\nWαβγ δ(x)\n)\n(30)\nwhere L is a scalar constructed out of the metric hαβ, constrained to be a metric of vanishing Weyl curvature, W δ\nαβγ ,\nand\n√\nh ≡ det1/2(hαβ) as usual.\nA variety of issues, some technical, have to be addressed in order t o define eq. 30 precisely: (i) The integration\nmeasure for integrating over metrics is notoriously nonlinear; (ii) So lving the constraint W = 0 , and then extracting\nthe volume of the group of diffeomorphisms will leave a Jacobian; (iii) Wh at is an appropriate choice for L, given the\nconstraint W = 0?\nBefore tackling the technical issues, consider the logic of what we w ish to do. Metrics with vanishing Weyl curvature\nare of the form\nhαβ(x) = ∂f δ\n∂xα\n∂f ǫ\n∂xβ e2φ(f (x)) δδǫ ≡ J δ\nαJ ǫ\nβe2φ(f (x)) δδǫ. (31)\nFor such metrics,\n√\nh = det(J)edφ. Now, following our steps in Sect. 2, we want to extremize (ignoring te chnicalities)\n\nˆS ≡\n∫\nddx det(J)edφL[φ(f (x))] + iλ\n( ∫\nddx det(J)edφ − 1\n)\n− N\n∫\nddx ˆP (x) [ln det J + φ] , (32)\nwhere N ˆP (x) ≡ ∑ δ(x − xi). Now, ˆP (x) is a density, so ddx ˆP (x) = ddf ˆP (f ), with ˆP (f ) det(J) = ˆP (x). We can\nrewrite eq. 32 as\nˆS =\n∫\nddf edφL[φ(f )] + iλ\n( ∫\nddf edφ − 1\n)\n− N\n∫\nddx ˆP (x) ln det J − N\n∫\nddf ˆP (f )φ , (33)\nIt follows that there are two independent functional variations, o ne with respect to φ(f ), and the other with respect\nto f (x), the latter giving\n∂\n∂xα\n[\nˆP (x) ∂xα\n∂f δ (x)\n]\n= 0. (34)\nEq. 34 is the analogue of eq. 21, and determines the co¨ ordinates f α in terms of the co¨ ordinates xα in which the data\nis presented, and is reparametrization-invariant because P (x) is a density, not a function. Consider the special case\nx = f : eq. 34 is the requirement that the density P in the co¨ ordinatesf is the constant density, exactly analogous to\nour result in one dimension, with φ a constant.\nIt is amusing to explicitly construct f α co¨ ordinates which satisfy\ndet ∂f α\n∂xβ = ˆP (x).\nWe construct these co¨ ordinates essentially iteratively as a one-d imensional distributions with parameters. If we\ncouldfind f :\nf 1(xα) = f 1(x1), f 2(xα) = f 2(x1, x2), . . .\nthen we would get\ndet ∂f α\n∂xβ =\n∏\nα\n∂f α\n∂xα ,\nso the problem is one of factoring ˆP appropriately. Define\nP 1(x1) ≡\n∫\nˆP (x1, x2, x3, . . .)\nd∏\nα=2\ndxα,\nP 2(x1, x2)P 1(x1) ≡\n∫\nˆP (x1, x2, x3, . . .)\nd∏\nα=3\ndxα,\n.\n.\n.\nP d(xα)\nd−1∏\nβ=1\nP β ≡ ˆP (xα), (35)\nthen\nf 1(x1) ≡\n∫ x1\nP 1(z)dz,\nf 2(x1, x2) ≡\n∫ x2\nP 2(x1, z)dz,\n.\n.\n. (36)\nis an explicit set of coordinates in which the given density ˆP is constant. Notice that each f α varies from 0 to 1, as\nbefits integrals of one-dimensional normalized probability distributio ns, P α.\n\nAs explained in Ref.’s [2,4], we are interested in extracting a continuum d istribution from some set of binned data,\nas independent as possible of the binning. In renormalization group t erminology, we are interested in the ‘infrared’\nproperties of the data, independent of the ‘cutoff’. The terms mos t relevant in the infrared are precisely the terms\nwith the fewest derivatives. Thus, since Einstein-Hilbert action is a t otal derivative for conformally Euclidean metrics,\nwe consider L ≡ − 1\n2 R∆ −1R, where ∆ is the Laplacian, and R is the Ricci scalar constructed from hαβ. This Lagrange\ndensity is non-local for general metrics in dimensions larger than tw o, but is local for metrics with W = 0,\n∫\nddx\n√\nh L = (d − 1)2\n∫\nddf e(d−2)φ(f )δαδ ∂φ\n∂f α\n∂φ\n∂f δ . (37)\nThe steps carried out in the previous section for determining the fin ite N corrections can be carried out in an exactly\nanalogous fashion for this action for the leading correction, using t he Laplacian in d dimensions in the f co¨ ordinates\nexplicitly constructed above. At higher orders in N, the nonlinearity of L leads to a distinctly different computation.\nI hope to address these issues, along with the technical issues men tioned above, elsewhere.\nIn conclusion, I have computed the finite sample size corrections to the geometric solution of the statistical inference\nproblem I gave in earlier work. The uniform convergence of these co rrections should be important for applications so\nit is of interest to test this theoretical construction in practice.\nI am grateful to V. Balasubramanian and C. Callan for conversation s. This work was supported in part by NSF\ngrant PHY96-00258.\n[1] J. Earman, Bayes or Bust?: A Critical Examination of Bayesian Confirmat ion Theory (MIT Press, Cambridge, 1992); from a\nphysics perspective, the inference problem has been discus sed by V. Balasubramanian, “Statistical Inference, Occam’ s Razor\nand Statistical Mechanics on the Space of Probability Distr ibutions”, Neural Computation (in press), cond-mat/96010 30;\n“Occam’s Razor for Parametric Families and Prior on the Spac e of Probability Distributions”, in Maximum Entropy and\nBayesian Methods, K.M. Hanson and R.N. Silver (eds.), (Kluw er Academic, Amsterdam, 1996); “A Geometric Formulation\nof Occam’s Razor for Inference of Parametric Distributions ”, adap-org/9601001; and references therein\n[2] W. Bialek, C.G. Callan and S.P. Strong, Phys. Rev. Lett. 77 4693(1996)\n[3] C. Callan explained to me the biophysical applications i n pattern recognition which motivated the investigation in Ref. [2].\nSee also D. Mumford, Pattern Theory: A Unifying Perspective , in Proceedings, First European Congress of Mathematicians ,\nvol. 1, eds. A. Joseph, F. Mignot, B. Prum and R. Rentschler (B irkh¨ auser Verlag, Basel, 1992), and references therein.\n[4] V. Periwal, Phys. Rev. Lett. 78 4671(1997)\n[5] S. Coleman, Ch. 7, Aspects of symmetry (Cambridge University Press, Cambridge, 1975)"}
{"id": "adap-org/9801003", "meta": {"categories": ["adap-org", "cond-mat.stat-mech", "nlin.AO", "q-bio"], "created": "1998-01-25", "extraction": {"body_chars": 66370, "cleaning": {"detected_repeated_margin_lines": ["1"], "page_count": 22, "removed_boilerplate_lines": 24}, "method": "pypdf_no_ocr", "source_pdf_bytes": 227386, "text_chars": 68004}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9801003", "primary_category": "adap-org", "source": "arxiv", "title": "Modelling coevolution in multispecies communities", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9801003"}, "text": "Modelling coevolution in multispecies communities\n\nAbstract\nWe introduce the Webworld model, which links together the ecological modelling of food web structure with the evolutionary modelling of speciation and extinction events. The model describes dynamics of ecological communities on an evolutionary timescale. Species are defined as sets of characteristic features, and these features are used to determine interaction scores between species. A simple rule is used to transfer resources from the external environment through the food web to each of the species, and to determine mean population sizes. A time step in the model represents a speciation event. A new species is added with features similar to those of one of the existing species and a new food web structure is then calculated. The new species may (i) add stably to the web, (ii) become extinct immediately because it is poorly adapted, or (iii) cause one or more other species to become extinct due to competition for resources. We measure various properties of the model webs and compare these with data on real food webs. These properties include the proportions of basal, intermediate and top species, the number of links per species and the number of trophic levels. We also study the evolutionary dynamics of the model ecosystem by following the fluctuations in the total number of species in the web. Extinction avalanches occur when novel organisms arise which are significantly better adapted than existing ones. We discuss these results in relation to the observed extinction events in the fossil record, and to the theory of self-organized criticality.\n\narXiv:adap-org/9801003v2 25 Jan 1998\nModelling coevolution in multispecies communities\nGuido Caldarelli 0,1,2, Paul G. Higgs 2 and Alan J. McKane 1\n1 Department of Theoretical Physics, University of Manchester,\nManchester M13 9PL, UK\n2School of Biological Sciences, University of Manchester,\nManchester M13 9PT, UK\nAbstract\nWe introduce the Webworld model, which links together the ec ological modelling of\nfood web structure with the evolutionary modelling of speci ation and extinction events.\nThe model describes dynamics of ecological communities on a n evolutionary timescale.\nSpecies are defined as sets of characteristic features, and t hese features are used to\ndetermine interaction scores between species. A simple rul e is used to transfer resources\nfrom the external environment through the food web to each of the species, and to\ndetermine mean population sizes. A time step in the model rep resents a speciation\nevent. A new species is added with features similar to those o f one of the existing\nspecies and a new food web structure is then calculated. The n ew species may (i) add\nstably to the web, (ii) become extinct immediately because i t is poorly adapted, or (iii)\ncause one or more other species to become extinct due to compe tition for resources.\nWe measure various properties of the model webs and compare t hese with data on\nreal food webs. These properties include the proportions of basal, intermediate and top\nspecies, the number of links per species and the number of tro phic levels. We also study\nthe evolutionary dynamics of the model ecosystem by followi ng the fluctuations in the\ntotal number of species in the web. Extinction avalanches oc cur when novel organisms\narise which are significantly better adapted than existing o nes. We discuss these results\nin relation to the observed extinction events in the fossil r ecord, and to the theory of\nself-organized criticality.\n0Present Address TCM, Cavendish Laboratory Madingley Road, Cam bridge CB3 0HE UK\n\n1 Introduction\nOur aim in this paper is to introduce a model for the evolution of many in teracting species in\nan ecosystem. A great deal of information has been assembled on t he properties of food webs\nfor naturally occurring ecosystems (Cohen, 1989; Martinez, 199 1; Polis, 1991; Hall & Raffaelli,\n1991; Goldwasser & Roughgarden, 1993), and a variety of models d escribing the structure of\nfood webs have been studied (Pimm, 1982; Briand & Cohen, 1984; Co hen et al., 1990; Cohen,\n1990; Pimm et al., 1991, Morin & Lawler, 1995). Most of these models c oncentrate on static\nproperties of the webs, such as the proportions of species at eac h of the trophic levels, and\nthe lengths of food chains. There have also been models for the ass embly of food webs by\ngradual addition of species (Luh & Pimm, 1993; Morton & Law, 1997) . One essential property\nof real food webs is that they have evolved. Species diversity will ac cumulate over time as\nnew species arise due to speciation. Species may also become extinct due to competition\nfrom better adapted species. Here we study a model which has the evolutionary dynamics of\nspeciation and extinction built into it. We will analyse the properties of the food webs which\narise in the model and compare these with ecological data.\nThe variation in the diversity of species seen in the fossil record is we ll documented (Sep-\nkoski, 1993). Attention has focused on the large-scale extinction events in which very large\nfractions of the existing species apparently become extinct almost simultaneously on a geolog-\nical timescale. Some of these large scale extinctions may be caused b y catastrophic external\nevents, or by major climate changes. However it has been shown (S neppen et al., 1995; Sol´ e\net al., 1997) that both the distribution of sizes of extinction events and the distribution of\nlifetimes of species in the fossil record are very broad and approxim ate to power-laws. It has\nbeen argued that the large fluctuations in species diversity which ar e seen may result from\nthe internal dynamics of the ecosystem. A newly-evolved species m ay cause extinction of its\nimmediate competitors. The extinction of one species may lead to the extinction of other\nspecies which prey upon it, and these extinctions may lead to furthe r extinctions, etc. Hence\nthere is the possibility of avalanches of extinction events spreading through an ecosystem.\nA link has been made between macroevolutionary dynamics and the th eory of self-organized\ncriticality (SOC). This theory was originally formulated in the language of sandpiles (Bak et\nal., 1988) and has since been applied to many other dynamical system s including earthquakes\nand forest fires. In the sandpile model one grain of sand is added at a time onto a pile of\nsand. This may either add stably to the pile, or may cause an avalanch e of one or more grains\nto tumble from the pile. The evolutionary analogy is clear: evolution lea ds to new species\nbeing added slowly to an ecosystem, and these may sometimes add st ably to the system or\nmay sometimes cause extinction events.\nThere is now a huge variety of evolutionary models inspired by the idea of SOC (Bak &\nSneppen, 1993; Paczuski et al., 1996; Kramer et al., 1996; Sol´ e & Bascompte, 1996; Sol´ e et\nal., 1996; Roberts & Newman, 1996). These models are deliberately e xtremely simple, and\nthey focus on the dynamical properties of extinction avalanches a nd species lifetimes. They\ndo not tell us anything about ecosystem structure, because the couplings between species\nare introduced either at random or in an ad-hoc manner (e.g. specie s are arranged on a line\nand neighbours interact). In our view it is essential to have a realist ic model of ecosystem\nstructure in order to study the way extinction events will propaga te through a food web.\nBy linking the ecological modelling of food web structure with the evolu tionary modelling of\nspeciation and extinction events we believe that the Webworld model described in this paper\n\nmakes a significant advance in both fields.\nAt this stage it is useful to introduce some notation used in the desc ription of food webs.\nIf one species preys on another then they are said to be linked. Basal species are those\nwith predators but with no prey and top species are those with prey but with no predators.\nIntermediate species have both predators and prey. We will refer to the percentages o f basal,\nintermediate and top species as B, I, and T . Two species are said to belong to the same\ntrophic species if they share the same set of prey and the same set of predators ( see Figure\n1). It is customary to group the original species in a web into trophic species and to analyse\nthe properties of the trophic species web. This goes some way towa rd standardizing webs\nobtained by different researchers using different criteria for inclus ion of species and different\ndegrees of resolution. The statistics presented in this paper will be for trophic species webs.\nOne food web statistic which we quote for the model webs is the ratio of prey to predator\nspecies, defined as ( B + I)/ (T + I). Another characteristic property of webs is the average\nnumber of links per species, defined simply as the total number of link s divided by the total\nnumber of species. It is also useful to classify the links according to the type of species which\nthey connect, i.e. we can measure the proportion of links between t op and basal ( T B), top\nand intermediate ( T I), intermediate and intermediate ( II ) and intermediate and basal ( IB )\nspecies.\nEcosystems are reliant on the input of resources from the extern al, non-living environment.\nWithin the model the environment is treated explicitly as a node in the f ood web. Species\nwhich are linked to the environment node are primary producers, i.e. they may survive in\nabsence of any other species. It follows that basal species (thos e having no prey) must be\nprimary producers, otherwise they would have no means of surviva l. However not all primary\nproducers are basal species, since a primary producer may also be a predator of other species.\nWe will use the following definition of trophic levels. All species linked to the environment\nnode (i.e. primary producers) are defined as level 1 species. All spe cies which have at least one\nlevel 1 prey are defined as level 2 species. All species with at least on e level 2 prey are defined\nas level 3, and so on. Hence, the level of a species is the length of th e shortest food chain\nfrom the external environment to that species. We could also have chosen to characterize\nspecies by the length of the longest chain linking them to the environm ent, or the average\nlength of all the possible chains, however these quantities are time c onsuming to calculate\nfor large webs, and also special rules are required to deal with the p ossibility of loops in the\nweb. The definition we use is rapid to calculate, and is unambiguous, ev en when loops are\npresent in the web. Also the majority of resources obtained by a sp ecies are likely to come\nvia the shortest route, since transfer of resources at each link o f the food chain is relatively\ninefficient. Therefore it seems that the definition of trophic level wh ich we use is meaningful\nfrom an ecological point of view.\n2 Definition of the Webworld Model\nFeatures and Species\nA species is defined in terms of a set of L characteristic features. Such features may\nbe either morphological or behavioral characteristics possessed by all the individuals of that\nspecies, e.g. sharp teeth, binocular vision, webbed feet, being noc turnal, forming social groups\netc. The L features for each species are picked from a total number of K possible features.\n\nIn the simulations presented here L = 10 and K = 500. The species are labelled by integers\nn, n ′ = 1, 2, ... .\nThe predator-prey relationships between species are determined by the features possessed\nby those species. The matrix, m, is a K × K matrix of scores representing the usefulness\nof any feature i of one species for predation against any feature j of another species. We\nsuppose that m is anti-symmetric, and that the elements are assigned randomly, su ch that\nmi,j is a random Gaussian variable with mean zero and variance 1 if i > j , mj,i = − mi,j, and\nthe diagonal elements mi,i are zero. This matrix of the scores is chosen at the beginning of a\nsimulation run and does not change during the run. The score Sn,n′ of one species n against\nanother species n′ is then defined as\nSn,n′ = max{0, 1\nL\n∑\ni∈n\n∑\nj∈n′\nmi,j} (1)\nwhere i runs over all the features of species n and j runs over all the features of species n′.\nThe max operator ensures that all scores are positive or zero. A positive s core indicates that\nn is adapted to be a predator of n′, and a zero score indicates that there is no interaction.\nThe factor of 1 /L has been chosen so that the scores have a root mean square value of order\n1.\nThe external environment is represented as an additional species 0 which is assigned a set\nof L features randomly at the beginning of a run and which does not chan ge. If Sn,0 > 0 the\nspecies n may be a primary producer, provided it is not out-competed (see be low).\nTransfer of Resources\nA total amount, R, of external resources is distributed amongst the primary produ cers, as\na function of the scores Sn,0, according to the rules for competition described below. These\nresources determine the population size of the primary producers . The population of species n\nis denoted N(n). For simplicity, we measure resources and population size in the sam e units,\nso that N(n) is equal to the amount of resources obtained by species n. Predator species\nobtain resources from their prey. We assume that a fraction λN (n) of the resources of a given\nspecies n is available to be passed on to predators of n. Thus λ is a parameter of the model\nwhich determines the relative sizes of predator and prey population s.\nCompetition for Resources\nThe resources obtained from species n are distributed between the predators of n in the\nfollowing way. (The same rules are used for the case n = 0, where the resources are the\nexternal resources R, and the ‘predators’ are the primary producers). The better ad apted the\npredator, the more resources it gets. We therefore define the main predator of n as the one\nwith the best score against n:\nSM\nn = max{Sn′,n} where n′ is a predator of n (2)\nThe predators of n will obtain a share of the available resources proportional to the qu antity:\nFn′,n = max\n{\n0,\n(\n1 − SM\nn − Sn′,n\nδ\n)}\n(3)\nwhere δ is a parameter of the model which determines the strength of comp etition. The\nsmaller δ the stronger the competition. We also define Fn′,n to be zero if Sn′,n = 0, so that\n\nonly species for which Sn′,n > S M\nn − δ, and Sn′,n > 0 successfully obtain any resources from\nn. The actual fraction of resources obtained is found by normalizing the Fn′,n:\nγn′,n = Fn′,n\n∑\nm Fm,n\n(4)\nPopulation Sizes\nWe will calculate the population sizes of the species by iteration of a se t of equations\nrepresenting transfer of resources between the species. Each iteration represents a small time\nperiod of order one generation time. If N(n, t ) is the population of species n at iteration t\nthen we may write\nN(n, t + 1) = γn,0R +\n∑\nn′\nγn,n′λN (n′, t ) + γn,nλN (n, t ). (5)\nThe first term represents external resources obtained by n, which may be zero if γn,0 = 0.\nThe second term represents resources obtained from prey ( n′ runs over all the prey of n). The\nthird term represents resources lost to predators. Here we hav e defined\nγn,n =\n{\n− 1, if n has at least one predator species\n0, if n has no predators (6)\nAfter several iterations the population sizes will converge to stat ionary values. We suppose\nthat the stationary population sizes determined by this method rep resent mean population\nsizes over many generations. The iteration procedure can be viewe d as an algorithm which\ndetermines the solution of the following set of linear equations for th e stationary values:\nN(n) =\n∑\nn′\nγn,n′λN (n′). (7)\nHere n′ runs over all the species, including n and 0, and for convenience we have defined\nN(0) = R/λ .\nThere have been many studies of population dynamics at short times cales using either\ndifference equations like (5), or differential equations like the Lotka -Volterra equations, or\ngeneralizations of them (Vida et al., 1990; Berryman et al., 1995; Ard iti & Michalski, 1996;\nZheng et al., 1997). Many of these equations have interesting dyna mical behaviour, such as\nperiodic solutions or chaos which may be relevant to real ecological p opulation dynamics. In\nfact we are not interested in the population dynamics at the scale of a few generations. We are\nonly interested in evolutionary time. We have deliberately chosen diffe rence equations which\nare as simple as possible, and which have only one stationary state. T hese equations converge\nrapidly, and always reach the same stationary state irrespective o f the initial conditions. It\nis a property of our equations that the amount of resources pass ed from a prey to a predator\nspecies is proportional to the prey population size only. This situatio n is usually called donor\ncontrol (Zheng et al., 1997). We will be interested in the way the pop ulation sizes change\nas the ecosystem evolves. Before describing the long timescale evo lutionary dynamics of\nour model it is useful to consider the following simple example which illust rates competition\nbetween species and transfer of resources between levels.\nSuppose there are two primary producers, 1 and 2, having scores S1,0 = 1. 0 and S2,0 = 0. 95.\nIn addition, species 3 is a predator of species 1, i.e. S3,1 > 0. Let the competition parameter\n\nδ be 0.1. From the above rules γ1,0 = 2 / 3 and γ2,0 = 1 / 3, and since 3 is the sole predator of\n1, γ3,1 = 1. The iterative equations are therefore\nN(1, t + 1) = 2 R/ 3 − λN (1, t )\nN(2, t + 1) = R/ 3 (8)\nN(3, t + 1) = λN (1, t )\nand the stationary states are N(1) = 2 R/ 3(1 + λ), N(2) = R/ 3, and N(3) = 2 λR/ 3(1 + λ).\nNotice that the sum of the populations is equal to R. It is a property of the model that\nthe sum of the populations at the stationary state is always equal t o the amount of external\nresources put in.\nGiven the lists of features representing any set of species it is poss ible to calculate the\nscores Sn,n′ and hence the steady state population sizes. We suppose that the re is a minimum\npopulation size necessary for survival of the species, and we set t his limit to 1. Any species\nfor which N(n) < 1 becomes extinct and is deleted from the list of species. The resultin g\necosystem is then stable, and properties of the web can be measur ed.\nFor the purposes of defining the food web structure, a link is assum ed to be present\nbetween n and n′ if n successfully obtains resources from n′, i.e. if γn,n′ > 0. Cannibalism\nhas been excluded from the model ( Sn,n is defined to be zero for all n). Also, since the mij\nmatrix is antisymmetric, then for any pair of species it is impossible for both Sn,n′ and Sn′,n\nto be non zero, i.e. there are no reciprocal pairs of links. Loops of t hree or more species can\noccur in the model, however.\nEvolutionary Dynamics\nEvolutionary time in the model proceeds in timesteps such that one s peciation event occurs\nin every timestep. At each step an existing species is chosen to unde rgo speciation with a\nprobability proportional to its population size. A new species is creat ed by copying the list\nof features of the parent species, randomly picking one these fea tures, and replacing it by\nanother randomly chosen feature from the complete list of possible features. The new species\nthus differs by only one feature from the old one. The stationary po pulation sizes of all the\nspecies are now recalculated, taking account of the presence of t he new species. There are\nseveral possible outcomes: (i) the new species may add to the web in a stable fashion, so that\nthe total number of species increases by one; (ii) the new species m ay be poorly adapted, and\nmay become extinct immediately; (iii) the new species may survive, and cause one or more\nother species to become extinct. A new species will often be in direct competition with the\nparent species, therefore a special case of outcome (iii) is that th e new species simply replaces\nits parent in the ecosystem.\nHaving eliminated any species which become extinct due to the new add ition, the web\nis once again in a stable state. This is the end of one complete timestep . This process is\nrepeated many times and the properties of the webs are recorded at the end of each step and\nsubsequently averaged. When choosing the new feature for each new species a restriction is\nmade that no feature may appear more than once in the list of featu res possessed by any one\nspecies. Also, when the new feature list is created, we check that it is not identical to the\nfeature list of any other species, and that it is not simply a permutat ion of any other list.\nThis prevents the occurrence of multiple copies of identical species .\nThe only remaining thing to be specified is the initial state of the web. O ne possibility is to\nconsider an ‘origin of life’ scenario, where the web begins with exactly one primary producer\n\nspecies. Another possibility is to begin with a small number of randomly chosen species. We\ncarried out runs where the initial state was composed of 1, 10 and 2 0 different species, and\nfound very little difference between these. All the results present ed here were started with a\nset of 10 random species. Since the features of these species are chosen at random, the species\nare not well adapted to survive together, hence there is a tenden cy for several extinctions to\noccur immediately on the first timestep.\n3 Properties of the Model\nIn this section we discuss some of the important properties of the m odel which are relevant to\nunderstanding the numerical results in the next section. The mode l contains three parameters,\nR, λ and δ, and these will affect the shapes of the food webs in various ways.\nWe would intuitively expect that increasing the amount of resources available should\nincrease the diversity of the ecosystem. In the model the sum of t he population sizes is\nequal to R, and since every species must have a population of at least one, the re can never\nbe more than R species. In fact the mean number of species observed is always muc h less\nthan R because resources are not distributed evenly between species. S pecies at lower levels\nhave much larger populations. The ratio of population sizes between predator and prey is\ncontrolled by λ. Together R and λ determine the maximum possible number of trophic levels\nin the system.\nConsider a web consisting of a single food chain of k species. The stationary population\nsizes obtained from the model for each level j except the top level are N(j) = λ j−1R/ (1 +λ)j,\nand the population of the top level is N(k) = λ k−1R/ (1 + λ)k−1. A chain of length k can only\nbe supported if N(k) > 1. This gives the condition\nk < 1 + log (R)\nlog ((1 + λ)/λ ) (9)\nand, since k is an integer, the maximum level is actually the largest integer below th is limit.\nThus the maximum food chain length only increases logarithmically with R. Although this\nlimit has been calculated assuming the web is a single chain, we believe tha t this is a strict\nlimit to the number of levels for any possible web generated by the mod el. This is because\nadding extra links and more species to the web always decreases the fraction of resources\nreaching the top level in comparison to the single chain calculation.\nThe number of levels observed in the model food webs is strongly sen sitive to λ. We have\nchosen to set λ = 0 . 1 in all the simulations reported here. This gives realistic numbers of\ntrophic levels in the model food webs and is also consistent with measu red values of predator\nprey population ratios, and estimates of the ecological efficiency (P imm, 1982).\nThe value of the parameter δ affects the properties of the web considerably. In order to\nchoose a sensible value of δ we note that the scores Sn,n′ are all of order 1, and that when\na single feature is changed, the score changes by an amount of ord er 1 /L . In other words,\nspecies competing for the same resources should have scores whic h differ by an amount of\norder 1/L . We should therefore set δ to to be roughly of this size. If we make δ >> 1/L , even\nvery uncompetitive species will be allocated some resources. Hence the number of predator\nspecies per prey species will be large, and this will lead to a highly conne cted web with a large\nnumber of links per species. As δ → 0, only the main predator will be allocated resources,\n\nand so in the limit the web will become a single food chain. For most of the results given in\nthis paper δ is in the range 0.05 - 0.2, which is comparable with 1 /L = 0. 1.\nIt is useful to mention several features which have deliberately be en excluded from the\nmodel. Firstly, there is no variation between individuals of a given spec ies, and there is\nno genetics. Species are simply represented by a list of phenotypic f eatures, which represent\naverage properties for all members of that species. A model which included both many species\nand many individuals per species, each with its own genotype and/or p henotype would require\nenormous computational resources. Secondly, even though spe ciation is an essential part of\nthe model, we do not attempt to consider the mechanism by which spe ciation occurs. We\nsimply suppose that species have an inherent tendency to diversify . Speciation involves the\nestablishment of reproductive isolation by some means or another, and we cannot deal with\nthis in the absence of a genetic description of the species. When a ne w species is created it\ndiffers by only one feature from the parent species. However, this is intended to represent\na major change to the phenotype, which would probably involve chan ges in more than one\ngene sequence, and possibly some considerable alteration to the de velopmental biology of the\norganism. Thus changing a single feature does not represent a sing le mutation, but is the\nresult of many changes at the genetic level. In addition we make no dis tinction between\nsexual and asexual methods of reproduction, and the model is int ended to apply equally well\nto either case.\n4 Results - Structure of Model Food Webs\nThe figures presented in the tables of results represent average s taken over many webs. For\neach set of parameters several independent simulation runs were performed using different\nrandomly generated m matrices and different random feature sets for the environment ( species\n0). Each run was for 250,000 timesteps, with the exception of the r uns with R = 10 10, which\nwere for 500,000 timesteps. In each case, average quantities wer e measured during the second\nhalf the run. The different runs for each parameter set were then averaged. Fluctuations in\nthe number of species and the number of links between different run s with the same parameter\nvalues were surprisingly large, with standard deviations being up to 5 0% of the mean. Since\nthese quantities increase in relation to one another, the fluctuatio ns in the number of links per\nspecies are much smaller (typically ± 0.2). Standard deviations in the percentages of B and\nI species are typically 7% − 10%. We have omitted error estimates from the tables for clarity.\nWe are principally interested in the main trends in the web properties a s the parameters are\nvaried, rather than in precise values. We have checked that these trends are significant. The\nquoted figures in the tables apply to webs of trophic species.\nTables 1 and 2 show results for R = 10 3, 10 6 and 10 10, for δ in the range 0.05-0.2. In all\ncases λ = 0 . 1. There is a clear tendency for the mean number of species to incre ase with R,\nfor fixed values of δ and λ. However, it should be noted that the number of species is very\nmuch less than the theoretical maximum of R. In fact the number of species only increases\nvery slowly with R: a change in R by seven orders of magnitude only causes a fivefold increase\nin the number of species. In contrast, the number of species incre ases rapidly with δ at fixed\nR. Also, changing δ has a large effect on the total number of links. The number of links\nincreases more rapidly than the number of species, so that the num ber of links per species\nincreases significantly with δ. The number of links per species increases slightly with R for\nfixed δ.\n\nR = 10 3\nδ = 0. 05 δ = 0. 1 δ = 0. 15 δ = 0. 2\nno. species 40 80 290 320\nno. links 55 150 800 1250\nlinks per species 1. 4 1. 9 2. 7 3. 8\naverage level 1. 7 1. 4 1. 2 1. 1\nmax level 3. 0 3. 0 3. 0 3. 0\nB species (%) 40 56 74 78\nI species (%) 59 44 26 22\nT species (%) 1 0 0 0\nIB links (%) 53 73 83 88\nII links (%) 43 27 17 12\nTI links (%) 4 0 0 0\nprey/predators 1. 6 2. 2 3. 8 4. 6\nTable 1: Results of a simulation of the model with λ = 0. 1 and R = 10 3 for four values of the\ncompetition parameter δ.\nR = 10 6 R = 10 10\nδ = 0. 05 δ = 0. 1 δ = 0. 15 δ = 0. 2 δ = 0. 05\nno. species 150 200 410 810 220\nno. links 240 510 1420 5930 350\nlinks per species 1. 6 2. 5 3. 5 7. 3 1.6\naverage level 2. 5 2. 4 2. 3 1. 8 3.2\nmax level 5. 2 5. 2 5. 3 5. 5 7.2\nB species (%) 10 25 32 60 3\nI species (%) 90 75 68 40 97\nT species (%) 0 0 0 0 0\nIB links (%) 22 47 51 70 10\nII links (%) 78 53 49 30 90\nTI links (%) 0 0 0 0 0\nprey/predators 1. 1 1. 3 1. 5 2. 5 1.0\nTable 2: Results of a simulation of the model with λ = 0 . 1, R = 10 6 or 10 10 and various\nvalues of δ.\n\nThese results imply that competition is a significant factor in determin ing the number\nof species and the number of links in the web. The competition parame ter δ determines the\ncut-off in the function Fn,n′ in equation (3), and hence controls the number of predator specie s\nwhich can successfully obtain resources from each prey. It is ther efore to be expected that\nthe number of links per species will increase with δ. Also, when the competition is weaker,\nit is easier for a species to find at least one prey for which it is sufficient ly well adapted\nto be a predator, therefore the total number of species should a lso increase with δ, as is\nobserved. The importance of δ in controlling species numbers also provides an explanation of\nwhy the number of species only increases very slowly with R. The number of level 1 species\nis limited by competition for external resources. Increasing R at fixed δ tends to increase\nthe populations of all level 1 species proportionately, rather than increasing the number of\nlevel 1 species. The same is also true of intermediate levels. However at the higher trophic\nlevels species numbers are limited by the criterion of the minimum popula tion size necessary\nfor viability ( N(n) > 1). The principal effect of increasing R is therefore to increase the\nmaximum number of trophic levels possible in the web, and to allow a few h igh level species\nto survive, without changing the number of low level species very mu ch. We have already\nshown that the maximum number of levels increases logarithmically with R, and this therefore\nsuggests that the total number of species should increase as log R, which is consistent with\nour results. Figure 2 shows a histogram of the mean number of spec ies at each level, and the\nway that this depends on R. This confirms the fact that increasing R leads principally to an\nincrease in species at higher trophic levels.\nTables 1 and 2 also show the change in the number of levels in the web wit h R and δ. The\naverage level is defined by calculating the level of each species, ave raging these for all species\nin each web, and then averaging these web averages over many web s for each parameter set.\nThe maximum level is defined by calculating the maximum level present in each web, and\naveraging these maximum values over many webs for each paramete r set. Both average and\nmaximum levels increase approximately logarithmically with R as expected. For R = 10 3, 106\nand 10 10 the maximum possible number of levels from equation (9) are 3, 6 and 1 0. For\nR = 10 3 the limiting value is always achieved. For R = 10 6 the mean value of the maximum\nlevel is only slightly below the limit, whereas for R = 10 10 it is considerably below. Two\nfactors contribute to this. Firstly, the theoretical limit was based on a single food chain since\nthis is the most efficient way of transferring resources to high levels . The webs generated\nby the model are multiply connected and contain omnivorous links and loops, hence the\nmaximum attainable number of levels is reduced. Secondly, the calcula ted limit takes only\necological efficiency into account, but ignores evolutionary constr aints, which may also limit\nthe number of levels in the Webworld model. Well adapted level 1 specie s may begin to evolve\nfrom the start of a simulation run, since the environment is fixed. Hig h level species depend\non lower level ones for their resources, hence well adapted high lev el species tend to evolve\nat later stages of the simulation when the properties of the lower lev el species are changing\nless rapidly. The properties of the runs with R = 10 10 still appeared to be changing after\n250,000 timesteps, hence they were continued for 500,000 timeste ps. This led to a significant\nincrease in the number of species and the number of links, and a slight increase in the number\nof levels. Most of the late evolving species are on higher levels, hence the proportion of basal\nspecies was found to decrease significantly between 250,000 and 50 0,000 timesteps.\nAlso from the tables it can be seen that the average level decrease s with increasing δ.\nThis is understandable, because the level is defined as the shortes t possible chain to the\n\nexternal environment node. If δ is large, then a species with a low score Sn,0, but a high\nscore Sn,n′ against another species n′ may successfully compete for the external resources,\nand will therefore count as level 1 (the link to n′ would not affect the calculation of the level).\nIf δ is smaller, the same species would only survive by being a predator of n′, and would\ntherefore count as a higher level species. The maximum number of le vels does not change\nmuch with δ. The apparent slight increase in the maximum level with δ in table 2 is probably\nnot significant.\nThe proportions of basal, intermediate and top species are given in t ables 1 and 2. In-\ncreasing δ at fixed R leads to an increase in B and a decrease in I and T . This is for the same\nreason that the average level decreases with δ. (However, in the following section we consider\na case with extremely large δ where this trend is no longer true). Increasing R at fixed δ\nleads to a decrease in B and an increase in I. Again this is because increasing R increases\nthe number of levels, and thus the proportion of species in level 1 go es down. The number of\ntop species is always very small: T ≤ 1. 5% when R = 10 3, and no top species were observed\nat all for R = 10 6 and R = 10 10. Thus increasing the number of levels does not mean that\nthe number of top species increases. In the model webs, for the p arameters discussed so far,\nalmost all species have predator species, even if they are at high tr ophic levels. This implies\nthe presence of large numbers of loops in the food web and large num bers of omnivores. The\nbehaviour of T is an important property which has been remarked upon in the study of real\nfood webs, and we will return to it in the following section.\nThe tables also give the fractions of links between top, basal, and int ermediate species,\nand the ratio of prey species to predator species. These quantitie s are given primarily for\ncomparison with the real food web data. Trends in these quantities can be understood in\nterms of the trends discussed above in B, I, and T . For example, as δ increases at fixed R,\nor R decreases at fixed δ, the percentage of species which are intermediate decreases and as a\nconsequence the fraction of the links which are between intermedia te species also decreases.\n5 Comparison with Real Food Webs\nThe problems in getting reliable data on real food webs are readily ack nowledged by ecologists\n(Cohen et al. 1993). These problems centre around the tremendo us amount of effort required\nto observe all species and all predator-prey interactions in a given ecosystem. Data from a\nlarge number of food web studies has been assembled in the EcoWeb d atabase (Cohen, 1989).\nOur analysis of the average properties of all the 181 webs in this dat abase is given in Table 3.\nIn addition we give data from several well studied webs which have be en published recently\n(see caption to Table 3).\nBefore these figures can be compared with the model it is necessar y to note several points.\nFor both the model and the real data, all statistics are for troph ic species. Many webs contain\ndetritus as a ‘species’, and we have followed the convention of treat ing this as a single basal\nspecies. Plants are often treated in a very aggregated way, e.g. t he St. Martin Web divides\nplants into fruit, nectar, leaves, wood and roots, each of which is t reated as a ‘species’. Several\nwebs split taxa into adult and juvenile ‘species’ where these have different diets. Again we have\nfollowed the conventions of the source article in these cases. In th e model webs the external\nresources are treated explicitly as a node in the web, and basal spe cies must be linked to this\nnode. Real webs have no such external resources node (althoug h evidently external resources\ndo enter the real ecosystems). Therefore, when counting the n umber of links in the model\n\nExperimental Data\nECOWEB Lovinkhoeve Coachella St. Martin Ythan Little Rock\nno. species 16 15 29 42 83 93\nno. links 33 30 262 203 398 1033\nlinks per species 2. 0 2. 0 9. 0 4. 8 4.8 11. 1\naverage level 2. 1 2. 5 2. 0 2. 1 2.5 2. 2\nmax level 3. 2 3 3 4 4 3\nB species (%) 21 13 10 14 5 13\nI species (%) 49 74 90 69 59 86\nT species (%) 30 13 0 17 36 1\nTB links (%) 10 3 0 3 1 0\nIB links (%) 29 10 13 19 10 9\nII links (%) 29 57 87 53 51 91\nTI links (%) 32 30 0 25 38 0\nprey/predators 0. 89 1. 0 1. 11 0. 97 0.67 1. 13\nTable 3: Data from experimentally studied food webs: the ECOWEB da tabase - average\nproperties of all webs, (Cohen, 1989); the Lovinkhoeve Experime ntal Farm soil food web (De\nRuiter et al, 1995); the Coachella Valley desert (Polis, 1991); St. Ma rtin Island (Goldwasser\n& Roughgarden, 1993); the Ythan Estuary (Hall & Raffaelli, 1991; H uxham et al., 1996); and\nLittle Rock Lake (Martinez, 1991).\nwebs, links to the external resources node were omitted, in order to allow comparison with the\nreal webs. The real webs contain some cannibalistic links (particular ly the Coachella web).\nFor simplicity we included these in the count of links, whereas cannibalis m does not occur in\nthe model webs. The statistics in the table differ very little according to whether one counts\nor discounts cannibalistic links.\nMost of the webs in the EcoWeb compilation are small, and they have be en criticised as\nbeing incomplete, and biased due to observational methods (Martin ez, 1991; Polis, 1991).\nWe have chosen the individual webs since in these cases attempts ha ve been made to be\nsystematic and inclusive. Nevertheless, one suspects that the lar ge difference in the number\nof trophic species between these data sets reflects the degree o f aggregation chosen by the\ndifferent workers, rather than the actual diversity of the ecosy stems. The five individual webs\nare listed in order of increasing number of trophic species. There ar e some trends apparent\nwhen comparing these webs with each other, although in many ways e ach must be considered\nas a special case. There is a general trend for the number of links p er species to be higher\nin larger webs. This would suggest that systematic study of a commu nity over a long period\nof time leads to a greater increase in the number of links observed th an in the number of\nspecies. It may be that the number of links is severely underestimat ed in some of the Ecoweb\nfood webs. The Coachella web breaks this trend, since it has a very la rge number of links per\nspecies and is only fairly small.\nThere is a strong similarity between the figures for the Coachella and Little Rock webs,\ndespite the difference in the number of species. They both have a hig h number of links per\nspecies, a low value of B, a high value of I and very low T . Martinez & Lawton (1995) have\nsuggested that, in the limit of extremely large webs which represent large geographical areas,\nI will increase to about 95%, B will decrease to about 5%, and T will decrease to zero. Our\n\ndata in tables 1 and 2 confirm that as R increases, the number of species increases and the\nfraction of intermediates becomes larger ( I = 97% in the run with R = 10 10, for which the\naverage number of trophic species is 200). The data from the Ytha n web appear to go against\nthis trend, however, since there is a very high value of T (36%), even though the web is almost\nas large as the Little Rock web. We have used the version of the Ytha n web without parasites\n(Huxham et al. 1996). Many of the top species become intermediate species when parasites\nare included, however the parasites themselves form a new top leve l which are not themselves\npreyed on. The number of top species is still very large in the Ythan w eb when parasites\nare included, hence this cannot account for the large difference be tween the Ythan and Little\nRock webs.\nWhen comparing the model results with the real data we view R as a parameter which\ncould change between ecosystems in different places, whereas δ is a fundamental property of\nthe competition interactions between species which should presuma bly not vary much between\nlocations. Therefore we would wish to choose a value of δ which best fits a range of real data.\nThe model results for δ = 0 . 05 have high I, low B and very low T , which seems to be\ncharacteristic of the well-characterised real webs. All the real w ebs have a maximum level of\n3 or 4. We obtain this number of levels in the model if R = 10 3 or 10 4. The very high value\nof 7 for the maximum level in the runs with R = 10 10 does not appear to be consistent with\nthe data, however it should be noted that all the model data is for λ = 0 . 1. If we reduce λ\nthen the number of levels in the model will reduce, and we will require la rger values of R in\norder to obtain 3 or 4 trophic levels. Low values of δ seem to be preferred for matching the\nvalues of B and I in the real data, however in this case the model predicts less than 2 links\nper species. The number of links per species in the real webs tend to be higher (as much as\n11 in Little Rock). It is possible to get high numbers of links per species in the model by\nincreasing δ (see e.g. R = 10 6, δ = 0. 2). However in this case we get much higher values of B\nthan are seen in real webs. We conclude that the properties of the model webs are generally\nof similar orders of magnitude to those of the real webs, but that w e have a not found a\nsingle set of parameters R, λ and δ which accurately match all the properties of the real data\nsimultaneously. We have certainly not done an exhaustive search of parameter space yet for\nthe model: we did not change λ at all. So it is possible that the agreement between the model\nand the data could be improved. Also, there are many refinements w hich we could make to\nthe model, such as changing the rules by which we distribute resourc es between species, or\nchanging the way the scores are calculated. Consideration of all th ese possibilities would be\njustified if there were an excellent set of data on real food web str ucture against which to\ntest the models. However, there are still problems with even the be st experimental data, and\nin view of this, it does not seem worthwhile worrying too much about th e precise values of\nthe food web statistics from the model. The model may in fact make s ome useful predictions\nwhich are difficult to test in real life. For example, we noted above tha t if δ is increased, so\nas to give high numbers of links per species, the value of B in the model appeared too high.\nReal webs tend to apply much higher degree of aggregation to low lev el species than high\nlevel ones, so maybe the values of B in the real webs should really be higher than they are.\nOver the range of δ of order 1 /L considered in the tables, I is a decreasing function of δ\nand B is an increasing function. However, as a limiting case, we also carried o ut a simulation\nwith R = 100 and δ = 25. With this very large δ there were almost no extinctions due to\ncompetition. After 500 speciations there were approximately 90 diff erent species which had\naverage population sizes just greater than 1. Extinctions occurr ed only due to the rule that\n\na minimum population size of 1 is necessary for viability. In this run we fo und that I = 99%\nand B = 1%. This implies that there is a reversal of the trend in B and I for very large values\nof δ. We do not think that this very large δ value is a reasonable one for comparison with\nreal webs, however this simulation does illustrate a limiting case of our model: in absence\nof competition very large highly connected webs arise which are almos t entirely intermediate\nspecies.\nWe wish to comment further on the meaning of trophic levels in the Web world model. The\nsimple picture of ecosystems which is often envisaged supposes tha t organisms can be grouped\nuniquely into trophic levels representing plants, herbivores, carniv ores, secondary carnivores\netc. This picture supposes that each level only interacts with the le vels immediately above\nand below it. Although it was argued from some early food web data th at omnivores (i.e.\nspecies feeding on more than one trophic level) were rare (Cohen et al., 1990), it has since\nbeen argued that omnivory in well-characterized webs is frequent, that organisms apparently\nassigned to the same level are far from equivalent in their dynamics a nd interactions, and\nhence that the concept of trophic levels is of little use (Polis & Strong , 1996). We found that\nomnivory is frequent in the model, and that it is not possible to assign s pecies to levels such\nthat interactions are only with the levels immediately above and below. However, we maintain\nthat the idea of level which we use here is a useful one, since it is a mea sure of food chain\nlength: the level of a species is the length of the shortest chain fro m the external resources\nto the species (including the link to the external node). This is a differ ent definition of chain\nlength and trophic level from that used in many other studies. For e xample, Martinez (1991)\ncalculates the average length of all food chains leading to each spec ies in the Little Rock web\nusing two algorithms which differ in the way they deal with loops (it is nec essary to exclude\nloops in some way or else chain length is infinite). He then defines the tr ophic level as the\nclosest integer to the average chain length + 1. Since there are hug e numbers of chains in\nlarge webs, algorithms which involve averaging all possible chains are s low. Martinez was\nlimited by computer time to looking at aggregated versions of his web u sing his algorithm. In\nanalysing the model data we have thousands of webs, each with hun dreds of species, rather\nthan just one real web, therefore we must use a rapid algorithm fo r practical reasons. The\nmaximum level is 3 in the Little Rock trophic species web using our definit ion, whereas\nusing Martinez’s algorithm the maximum level is 9. It is clear that the Lit tle Rock web is\nextremely complex and that long chains occur frequently, however , this is a statement about\nthe combinatorial properties of highly connected graphs, and may not be significant from an\necological point of view. Our algorithm shows that there is no species in the Little Rock web\nwhich does not have a chain of length 3 or shorter. We believe that sh orter chains are much\nmore important energetically than longer ones. This is certainly true in the model webs, since\nthere is an efficiency factor of λ = 0 . 1 associated with every additional link. It is probably\ntrue also in real webs, and could be tested using data which measure s energy flow along each\nlink. Therefore we believe that our algorithm for chain length and tro phic levels is justified\necologically as well as on grounds of practicality.\nSince real webs are likely to be incomplete due to observational difficu lties, Goldwasser &\nRoughgarden (1997) investigated the effect of incompleteness of data by deliberately omitting\nlinks from the original St. Martin island web. They have shown that mo st web properties vary\nstrongly as links are removed. We carried out the following procedur e in order to investigate\nthe effect that incompleteness of data would have on the food webs in our model. Beginning\nwith a set of webs generated from the model with R = 10 6 and δ = 0. 05, we deleted a certain\n\nδ = 0. 05, R = 10 6\nall links 10% removed 30% removed 50% removed\nno. species 150 150 140 130\nno. links 240 220 170 125\nlinks per species 1. 6 1. 5 1. 2 1. 0\naverage level 2. 5 2. 3 2. 3 2. 1\nmax level 5. 2 4. 5 4. 2 4. 0\nB species (%) 10 16 24 30\nI species (%) 90 79 57 43\nT species (%) 0 5 19 25\nTB links (%) 0 2 8 16\nIB links (%) 22 25 29 31\nII links (%) 78 68 47 31\nTI links (%) 0 5 15 21\nprey/predators 1. 1 1. 2 1. 1 1. 0\nTable 4: The effect on the web properties of removing a fraction of t he links at random.\nproportion of links randomly. Deleted links represent real predato r-prey interactions which\nwere not observed due to poor experimental resolution. After re moval of the links, if there\nwere any species which were left unconnected to the rest of the we b these species were also\nremoved from the web. The deletion of the links was carried out on th e original species web,\nand the trophic species web for the new set of original species inter actions was then found.\nAs with all the tables in this article, figures refer to the trophic spec ies webs. Table 4 shows\nthe results of this process. T increases rapidly as links are removed, I decreases rapidly, and\nB increases more slowly. There are consequent changes in the propo rtions of T B, IB , II\nand T I links. The number of links per species, the average level and the max imum level all\ndecrease as links are removed. All these trends are the same as th ose observed when links\nare removed from the St. Martin web (Goldwasser & Roughgarden, 1997). The proportions\nof B, I and T species after removal of a substantial fraction of links are similar t o those in\nthe EcoWeb data, whereas before removal of the links they are clo se to the values of the\nmore highly resolved webs like Little Rock. This supports the hypothe sis that many of the\nwebs in the EcoWeb collection suffer substantially from incompletenes s. Real webs are also\ninfluenced by the degree to which species are aggregated into clust ers. Martinez (1991) has\nshown that aggregation of species within the original Little Rock web results in smaller webs\nwith statistics which are closer to those of the EcoWeb compilation th an the full web, which\nsuggests that the smaller webs are also influenced significantly by ag gregation. We have not\nyet considered aggregation effects in our model webs, but we expe ct that the changes will be\nsimilar to those observed with the Little Rock web.\n6 Results - Evolutionary Dynamics\nFigure 3 shows the number of species as a function of evolutionary t ime for three runs of\nthe simulation with the same parameter values. The numbers of spec ies differ significantly\nbetween different random assignments of the mij matrix. In each case the number of species\n\ntends to increase fairly rapidly at first, but after a certain time the system tends to a steady\nstate with fairly constant species number. Extinction avalanches a re visible as sudden drops\nin the species number. The figure shows the number of original spec ies, but the curves for\nthe number of trophic species follow the same pattern with slightly sm aller numbers, and\nshow extinction events occurring in the same places. In previous mo dels of macroevolution,\nattention has focused on the dynamics of extinction events when t he model is in a stable state.\nThe avalanche size distribution has been measured and has been obs erved to have a power\nlaw shape for some of these models (Bak & Sneppen, 1993; Sol´ e & B ascompte, 1996). In the\nWebworld model we may define the avalanche size at a given timestep a s the number of species\nwhich become extinct due to the addition of one new species (if the ne w species adds stably to\nthe web this is an avalanche of size zero). We intend to discuss the siz es of extinction events\nin our model in more detail in a subsequent paper. However, our initia l results suggest that\nthe size of avalanches tends to decrease with time in the Webworld mo del. Once the system\nhas reached a steady state, fluctuations in species number tend t o be small, whilst in the\ninitial stages of the simulation large avalanches can occur. Within the model large avalanches\nare associated with evolutionary progress. Initially there are relat ively few species, and these\nare relatively poorly adapted. If we consider only level 1 species initia lly, then we expect that\nthe scores Sn,0 of species will gradually increase as better adapted primary produc ers evolve.\nIn the early stages of the simulation there is a reasonable chance th at a new species will be\nsubstantially better than existing ones, and an extinction avalanch e may occur when the new\nspecies evolves. As time goes on it will become increasingly more difficult to evolve species\nwhich are better adapted, and when improvements do occur, scor es are likely to increase by\nsmaller amounts. Hence avalanches are likely to decrease in size and in frequency as time\ngoes on. Since the level 1 species tend to change less rapidly as they become better adapted\nit follows that conditions for higher level species become more stable . Hence level 2 species\ncan become increasingly better predators of the existing level 1 sp ecies, and so on throughout\nthe web. Therefore evolutionary changes on all levels of the web ar e likely to slow down in\nthe same way.\nSelf-organized models of evolution are designed in such a way that th e stationary state is\ncritical and has interesting dynamics. In these models the avalanch e size distribution and the\nspecies lifetime distribution are power laws. We believe that after a ve ry long time our model\nwould reach an absolutely stationary state where it would be impossib le for any new species\nto evolve. None of our simulations ever reached this point entirely, a lthough the probability\nthat a new species survives on the first timestep in which it evolved be came very small toward\nthe end of the runs. The time to reach this point of evolutionary ‘sta gnation’ should depend\non the total number of possible species in the model. The number of c ombinations of 10\nfeatures out of 500 possible ones is approximately 2 . 5 × 1020, which is large but still finite. We\nintend to investigate the effect of changing the total number of po ssible features K in future\nwork.\nThe initial period of evolutionary progress and large avalanche sizes is an interesting feature\nof our model, and we believe it may tell us something about the real wo rld. Is the real world\nin a stationary state? Are the properties of ecosystems today st atistically equivalent to those\nin earlier geological periods? We do not know how to answer these que stions, but we believe\nthey are important questions to ask. It is clear that the model has a finite number of possible\nspecies, but it is not clear whether this is true for the real world. Th e earth is obviously much\nmore complex than any computer model, however it is still finite in term s of the available\n\nspace, energy and raw materials. If the earth were unchanging, it does not seem unreasonable\nto suppose that real ecosystems would gradually perfect themse lves to the external conditions,\nand that the consequent rate of evolutionary change would decre ase, just as in the model.\nThe origin of life probably occurred about 3 . 5 − 3. 8 × 109 years ago. It is not clear whether\nthis is a long or a short period measured on the timescale of what is evo lutionarily possible.\nMeasures of diversity in the fossil record (Sepkoski, 1993) show a general increasing trend in\nspecies numbers between the Cambrian and the present, despite s everal very large extinction\nevents, and there is no real indication of any levelling off in diversity.\nOf course, the conditions on earth are not fixed: climatic change oc curs on a wide range\nof timescales. If we view the non-living world as continually changing, t hen the living world\nmust also continually change, and never has the opportunity to rea ch evolutionary stagna-\ntion. There is also the possibility that the external conditions might c hange by sudden rare\ncataclysmic events, such as meteorite strikes, rather than by sm ooth gradual change. Such a\nlarge scale external event might cause a large scale extinction, and might change conditions\nsufficiently that the evolutionary clock would effectively be set back t o an early stage where\necosystems were poorly adapted to the environment. This would en able a new burst of evo-\nlution, with many novel species arising. This suggests a picture of th e real world where there\nis a considerable rate of inherent evolutionary change and consider able fluctuation in species\nnumbers, and where external events causing major changes hap pen sufficiently often to pre-\nvent evolutionary stagnation from occurring. The data in the foss il record may some-day be\ncomplete enough to give some answers these questions. In the mea ntime there are still many\nuseful issues which can be addressed by studying models such as ou rs.\n7 Conclusions\nThe Webworld model describes the interactions of coevolving specie s. The model makes\npredictions concerning the structure of food webs which can be co mpared with data on real\nwebs. There are only three parameters in the model — R, λ, and δ — hence, the number of\ntestable predictions is much larger than the number of parameters . The measured quantities\nsuch as the number of links per species, the number of trophic levels , and the proportions of\ntop, basal and intermediate species, are not far from the values o bserved in real food webs in\nmost cases. It would be possible to choose parameters so that the results match a particular\nset of real food web data as closely as possible. However, given the uncertainties in most of\nthe present food web data, we have not tried to match these data too closely. Instead we have\ntried to point out the major qualitative trends in food web propertie s which occur when the\nparameters are changed. These trends make sense from an ecolo gical point of view. We would\nlike to contrast our model with the cascade model of food web stru cture (Cohen, 1990; Cohen\net al., 1990). Even though the cascade model successfully describ es many food web properties,\nit is basically just a set of probabilistic rules for assigning links between nodes in a graph.\nThe justification of these rules comes entirely from comparison mod el webs with real data.\nIn contrast, the parameters in our model have an ecological mean ing: the available external\nresources, the fraction of resources transferred from prey t o predator, and the strength of\ncompetition are all meaningful quantities in real webs.\nAn important question in food web theory which has had considerable attention recently\nis the issue of food web assembly (Luh & Pimm, 1993; Morton & Law, 19 97). In assembly\nmodels species are invading the ecological community from an extern al species pool and are\n\ntherefore unrelated to existing species, whereas in the Webworld m odel new species are arising\nby evolution and are therefore similar to the species from which they evolve. We intend to\ndevelop the model to compare properties of webs generated by inv asion and by evolution. One\nobservation in assembly models is that the probability of successful invasion of a community\ntends to decrease with time. This seems to have a parallel in Webworld , where the survival\nprobability of a newly generated species tends to decrease with time .\nThis work was motivated in part by the wish to explore recent claims th at models of\nevolution may have a tendency to “self-organise” into a critical non -equilibrium state which\nhas avalanches on all scales. We approached the problem by trying t o design a realistic model\nfor co-evolution in which the evolutionary dynamics can be studied, r ather than by deliberately\ndesigning a very simple model with interesting dynamics, but which is mo re difficult to relate\nto real evolutionary and ecological phenomena. This article has con centrated on the ecological\nproperties of the food webs, and we are currently investigating th e evolutionary properties of\nWebworld in more detail.\nAcknowledgements\nWe thank Mark Huxham for clarifying some details regarding the Ytha n web. This work\nwas supported in part by a grant from the University of Mancheste r and by EPSRC grant\nGR/K/79307.\nReferences\nArditi R. & Michalski, J. (1996) Nonlinear Food Web Models and their Re sponses to Increased\nBasal Productivity. Food Webs: Integration of Patterns and Dynamics pp 122-133. Eds. G.\nA. Polis and K. O. Winemiller. Chapman and Hall, New York.\nBak, P., Tang, C. & Wiesenfeld, K. 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American Naturalist 149, 125-\n144.\n\n/0/0/0\n/0/0/0\n/0/0/0\n/0/0/0\n/1/1/1\n/1/1/1\n/1/1/1\n/1/1/1\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/0/0/0/0/0/0/1/1/1/1/1/1\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\nExternal Resources\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/0\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/1\n/0/0/0/0\n/0/0/0/0\n/0/0/0/0\n/0/0/0/0\n/0/0/0/0\n/0/0/0/0\n/0/0/0/0\n/0/0/0/0\n/0/0/0/0\n/0/0/0/0\n/1/1/1/1\n/1/1/1/1\n/1/1/1/1\n/1/1/1/1\n/1/1/1/1\n/1/1/1/1\n/1/1/1/1\n/1/1/1/1\n/1/1/1/1\n/1/1/1/1\n/0/0/0/0/0\n/0/0/0/0/0\n/0/0/0/0/0\n/0/0/0/0/0\n/0/0/0/0/0\n/0/0/0/0/0\n/0/0/0/0/0\n/0/0/0/0/0\n/0/0/0/0/0\n/0/0/0/0/0\n/1/1/1/1/1\n/1/1/1/1/1\n/1/1/1/1/1\n/1/1/1/1/1\n/1/1/1/1/1\n/1/1/1/1/1\n/1/1/1/1/1\n/1/1/1/1/1\n/1/1/1/1/1\n/1/1/1/1/1\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\n/1/1/1/1/1/1/1/1\nlevel 2\nlevel 3\nlevel 1\nFigure 1: An illustrative example of a food web. Arrows indicate the dir ection of flow\nof resources. Input of external resources is specifically indicate d. Basal species are black,\nintermediate species are white, and top species are patterned. Bo xed species form part of the\nsame trophic species. The trophic level of a species is the length of t he shortest food chain\nfrom the external resources to that species.\n0 1 2 3 4 5 6 7 8 9 10\nLevel\n0.00\n20.00\n40.00\n60.00\n80.00\nNumber of species\nR=10\nR=10\nR=10\nFigure 2: The mean number of species on each trophic level with λ = 0 . 1, δ = 0 . 05 and\nR = 10 3, 106 and 10 10.\n\n0.0 50000.0 100000.0 150000.0 200000.0\ntime\n0.0\n100.0\n200.0\n300.0\nnumber of species\nFigure 3: The number of species in the web is shown as a function of tim e for three different\nruns of the Webworld model with the same parameters R = 10 6, δ = 0. 05 and λ = 0. 1."}
{"id": "adap-org/9801004", "meta": {"categories": ["adap-org", "nlin.AO"], "created": "1998-02-04", "extraction": {"body_chars": 15125, "cleaning": {"detected_repeated_margin_lines": ["1"], "page_count": 4, "removed_boilerplate_lines": 4}, "method": "pypdf_no_ocr", "source_pdf_bytes": 105902, "text_chars": 15462}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9801004", "primary_category": "adap-org", "source": "arxiv", "title": "Invariant closure for the Fokker-Planck equation", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9801004"}, "text": "Invariant closure for the Fokker-Planck equation\n\nAbstract\nWe develop the principle of dynamic invariance to obtain closed moment equations from the Fokker-Planck kinetic equation. The analysis is carried out to explicit formulae for computation of the lowest eigenvalue and of the corresponding eigenfunction for arbitrary potentials.\n\narXiv:adap-org/9801004v3 4 Feb 1998\nInvariant closures for the Fokker–Planck equation\nIliya V. Karlin ∗\nComputing Center RAS, Krasnoyarsk 660036, Russia\nV. B. Zmievskii †\nLMF/DGM, Swiss Federal Institute of Technology, CH-1015 Lausa nne, Switzerland\nWe develop the principle of dynamic invariance to obtain clo sed moment equations from the\nFokker–Planck kinetic equation. The analysis is carried ou t to explicit formulae for computation\nof the lowest eigenvalue and of the corresponding eigenfunc tion for arbitrary potentials.\n05.20.Dd, 05.70.Ln, 83.10.-y\nThe Fokker–Planck equation (FPE) is a familiar model in various proble ms of nonequilibrium statistical\nphysics [1]. In this paper we consider the FPE of the form\n∂tW = ∂x ·{D · [W ∂xU + ∂xW ]} . (1)\nHere W (x, t ) is the probability density over the configuration space x, at the time t, while U (x) and D(x) are\nthe potential and the positively semi-definite ( y · D · y ≥ 0) diffusion matrix. The dot denotes convolution in\nthe configuration space. The FPE (1) is particularly important in stu dies of polymer solutions [2]. Let us recall\nthe two properties of the FPE (1), important to what will follow: (i). Conservation of the total probability:∫\nW (x, t )dx = 1 . (ii). Dissipation: The equilibrium distribution, Weq ∝ exp(−U ), is the unique stationary\nsolution to the FPE (1). The entropy,\nS[W ] = −\n∫\nW (x, t ) ln\n[ W (x, t )\nWeq(x)\n]\ndx, (2)\nis a monotonically growing function due to the FPE (1), and it arrives a t the global maximum in the equilibrium.\nThese properties are most apparent when the FPE (1) is rewritten as follows:\n∂tW (x, t ) = ˆMW\nδS[W ]\nδW (x, t ) , (3)\nwhere ˆMW = −∂x · [W (x, t )D(x) · ∂x] is a positive semi–definite symmetric operator with kernel 1. The fo rm\n(3) (the dissipative vector field is a metric transform of the entrop y gradient) is an example of the dissipative\npart of a structure termed GENERIC in a recent series of papers [3 ].\nUsually one is interested in dynamics of moments of the distribution fu nction W rather than in the dynamics\nof the W itself. Except for simplest potentials U and diffusion matrices D, the moment equations, as they\nfollow from the FPE (1), are not closed. Therefore, closure proce dures are required.\nIn this paper we address the problem of closure for the FPE (1) in a g eneral setting. First, we review the\nmaximum entropy principle (MEP) as a source of suitable initial approx imations for the closures. We also\ndiscuss a version of the MEP, valid for a near–equilibrium dynamics, an d which results in explicit formulae for\narbitrary U and D.\nThe MEP closures are almost never invariants of the true moment dy namics, and corrections to the MEP\nclosures is the central issue of this paper. For this purpose, we ap ply the method of invariant manifold [4],\nwhich is carried out (subject to certain approximations explained be low) to explicit recurrence formulae for\none–moment near–equilibrium closures for arbitrary U and D. These formulae give a method for computing\nthe lowest eigenvalue of the problem, and which dominates the near– equilibrium FPE dynamics.\nMEP closures [5]. Let M = {M0, M 1, . . . , M k} be linearly independent moments of interest, Mi[W ] =∫\nmi(x)W (x)dx, and where m0 = 1. We assume existence a function W ∗ (M, x ) which extremizes the entropy\nS (2) under the constraints of fixed M . This MEP distribution function may be written\nW ∗ = Weq exp\n[ k∑\ni=0\nΛ imi(x) − 1\n]\n,\nwhere Λ = {Λ 0, Λ 1, . . . , Λ k} are Lagrange multipliers. Closed equations for moments M are derived in two\nsteps. First, the MEP distribution is substituted into the FPE (1) or (3) to give a formal expression: ∂tW ∗ =\nˆMW ∗ (δS/δW )\n⏐\n⏐\nW =W ∗ . Second, applying a projector Π ∗ ,\nΠ ∗ • =\nk∑\ni=0\n(∂W ∗ /∂M i)\n∫\nmi(x) • dx,\n\non both sides of this formal expression, we derive closed equations for M in the MEP approximation. Further\nprocessing requires an explicit solution to the constraints, ∫ W ∗ (Λ , x )mi(x)dx = Mi, to get the dependence of\nLagrange multipliers Λ on the moments M . Though typically the functions Λ( M ) are not known explicitly,\none general remark about the moment equations is readily available. Specifically, the moment equations in the\nMEP approximation have the form:\n˙Mi =\nk∑\nj=0\nM ∗\nij(M ) ∂S ∗(M )\n∂M j\n, (4)\nwhere S∗ (M ) = S[W ∗ (M )] is the macroscopic entropy, and where M ∗\nij is an M -dependent ( k + 1) × (k + 1)\nmatrix:\nM ∗\nij =\n∫\nW ∗ (M, x )[∂xmi(x)] · D(x) · [∂xmj(x)]dx.\nThe matrix M ∗\nij is symmetric, positive semi–definite, and its kernel is the vector δ0i. Thus, the MEP closure\nreproduces the GENERIC structure on the macroscopic level , the vector field of macroscopic equations (4) is a\nmetric transform of the gradient of the macroscopic entropy.\nTriangle MEP closures [6]. The following version of the MEP makes it possible to derive more exp licit results\nin a general setting: In many cases, one can split the set of moment s M in two parts, MI = {M0, M 1, . . . , M l}\nand MII = {Ml+1, . . . , M k}, in such a way that the MEP distribution can be constructed explicitly for MI\nas W ∗\nI (MI , x ). The full MEP problem for M = {MI, M II } in the ”shifted” formulation reads: extremize the\nfunctional S[W ∗\nI + ∆ W ] with respect to ∆ W , subject to the constraints MI [W ∗\nI + ∆ W ] = MI and MII [W ∗\nI +\n∆ W ] = MII . Let us denote as ∆ MII deviations of the moments MII from their values in the state W ∗\nI . For\nsmall deviations, the entropy is well approximated with a quadratic f unctional ∆ S[∆ W ] which is an expansion\nof the functional (2) in the state W ∗\nI up to the terms of the order ∆ W 2. With MI [W ∗\nI ] = MI , we come\nto the following problem: extremize the functional ∆ S[∆ W ], subject to the constraints MI [∆ W ] = 0, and\nMII [∆ W ] = ∆ MII . The solution to the latter problem is always explicitly found from a ( k + 1)× (k + 1) system\nof linear algebraic equations for Lagrange multipliers.\nIn the remainder of this paper we deal with one–moment near–equilib rium closures: MI = M0, (i. e. W ∗\nI =\nWeq), and the set MII contains a single moment M =\n∫\nmW dx, m(x) ̸= 1. We will specify notations for the\nnear–equilibrium FPE , writing the distribution function as W = Weq(1 + Ψ), where the function Ψ satisfies an\nequation:\n∂tΨ = W − 1\neq ˆJΨ , (5)\nwhere ˆJ = ∂x · [WeqD · ∂x]. The triangle one–moment MEP function reads:\nW (0) = Weq\n[\n1 + ∆ M m(0)\n]\n(6)\nwhere ∆ M = M − ⟨m⟩, and\nm(0) = [⟨mm⟩ − ⟨m⟩2]− 1[m − ⟨m⟩]. (7)\nBrackets ⟨. . . ⟩ =\n∫\nWeq . . . dx denote equilibrium averaging. The superscript (0) indicates that th e triangle MEP\nfunction (6) will be considered as an initial approximation to a proced ure which we address below. Projector\nfor the approximation (6) has the form\nΠ (0)• = Weq\nm(0)\n⟨m(0)m(0)⟩\n∫\nm(0) • dx. (8)\nsubstituting the function (6) into the FPE (5), and applying the pro jector (8) on both the sides of the resulting\nformal expression, we derive an equation for M : ˙M = −λ 0∆ M , where 1 /λ 0 is the inverse effective time of\nrelaxation of the moment M to its equilibrium value, in the MEP approximation (6):\nλ 0 = ⟨m(0)m(0)⟩− 1⟨∂xm(0) · D · ∂xm(0)⟩. (9)\nInvariant closures. Both the MEP and the triangle MEP closures are almost never invarian ts of the FPE\ndynamics. That is, the moments M of solutions to the FPE (1) vary in time differently from the solutions t o the\nclosed moment equations like (4), and these variations are generally significant even for the near–equilibrium\ndynamics. Therefore, we ask for corrections to the MEP closures to finish with the invariant closures [4].\nFirst, the invariant one–moment closure is given by an unknown distr ibution function W (∞ ) = Weq[1 +\n∆ M m(∞ )(x)] which satisfies an equation\n\n[1 − Π (∞ )] ˆJ m(∞ ) = 0. (10)\nHere Π (∞ ) is a projector, associated with an unknown function m(∞ ), and which is also yet unknown. Eq. (10)\nis a formal expression of the invariance principle for a one–moment n ear–equilibrium closure: considering W (∞ )\nas a manifold in the space of distribution functions, parameterized w ith the values of the moment M , we require\nthat the microscopic vector field ˆJ m(∞ ) be equal to its projection, Π (∞ ) ˆJ m(∞ ), onto the tangent space of the\nmanifold W (∞ ).\nNow we turn our attention to solving the invariance equation (10) ite ratively, beginning with the triangle\none–moment MEP approximation W (0) (6). We apply the following iteration process to the Eq. (10):\n[1 − Π (k)] ˆJ m(k+1) = 0, (11)\nwhere k = 0, 1, . . . , and where m(k+1) = m(k)+µ (k+1), and the correction satisfies the condition ⟨µ (k+1)m(k)⟩ = 0.\nProjector is updated after each iteration, and it has the form\nΠ (k+1)• = Weq\nm(k+1)\n⟨m(k+1)m(k+1)⟩\n∫\nm(k+1)(x) • dx. (12)\nApplying Π (k+1) to the formal expression, Weqm(k+1) ˙M = ∆ M [1 − Π (k+1)] ˆJ m(k+1), we derive the macroscopic\nequation, ˙M = −λ k+1∆ M , where λ k+1 is the ( k + 1)th update of the inverse effective time (9):\nλ k+1 = ⟨∂xm(k+1) · D · ∂xm(k+1)⟩\n⟨m(k+1)m(k+1)⟩ . (13)\nSpecializing to the one–moment near–equilibrium closures, and followin g a general argument [4], solutions to\nthe invariance equation (10) are eigenfunctions of the operator ˆJ, while the formal limit of the iteration process\n(11) is the eigenfunction which corresponds to the eigenvalue with t he minimal nonzero absolute value.\nDiagonal approximation. To obtain more explicit results in the iteration process (11), we intro duce an\napproximate solution on each iteration . The correction µ (k+1) satisfies the condition ⟨m(k)µ (k+1)⟩ = 0, and can\nbe decomposed as follows: µ (k+1) = α ke(k) + e(k)\nort. Here e(k) = W − 1\neq [1 − Π (k)] ˆJ m(k) = λ km(k) + R(k) is the\nvariance of the kth approximation, where\nR(k) = ∂x · [D · ∂xm(k)] − ∂xU · D · ∂xm(k). (14)\nThe function e(k)\nort is orthogonal to both e(k) and m(k): ⟨e(k)e(k)\nort⟩ = 0, and ⟨m(k)e(k)\nort⟩ = 0. Our diagonal\napproximation (DA) consists in disregarding the part e(k)\nort. Specifically, we consider the following ansatz at the\nkth iteration:\nm(k+1) = m(k) + α ke(k). (15)\nSubstituting the ansatz (15) into the Eq. (11), and integrating th e latter expression with the function e(k), we\nevaluate the coefficient α k:\nα k = Ak − λ 2\nk\nλ 3\nk − 2λ kAk + Bk\n, (16)\nwhere parameters Ak and Bk represent the following equilibrium averages:\nAk = ⟨m(k)m(k)⟩− 1⟨R(k)R(k)⟩ (17)\nBk = ⟨m(k)m(k)⟩− 1⟨∂xR(k) · D · ∂xR(k)⟩.\nFinally, putting together Eqs. (13), (14), (15), (16), and (17), we arrive at the following DA recurrency\nsolution, and which is our main result:\nm(k+1) = m(k) + α k[λ km(k) + R(k)], (18a)\nλ k+1 = λ k − (Ak − λ 2\nk)α k\n1 + (Ak − λ 2\nk)α 2\nk\n. (18b)\nTo test the convergency of the DA process (18) we have consider ed two potentials U in the FPE (1) with a\nconstant diffusion matrix D. The first test was with the square potential U = x2/ 2, in the three–dimensional\nconfiguration space, since for this potential the detail structur e of the spectrum is well known. We have\nconsidered two examples of initial one–moment MEP closures with m(0) = x1 + 100(x2 − 3) (example 1), and\n\nm(0) = x1 + 100x6x2 (example 2), in the Eq. (7). The result of performance of the DA fo r λ k (18b) is presented\nin the Table I, together with the error δk which was estimated as the norm of the variance at each iteration:\nδk = ⟨e(k)e(k)⟩/ ⟨m(k)m(k)⟩. In both examples, we see a good monotonic convergency to the min imal eigenvalue\nλ ∞ = 1, corresponding to the eigenfunction x1. This convergency is even striking in the example 1, where the\ninitial choice was very close to a different eigenfunction x2 − 3, and which can be seen in the non–monotonic\nbehavior of the variance. Thus, we have an example to trust the DA as converging to the stationary point of\nthe original iteration procedure (11).\nFor the second test, we have taken a one–dimensional potential U = −50 ln(1 − x2), the configuration\nspace is the segment |x| ≤ 1. Potentials of this type (so–called FENE potential) are used in applic ations of\nthe FPE to models of polymer solutions [2]. Results are given in the Table II for the two initial functions,\nm(0) = x2 + 10x4 − ⟨x2 + 10x4⟩ (example 3), and m(0) = x2 + 10x8 − ⟨x2 + 10x8⟩ (example 4). Both the examples\ndemonstrate a stabilization of the λ k at the same value after some ten iterations.\nIn conclusion, we have developed the principle of invariance to obtain moment closures for the Fokker–Planck\nequation (1), and have derived explicit results for the one–moment near–equilibrium closures, particularly\nimportant to get information about the spectrum of the FP operat or.\nI. V. K. is thankful to M. Grmela for providing a preprint of their pap er [3], and to H. C. ¨Ottinger for\nstimulating discussions. V. B. Z. is thankful to J.–C. Badoux and M. D eville for the possibility of the research\nstay at the EPFL. The work of I. V. K. was supported by CNR, and b y RFBR through grant No. 95-02-03836-a.\n∗ The author to whom correspondence should be sent. Present ad dress: Istituto Applicazioni Calcolo CNR, V. del\nPoliclinico, 137, Roma I-00161, Italy. E-mail: iliya@momi x.iac.rm.cnr.it\n† Permanent address: Computing Center RAS, Krasnoyarsk 6600 36, Russia\n[1] N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam 1981); H. Risken,\nThe Fokker–Planck Equation (Springer, Berlin 1984).\n[2] R. B. Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids , 2nd edn., (Wiley, New\nYork, 1987); M. Doi and S. F. Edwards, The Theory of Polymeric Dynamics (Clarendon Press, Oxford, 1986); H. C.\n¨Ottinger, Stochastic Processes in Polymeric Fluids (Springer, Berlin, 1996).\n[3] M. Grmela and H. C. ¨Ottinger, Phys. Rev. E (1997) to appear.\n[4] A. N. Gorban and I. V. Karlin, Transp. Theory Stat. Phys. 23, 559 (1994).\n[5] The MEP was used by many authors for various systems, e. g. for the Boltzmann equation: A. M. Kogan, Prikl.\nMath. Mech. 29, 122 (1965), for a recent thorough study see C. D. Levermore, J. Stat. Phys. 83, 1021 (1996); for\nMarkov chains: A. N. Gorban, Equilibrium Encircling. Equations of Chemical Kinetics an d their Thermodynamic\nAnalysis (Nauka, Novosibirsk, 1984); for the BBGKY hierarchy: J. Kar kheck and G. Stell, Phys. Rev. A 25, 3302\n(1984); A very general discussion of the MEP is given in: R. Ba lian, Y. Alhassid and H. Reinhardt, Phys. Rep. 131,\n1 (1986). We, however, failed to find a reference for a direct a pplication of the MEP to the FPE.\n[6] I. V. Karlin, in: Mathematical Problems of Chemical Kinetics , eds. K. I. Zamaraev and G. S. Yablonskii, (Nauka,\nNovosibirsk, 1989), p. 7; A. N. Gorban and I. V. Karlin, Phys. Rev. E 54, R3109 (1996).\nTABLE I. Iterations λ k and the error δk for U = x2/ 2.\n0 1 4 8 12 16 20\nEx. 1 λ 1.99998 1.99993 1.99575 1.47795 1.00356 1.00001 1.00000\nδ 0. 16 ·10− 4 0. 66 ·10− 4 0. 42 ·10− 2 0. 24 0. 35 ·10− 2 0. 13 ·10− 4 0. 54 ·10− 7\n0 1 2 3 4 5 6\nEx. 2 λ 3.399 2.437 1.586 1.088 1.010 1.001 1.0002\nδ 1. 99 1. 42 0. 83 0. 16 0. 29 ·10− 1 0. 27 ·10− 2 0. 57 ·10− 3\nTABLE II. Iterations λ k for U = − 50 ln(1 − x2).\n0 1 2 3 4 5 6 7 8\nEx. 3 λ 213.1774 212.1864 211.9148 211.8619 211.8499 211.8453 211.8433 211.8422 211.8417\nEx. 4 λ 216.5856 213.1350 212.2123 211.9984 211.9295 211.8989 211.8838 211.8757 211.8713"}
{"id": "adap-org/9802002", "meta": {"categories": ["adap-org", "nlin.AO", "q-bio"], "created": "1998-02-13", "extraction": {"body_chars": 73281, "cleaning": {"detected_repeated_margin_lines": ["1"], "page_count": 40, "removed_boilerplate_lines": 99}, "method": "pypdf_no_ocr", "source_pdf_bytes": 1321737, "text_chars": 74260}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9802002", "primary_category": "adap-org", "source": "arxiv", "title": "Emergence of Rules in Cell Society: Differentiation, Hierarchy, and Stability", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9802002"}, "text": "Emergence of Rules in Cell Society: Differentiation, Hierarchy, and Stability\n\nAbstract\nA dynamic model for cell differentiation is studied, where cells with internal chemical reaction dynamics interact with each other and replicate. It leads to spontaneous differentiation of cells and determination, as is discussed in the isologous diversification. Following features of the differentiation are obtained: (1)Hierarchical differentiation from a ``stem'' cell to other cell types, with the emergence of the interaction-dependent rules for differentiation; (2)Global stability of an ensemble of cells consisting of several cell types, that is sustained by the emergent, autonomous control on the rate of differentiation; (3)Existence of several cell colonies with different cell-type distributions. The results provide a novel viewpoint on the origin of complex cell society, while relevance to some biological problems, especially to the hemopoietic system, is also discussed.\n\narXiv:adap-org/9802002v1 13 Feb 1998\nEmergence of Rules in Cell Society:\nDifferentiation, Hierarchy, and Stability\nChikara Furusawa and Kunihiko Kaneko\nDepartment of Pure and Applied Sciences\nUniversity of Tokyo, Komaba, Meguro-ku, Tokyo 153, JAPAN\nAbstract\nA dynamic model for cell differentiation is studied, where cel ls with\ninternal chemical reaction dynamics interact with each oth er and repli-\ncate. It leads to spontaneous differentiation of cells and det ermination,\nas is discussed in the isologous diversification. Following features of\nthe differentiation are obtained: (1)Hierarchical differenti ation from a\n“stem” cell to other cell types, with the emergence of the int eraction-\ndependent rules for differentiation; (2)Global stability of an ensemble\nof cells consisting of several cell types, that is sustained by the emer-\ngent, autonomous control on the rate of differentiation; (3)E xistence of\nseveral cell colonies with different cell-type distribution s. The results\nprovide a novel viewpoint on the origin of complex cell socie ty, while\nrelevance to some biological problems, especially to the he mopoietic\nsystem, is also discussed.\n1 Introduction\nA multicellular organism is an ordered clone of a fertilized egg. All the\ncells contain the same genome set but are specialized in different ways . The\nemergence of different cell types is determined rather precisely, w hile the\ndevelopmental process of the cells, viewed as a cell society, has ro bustness\nagainst perturbations.\nIn molecular biology, the differentiation processes are often regar ded as\non-off switching processes. Switch depends on inputs by signal mole cules,\nwhich leads to a variety of cell types as outputs. A large number of r eac-\ntions between inputs and outputs are represented as a “cascade ”, where the\nreactions are assumed to be approximately independent of the oth er reac-\ntion processes in the cell. The switching behavior (given by the sigmoid al\nfunction) is assumed to be generated from a chain of these reactio ns. With\nthis viewpoint, one can decompose the differentiation by successive local el-\nementary processes. It enables us to elucidate the differentiation processes\nby experimental methods, where several signal molecules and ess ential genes\nfor differentiations are identified.\nOf course, the development progresses through cooperation of several pro-\ncesses. Successive differentiation processes are often express ed as “canaliza-\ntion”, where differentiations are captured as a result of dynamics o f complex\n\nchemical networks, in contrast with a linear combination of simple pat hways\nof chemical reactions. The pioneering study of Kaufmann (1969) d emon-\nstrated that the Boolean network of genes gives a variety of final states de-\npending on the initial conditions, and he has suggested that each fin al state\ncorresponds to each cell type. However, needless to say, a single initial state\nembedded in a fertilized egg can produce several different cell type s. Thus,\nthe following questions remain unanswered about the gene network : How\ndo the different initial states leading to the different cell types arise in the\nprocess of development? How does a selection of specific initial cond itions\nlead to precise rules of differentiations?\nIt should be noted that in the gene network picture cellular interact ions\nare not explicitly taken into account, which should be important in the course\nof development. A pioneering study for the pattern formation is du e to\nTuring (1952), where dynamic instability by the cellular interactions le ads to\nthe pattern formation (Newman, 1990). However, it remains still u nsolved\nhow such cell-to-cell interactions are incorporated with internal d ynamical\ncomplexity including the gene networks (see also Bignone, 1993; Mjo lsness\net al., 1991; and Thomas et al, 1995).\nHence it is necessary and important to consider a system of interna l dy-\nnamics with suitable cell-cell interactions. One of the authors (KK) a nd\nYomo have performed several simulations of interacting cells with int ernal\nbiochemical networks and cell divisions that lead to the change in the num-\nber of degrees of freedom. The “isologous diversification theory” is proposed\nas a general mechanism of spontaneous differentiation of replicatin g biologi-\ncal units (Kaneko & Yomo, 1994, 95, 97). In the theory, the follow ing three\npoints are essential.\n• Spontaneous differentiation: The cells, which have oscillatory chem-\nical reactions within, differentiate through interaction with other c ells.\nThis differentiation is provided by the separation of orbits in the phas e\nspace. The dynamics of cells first split into groups with different phas es\nof oscillations, and then to groups with different compositions of che m-\nicals. These differentiations are not caused by specific substances , but\nare triggered by the instability brought about by nonlinear systems .\nThe background of this lies in the dynamic clustering in globally cou-\npled chaotic systems (Kaneko, 1990, 91, 92).\n• Inheritance of the differentiated state to the offspring: Each\ndifferentiated state of a cell is preserved by the cell division and tra ns-\nmitted to its offspring. Chemical composition of a cell is recursively\nkept with respect to divisions. Thus a kind of “memory” is formed,\nthrough the transfer of initial conditions (e.g., of chemicals). By re pro-\nduction, the initial condition of a cell is chosen so that the same cell\ntype is produced at the next generation.\n• Global stability: Multicellular organism often shows a robustness\nagainst some perturbation, such as somatic and other mutations. An\nextreme example is seen in a mutation to triploid in newt, where the cell\nsize becomes three times, but the total cell number is reduced to o ne\n\nthird, and the final body remains not much affected by the mutation\n(Fankhauser, 1995).\nThe distribution of cell types obtained is robust against external p ertur-\nbations. For instance, when the number of one type of cells is decre ased\nby external removal, the distribution is recovered by further diffe renti-\nations to generate the removed cell-type. In this theory, althoug h the\ninstability triggers the differentiations, the cell society as a whole is\nstabilized through cell-to-cell interactions.\nIn the present paper we extend these previous studies (Kaneko & Yomo,\n1997) to incorporate the formation of a complex cell society. We ex tend our\nmodel to allow for complex internal dynamics, in particular, focusing on the\nfollowing three topics.\n• (i) The hierarchical organization and the emergence of stoc has-\ntic rules; The cell differentiation process in nature follows a hierar-\nchical organization. For example, the pluripotent cells like stem cells\ngive rise to committed cells, which further differentiate to terminally\ndifferentiated cells. Here the rules of differentiation are written as e x-\npressions of DNA in principle, but it should be noted that the differ-\nentiation is often interaction dependent. Furthermore, in the hem opoi-\netic system, the differentiation process appears to be stochastic , and\nthe probability of each choice seems to depend also on the distribu-\ntion of cell types (Ogawa, 1993). Hence it is interesting how such\ninteraction-dependent rules of hierarchical differentiation are fo rmed\nnaturally through the interplay between internal dynamics and cell- to-\ncell interactions.\n• (ii) Stability of cell types and cell groups; Cells belonging to the\nsame cell-type also slightly differ each other. Hence discretization of\nstates to types and their continuous change coexist among cells. T he\ndifferentiation rules of cell types are written for the discrete type s.\nWhen cell differentiation is determined, memory of the discrete stat e is\nstable against cell division. Then, the state has to be dynamically sta -\nble (like an attractor), while for stem cells or undetermined cells, the ir\nstate must have both stability and variability (differentiability) by divi-\nsions. Here we are interested in how such stability and differentiability\nare compatible as a state of dynamical systems. Besides the stabilit y of\na cellular state, stability about the distribution of cell types has to b e\nattained through the developmental process. For example, the d istri-\nbution of cell types in the hemopoietic system is robust against exte rnal\nperturbations. Here we try to answer the question of stability thr ough\nan interplay of internal dynamics and cell-cell interaction, that lead s to\nmodulation of internal cellular states and to stability of distribution o f\nthe cells at the ensemble level.\n• (iii) Differentiation of cell colony; In an organism, there often\nappears a higher level of differentiations, leading to several distinc t\ntypes of tissues. They consist of different types of cells and/or diff erent\ndistribution of cell types. Indeed in the hemopoietic system, sever al\n\ncolonies consisting of different cell types appear from same stem ce lls\n(Nakahata et al., 1982). It is an important question how a single cell\ncan form such different cell colonies. This is a higher level question\nthan cell differentiation, since the population of cell types has to be\ndifferentiated.\nIn the present paper we study the above three problems by exten ding the\nprevious model of cell differentiation, to allow for complex internal d ynamics.\nHere the cellular states are given by a set of chemical concentratio ns, while\nthe internal dynamics is given by mutually catalytic reaction network s. In\ncontrast with the previous model, the internal dynamics allows for c haos and\nalso coexistence of multiple attractors. Interaction among cells is g iven by\nthe diffusive transport of chemicals between each cell and a homoge neous\nenvironment. Cell volume is increased through the transport of ch emicals\nfrom the environment, which leads to the cell division when it is larger t han\na given threshold.\nBy allowing complex dynamics at the internal cell level, we will show\nthat the above three problems are answered from our standpoint . First, hi-\nerarchical differentiation of several cell types is formed. There a ppears a cell\ntype that plays the role of “stem cell”, from which different cell type s are\ndifferentiated. The probabilistic switch of cell types is given through the in-\nternal dynamics, whose rate is dependent on the interaction, and accordingly,\non the distribution of other cell types. Second, the stability of cell types is\ngiven as a “partial attractor” (to be discussed) of the internal d ynamics, sta-\nbilized through interactions. Third, stochastic population dynamics of cell\ntypes emerges as a higher level. It is found that this dynamics has se veral\nattracting states, which supports different stable cell colonies (t issues).\nThe organization of the paper is as follows. In §2, our model is pre-\nsented. Although a specific type of catalytic reaction network is ad opted\nin the present paper, it should be noted that the results are gener ally seen\nin a variety of reaction networks. Although the interaction we adop t here is\nglobal, in the sense that all cells interact with each other, our main co nclusion\non the differentiation and the formation of cell society is invariant ev en if the\n“spatial” effect is explicitly taken into account as local diffusion proce ss. In\n§3, we will show numerical results of the evolution of cell society from a single\ncell, where the emergence of distinct cell types is given. Differentiat ion of\nthese cells is found to obey a specific rule, that emerges as a higher le vel than\nthe chemical reaction rules we have adapted. The mechanism of disc retiza-\ntion of states, and the formation of (interaction-dependent) ce ll memory is\ndiscussed in §4. The rule of “stochastic” differentiation from a stem-type\ncell is studied in §5, where the stability of the population of cell types is\nnoted. In §6, the diversity of cell colonies is shown in relation with several\nattracting states of a higher-level dynamics, i.e., the population dy namics of\ncell types. Summary and discussions are given in §7 and §8, where relevance\nof our results to cell biology is discussed, which covers origin of stem cells, in\nparticular stochastic branching, stability and diversity of cell colon ies in the\nhemopoietic system, and the origin of multicellular organism.\n\n2 Model\nOur model for differentiation consists of\n• Internal dynamics by biochemical reaction network within each cell\n• Interaction with other cells through media: Inter-cellular dynamics\n• Cell division\nThe basic strategy of the modeling follows the previous works (Kane ko &\nYomo, 1997), although we take different dynamics for each of the a bove three\nprocesses. In essence we assume a network of catalytic reaction s for internal\ndynamics that allows for a periodic and/or chaotic oscillations of chem icals,\nwhile the interaction process is just a diffusion of chemicals through m edia.\nWe represent the internal state of a cell by k chemicals’ concentrations as\ndynamical variables. Cells are assumed to be in surrounding media, wh ere\nthe same set of chemicals is given. Hence the dynamics of the interna l state\nis represented by a set of variables x(m)\ni (t), the concentration of the m-th\nchemical species at the i-th cell, at time t. The corresponding concentration\nof the species in the medium is represented by a set of variables X (m)(t). We\nassume that the medium is well stirred by neglecting the spatial varia tion\nof the concentration, so that all cells interact each other throug h identical\nenvironment.\n2.1 Internal chemical reaction\nWithin each cell, there is a network of biochemical reactions. The net work\nincludes not only a complicated metabolic network but also reactions a ssoci-\nated with genetic expressions, signaling pathways, and so on. In th e present\nmodel, a cellular state is represented by the concentrations of k chemicals.\nAs internal chemical reaction dynamics we choose a catalytic netwo rk\namong the k chemicals. Each reaction from the chemical i to j is assumed\nto be catalyzed by the chemical ℓ, which is determined randomly. To repre-\nsent the reaction-matrix we adopt the notation Con(i, j, ℓ) which takes unity\nwhen the reaction from the chemical i to j is catalyzed by ℓ, and takes 0\notherwise. Each chemical has several paths to other chemicals, w hich act\nas a substrate to create several enzymes for other reactions. Thus these\nreactions form a complicated network. This matrix is generated ran domly\nbefore simulations, and is fixed throughout the simulation. We use th e same\nreaction-matrix throughout a series of simulations in this paper (se e also §3\nand §7 for dependence on the reaction-matrix).\nUsually, chemical kinetics with enzymes is solved under some approxi-\nmations, like Michaelis-Menten form. In this paper, we assume quadr atic\neffect of enzymes. Thus the reaction from the chemical m to ℓ aided by the\nchemical j leads to the term e1x(m)\ni (t)(x(j)\ni (t))2, where e1 is a coefficient for\nchemical reactions, which is taken identical for all paths. The quad ratic ef-\nfect of enzymes is not essential to our scenario of cell differentiat ions. Several\nother forms on the internal dynamics lead to qualitatively the same b ehavior,\nas long as nonlinear oscillation is included. The scenario of the different iation\n\nwhich we propose here is independent of the details of this specific ch oice of\nbiochemical dynamics.\nBesides the change of chemical concentrations, we have to take in to ac-\ncount the change of the volume of cell. The volume is now treated as a\ndynamical variable, which increases as a result of transportation o f chem-\nicals into the cell from the environment. Of course, the concentra tions of\nchemicals are diluted according to the increase of the volume of the c ell.\nFor simplicity, we assume that the volume of cell is proportional to th e sum\nof chemicals in the cell. Under this assumption, the operation which co m-\npensates the concentration of chemicals with the volume change is id entical\nto imposing the restriction ∑\nℓ x(ℓ)\ni = 1, namely normalizing the chemical\nconcentrations at each step of the calculation, while the volume cha nge is\ncalculated from the transport as will be given later.\n2.2 Interaction with other cells through media\nEach cell communicates with its environment through transport of chemicals.\nInteractions between cells, thus, occur through the environmen t. Here, the\nenvironment does not mean external environment for individual or ganism,\nbut is intended as interstitial environment of each cell. In this model, we\nconsider only diffusion process through the cell membrane. Thus, t he rates\nof chemicals transported into a cell are proportional to difference s of chemical\nconcentrations between the inside and the outside of the cell. Of co urse, the\ntransport through the membrane is not so simple, including several mecha-\nnisms such as channel proteins and endocytosis. We omit these com plicated\nmechanisms for simplicity.\nThe transportation or diffusion coefficient should be different for diff erent\nchemicals. Here we assume that there are two types of chemicals, t hose which\ncan penetrate the membrane and which can not. We use the notatio n σm,\nwhich takes 1 if the chemical x(m)\ni is penetrable, and 0 otherwise.\nTo sum up all these process, the dynamics of chemical concentrat ion in\neach cell is represented as follows:\ndx(ℓ)\ni (t)/dt = δx(ℓ)\ni (t) − (1/k)\nk∑\nl=1\nδx(ℓ)\ni (t) (1)\nwith\nδxℓ\ni(t) =\n∑\nm,j\nCon(m, ℓ, j) e1 x(m)\ni (t) (x(j)\ni (t))2\n−\n∑\nm′,j′\nCon(ℓ, m′, j′) e1 x(ℓ)\ni (t) (x(j′)\ni (t))2\n+σℓD(X (ℓ)(t) − x(ℓ)\ni (t)) (2)\nwhere the term with\n∑\nCon(· · ·) represents paths coming into ℓ and out of\nℓ respectively. The term δx(ℓ)\ni gives the increment of chemical ℓ, while the\nsecond term in eq.(1) gives the constraint of ∑\nℓ x(ℓ)\ni (t) = 1 due to the growth\nof the volume. The third term in eq.(2) represents the transport b etween\nthe medium and the cell, where D denotes a diffusion constant, which we\nassume to be identical for all chemicals. Since the penetrable chemic als in\n\nthe medium can be consumed with the flow to the cells, we need some flo w\nof chemicals (nutrition) into the medium from the outside. By denotin g the\nexternal concentration of these chemicals by X and its flow rate per volume\nof the medium by f , the dynamics of penetrable chemicals in the medium is\nwritten as\ndX (ℓ)(t)/dt = f σℓ(X (ℓ) − X (ℓ)(t)) − (1/V )\nN∑\ni=1\nσℓD(X (ℓ)(t) − x(ℓ)\ni (t)) (3)\nwhere N denotes the number of the cells in the environment, and V denotes\nthe volume of the medium in the unit of a cell.\n2.3 Cell division\nEach cell takes penetrable chemicals from the medium as the nutrien t, while\nthe reactions in the cell transform them to unpenetrable chemicals which\nconstruct the body of the cell such as membrane and DNA. As a res ult of\nchemical flow, the volume of the cell is increased by the factor (1 +\n∑\nℓ δxℓ\ni (t))\nper dt. In the present paper, the cell is assumed to divide into two almost\nidentical cells when the volume of the cell is doubled. 1 2\nThe concentrations of chemicals in the daughter cells are almost equ al\nto the concentrations of the mother cell. “Almost” here means tha t the\nconcentrations of chemicals in a daughter cell are slightly different f rom the\nmother’s. Each cell has (1 + ǫ)x(l) and (1 − ǫ)x(l) respectively with a small\n“noise” ǫ, a random number with a small amplitude, say over [ − 10−6, 10−6].\nAlthough the existence of imbalance is essential to the differentiatio n in our\nmodel and in nature, the degree of imbalance itself is not essential t o our\nresults to be discussed. The important feature of our model is the amplifi-\ncation of microscopic differences between the cells through the inst ability of\nthe internal dynamics.\n2.4 Internal dynamics in single cell\nBefore studying the dynamics of cell society, we demonstrate a ty pical be-\nhavior of our model by taking only one cell and medium. In our theory , the\nfundamental assumption is that the internal dynamics of chemicals in the\ncell shows oscillation as in Fig.1. In real biological systems, oscillations are\n1In other words, the cell divides at the time t when\n∫ t\ntb\nexp(1 +\n∑\nℓ\nδxℓ\ni (t′))dt′ = 2 (4)\nis satisfied since the previous division time tb.\n2 Embryos fall into two general categories: those in which cell division is accompanied\nby growth of the cells back to their former volume (as mammals and bir ds); those in which\ncell division results in cells 1/2 the previous volume (as amphibians). Alt hough our model\nhere adopts the division process as in mammals and birds, we have also confirmed that\nthe present differentiation mechanism also holds for a model with amp hibians-like rules,\nwhere cell division makes cell volume 1/2, and each cell interacts with neighborhood cells\nlike gap junctions.\n\nobserved in some chemical substrates such as Ca, NADH, cyclic AMP , and\ncyclins (Tyson et al., 1996; Hess et al., 1971; Alberts et al., 1994). He nce it is\nnatural to postulate such oscillatory dynamics to our model. The imp ortance\nof oscillatory dynamics in cellular systems has been pointed out by Goo dwin\n(1963).\nThe nature of internal dynamics by eqs.(1)-(2) depends on the ch oice of\nthe reaction network, in particular on the number of paths in the re action\nmatrix. When the number of reaction paths is small, cellular dynamics f alls\ninto a steady state without oscillation, where a small number of chem icals is\ndominant while other chemicals’ concentrations vanish. On the othe r hand,\nwhen the number of reaction paths is large, many chemicals generat e each\nother. Then chemical concentrations take constant values (whic h are of-\nten almost equal). Only for medium number of reaction paths, non-t rivial\noscillations of chemicals appear as in Fig.1. We use such network for ou r\nsimulation. It is not easy to estimate the number of paths in real bioc hem-\nical data, although they may suggest the medium number (3-6) of p aths as\nrequired in our simulation.\nFurthermore, the behavior of dynamics depends on the number of pene-\ntrating chemicals. The number of penetrating chemicals is another c ontrol\nparameter for the capacity of the oscillation or differentiation. Whe n the\nnumber of penetrating chemicals is small, e.g., only one, the rate of ra n-\ndomly chosen reaction networks which show oscillatory dynamics is sm all.\nOn the other hand, when the number of penetrating chemicals is too large,\nit is also difficult to obtain the network with oscillatory dynamics.\nAnother relevant factor to the nature of internal dynamics is the frequency\nof auto-catalytic paths. Indeed, the oscillatory dynamics is rathe r common\nas the number of auto-catalytic paths is increased (see §7).\n3 Differentiation Process: Numerical Results\nWe have performed several simulations of our model with different c hemical\nnetworks and different parameters. Since typical behaviors are r ather com-\nmon, we present our results by taking a specific chemical network w ith the\nnumber of chemicals k = 20. 3\nAs an initial condition, we take a single cell, with randomly chosen chem-\nical concentrations of x(ℓ)\ni satisfying ∑\nℓ x(ℓ)\ni = 1. In Fig.1, we have plotted\na time series of concentration of the chemicals in a cell, when only a sing le\ncell is in the medium. This attractor of the internal chemical dynamic s is a\nlimit cycle, whose period is longer than the plotted range in Fig.1. We call\nthis state “attractor-0” or “type-0” in this paper. This is the only attractor\nthat is detected from randomly chosen initial conditions 4.\n3 In this example, we do not choose reaction paths equivalently among all chemicals,\nbut select two class of reactions randomly. One class of reactions is paths from penetrable\nto any other chemicals, and another is paths from any of chemicals t o penetrable ones. The\npurpose of this selection is to enhance auto-catalytic reaction loop , and to get oscillatory\nreaction dynamics easily. Of course, reaction networks chosen eq uivalently and randomly\ncan also show the same type of behavior to be discussed in this paper . As for relationship\nbetween auto-catalytic reactions and our scenario for differentia tion, see also §7.\n4As will be seen later, there is another attractor as a single cell stat e. However, this\n\nWith the diffusion term, external chemicals flow into the cell, which lead s\nto the increase of the volume of the cell. Thus the cell is divided into tw o,\nwith almost identical chemical concentrations. Chemicals of the two daughter\ncells oscillate coherently, with the same dynamical behavior as the mo ther\ncell (i.e., attractor-0). Successive cell divisions occur simultaneou sly, and the\ncell number increases as 1 − 2 − 4 − 8 · · ·, up to some threshold number. At\nthis stage, internal dynamics of each cell belongs to the same attr actor (i.e.,\nattractor-0), but the oscillations are no longer synchronized. Th e microscopic\ndifferences introduced at each cell division are amplified to a macrosc opic\nlevel through the interaction, which destroys the phase coheren ce.\nWhen the number of cells exceeds this threshold value, some cells sta rt\nto show a different type of dynamics. The threshold number depend s on\nthe parameters of our model. In the present example, 2 cells start to show\na different dynamical behavior (as plotted in Fig.2(a)), when the tot al cell\nnumber becomes 16. In Fig.2(a), the time series of the chemicals in th is cell\nare plotted. We call the state as “partial attractor-1” (or “typ e-1” cell). We\ndo not call it an attractor, since the state does not exist as an att ractor of\ninternal dynamics of a single cell. As will be discussed later, the stabilit y\nof the state is sustained only through the interaction. In Fig.3(a), orbits of\nchemical concentrations are plotted in the phase space during the transition\nform type-0 to type-1. It shows that each attractor occupies d istinct regimes\nin the phase space. These two types of cells are clearly distinguishab le as\ndigitally distinct states. Hence we interpret this phenomenon as diffe rentia-\ntion.\nAs the cell number further increases, another type of cell appea rs, which\nwe call type-2 here. It is again differentiated from the type-0 cell ( see Fig.2(b)\nand Fig.3(b)). The type-0 cells have potentiality to differentiate to e ither “1”\nor “2”, while some of the type-“0” cells remain to be of the same type by the\ndivision.\nFor some simulations (i.e., for some initial conditions), the differentiat ion\nprocess stops at this stage, and only three types of cells coexist. In many\nother simulations, however, the differentiation process continues . At this\nstage, hierarchical differentiation occurs. The cell type “1” furt her differen-\ntiates into either of three groups represented as “3”, “4”, or “5 ”. The time\nseries of these three types are shown in Fig.2(c)-(e). The interna l dynamics of\neach type is plotted in a projected phase space in Fig.4 . The orbit of t ype-1\ncell itinerates over the three regions corresponding to “3”, “4”, and “5”. For\nexample, Fig.3(c) shows a switch from type-1 to type-3 in the phase space\nby taking a projection different from that in Fig.3(a)(b) (note the d ifference\nof scales). It is also noted that the difference by cell types is more c learly\ndistinguishable by chemicals with lower concentrations.\nIn the normal course of cell differentiation process (without exte rnal oper-\nation), cells of the types “2” and “1” reproduce themselves or fur ther differ-\nentiate to the other cell types, but the offspring never go back to the type-0\ncell. Besides the cell type-2, the cell types-3, 4, and 5 reproduce themselves\nwithout any further differentiation. Among these three types, on ly the cell\nattractor is not observed when the initial condition is randomly chos en; in other words\nthe basin volume for it is very small.\n\ntype-5 is an attractor by itself, while others replicate only under th e pres-\nence of different types of cells. Indeed, the type-5 is rather spec ial, whose\nappearance destabilizes the cell society consisting of “0”, “1”, an d “2”. Once\nthe type-5 cell appears, all the cells will finally be transformed to th is type.\nWhether the type-5 cell appears or not depends on the initial cond ition, while\nthe cell society without the type keeps diversity of cell types (see §6).\nAt this stage the differentiation is determined, and cellular memory is\nformed as is first discussed in (Kaneko and Yomo, 1997). According ly we\ncan draw the cell lineage diagram as shown in Fig.5, where the division\nprocess with time is represented by the connected line between mot her and\ndaughter cells while the color in the figure shows the cell type.\nThe switch of types by differentiations turns out to obey a specific r ule.\nIn Fig.6, we write down an automaton-like representation of the rule of dif-\nferentiation. The node “0” has three paths; one to itself, and the others to\nthe nodes “1” and “2”. The path to itself means replication of the sa me cell\ntype through division, while the other paths give the differentiation t o the\ncorresponding cell types. Fig.6 represents the potentiality of the se differen-\ntiations.\nNote that this differentiation is not induced directly by the tiny differ-\nences introduced at the division. The switch from one cell-type to an other\ndoes not occur simultaneously with the division, but occurs later thr ough\nthe interaction among the cells. This phenomenon is caused by dynam ical\ninstability in the total system consisting of all cells and medium. The tin y\ndifference between two daughter cells is amplified to yield macroscopic differ-\nence through the interaction. Our results show that these trans itions are not\naccompanied by the cell division but occur through cell-to-cell inter actions.\nThis conclusion is consistent with experimental data, where the ons et of new\ngene expression is not always accompanied by the cell division. Accor ding\nto our theory and simulations, the time lag between the cell division an d\nthe onset of new gene expressions depends on the cell-to-cell inte raction, i.e.,\nthe surrounding cells. On the other hand, change in the number of d egrees\nof freedom by division amplifies the instability in the dynamics of the tot al\nsystem. When the instability exceeds some threshold, the different iations\nstart. Then, the emergence of another cell type stabilizes the dy namics of\neach cell again. The cell differentiation process in our model is due to the\namplification of tiny differences by orbital instability (transient chao s), while\nthe coexistence of different cell types stabilizes the system.\n4 State Discretization, Hierarchical Organi-\nzation and Dual Memory\nOne might wonder that our definition of types is rather ambiguous an d is\nnot clearly defined. Indeed one can clearly distinguish them by plottin g and\ncomparing the time series and check how these orbits are separate d. To\nconfirm that the state in each type is clearly separated, we introdu ce the\ndistance between cells in the k-dimensional phase space.\nSince a cell’s state is determined by chemical concentrations in the pr esent\nmodel, the cellular state is represented by an orbit in the k-dimensional phase\n\nspace. Here we first consider the average position of an orbit for s implicity;\nx(ℓ)\ni = (1/T )\n∫\nx(ℓ)\ni (t)dt (5)\nAs the difference between two cells we adopt the Euclid distance\nDi,j ≡\n√ ∑\nℓ\n(x(ℓ)\ni − x(ℓ)\nj )2 (6)\nThe distance between two cell types is plotted in Table I. Note that t here\nremains some difference in the same type of cell as mentioned. Howev er,\nthis difference is clearly much smaller than that between different cell types.\nThis demonstrates that the differentiated cell types (from “0” to “5”) are\nwell-defined as “digitally” distinct states. Then one might suspect th at these\ndifferent states may be just a different attractor in each dynamics . This is not\nthe case. Except the type-0 and type-5 cells, the state of differe ntiated cells\nis unstable by itself. When we start the simulation of a single cell with th e\nstate of cell type “1”,“2”,“3”,or “4” with the same media (but withou t any\nother cells), the cell is de-differentiated back to the attractor-0 . The states\nfor types “1”,“2”,“3”, and “4” are stabilized only through the inter action\namong other cells. For example, the existence of type-0 cells is nece ssary to\nkeep the stability of cell types “1” and “2”.\nIt is also interesting to compare the bifurcation rule of cell types (in Fig.6)\nwith the distance. If the history of cell lineage reflects on the dista nce of cell\nfeatures, it is expected that for j = 3 , 4, 5 D1,j < D 0,j or D1,j < D 2,j since\nthe types 3,4,5 are derived from the type-1 cell. This is not necessar ily true\nin Table I. The reason for this discrepancy is due to the insufficiency in the\nrepresentation for the distance measured after taking the aver age. As is seen\nin Fig.4, the orbit of the type-1 cell itinerates over the states close to the\ntype 3,4, and 5. Hence it is useful to define the minimal distance by\nDmin\ni,j ≡ mint\n\n\n√\n∑\nℓ\n(x(ℓ)\ni (t) − x(ℓ)\nj (t))2\n\n (7)\nwhere mint means the minimum over time. The distance is given in Table II,\nwhere one can clearly see the hierarchical organization of cell type s according\nto the bifurcation rule of Fig.6. The distance between two of cell typ es\n“0”,“1”, and “2” is smaller than that between “0” or “2” and “3”,“4” ,or “5”.\nThe distance between “1” and “3”,“4”, or “5” is much smaller than ot hers.\nLet us reconsider the form of memory using the distance. First, th e\nmemory of cell types is sustained in the internal dynamics modulated by\nthe interaction. The memory corresponds to a partial attractor stabilized\nby the interaction. Here, the information on the distribution of cell types is\nembedded in each internal dynamics.\nFor example, each internal dynamics is gradually modified with the cha nge\nof distribution of other cells. In Fig.7 we have studied how the dynamic s of\nthe type-2 cell changes when the rate of type-0 cell is varied. In t he sim-\nulation, we choose (a stable) cell society consisting of types “0”, “ 2”, and\n“3”, and successively replace a cell of type-0 by type-3. To avoid t he per-\nturbation due to the change in the number of cells, we remove the ru le of\n\ndivision in the present simulation, to fix the number. As a result of cha nge\nof the distribution of cell types (i.e. the fraction of type-0 cells), t he dy-\nnamics of each cell (e.g., of the type 2) is modulated. We have plotted the\ndistance D2,20 where the 2 0 denotes the cell when the distribution of cells\nsatisfies ( n0, n2, n3) = (23 , 50, 27), the condition at the left-end point of the\naxis, where nk represents the number of the type- k cell. The distance D2,20\nincreases (roughly linearly) with the decrease of n0, until the further decrease\ndestabilizes the cell society and the switching of cells to type-5 star ts. The\ngradual change of D2,20 means that the internal cell state varies according to\nthe cell distribution. Hence the global information on the cell distrib ution is\nembedded in the internal cellular state. We note that this informatio n adopts\n“analogue” representation, instead of digital one adopted for th e distinct cell\ntype. Hence our cellular system has both analogue and digital memor ies.\n5 Interaction-dependent rules and stability of\ncell society\nIn Fig.6, we have shown that the automaton-like rule has emerged wit hout\nexplicit implementation. The rule is not solely determined by its cell type .\nWhen there are multiple choices of differentiation process (as in “0” → “0”,\nor “0” → “1”, and “0” → “2”) the rate of each path is neither fixed nor\nrandom, but depends on the number distribution of cell types in the system,\nembedded in the internal dynamics. This implies that a higher level dyn amics\nemerges, which controls the rate of cell division and differentiation a ccording\nto the number of each cell type. In other words, the dynamics on t he number\nof each cell type n0,n1,.., and n5 can be represented by {nk} (k = 0 , · · ·, 5).\n(This dynamics should be stochastic, since we have neglected the inf ormation\non each cellular state and reduced it to only the number of cell types ).\nThis dynamics allows for stability at the level of ensemble of cells. The\nvariety and the population distribution of cell types are robust aga inst ex-\nternal perturbations. As an example, let us consider the case with three cell\ntypes (“0” ,“1” ,“2” in Fig.6). When the type-2 cells are removed to de crease\ntheir population, events of differentiations from “0” to “2” are enh anced, and\nthe original cell-type distribution is recovered.\nIn Fig.8, the rate of differentiation from the type-0 cell to others is plotted.\nIn this simulation, to capture the dynamics of the number of each ce ll type,\nthe total number of cells in the medium is fixed (to N=100 in the present\ncase), by removing the division rule. As the initial condition, N cells are\nplaced in the medium, where the concentration of chemicals in each ce ll is\nselected so that they give type-0, 1, or 2 cell. The switch of cell typ es is\nmeasured when the system settles down to a stable distribution of c ell types.\nThe simulations are repeated by changing the initial distribution of ce ll types\n(n0, n1, n2), to plot the number of the switches from 0 to others, while the\nfinal number of cells for each type is also plotted. As in Fig8, the freq uency\nof switches from the cell type-0 increases almost linearly with n0 when it\nis larger than approximately 40%. With this switch, the stability of cell\ndistribution around approximately ( n0, n1, n2) = (40 , 30, 30) is attained.\nThis kind of robustness at an ensemble level is expected from our iso lo-\n\ngous diversification theory, since the stability of macroscopic char acteristics\nis attained in coupled dynamical systems (Kaneko 1992, 94). In our case,\nthe macroscopic stability is sustained by the change of the rate of d ifferen-\ntiation from “0” to other types. Recall that, the differentiations f rom “1”\nor “2” to “0” does not occur (see Fig.6), even if some of the type “0 ” cells\nare removed 5. In the hierarchical structure represented in Fig.6, the cell\nat an upper node behaves as a stem cell, and regulates the distribut ion of\nthe cells at a lower node. This type of regulation system is often adop ted in\nthe real multicellular organism (e.g. in the hemopoietic system)(Scho field et\nal., 1980). An important point of our result is that the dynamical diffe ren-\ntiation process always accompanies this kind of regulation process, without\nany sophisticated programs implemented in advance. This robustne ss pro-\nvides a novel viewpoint to understand how the stability of the cell so ciety is\nmaintained in the multicellular organism.\n6 Differentiation of Colonies\nThe automaton rule of Fig.6 does not necessarily mean that all of the se six\ntypes of cells coexist in a cell society emerged in the course of the de velop-\nment. Cell groups consisting only of two or three cell types can app ear: For\nexample, cell groups only of “0”,“1”, and “2” types and of “0”, “2” , and “4”\ntypes are observed.\nThis implies that the dynamics on the number of cells of each type has\nalso several stable attractors due to the autonomous control o f the rate of\ndifferentiation. They correspond to stable distributions of cell typ es in each\ncell group. In other words, there are several possible distributio ns of cell\ntypes when cells are developed from a single cell. To confirm it, we have\nperformed the following simulations. First, we initially put one cell whos e\ninternal chemical concentrations are chosen randomly. Then the cell society\nis evolved following the rules of the present model, until the total ce ll number\nreaches a given threshold value, when we stop the simulation and mea sure\nthe distribution of cell types. We have repeated this course of simu lations\nfor hundred times, starting from different initial conditions.\nIn Fig.9, the number of initial configurations leading to a cell-type dist ri-\nbution with a given range of n2 is plotted as a histogram, where the number\nof type-2 cells n2 is measured when the total cell number has reached 300.\nFour peaks are clearly visible at n2 = 0 , ∼ 100, ∼ 150, and ∼ 220, which\ncorrespond to possible distinct sets of cell distributions. As mentio ned, the\npossible set of cell types (from “0” to “5”) and the temporal orde ring of\ntheir appearance (e.g., 0 → 1 → 2) are independent of the initial conditions.\nHowever, at the later stage, several types of cell groups emerg e depending on\nthe initial conditions.\nThe most relevant factor to the choice of cell groups is the ratio of the\nnumbers of differentiated cells (i.e. type-1 and type-2 cells) to undiff erenti-\nated cells (i.e. type-0 cells) at an early stage of development, when t he first\n5Transformation from type-2 to type-0 cells occurs as a transient process to type-5 cell,\nwhich is seen only in the case when the type-5 cell appears and start s to dominate the\nsociety.\n\ndifferentiations from “0” to “1” and “2” occur. Thus the fate of ce ll groups\nis determined at a rather early stage.\nRecall that the differentiation rate of cells (each arrow in Fig.6), and\naccordingly the higher-level dynamics of nk depend on the distribution of\ncells nk. The result of Fig.9 implies that there are several attractors on th is\nhigher-level dynamics on nk. As discussed in §5, an “attractor” of this higher\nlevel dynamics is stable against perturbations to change the numbe r of cells\nof each type.\nIn Fig.10 we have shown the flow chart of the change of ( n0, n1, n2), where\nthe direction of change of n0 and n2 is represented by the arrow, starting\nfrom the initial distribution given by ( n0, n1, n2) of the corresponding site.\nTo draw the figure, we adopt the same rules as in Fig.8, where the tot al\ncell number ( n0 + n1 + n2) is fixed to 100, and the division rule is removed.\nFrom the 2-dimensional plane, the number of cells of types 1,3,4, and 5 are\ngiven by 100 − n0 − n2. The chart shows that cell colonies on the cell-\ntype distribution {n0, · · ·n5} have at least 5 stable states around ( n0, n2)\n=(0,0), (38,32), (30,50), (18,58), and (0,78) respectively. Each s tate has a\nbasin of attraction, and the corresponding cell-type distribution is stable\nagainst external perturbations, as is supported by the higher-le vel dynamics\non {n0, · · ·n5}.\nThe fixed point at ( n0, n2) = (0 , 0) (“A” in the figure) corresponds to\na colony consisting only of type-5 cells, while the fixed point denoted b y\n“B” corresponds to a colony only of 0,1,2, “C” of 0,2,3, and “D” of 0,2,4 ,\nrespectively. Indeed, these cell-type distributions correspond t o the peaks of\nFig.9, respectively.\nStill there is a clear difference between the developmental process from\none cell (Fig.9) and the present simulation (Fig.10) with a fixed cell num ber.\nSome region in the plane of Fig.10 cannot be reached by the simulation f rom\na single cell. For example, the state “E” consisting of types 2 and 4 ca nnot be\nobtained from the developmental process from a single cell. Furthe rmore, the\nstate “B”, which does not have a large attraction volume in Fig.10, ha s the\nlargest probability to be reached from the developmental process (see Fig.9).\nThis discrepancy is caused by the conjunction of cell number chang e with\nthe population dynamics of cell types. Through the change of the n umber of\ncells, the population dynamics shifts from one flow chart of Fig.10 to a nother\nwith different number of cells. The organized cell colony from a single c ell\nhas such developmental constraints.\nNow the coexistence of several stable cell colonies is clear. Depend ing\non the initial cell condition, different cell colonies are obtained. The r esult\nhere means that several types of tissues can appear through th e interactions\namong cells. This kind of diversity is often observed in a cultivation sys tem\nof a colony of blood cells starting from a stem cell (Nakahata et al., 19 82).\n7 Summary\nIn the present paper, we have studied a dynamical model to show t hat a\nprototype of cell differentiation occurs as a result of internal dyn amics, in-\nteraction, and division. We have made several simulations choosing s everal\n\nchemical networks, also with a different number of chemical species , and the\nsame scenario for cell differentiation is obtained. Under the same pa rameters\nused in the previous example, approximately 5% of randomly chosen c hem-\nical networks show oscillatory behavior, while others fall into fixed p oints.\nFurthermore, approximately 20% of these oscillatory dynamics are destabi-\nlized through the cell division, where some of the cells differentiate fo llowing\na specific rule like Fig.6.\nSome may cast a question why we can select such oscillatory dynamics\nto draw a general mechanism for differentiation, even if only a few ra ndomly\nchosen chemical networks are oscillatory. One reason why only a fe w reaction\nnetworks are oscillatory is that we choose reaction paths randomly and with\nthe identical coefficients. On the other hand, the chemical reactio n network of\nthe real biological system is more sophisticated through evolutiona ry process.\nFor example, there are positive and negative feedback reactions u biquitously.\nThis feedback mechanism, in particular auto-catalytic reaction, is im portant\nto provide oscillatory dynamics which are observed in the real biologic al\nsystem (e.g. Ca oscillation).\nIn the present model with randomly chosen networks, only few rea ctions\nhave auto-catalytic effects. By increasing the rate of auto-cata lytic reaction\npaths, the probability of the network with oscillatory dynamics and d iffer-\nentiation gets much higher. For comparison, we have also studied a c lass of\nmodels where each chemical can catalyze a reaction to generate its elf from\nanother chemical, besides the ordinary reaction paths determined randomly.\nBy sampling several reaction networks, we have found that 40% of the reac-\ntion networks has oscillatory dynamics and more than 20% of these d ynamics\nare destabilized to show cell differentiation by cell divisions.\nThen, why are such auto-catalytic reactions common? To make rep lica-\ntions efficiently, some mechanism to amplify reaction by its product is g ener-\nally expected at the first stage of life (Eigen and Schuster, 1979). Also, auto-\ncatalytic reactions are necessary to add new metabolites in the met abolic\nnetwork through the evolutionary process. Indeed, when novel chemicals\nare included in the evolutionary process of metabolic network, their con-\ncentrations must be amplified by the reactions. This implies that thes e new\nchemicals must constitute an auto-catalytic set (see Appendix of K aneko and\nYomo, 1997).\nLet us summarize the consequences of our simulations. First, we ha ve\nprovided a further support for isologous diversification previously proposed.\nCells are differentiated through the interplay between intra-cellular chemical\nreaction dynamics and the interaction among cells through media. As the\ncell number is increased, the oscillatory dynamics in each cell is desta bilized\nand loses synchrony. Then, some of cells change their internal dyn amics,\nwhich form a group with a different stable dynamics. Discrete, differe ntiated\nstates appear, which are transmitted to their daughter cells as a m emory.\nWe interpret this phenomenon as (determined) differentiation.\nThe differentiation in our theory is caused by the instability in internal\ndynamics triggered by cell-to-cell interactions. Microscopically spe aking in\nbiological terms, this may be regarded as a switching process followin g signal\nmolecules from outside of the cell. Our theory is not inconsistent with such\nbiological knowledge, but the point in our theory lies in that such local\n\ntransition of internal states has also the information on macrosco pic states,\ni.e., the distribution of cell types. With this, the robustness of cell s ociety\nemerges in spite of instability in each internal dynamics.\nBesides this further support for the isologous diversification, we h ave\ndemonstrated the hierarchical cell differentiation, generation of interaction-\ndependent rules, and the existence of distinct cell groups.\n1) hierarchical organization\nDifferentiation from a stem cell to two different types, and then to t hree\ntypes from one of them are observed. Hierarchical rule of differen tiation is\nthus generated. Although the number of cell types and the rule of differenti-\nation depend on the choice of chemical networks, generation of a h ierarchical\nrule (written by the tree-type diagram as in Fig.6) is generally observ ed.\n2) generation of rules and internal memory reflecting on the\nenvironment (that is the distribution of other cell types)\nThese differentiations obey a specific rule, which emerges from inter - and\nintra- dynamics. It is often believed that the rules of the differentia tion,\nwhich determine when, where, and what type of a cell appears in a mu lticel-\nlular organism, should be pre-specified as the information on DNA. We do\nnot deny such role of DNA, but it should be stressed that the rules o f differ-\nentiation and the higher level dynamics emerge through interaction of cells\nwith internal dynamics. As a consequence of our interaction-base d approach,\nthe diversity of cells and the stability of cell society naturally follow.\nThe rate of differentiation and reproduction vary with the distribut ion\nof cell types. The global stability of the whole system is obtained, wh ich is\nsustained by regulating the rates of the differentiations.\nAs a coupled dynamical system, the memory of cell types is given in a\nstate stabilized by interactions. This state is not necessarily an att ractor\nas a single cell dynamics, but is a “partial attractor” stabilized only in the\npresence of suitable interactions provided by the distribution of ot her cells.\nThrough the cell divisions and the evolution of the cell society, the c ells\nchoose suitable interactions so that the memory of their types is pr eserved.\nThis is the mechanism of how the recursivity of cell types is attained, while\nthe global stability of cell society is assured through the interactio n. Indeed\nsuch partial attractors lose stability and switch to other cell type s when the\ninteraction by the cell distribution is not “suitable”.\nIt should be noted that two types of memory coexist, analogue and digital.\nThe former gives information on the cell society, i.e., the distribution of cell\ntypes, while the latter gives a distinct internal state on cell differen tiation.\nWe believe that such dual memory structure is a general feature in a biological\nsystem. In cell biology, the “analogue” difference reflecting on the interaction\nis known as modulation (Alberts et al., 1994).\n3) formation of higher level dynamics and diversity of cell g roups\nThe rule of differentiation depends on the number distribution of oth er cell\ntypes, for which stochastic dynamics at a higher level is formed. Th e result\nprovides the first example that a 2-step higher-level dynamics is fo rmed, that\nis a colony level from cellular one, that is formed from a chemical netw ork\nlevel. Here the dynamics of a colony level (i.e., the change of the numb er\nof each cell type) is “stochastic”, because the information on the number of\ncell types is not complete, where the lower-level information on the internal\n\nstate (of chemical concentrations) is discarded. It is interesting to note that\nthe macroscopic flow chart on the number of cell types is formed in s pite of\nthe stochasticity.\nOur result shows that there are several attracting states for t his higher-\nlevel dynamics. In biological term, this corresponds to the existen ce of several\ncell colonies, distinguishable by the number distribution of cell types . These\ndiverse colonies appear from a single stem cell. Each cell colony is stab le, in\nthe sense that the original distribution is recovered after pertur bations (of\nnot too large size) are added on the cell colony, such as elimination of few\ncells of one type.\n8 Discussion\nOf course, there has been preceding theories for cell differentiat ion. The idea\nto regard the differentiation as the transition in cellular multi-station arity is\ntraced back to Delbr¨ uck (1949), who proposed a simple bistable reaction net-\nwork with two metabolic chains that are cross-inhibited by their prod ucts.\nIndeed the epigenetic transmission of such stationary states hav e been re-\nported in unicellular organisms (Novick and Wiener, 1957; Sonneborn , 1964).\nIn general, this multi-stationarity results from positive and negativ e feed-\nbacks in metabolic reaction networks. This leads to the viewpoint tha t each\ndifferentiated cell state is represented by an attractor of intra- cellular dynam-\nics, as has been demonstrated by Kauffman (1969) in his Boolean net work. It\nleads to a variety of stable states (attractors), depending on th e initial condi-\ntions. Here, each cell type corresponds to an attractor of inter nal networks,\nwhile an external mechanism is required to have the transition betwe en these\nattractors.\nSuch mechanism is supported by cell-to-cell interaction. Indeed, a mech-\nanism of interaction-induced differentiation has been proposed by T uring’s\npioneering study (Turing, 1952). Now, a well-known mechanism of ex ter-\nnal regulation is gradient of morphogen, in which the transition depe nds\non the concentration of chemical substances. (See, however, K aneko and\nYomo (submitted), for instability due to stochastic fluctuation in th e thresh-\nold mechanism on the gradient of chemicals). Another possible mecha nism\nfor interaction-induced differentiation is proposed by Gordon, whe re the me-\nchanical wave transmits among the cells and controls the cell state (Gordon\net al., 1994).\nThen, combination of the multi-stationary reaction network and th e ex-\nternal regulation mechanism might be relevant to explain the local diff er-\nentiations. However, to understand the complexity and the stabilit y of cell\nsociety, an important question still remains: How is such external r egulation\nmechanism regulated? Is another external mechanism required?\nOur results provide a distinct, and plausible standpoint for this prob lem.\nNoticing the interplay between intra-cellular dynamics and interactio n, we\nhave proposed a novel concept “partial attractor”, which is sta bilized only\nby cell-to-cell interaction. Thus, a cell state is not always determin ed by\nthe attractor of internal dynamics, but it also depends on the oth er cells.\nAn important consequence of our results is that there is no distinct ion be-\n\ntween internal dynamics that determines the cell state and the re gulation\nmechanism of differentiation. Rather, the mechanism for regulation is spon-\ntaneously accompanied by the multi-stationarity, because the num ber of cells\nwith each partial attractor is found to depend on the circumstanc e of cell so-\nciety which sustains them.\nA consequence of our theory is ‘relativity’ of determination of a diffe r-\nentiated cell. Since our cellular state reflects the interaction, the r ule of\ndifferentiation as well as the recursivity may be affected by the cells a round\nthe cell in concern. A suitable experiment system to distinguish our t heory\nfrom previous theories is provided by a hemopoietic stem cell system , where\nspatial pattern mechanism a l’a Turing no longer works. Now we will disc uss\nbriefly relevance of our results to the cell biology.\nApplication to biological problems\nSince our model reaction process does not have one-to-one corr espondence\nto any existing biochemical network, one cannot make a detailed pre diction\non a specific example in biology. However, the present scenario shed s a new\nlight on some open questions in biology, by providing a coherent viewpo int\non them. Let us discuss two of them.\nSince our results provide hierarchical differentiation, it is interestin g to\ncompare them with such an example in cell biology. A well known example is\na hemopoietic system (Ogawa, 1993). The blood contains many type s of cells\nwith different functions, while a pluripotent stem cell in the bone marr ow\ngives rise to all classes of blood cells. The hemopoietic system can be v iewed\nas a hierarchy of cells, where pluripotent stem cells differentiate to p rogenitor\ncells determined as ancestors of one or a few terminal differentiate d blood cell\ntypes. In general, these terminal blood cells have limited lifespans an d are\nproduced throughout the life of the animal. Thus, to keep a variety of blood\ncells, it is important to control the differentiation and proliferation o f the\nstem cell at the higher level of hierarchy. However, because of th e difficulty of\nidentifying the stem cells in the bone marrow, the behavior of the plur ipotent\nstem cells in vivo remains especially elusive. In the experimental result in\nvitro, even if the cells have been selected to be as homogeneous as p ossible,\nthere is a remarkable variability in the sizes and often in the characte rs of\nthe developed colonies (Nakahata et al., 1982). Even if two sister ce lls are\ntaken immediately after a cell division and cultured apart under ident ical\nconditions, they frequently give rise to colonies that contain differe nt types\nof blood cells or different distribution of the types of cells.\nIt is often interpreted that the differentiation of a hemopoietic ste m cell is\nstochastic whose probability is controlled by some other control me chanism,\nby which the multicellular system as a whole regulates the distribution o f\ncell types(Till et al., 1964; Ogawa, 1993). Results of our model pro vide a\nnovel interpretation of these experiments. First, the rules of diff erentiation\nare generated through interactions. Second, the stochastic diff erentiation of\nthe cells and regulation of the probability of the differentiations natu rally\nemerge from the cell-to-cell interaction, without imposing any rand om event\nor external regulation mechanism. Third, the diversity of the colon ies which\nhave developed from a same type of cell is a natural consequence, as a multi-\nstability of higher-level dynamics. We note that a single cell with a sligh tly\ndifferent initial condition can lead to a different colony in our simulation.\n\nThrough the chemical substrates such as the Interleukins, the c omplicated\ninteractions among the blood cells are observed experimentally. It is plausible\nto assume the dynamical interaction adopted in our theory.\nHere, we propose an experiment on the hemopoietic system to make some\npredictions. As mentioned, one of the important consequences of our results\nis that the states of cells are not always determined by the attract or of inter-\nnal dynamics, but often are sustained by interaction with other ce lls. This\nimplies that some types of cells in the hemopoietic system do not corre spond\nto a stable attractor of internal dynamics, but are stabilized by ot her cells.\nThe pluripotent stem cell and the terminal differentiated cell, the to p and\nbottom of the hierarchy respectively, seem to correspond to a st able attrac-\ntor, because they can stand independently of other blood cells. On the other\nhand, the progenitor cells can be observed only in the colonies of bloo d cell.\nWe expect that the internal dynamics of these progenitor cells is re presented\nby a partial attractor. Then, these cells can differentiate back to the cell of\nhigher hierarchy when these cells are separated and cultured indep endently.\nTo confirm this hypothesis, differences between an isolated blood ce ll and\nthat surrounded by other cells should be tested experimentally. We predict\nthat the potentiality of blood cells to differentiate and to proliferate is quite\ndifferent these two situations, which confirms the importance of th e cell-cell\ninteraction in the hemopoietic system.\nIn general, there is a level of differentiations in cell. The determined d if-\nferentiation keeps the memory even if a cell is transplanted, while so me cells\ncan be transdifferentiated (Alberts et al., 1994). In our dynamical systems\nrepresentations, such difference can be expressed as the distinc tion between\nan attractor by the cell itself and the partial attractor stabilized by the inter-\naction. The merit in our approach is that such levels of differentiation appear\nwithout external implementation, which is important when one consid ers the\norigin of multicellular organisms.\nThere are several types of cells in a multicellular organism. In particu lar,\nalmost all organisms have distinction between the germ cells and the s omatic\ncells. The appearance of these two types are controlled elaborate ly by com-\nplicated interaction among cells in the contemporary multicellular orga nisms.\nHowever, it is hard to postulate that such a mechanism appeared at the same\ntime with the emergence of the multicellular organisms. Our theory pr ovides\none possible solution to this problem. According to our results, differ enti-\nation in a group of identical cells occurs through the dynamical inter action\namong cells, as long as the intra-cellular reaction dynamics can show n onlin-\near oscillations. The differentiation, as well as the stability of such div erse\ncells, is a rather natural consequence of interacting cells. At the n ext stage in\nthe evolution, more complex cell society must have appeared, wher e several\ntypes of tissues exist as a higher hierarchy, which interact each ot her. Our re-\nsult about the diversity of cell group shows potentiality that sever al types of\ntissues appear at this stage, based on the dynamical interaction a mong cells.\nIn our model, we do not take into account of the spatial variation. S election\nof each cell group, thus, depends on the choice of different initial c onditions.\nOn the other hand, when the spatial information is included in this sys tem,\nit is possible that several types of cell group coexist at a (spatially) different\nregion.\n\nOf course, contemporary multicellular organisms such as mammal of ten\nhave hundreds of cell types, though our results in this paper can s how coex-\nistence with only few cell types. The number of cell types in our dyna mical\ndifferentiation model does not show clear increase with the number o f chem-\nical substances. We suppose that the reason for these few type s of cells is\ndue to our random choice of chemical reaction network, where the reaction\npaths are chosen equivalently. In real biological system, the chem ical reaction\nnetwork is more organized, possibly in a hierarchical manner.\nTo choose such suitable network, evolutionary aspect of chemical reaction\nnetwork should be taken into account for our model. This problem als o con-\ncerns with the emergence of gene expression system, and is under our current\ninvestigation.\nacknowledgements\nThe authors are grateful to T. Yomo, T. Ikegami, S. Sasa, and T. Yamamoto\nfor stimulating discussions. The work is partially supported by Grant -in-Aids\nfor Scientific Research from the Ministry of Education, Science, an d Culture\nof Japan.\nReferences\n[1] B. Alberts, D.Bray, J. Lewis, M. Raff, K. Roberts, and J.D. Watso n,\nThe Molecular Biology of the Cell , 1983, 1989, 1994\n[2] F. Bignone, J. Theor. Biol. 161 (1993) 231\n[3] M. Delbr¨ uck, Discussion in: Unit´ es Biologiques Dou´ ees de Continuit´ e\nG´ en´ etiqueEditions du CNRS (Lyon) 33\n[4] M. Eigen, and P. Schuster, The Hypercycle: A Principle of Natural\nSelf-Organization, Springer-Verlag, Berlin (1979)\n[5] G. Fankhauser, Analysis of Development, B. H. Willier et al. eds,\nPhiladelphia: Saunders, 1955, pp. 126-150\n[6] B. Goodwin, “Temporal Organization in Cells” Academic Press, Lon -\ndon (1963)\n[7] B. Goodwin, J. Theor. Biol. 97 (1982) 43\n[8] A. Novick and M. Weiner, Proc. Natl. Acad. Sci. USA. 43 (1957) 55 3\n[9] T. M. Sonneborn, Proc. Natl. Acad. Sci. USA. 51 (1964) 915\n[10] R. Gordon, N.K. Bj¨ oklund and P.D. Nieuwkoop, Int. Rev. Cytol. 150\n(1994) 373\n[11] K. Kaneko, Physica 41 D (1990) 137\n[12] K. Kaneko, Physica 54 D (1991) 5\n\n[13] K. kaneko, Physica 55 D (1992) 368\n[14] K. Kaneko and T. Yomo, Physica 75 D (1994) 89\n[15] K. Kaneko and T. Yomo, “ A Theory of Differentiation with Dy-\nnamic Clustering”, in Advances in Artificial Life , F. Moran et al. eds.,\nSpringer, 1995, pp. 329-340\n[16] K. Kaneko and T. Yomo, Bull. Math. Biol. 59 (1997) 139\n[17] S.A. Kauffman, J. Theor. Biol. 22, (1969) 437\n[18] B. Hess and A. Boiteux Ann. Rev. Biochem. 40 (1971) 237\n[19] E. Mjolsness, D. H. Sharp, J. Reinitz, J. Theor. Biol. 152 (1991 ) 429\n[20] T. Nakahata, A. J. Gross, M. Ogawa, J. Cell. Phy. 113 (1982) 4 55\n[21] S. A. Newman and W. D. Comper, Development 110 (1990) 1\n[22] M. Ogawa, Blood 81 (1993) 2844\n[23] R. Schofield, S. Load, S. Kyffin, C. M. Glibert, J. Cell. Phy. 103 (1 980)\n[24] J. E. Till, E. A. McCulloch, L. Siminovitch, Proc. Natl. Acad. Sci. US A\n51 (1964) 29\n[25] R. Thomas, D. Thieffry, M. Kaufman, Bull. Math. Biol. 57(2) (199 5)\n[26] A. M. Turing, Phil. Trans. Roy. Soc. 237 (1952) 5\n[27] J. J. Tyson, et al. TIBS 21 (1996) 89\n\nFigure caption\nFig.1: Overlaid time series of x(m)(t) of a single cell in medium, obtained\nform a network with 16 chemicals and three connections in each chem ical.\nOnly the time series of 5 chemicals are plotted out of 16 internal chem icals.\nEach line with the number m=2,9,10,11,12 gives the time series of the con-\ncentrations of the corresponding chemical x(m)(t). This oscillatory behavior\nis a limit cycle, whose period T is longer than the plotted range of the fig-\nure ( T ∼= 16000 time steps). The parameters are set as e1 = 1, D = 0 .01,\nf = 0 .01,X (ℓ) = 0 .1 for all ℓ, and V = 100. Chemicals x(ℓ)(t) for m ≤ 3 are\npenetrable(i.e., σℓ = 1), and others are not. The reaction network Con(i, j, ℓ)\nis randomly chosen initially, and is fixed throughout the simulation resu lts\nof the present paper.\nFig2: Time series of x(m)(t), overlaid for the 5 chemicals (as given in Fig.1)\nin a cell. (a)-(e) represent the course of differentiation to type-1 , 2, 3, 4, and\n5 cells respectively. The differentiation to type-3, 4, and 5 cells alway s occurs\nfrom type-1 cells.\nFig.3: Orbits of internal chemical dynamics in the phase space. (a) a nd (b)\nshow the orbits of chemical concentrations for a switching proces s from type-0\nto type-1,2 cells, respectively, plotted in the projected space ( x(2)(t), x(13)(t)).\nFig.3(c) gives a plot of ( x(1)(t), x(8)(t)), which shows a switch from type-1 to\ntype-3 cells. (Note the difference of scales.) Each cell type is clearly distinct\nin the phase space.\nFig.4: Orbits of internal dynamics for each cell type. The dynamics o f\neach cell type is plotted in the same projected space as in Fig.3(a),(b ), i.e.,\n(x(2)(t), x(13)(t)). Color corresponds to each cell type.\nFig.5: Cell lineage diagram. Differentiation of cells (whose indices are giv en\nby the horizontal axis) is plotted with time as the vertical axis. In th is dia-\ngram, each bifurcation of lines through the horizontal segments c orresponds\nto the division of the cell, while the color indicates the cell type ( red fo r\ntype 0, green for type 1, blue for type 2 respectively.). We have plo tted up\nto the stage of the differentiation to three types, while the bifurca tion to 3,4,\nand 5 from 1 occurs at a later course.\nFig.6: Automaton-like representation of the rule of differentiation. The path\nreturning to the node itself represents the reproduction of its ty pe, while\nthe paths to other nodes represent the potentiality to differentia tion to the\ncorresponding cell types. The dotted line from type-2 to type-0 g ives an ex-\nceptional case: Indeed the differentiation from “2” to “0” never o ccurs when\nseveral types of cells such as “0”,“1” and “2” coexist. It occurs e xceptionally\nonly if “5” cells dominate the system, when all cells are finally differentia ted\nto type-”5”. In this case the type-“2” cells de-differentiate to “0 ” ( and fi-\nnally to “5”).\nFig.7: Variation of dynamics of type-2 cells with the change of the rat e\n\nof type-0 cell. The distance D2,20 of eq. (6) is measured between two\ntype-2 cells from the conditions ( n0, n2, n3) = (23 , 50, 27) and ( n0, n2, n3) =\n(n0, 50, 50 − n0),as one of type-3 cells is successively switched to type-0 ex-\nternally by 2 × 104 step. Besides the distance, the number of type-0 cells is\nplotted against time.\nFig.8: Rate of the differentiation from type-0 to other cell types. T he total\ncell number is fixed to 100 (without division process), while we take th e ini-\ntial cell distribution of three types as ( n0, n1, n2) = ( n0, 30, 70 − n0). Starting\nthe simulation with this initial condition, the final number of each cell t ype,\nas well as the number of differentiations from type-0 to others, is p lotted, as\na function of the initial number of type-0 cells ( n0). Within this range of cell\ntype distribution, none of type-3,4,5 cells appear(see section 6).\nFig.9: Histogram about the number of type-2 cells. Starting from a s ingle\ncell with randomly chosen chemical concentrations, the simulation is carried\nout until the total cell number reaches 300, when the number n2 of type-2\ncells is measured. Repeating the runs 347 times, we have counted th e num-\nber of such initial conditions that n2 falls onto a given bin (with the size 5).\nThe histogram of n2 is obtained from the count. There are four peaks at\nn2=0,100,150,220, each of which corresponds to a stable distribution o f the\ncell colony.\nFig.10: Flow chart of the change of ( n0, n1, n2). We have carried out the\nsimulations starting from the initial condition at each ( n0, 100 − n0 − n2, n2)\nby fixing the total cell number to 100 (by removing the cell division pr ocess).\nChange of the number of cell types is measured from simulations, fr om which\nthe direction of changes of ( n0, n2) is shown as an arrow in the ( n0, n2) space.\nAs is seen, there are 5 fixed points, each of which corresponds to t he stable\npopulation distribution of cell types.\nTable Caption\nTable I: The average distance in phase space between each cell typ e: Di,j\nin eq.(6) is estimated by taking the average over 5 × 104 time steps. Each\ncell type is sampled from a course of the evolution starting from one cell.\nTable II: Minimum distance in phase space between each cell type. Dmin\ni,j\nin eq.(7) is estimated from 5 × 104 time steps. Each cell type is sampled from\na course of the evolution starting from one cell.\n\n0.05\n0.1\n0.15\n0.2\n0.25\n0.3\n0.35\n0.4\n94000 96000 98000 100000 102000 104000 106000\nconcentration\ntime\nx2\nx9\nx10\nx11\nx12\n\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n100000 120000 140000 160000 180000 200000\nconcentration\ntime\ntype-0 type-1\n\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n110000 115000 120000 125000 130000 135000 140000 145000 150000 155000\nconcentration\ntime\ntype-2type-0\n\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n100000 110000 120000 130000 140000 150000 160000 170000 180000 190000\nconcentration\ntime\ntype-0 type-1 type-3\n\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n100000 110000 120000 130000 140000 150000 160000 170000\nconcentration\ntime\ntype-0 type-1 type-4\n\ntype-0 type-1 type-5\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n100000 150000 200000 250000 300000\nconcentration\ntime\n\n0.05\n0.1\n0.15\n0.2\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\nchemical13\nchemical2\ntype-0\ntype-1\n\n0.05\n0.1\n0.15\n0.2\n0 0.1 0.2 0.3 0.4 0.5 0.6\nchemical13\nchemical2\ntype-0\ntype-2\n\n0.001\n0.002\n0.003\n0.004\n0.005\n0.006\n0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008\nchemical8\nchemical1\ntype-1\ntype-3\n\n0.05\n0.1\n0.15\n0.2\n00.10.20.30.40.50.60.7\nchemical13\nchemical2\n0.001\n0.002\n0.003\n0.004\n0.005\n0.006\n0.007\n0.008\n00.10.20.30.40.50.60.7\ntype-0\ntype-1\ntype-2\ntype-5\ntype-4\ntype-3\nexpansion\n\n1 2\n\n200000 250000 300000 350000 400000 450000 500000 550000 600000 650000200000 250000 300000 350000 400000 450000 500000 550000 600000 650000\ndistance\ntype-0\nnumber of type-0 cell\ntime\ndistance\n\n-10\n10 20 30 40 50 60 70\nnumber\ninitial number of type-0 cell\ntype-2\ntype-1\ntype-0\ndifferentiation\n\n0 50 100 150 200 250 300\ntrial\nnumber of type-2 cell\n\n20 40 60 800\nA\nB\nC\nD\nE\nnumber of type-0 cell\nnumber of type-2 cell\n\n/99/101/108/108/45/48 /99/101/108/108/45/49 /99/101/108/108/45/50 /99/101/108/108/45/51 /99/101/108/108/45/52 /99/101/108/108/45/53\n/99/101/108/108/45/48 /49 /58 /51 /2 /49/48\n/49\n/49 /58 /56 /2 /49/48\n/51\n/49 /58 /54 /2 /49/48\n/51\n/50 /58 /57 /2 /49/48\n/51\n/51 /58 /48 /2 /49/48\n/51\n/51 /58 /50 /2 /49/48\n/51\n/99/101/108/108/45/49 /42 /53 /58 /51 /2 /49/48\n/50\n/49 /58 /50 /2 /49/48\n/51\n/49 /58 /56 /2 /49/48\n/51\n/50 /58 /51 /2 /49/48\n/51\n/50 /58 /52 /2 /49/48\n/51\n/99/101/108/108/45/50 /42 /42 /50 /58 /48 /2 /49/48\n/50\n/49 /58 /53 /2 /49/48\n/51\n/49 /58 /53 /2 /49/48\n/51\n/51 /58 /54 /2 /49/48\n/51\n/99/101/108/108/45/51 /42 /42 /42 /49 /58 /50 /2 /49/48\n/49\n/54 /58 /51 /2 /49/48\n/50\n/52 /58 /49 /2 /49/48\n/51\n/99/101/108/108/45/52 /42 /42 /42 /42 /50 /58 /51 /2 /49/48\n/49\n/52 /58 /54 /2 /49/48\n/51\n/99/101/108/108/45/53 /42 /42 /42 /42 /42 /49 /58 /51 /2 /49/48\n/49\n/99/101/108/108/45/48 /99/101/108/108/45/49 /99/101/108/108/45/50 /99/101/108/108/45/51 /99/101/108/108/45/52 /99/101/108/108/45/53\n/99/101/108/108/45/48 /42 /50 /58 /56 /2 /49/48\n/50\n/49 /58 /49 /2 /49/48\n/50\n/49 /58 /50 /2 /49/48\n/51\n/49 /58 /50 /2 /49/48\n/51\n/49 /58 /49 /2 /49/48\n/51\n/99/101/108/108/45/49 /42 /42 /49 /58 /50 /2 /49/48\n/50\n/56 /58 /48 /2 /49/48\n/49\n/56 /58 /55 /2 /49/48\n/49\n/50 /58 /56 /2 /49/48\n/50\n/99/101/108/108/45/50 /42 /42 /42 /52 /58 /49 /2 /49/48\n/50\n/52 /58 /56 /2 /49/48\n/50\n/50 /58 /54 /2 /49/48\n/51\n/99/101/108/108/45/51 /42 /42 /42 /42 /52 /58 /57 /2 /49/48\n/49\n/50 /58 /54 /2 /49/48\n/51\n/99/101/108/108/45/52 /42 /42 /42 /42 /42 /50 /58 /57 /2 /49/48\n/51\n/99/101/108/108/45/53 /42 /42 /42 /42 /42 /42"}
{"id": "adap-org/9803001", "meta": {"categories": ["adap-org", "nlin.AO", "physics.bio-ph", "q-bio"], "created": "1998-03-10", "extraction": {"body_chars": 31252, "cleaning": {"detected_repeated_margin_lines": ["0", "∫ x"], "page_count": 8, "removed_boilerplate_lines": 8}, "method": "pypdf_no_ocr", "source_pdf_bytes": 202937, "text_chars": 31885}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9803001", "primary_category": "adap-org", "source": "arxiv", "title": "Large-scale evolution and extinction in a hierarchically structured environment", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9803001"}, "text": "Large-scale evolution and extinction in a hierarchically structured environment\n\nAbstract\nA class of models for large-scale evolution and mass extinctions is presented. These models incorporate environmental changes on all scales, from influences on a single species to global effects. This is a step towards a unified picture of mass extinctions, which enables one to study coevolutionary effects and external abiotic influences with the same means. The generic features of such models are studied in a simple version, in which all environmental changes are generated at random and without feedback from other parts of the system.\n\narXiv:adap-org/9803001v1 10 Mar 1998\nLarge-scale evolution and extinction in a hierarchically s tructured\nenvironment\nC. Wilke, S. Altmeyer and T. Martinetz\nRuhr-Universit¨ at Bochum\nD-44780 Bochum, Germany\ne-mail: Claus.Wilke@neuroinformatik.ruhr-uni-bochum.de\nAbstract\nA class of models for large-scale evolution and mass\nextinctions is presented. These models incorporate en-\nvironmental changes on all scales, from influences on a\nsingle species to global effects. This is a step towards\na unified picture of mass extinctions, which enables\none to study coevolutionary effects and external abi-\notic influences with the same means. The generic fea-\ntures of such models are studied in a simple version,\nin which all environmental changes are generated at\nrandom and without feedback from other parts of the\nsystem.\nIntroduction\nIn the history of the Earth, there have been several\ncatastrophic events which in a short period of time\nhave wiped out large parts of the existing species. The\namount of species annihilated in such events has been\nup to 96% of the biodiversity at that time (Raup 1986).\nIt has often been argued that these mass extinctions\nmust have been caused by some disastrous abiotic inci-\ndences like extraterrestrial impacts. Evidence in favor\nof that has been put forward (Alvarez 1987), but on\nthe other hand, only 5% of the total loss of biodiversity\nin the fossil record can be connected to mass extinc-\ntions. The rest are the so-called background extinc-\ntions, which happen on much smaller scales. Interest-\ningly, the two types of extinction cannot clearly be dis-\ntinguished from another in the frequency distribution\nof extinction event sizes. The event sizes’ distribution\nforms a smooth curve, very close to a power-law (Sol´ e\n& Bascompte 1996).\nIn order to explain a single smooth distribution,\nthe idea of coevolutionary avalanches has been de-\nveloped (Kauffman 1992). The extinction of a single\nspecies might cause another species to die out, which\nmight drive a third species into extinction and so on,\nproducing an avalanche that in principle could span\nthe whole system. Because of the diverging mean\navalanche size, the distribution of extinction events\nwould then be a power-law, similar to the situation\nof thermodynamical systems at the point of a phase-\ntransition. Nevertheless, this mechanism, called self-\norganized criticality, completely neglects external in-\nfluences that certainly are present.\nOn the contrary, as it has recently been shown, a\npower-law distribution of extinction events can appear\neven in a system in which species are wiped out solely\nbecause of external influences (Newman 1996). How-\never, this effect depends crucially on influences that\nare imposed on all species coherently.\nFrom the point of view of a single species it does\nnot really matter whether it has to struggle with bad\nconditions imposed externally, e.g., a global shift in\ntemperature, or with bad conditions due to heavy com-\npetition with other species. All that counts for a single\nspecies is whether it can keep up with its environment\nor not.\nA species goes extinct when its population decreases\nto zero. This can happen for several reasons. One is a\nloss of habitat. Climatic or tectonic changes affect the\nlocation and the size of a species’ habitat. If the size\ndecreases rapidly, the species may not be able to adapt\nfast enough to find a new niche. Then the population\nwill drop below a level at which it can sustain itself and\nthe species will die out. Another reason for species’\nextinction is the invasion of new competitors or new\npredators. Competitors that invade a territory may be\nbetter adapted to a niche than the species originally\noccupying this niche. In this case, the population of\nthe native species can be decimated so effectively that\nit is wiped out. The same thing can happen because\nof an invading predator superior to the defense mech-\nanisms of the species. Similarly, new parasites can sig-\nnificantly reduce the population of a species and drive\nit to extinction.\nFrom the species point of view, all the above cases\ncan be subsumed under the notion of stress. A species\nsuffers stress of various kinds, stress because of climatic\nchanges, stress because of competition and predation\n\n. . .\n. . .\nglobal stress\nstress on large regions,\nlike continents\nstress on smaller regions\nstress on a single species\nFigure 1: Stress is generated in a tree structure.\netc. If the stress exceeds the level a species can sustain,\nit will go extinct.\nWe are going to develop a model in which all causes\nfor the extinction of a species will be regarded as stress.\nEvery species i has a threshold xi, or in general a vector\nxi, against stress. If a species suffers a stress ηi > x i,\nor in the general case a vector of stresses η i, where at\nleast one component exceeds the corresponding com-\nponent of the threshold vector xi, it dies out. So far,\nthis is a very general approach for a model in which\nspecies are the smallest units considered, i.e. a model\nthat does not work with individuals or populations.\nClearly, in such a model there will be stress on sev-\neral scales. We have global stress like a global shift in\ntemperature due to a slight change in the orbit of the\nEarth around the sun, or the impact of a very large\nmeteor. Then we have stress that spans large parts of\nthe Earth, e.g. a continent or a hemisphere, like the El\nNi˜ no phenomenon that roughly spans the region about\nthe tropical Pacific Ocean. And finally, we have stress\nthat affects smaller regions, or only a single species.\nThis leads us to a hierarchically ordered system of en-\nvironmental stresses. The simplest way to model it is\nto generate stress in a tree structure, as it is shown in\nFig. 1.\nOn all scales, the stress can be abiotic or biotic.\nThis may sound a bit counter-intuitive, since abiotic\nchanges are usually taken as large-scale phenomena,\nand biotic factors are usually taken as local phenom-\nena. Abiotic changes happen often on a global scale,\nlike the above mentioned examples of the orbit shift\nof the Earth or the meteor impact. But clearly there\nare more localized events. A small meteor or a small\nvulcano may affect only a limited number of species. If\na species happens to live only in a very small territory,\nand this territory gets destroyed by a meteor impact,\nthe species may be the only one that goes extinct be-\ncause of the impact.\nOn the other hand, biotic phenomena are not nec-\nessarily localized. Although direct species competition\nwill usually be a local phenomenon, there can be also\nglobal biotic phenomena. The composition of the at-\nmosphere, for example, depends strongly on biotic fac-\ntors, and it can change significantly due to biotic ef-\nfects.\nSo far, we have a model which represents the bio-\nsphere as a tree, with species situated at the leafs, and\nenvironmental stress generated at the nodes. Now we\nhave to choose the rules that determine how stress\nis generated and what thresholds against stress the\nspecies are given. This is the crucial part where we\ndecide what mechanisms we want to investigate. If\nwe were interested mainly in coevolutionary effects, we\nwould choose rules that link the properties and actions\nof the species directly to the generation of the stress.\nIn such a model, for example, the global stress at time\nt could be some sort of a sum over all the adaptive\nmoves of the species at time t − 1. In this work, how-\never, we are mainly interested in the generic features\nwe can expect from the hierarchical structure of the\nbiosphere. Therefore, we will focus on a version of\nthe model where the stresses and the species’ thresh-\nolds are simply random variables. Species’ interactions\nand abiotic effects can be so complicated and so un-\npredictable that in a first approximation we want to\nassume them to be completely random.\nThe model we study here is probably the simplest\npossible. Yet it has some intriguing features which are\nvery similar to characteristics seen in the fossil record.\nTo keep our model simple, we choose a homogenous\ntree, with l layers and n subtrees per node. In general,\nof course, one has to deal with inhomogenous trees.\nTo each node of the final layer we connect exactly one\nleaf, where we put m species. An example of such a\ntree with l = 4 and n = 2 is displayed in Fig. 2. The\ntotal number of stresses that have to be generated in\none time step is\nNstress =\nl−1∑\ni=0\nni , (1)\nand the total number of species in the model is\nNspecies = mnl−1 . (2)\nEvery species i has a single threshold xi, chosen at\nrandom from the uniform distribution on the intervall\n[0; 1). At every node j, the stress ηj generated in one\ntime step is a positive, real random variable drawn\nfrom a distribution with probability densitiy function\n(pdf) pj(x). It is a reasonable assumption to expect\nsmaller stresses to happen much more often than larger\n\nstresses. Therefore, we use pdf’s that fall off relatively\nfast with x → ∞ . An Exponential or Gaussian de-\ncrease should be a good choice, but the exact form of\nthe pdf is not really important. We choose the pdf’s\npj(x) at the beginning of the simulation at random\nfrom some family of distribution functions and keep\nthis choice fixed throughout the course of the simula-\ntion.\nFinally, we have to fix the way a species is affected\nby stress generated on different levels of the tree. We\nsimply take the maximum of all the stress values gen-\nerated at nodes that lie above the species in the tree:\nif at any of these nodes a stress ηj is generated which\nexceeds the species threshold xi, this species goes ex-\ntinct. It is then immediately replaced by a new species\nwith new random threshold.\nIn addition to the extinction dynamic, we introduce\nsome sort of adaption. In agreement with our idea of a\nfirst, simple model, the adaption is a random walk: in\nevery time step, a fraction f of the species is selected\nat random and given new thresholds.\nThere are certainly some oversimplifications in this\nmodel, such as the fixed number of species or the fact\nthat all species have only one trait. We will return to\nthis later and explain why we can still expect to cover\nthe basic features of the extinction dynamic.\nAnalysis\nThe behaviour of the above introduced model can be\nunderstood to a large extent from analytical calcula-\ntions. But before we begin with our analysis, we note\nthat the mechanism for species extinction and adap-\ntion presented here is similar to the one of the so-\ncalled ’coherent-noise’ models introduced by Newman\nand Sneppen (Newman & Sneppen 1996). These mod-\nels display a distribution of extinction events that fol-\nlows a power-law with exponent ≈ − 2, which is in good\nagreement with the fossil record. For this reason, they\nhave already been used to study macroevolutionary\nphenomena (Newman 1996; Wilke & Martinetz 1997).\nThe difference to our actual approach lies in the fact\nthat we use a multitude of stresses in a hierarchically\nordered system, whereas in the coherent-noise models\nthere is only a single stress, acting on the whole system\nat once. Therefore, in the previous works the idea of\nstress imposed on the species has been linked to exter-\nnal influences like meteor impacts and was opposed to\ncoevolutionary effects.\nNote that we have effectively a coherent-noise model\nat every leaf of the tree if the number m of species\nlocated at one leaf is large.\nThe effective stress-distribution at a leaf\nof the tree\nEvery leaf of the tree feels a stress-distribution which\ndepends on the distributions of the nodes above it. Let\nthere be N nodes above a leaf. Then the N stress\nvalues having influence on this leaf are N random\nvariables X1, . . . , X N with pdf’s p1(x), . . . , p N (x). We\nhave to calculate the pdf pmax(x) of the random vari-\nable Xmax = max{X1, . . . , X N }, i.e.,\npmax(x) dx = P (x ≤ max{X1, . . . , X N } < x + dx) .\n(3)\nWith the partition theorem we can write the prob-\nability on the right-hand side as a weighted sum of\nconditional probabilities:\nP (x ≤ max{X1, . . . , X N } < x + dx)\n=\nN∑\ni=1\nP (x ≤ max{X1, . . . , X N } < x + dx\n⏐\n⏐\n⏐x ≤ Xi < x + dx)\n× P (x ≤ Xi < x + dx) . (4)\nThe conditional probabilities read\nP (x ≤ max{X1, . . . , X N } < x + dx⏐\n⏐\n⏐x ≤ Xi < x + dx)\n= 1\nP (x ≤ Xi < x + dx)\n× P (x ≤ max{X1, . . . , X N } < x + dx\n∧ x ≤ Xi < x + dx)\n=\nP (x ≤ Xi < x + dx) ∏ N\nj=1,j ̸=i P (x > X j)\nP (x ≤ Xi < x + dx)\n=\nN∏\nj=1,j ̸=i\nP (x > X j) . (5)\nAfter inserting Eq. (5) into Eq. (4) we find\nP (x ≤ max{X1, . . . , X N } < x + dx)\n=\nN∑\ni=1\nP (x ≤ Xi < x + dx)\nN∏\nj=1,j ̸=i\nP (x > X j) .\n(6)\nConsequently, for the pdf pmax(x) we have\npmax(x) =\nN∑\ni=1\npi(x)\nN∏\nj=1,j ̸=i\nP (x > X j)\n=\nN∑\ni=1\npi(x)\nN∏\nj=1,j ̸=i\nx∫\npj(x′) dx′ . (7)\n\nWe are interested in the tail of pmax(x). For coherent-\nnoise models we know that a power-law distribution\nof event-sizes will appear if the stress-distribution\npstress(x) satisfies\n∞∫\nη\npstress(x) dx ≈ Cp α\nstress(η) for η → ∞ , (8)\nwhere C and α are positive constants which depend on\npstress(x) (Sneppen & Newman 1997). Therefore, we\nassume this condition to hold also for the distributions\npj(x) in Eq. (7), with constants Cj and α j, respec-\ntively. Then we can approximate the tail of pmax(x)\nby\npmax(x) ≈\nN∑\ni=1\npi(x)\nN∏\nj=1,j ̸=i\n(\n1− Cjpα j\nj (x)\n)\nfor x → ∞ .\n(9)\nWe proceed further by taking only linear terms in pi(x)\nand obtain\npmax(x) ≈\nN∑\ni=1\npi(x) for x → ∞ . (10)\nFor large x, this sum will be dominated by the pi(x)\nthat is falling off slowest. We say that distribution\npi(x) falls off slower than distribution pj(x) if there\nexists a x0 such that\npi(x) > p j(x) for all x > x 0. (11)\nFor a set of reasonable stress-distributions it is always\npossible to identify one that is falling off slowest ac-\ncording to this definition.\nThe fact that the sum in Eq. (11) will asymptotically\nbe dominated by a single term leads to the situation\ndepicted in Fig. 2. The tree breaks down into several\nindependent subsystems. The meaning of the numbers\nin the figure will be explained in detail later. In a\nnutshell, they indicate how slow a stress distribution\nis falling off. What interests us here is the breakup of\nthe tree into several independent parts in the regime\nof large stresses. If these parts are not too small, they\nwill behave like independent coherent-noise systems.\nAn ensemble of a finite number of\nindependent coherent-noise systems\nIf the stress-distributions close to the root dominate\nthe behaviour of the system, the tree will break down\ninto independent coherent-noise systems, as we have\nmentioned above. Consequently, we proceed with the\ncalculation of the distribution of extinction events in\na system consisting of independent coherent-noise sub-\nsystems. In the calculation, however, we will deviate\n3 14\n12 5 1 11\n9 2 13 8 15 10 4 2\n12 12 13 8 15 14 14 14\nFigure 2: The tree breaks down into virtually indepen-\ndent parts in the limit of large stresses.\nslightly from the actual situation in the tree model by\nassuming the subsystems to have each an infinite size.\nThis allows for an easy calculation, and the main re-\nsults should also hold for large but finite sizes.\nIn the case of an infinite system size, the event distri-\nbution of a coherent-noise model possesses a power-law\ntail that extends to arbitrary large events. Therefore,\nthe task of calculating the event distribution of the\ncompound system equals to the task of calculating the\nsum of a finite number of nonidentically distributed\nrandom variables with power-law tail. The latter can\nbe treated mathematically exact under relatively weak\nassumptions (Wilke unpublished). But since the ex-\nact calculations are too extensive to be included in\nthis work, we will here give only an intuitive argument\nabout the tail behaviour of the sum.\nWe begin with the sum of two positive, real random\nvariables X1 and X2, where the pdf’s p1(x) and p2(x)\nhave a power-law tail x−τ1 and x−τ2 , respectively. We\nassume the pdf’s to be continuous, non-singular, and\nreasonably smooth. Under these conditions, we can\nwrite p1(x) and p2(x) in the form\np1(x) = f1(x)\n(x + 1)τ1\n, (12)\np2(x) = f2(x)\n(x + 1)τ2\n, (13)\nwhere f1(x) and f2(x) are continuous, non-singular,\nand reasonably smooth functions which tend towards\na positive constant for x → ∞ . The pdf psum(x) of\nthe sum X = X1 + X2 is the convolution of p1(x) and\np2(x):\npsum(x) =\np1(x′)p2(x − x′) dx′\n\n=\nf1(x′)\n(x′ + 1)τ1\nf2(x − x′)\n(x − x′ + 1)τ2\ndx′ .\n(14)\nAfter a change of the integration variable to z = x′/x\nwe obtain\npsum(x) =\n∫ 1\nf1(xz)\n(xz + 1)τ1\nf2(x(1 − z))\n(x(1 − z) + 1)τ2\nx dx′\n= x1−τ1−τ2\n∫ 1\nf1(xz)\n(z + 1\nx )τ1\nf2(x(1 − z))\n(1 − z + 1\nx )τ2\ndx′ .\n(15)\nFor large x, there are two main contributions to this\nintegral, at z ≈ 0 and at z ≈ 1, which stem from the\nfirst and from the second term in the denominator.\nSince the denominators will become arbitrarily large\nfor large x, we can assume the other terms to be con-\nstant in the regions where the main contributions come\nfrom. Therefore, we find\npsum(x) ≈ x1−τ1−τ2\n[\nC1xτ2−1 + C2xτ1−1\n]\n, (16)\nwhere C1 and C2 are positive constants. Obviously for\nlarge x the term with the largest exponent will domi-\nnate. Hence we have\npsum(x) ∼ x− min{τ1,τ 2} . (17)\nThis result can be easily extended to the case of\nan arbitrary finite number of random variables with\npower-law tail by iteration. Asymptotically, the tail of\npsum(s) will always be dominated by the contribution\nfrom the term with the smallest exponent.\nBack to the ensemble of infinitely large coherent-\nnoise systems, we find that it will display power-law\ndistributed event sizes, as its single constituents do.\nIf the subsystems’ stress-distributions are functionally\ndifferent, the exponent of the compound system’s event\ndistribution will be the smallest of the subsystems’ ex-\nponents.\nThe above result should also hold in the situation\nof finite coherent noise systems, as long as their total\nnumber is small compared to their typical size.\nTrees with random stress distributions\nWe argue above that in the limit of large stresses the\ntree will break down into subsystems, virtually inde-\npendent of each other. The behaviour of our model\ndepends heavily on the size of the parts we find. If\nthe different parts are all very small, the system will\nloose its coherent-noise characteristics. Instead of a\npower-law distribution the extinction events will then\nfollow a gaussian distribution because of the central-\nlimit theorem. Therefore, in this section we will study\nthe distribution of the subsystems’ sizes that arises if\nwe randomly assign stress distributions to the tree’s\nnodes.\nWe assume that the propability for a certain stress\ndistribution to be assigned to a certain node does not\ndepend on the position of the node in the tree. In other\nwords, we use the same set of stress distributions on all\nlevels of the tree. Furthermore, we assume that for any\ntwo stress distributions we use we can identify one of\nthe two that falls off faster than the other one. Under\nthese conditions, we can study the structure of such\ntrees by simply assigning integers to the nodes of the\ntree, where larger integers stand for distributions that\nare falling off slower. If the set of possible stress distri-\nbutions is infinite, the probability of finding two nodes\nwith the same distribution is zero. Consequently, in\na tree with n nodes, we will assign every integer from\n1 . . . n to exactly one node. This is displayed in Fig. 2\nfor a tree with 15 nodes. For every leaf i of the tree\nwe can then define a characteristic number ai. This\nnumber is the maximum of the nodes’ numbers en-\ncountered on the way from the leaf up to the root. All\nthe leafs with the same characteristic number belong\nto the same subsystem. In the example of Fig. 2, we\nhave five subsystems in total. Three of them contain\nonly one leaf, one contains two and one contains three\nleafs.\nIn general, we are interested in the distribution of\nsubsystems arising in large trees. Therefore, we have\ndone simulations in which we have several thousand\ntimes assigned random integers to the nodes of a large\ntree. For every single realization of the tree, we have\ncomputed a histogram of the frequency of the differ-\nent parts’ sizes. Finally, we have calculated the aver-\nage of all the histograms. Fig. 3 shows the result of\nsuch simulations for two different trees with 10000 his-\ntograms each. We find the expected frequency f (k) of\nlarge independent parts in the tree decreasing as a saw-\ntooth function that follows approximately a power-law\nwith exponent − 2, independent of l and n. The sharp\npeaks in the distribution arise whenever the size of a\ncomplete subtree is reached. Therefore, we observe in\nFig. 3, e.g., the peaks in the distribution of the tree\nwith n = 10 appearing at powers of 10.\nThe power-law can be explained easily if we assume\nthe main contributions to come from complete sub-\ntrees. The expected frequency f (k) to find an inde-\npendent subtree with b layers, which corresponds to a\nsubsystem of size k = nb, can be written as the number\nof such subtrees in the whole system, N (b), times the\nprobability that any of these subtrees will be indepen-\n\ndent of the rest, P (b). Hence we write\nf (nb) = N (b)P (b) . (18)\nThe number of subtrees of size nb is N (b) = nl−b. For\nthe probability P (b) we find\nP (b) =\n(\nl − b +\nb−1∑\ni=0\nni\n) −1\n, (19)\nwhich is simply the probability for the integer assigned\nto the node at the root of the subtree to be larger\nthan all the other integers which are assigned to the\nremaining nodes of the subtree and to the nodes above\nthe subtree. If we increase b by one, we get N (b + 1) =\nnl−b−1 = N (b)/n . With slightly more effort, we find\nalso\nP (b + 1) =\n(\nl − b − 1 +\nb∑\ni=0\nni\n) −1\n=\n(\nl − b + n\nb−1∑\ni=0\nni\n) −1\n≈ 1\nn P (b) . (20)\nTherefore, we can write\nf (nk) ≈ N (k)\nn\nP (k)\nn = n−2f (k) , (21)\nwhich implies f (k) ∼ k−2.\nThe peaks in Fig. 3 appear whenever the size of a\ncomplete subtree is reached, as we have noted above.\nThis means they are connected to the extremely regu-\nlar structure of the trees we use in this work. There-\nfore, we are currently investigating trees with irreg-\nular structure. For these trees, the spikes disappear\nand, in log-log plot, the function f (k) becomes almost\na straight line with slope -2. From the simulations we\nhave done so far, we can say that this result is very gen-\neral and seems to be independent of the special trees’\nproperties.\nSimulation results\nSince we are interested in the typical behaviour of our\nmodel, we have to do many simulation runs with dif-\nferent tree sizes and different stress distributions at the\ntree’s nodes. But the simulation of large trees is very\nslow, and therefore it is hard to get a good sample of\nthe parameter-space. To overcome this difficulty we\nhave also done simulations based on the arguments of\nthe previous sections. As we have seen there, in the\nlimit of large stresses it is possible to map the leafs of\nthe tree onto a system consisting of several indepen-\ndent coherent-noise models, with the sizes k of these\nsubsystems distributed according to k−2.\n/1/0\n/BnZr /4\n/1/0\n/BnZr /3\n/1/0\n/BnZr /2\n/1/0\n/BnZr /1\n/1/0\n/0\n/1/0\n/1\n/1/0\n/2\n/1/0\n/3\n/1/0\n/4\n/1/0\n/5\n/1/0\n/6\n/1/0\n/0\n/1/0\n/1\n/1/0\n/2\n/1/0\n/3\n/1/0\n/4\n/1/0\n/5\n/1/0\n/6\nfrequency\nsize\nFigure 3: The expected frequency for the occurence of\nlarge independent subsystems decreases as a sawtooth\nfunction that follows approximately a power-law with\nexponent − 2. The upper curve stems from a tree with\nl = 18 and n = 2. It has been rescaled by a factor\nof 100 so as not to overlap with the lower curve. The\nlower curve stems from a tree with l = 6 and n = 10.\nIn Fig. 4 we show a comparison between the full\nsimulation and the approximation. To come as close\nas possible to the full simulation, we use the maxi-\nmum of 5 independent, exponentially distributed ran-\ndom variables as stresses for the independent coherent-\nnoise models, since for the tree we have likewise chosen\nl = 5 and exponential stress-distributions. Clearly the\nbehaviour of the approximation is close to the one of\nthe full simulation, which verifies the analytical reason-\ning of the previous sections. Both simulations display\npower-law distributed extinction events. For the full\ntree, we find an exponent τtree = 2. 35 ± 0. 05, while for\nthe approximation, we find τapprox = 2. 30 ± 0. 05. If we\nconsider the high level of abstraction from the tree to\nan ensemble of coherent-noise systems, this agreement\nis excellent.\nNote that in comparison to a normal coherent-noise\nmodel with only a single stress variable, the tree model\nproduces a significantly larger exponent τ (If we run a\nnormal coherent-noise model with the stress distribu-\ntion of the approximaton in Fig. 4, we get an exponent\nτ ≈ 1. 8). The increased exponent τ has its origin in the\ndistribution of the subsystems’ sizes. The sizes scale\nthemselves, thus modifying the scale-invariant behav-\nior of the ensemble, compared to the one of a single\ncoherent-noise system.\nDiscussion\nWe have presented a model of large-scale evolution and\nextinction that combines biotic and abiotic causes for\nextinction within a single mathematical framework.\n\n/1\n/1/0\n/1/0/0\n/1/0/0/0\n/1/0/0/0/0\n/1/0/0/0/0/0\n/1/0 /1/0/0 /1/0/0/0\nfrequency p /( s /)\nextinction ev en t size s\nFigure 4: The frequency of extinction events in the\ntree model and in an ensemble of coherent-noise mod-\nels. The lower curve stems from the simulation of a\ntree with l = 5, n = 10 and m = 1, which amounts\nto a total of 10 5 species. Stress distributions were as-\nsigned at random to the nodes of the tree. We used\nexponentially distributed stress with σ between 0.03\nand 0.05. The upper curve corresponds to the simu-\nlation of an ensemble of coherent-noise models with a\ntotal of 10 5 species, and with the sizes k of the sub-\nsystems distributed according to k−2. As stresses we\nused the maximum of 5 exponentially distributed ran-\ndom variables.\nFurthermore, the model takes into account the hi-\nerarchical structure of the biosphere. To the best\nof our knowledge, the implications of environmental\nchanges happening on different scales have not been\nstudied previously in macroevolutionary models. De-\nspite the choice of completely random environmental\nchanges, the model has some interesting features. The\ndistribution of extinction events follows a power-law\nwith exponent in the region of 2 (note that the ex-\nponent depends on the choice of the stress distribu-\ntion, as it is the case with coherent-noise models).\nFrom the fossil record, a power-law distribution with\nexponent τ ≈ 2 is reported for the extinction event\nsizes of taxonomical families (Sol´ e & Bascompte 1996;\nNewman 1996). Moreover, it is interesting to observe\nthe breakup of the tree into subsystems with sizes k dis-\ntributed according to k−2. The power-law distribution\nof the subsytem sizes implies that even in very large\ntrees we will find large subsystems, governed mainly by\nonly a single stress distribution. Intuitively, we would\nexpect the subsystems to have roughly similar sizes,\nand to enter the dynamic of the whole system on an\nequal basis. But we observe exactly the opposite. The\nsubsystems’ sizes are scale-invariant, thus producing\na scale-invariant distribution of contributions to the\noverall system’s behavior. In particular, only a small\nnumber of large subsystems produces events on large\nscales. This might be an explanation for the fact that\nin such large and complex systems like the biosphere\nwe find usually smooth frequency distributions of typ-\nical objects or events.\nThe model we have studied in this work is certainly\noversimplified. For that reason, we will close this paper\nwith some remarks about extensions to the model that\nshould be examined in a next step closer to biologi-\ncal reality. First of all, it is certainly a severe restric-\ntion to keep the number of species fixed throughout\nthe simulation. Nevertheless, this is a restriction used\nvery often in models of macroevolution (Peliti 1997).\nOnly recently, work has been done where a change\nin biodiversity is considered (Head & Rodgers 1997;\nWilke & Martinetz 1997). The behaviour of the model\nwe study here is governed by the coherent-noise dy-\nnamic. For this dynamic, it has been shown that it\ncan be generalized to include a variable system size\nwithout loss of it’s main features (Wilke & Martinetz\n1997). Therefore, we believe a fixed system-size can be\njustified in the present work. It should be possible to\nextend our tree model to a model with variable system\nsize. Another severe restriction is the usage of only one\ntrait. But here a similar argument holds as in the case\nof the fixed number of species. A multi-trait version\nof the original coherent-noise model has already been\nstudied (Newman in press). It behaves very similar to\nthe single-trait version.\nFinally, we want to discuss the way we compute the\nstress on a single species out of the multitude of stress\nvalues, generated at the different levels of the tree.\nThroughout this paper, we have used the maximum of\nthe stress values. This allows for an easy and very gen-\neral analytical investigation. Another natural choice,\nhowever, would be to sum up all the stresses. We have\nalso done some simulations in this fashion. The be-\nhavior of the system remains roughly the same. This\nhappens because in a finite sum of non-identically dis-\ntributed random variables, we expect large values to be\ndominated by a single term of the sum, similar to the\ncase of the maximum of several random variables. For\nthe sum of exponentially distributed random variables,\nan easy calculation shows that this conjecture is indeed\ntrue. With some more effort, we can prove the same\nfor the sum of power-law distributed random variables,\nas we have already done in this paper. Nevertheless,\nin the general case with arbitrary distributions, the\nconjecture is hard to demonstrate.\n\nReferences\n[Alvarez 1987] Alvarez, L. W. 1987. Mass extinctions\ncaused by large bolide impacts. Physics Today pp\n24–33, July 1987.\n[Head & Rodgers 1997] Head, D. A., and Rodgers,\nG. J. 1997. Speciation and extinction in a simple\nmodel of evolution. Phys. Rev. E 55:3312.\n[Kauffman 1992] Kauffman, S. A. 1992. The Origins\nof Order . Oxford: Oxford University Press.\n[Newman & Sneppen 1996] Newman, M. E. J., and\nSneppen, K. 1996. Avalanches, scaling and coher-\nent noise. Phys. Rev. E 54:6226.\n[Newman 1996] Newman, M. E. J. 1996. Self-\norganized criticality, evolution, and the fossil extinc-\ntion record. Proc. R. Soc. London B 263:1605.\n[Newman in press] Newman, M. E. J. (in press). A\nmodel of mass extinction. J. Theor. Biol. also adap-\norg/9702003.\n[Peliti 1997] Peliti, L. 1997. Introduction to the\nstatistical theory of darwinian evolution. cond-\nmat/9712027.\n[Raup 1986] Raup, D. M. 1986. Biological extinction\nand earth history. Science 231:1528.\n[Sneppen & Newman 1997] Sneppen, K., and New-\nman, M. E. J. 1997. Coherent noise, scale invari-\nance and intermittency in large systems. Physica D\n107:292.\n[Sol´ e & Bascompte 1996] Sol´ e, R. V., and Bascompte,\nJ. 1996. Are critical phenomena relevant to large-\nscale evolution? Proc. R. Soc. Lond. B 263:161.\n[Wilke & Martinetz 1997] Wilke, C., and Martinetz,\nT. 1997. Simple model of evolution with variable\nsystem size. Phys. Rev. E 56:7128.\n[Wilke unpublished] Wilke, C. unpublished. On the\nsum of random variables with power-law tail."}
{"id": "adap-org/9804002", "meta": {"categories": ["adap-org", "nlin.AO", "q-bio.PE"], "created": "1998-04-02", "extraction": {"body_chars": 31322, "cleaning": {"detected_repeated_margin_lines": [], "page_count": 7, "removed_boilerplate_lines": 2}, "method": "pypdf_no_ocr", "source_pdf_bytes": 215468, "text_chars": 32141}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9804002", "primary_category": "adap-org", "source": "arxiv", "title": "Critical Exponent of Species-Size Distribution in Evolution", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9804002"}, "text": "Critical Exponent of Species-Size Distribution in Evolution\n\nAbstract\nWe analyze the geometry of the species- and genotype-size distribution in evolving and adapting populations of single-stranded self-replicating genomes: here programs in the Avida world. We find that a scale-free distribution (power law) emerges in complex landscapes that achieve a separation of two fundamental time scales: the relaxation time (time for population to return to equilibrium after a perturbation) and the time between mutations that produce fitter genotypes. The latter can be dialed by changing the mutation rate. In the scaling regime, we determine the critical exponent of the distribution of sizes and strengths of avalanches in a system without coevolution, described by first-order phase transitions in single finite niches.\n\narXiv:adap-org/9804002v1 2 Apr 1998\nCritical Exponent of Species-Size Distribution in Evoluti on\nChris Adami 1, Ryoichi Seki 1,2 and Robel Yirdaw 2\n1California Institute of Technology, Pasadena, CA 91125\n2California State University, Northridge, CA 91330\nAbstract\nWe analyze the geometry of the species– and genotype-\nsize distribution in evolving and adapting populations of\nsingle-stranded self-replicating genomes: here programs\nin the Avida world. We find that a scale-free distribution\n(power law) emerges in complex landscapes that achieve\na separation of two fundamental time scales: the relax-\nation time (time for population to return to equilibrium\nafter a perturbation) and the time between mutations\nthat produce fitter genotypes. The latter can be dialed\nby changing the mutation rate. In the scaling regime,\nwe determine the critical exponent of the distribution\nof sizes and strengths of avalanches in a system without\ncoevolution, described by first-order phase transitions in\nsingle finite niches.\nIntroduction\nPower law distributions in Nature usually signal the\nabsence of a scale in the region where the scaling is\nobserved, and sometimes point to critical dynamics.\nIn Self-Organized-Criticality (SOC) (Bak, Tang, and\nWiesenfeld 1987, 1988), for example, power law distribu-\ntions reveal the dynamics of an unstable critical point,\nbrought about by slow driving and a feed-back mecha-\nnism between order parameter and critical parameter.\nThe critical dynamics is usually described within the\nlanguage of second-order phase transitions in condensed\nmatter systems (Sornette, Johansen and Dornic 1996),\nbut it can be shown that SOC-type behavior also oc-\ncurs within a dual description in terms of the Landau-\nGinzburg equation as first-order transitions (Gil and Sor-\nnette 1996). Indeed, it was shown that a power law\ndistribution of epoch-lengths, that is, the time a par-\nticular species dominates the dynamics of an adapting\npopulation, is explained by a self-organized critical sce-\nnario (Adami 1995) that carries the hallmark of first-\norder phase transitions. Here, we measure the distribu-\ntion of abundances of species and genotypes in an artifi-\ncial chemistry, (the Avida Artificial Life system, Adami\nand Brown 1995, Ofria, Brown and Adami 1998) and\nshow that the distribution is scale-free under a broad\nclass of circumstances, confirming the results reported in\nAdami (1995). In the next section, we discuss the first-\norder dynamics in more detail and examine “avalanches\nof invention” from the point of view of a thermodynam-\nics of information. In Section III, we measure the critical\nexponent of the power law of genotype abundances in\nthe limit of infinitesimal driving, i.e., infinitesimal mu-\ntation rate, and discuss the role of the fitness landscape\nin shaping the distribution. In Section IV, we repeat the\nanalysis for a higher taxonomic level (that of species)\nand discuss its relation to the geometric distributions\nfound by Burlando (1990, 1993). Conclusions about the\nevolutionary process drawn from the data obtained in\nthis paper are presented in Section V.\nSelf-Organization in Evolution\nThe idea that the evolutionary process occurs in spurts,\njumps, and bursts rather than gradual, slow and contin-\nuous changes has been around for over 75 years (Willis\n1922), but has gained prominence as “punctuated equi-\nlibrium” through the work of Gould and Eldredge (1977,\n1993). The general idea is that evolutionary innovations\nare not bestowed upon an existing species as a whole,\ngradually, but rather by the emergence of one better\nadapted mutant which, by its superiority, serves as the\nseed of a new breed that sweeps through an ecological\nniche and supplants the species previously occupying it.\nThe global dynamics thus has a microscopic origin, as\nshown experimentally, e.g., in populations of E. Coli by\nElena, Cooper and Lenski (1996).\nSuch avalanches can be viewed in two apparently con-\ntradictory ways. On the one hand we may consider the\nwave of extinction touching all species that are connected\nby their ecological relations, a process akin to percola-\ntion and therefore suitably described by the language\nof second-order critical phenomena (Bak and Sneppen\n1993). Such a scenario relies on the coevolution of species\n(to build their ecological relations) and successfully de-\nscribes power-law distributions obtained from the fos-\nsil record (Sol´ e and Bascompte 1996, Bak and Paczuski\n1996). There is, on the other hand, a description in\nterms of informational avalanches that does not require\ncoevolution and leads to the same statistics, as we show\n\nhere. Rather than contradicting the aforementioned pic-\nture (Newman et al. 1997), we believe it to be comple-\nmentary.\nIn the following, we set up a scenario in which informa-\ntion is viewed as the agent of self-organization in evolving\nand adapting populations. Information is, in the strict\nsense of Shannon theory, a measure of correlation be-\ntween two ensembles: here a population of genomes and\nthe environment it is adapting to. As described else-\nwhere (Adami 1998), this correlation grows as the popu-\nlation stores more and more information about the envi-\nronment via random measurements, implementing a very\neffective natural Maxwell demon . Any time a stochas-\ntic event increases the information stored in the pop-\nulation, a wave of extinction removes the less adapted\ngenomes and establishes a new era. Yet, information\ncannot leave the population as a whole, which there-\nfore may be thought of as protected by a semi-permeable\nmembrane for information, the hallmark of the Maxwell\ndemon. Let us consider this dynamics in more detail.\nThe simple living systems we consider here are popu-\nlations of self-replicating strings of instructions, coded in\nan alphabet of dimension D with variable string length\nℓ. The total number of possible strings is exponentially\nlarge. Here, we consider the subset of all strings cur-\nrently in existence in a finite population of size N , har-\nboring Ng different types, where Ng ≪ D ℓ. Each geno-\ntype (particular sequence of instructions) is character-\nized by its replication rate ǫi, which depends on the se-\nquence only, while its survival rate is given by ǫi/⟨ǫ⟩, in\na “stirred-reactor” environment that allows a mean-field\npicture. This average replication rate ⟨ǫ⟩ characterizes\nthe fitness of the population as a whole, and is given by\n⟨ǫ⟩ =\nNg∑\ni\nni\nN ǫi , (1)\nwhere ni is the occupation number, or frequency, of geno-\ntype i in the population. As Ng is not fixed in time, the\naverage depends on time also, and is to be taken over\nall genotypes currently living. The total abundance, or\nsize, of a genotype is then\nsi =\n∫ ∞\nni(t) dt =\n∫ Te\nTc\nni(t) dt , (2)\nwhere Tc is the time of creation of this particular geno-\ntype, and Te the moment of extinction. Before we obtain\nthis distribution in Avida, let us delve further into the\nstatistical description of the extinction events.\nAt any point in time, the fate of every string in the\npopulation is determined by the craftiness of the best\nadapted member of the population, described by ǫbest.\nIn this simple, finite, world, which does not permit\nstrings to affect other members of the population ex-\ncept by replacing them, not being the best reduces a\nFigure 1: “Energies” (inferiorities) of strings in a first-\norder phase transition with latent heat ∆ ǫ.\nold= 0\nE new= 0\nE\nE\n∆ε\nstring to an ephemeral existence. Thus, every string is\ncharacterized by a relative fitness, or inferiority\nEi = ǫbest − ǫi (3)\nwhich plays the role of an energy variable for strings of\ninformation (Adami 1998). Naturally, ⟨E⟩ = 0 charac-\nterizes the ground state , or vacuum, of the population,\nand strings with Ei > 0 can be viewed as occupying\nexcited states, soon to “decay” to the ground state (by\nbeing replaced by a string with vanishing inferiority).\nThrough such processes, the dynamics of the system tend\nto minimize the average inferiority of the population, and\nthe fitness landscape of replication rates thus provides a\nLyapunov function. Consequently, we are allowed to pro-\nceed with our statistical analysis. Imagine a population\nin equilibrium, at minimal average inferiority as allowed\nby the “temperature”: the rate (or more precisely, the\nprobability) of mutation. Imagine further that a muta-\ntion event produces a new genotype, fitter than the oth-\ners, exploiting the environment in novel ways, replicating\nfaster than all the others. It is thus endowed with a new\nbest replication rate, ǫnew\nbest, larger than the old “best” by\nan amount ∆ ǫ, and redefining what it means to be infe-\nrior. Indeed, all inferiorities must now be renormalized:\nwhat passed as a ground state ( E = 0) string before\nnow suddenly finds itself in an excited state. The seed\nof a new generation has been sown, a phase transition\nmust occur. In the picture just described, this is a first-\norder phase transition with latent heat ∆ ǫ (see Fig. 1),\nstarting at the “nucleation” point, and leading to an ex-\npanding bubble of “new phase”. This bubble expands\nwith a speed given by the Fisher velocity\nv ∼\n√\nD∆ ǫ , (4)\nwhere D is the diffusion coefficient (of information) in\nthis medium, until the entire population has been con-\nverted (Chu and Adami 1997). This marks the end of\n\nthe phase transition, as the population returns to equi-\nlibrium via mutations acting on the new species, creating\nnew diversity and restoring the entropy of the population\nto its previous value. This prepares the stage for a new\navalanche, as only an equilibrated population is vulner-\nable to even the smallest perturbation. The system has\nreturned to a critical point, driven by mutations, self-\norganized by information.\nThus we see how a first-order scenario, without coevo-\nlution, can lead to self-organized and critical dynamics.\nIt takes place within a single, finite, ecological niche, and\nthus does not contradict the dynamics taking place for\npopulations that span many niches. Rather, we must\nconclude that the descriptions complement each other,\nfrom the single-niche level to the ecological web. Let us\nnow take a closer look at the statistics of avalanches in\nthis model, i.e., at the distribution of genotype sizes.\nExponents and Power Laws\nThe size of an avalanche in this particular system can be\napproximated by the size s of the genotype that gave rise\nto it, Eq. (2). We shall measure the distribution of these\nsizes P (s) in the Artificial Life system Avida, which im-\nplements a population of self-replicating computer pro-\ngrams written in a simple machine language-like instruc-\ntion set of D = 24 instructions, with programs of varying\nsequence length. In the course of self-replication, these\nprograms produce mutant off-spring because the copy\ninstruction they use is flawed at a rate R errors per in-\nstruction copied, and adapt to an environment in which\nthe performance of logical computations on externally\nprovided numbers is akin to the catalysis of chemical\nreactions (Ofria, Brown and Adami 1998). In this ar-\ntificial chemistry therefore, successful computations ac-\ncelerate the metabolism (i.e., the CPU) of those strings\nthat carry the gene (code) necessary to perform the trick,\nand any program discovering a new trick is the seed of\nanother avalanche.\nAvida is not a stirred-reactor environment (although\none can be simulated). Rather, the programs live on a\ntwo-dimensional grid, each program occupying one site.\nThe size of the grid is finite, and chosen in these experi-\nments to be small enough that avalanches are generally\nover before a new one starts. As is well-known, this is\nthe condition sine qua non for the observation of SOC\nbehavior, a separation of time scales which implies that\nthe system is driven at infinitesimal rates.\nLet τ denote the average duration of an avalanche.\nThen, a separation of time scales occurs if the average\ntime between the production of new seeds of avalanches\nis much larger than τ . New seeds, in turn, are produced\nwith a frequency ⟨ǫ⟩P , where ⟨ǫ⟩ is again the average\nreplication rate, and P is the mutation probability (per\nreplication period) for an average sequence of length ℓ,\nP = 1 − (1 − R)ℓ . (5)\nFigure 2: Fitness of the dominant genotype in the pop-\nulation, ǫbest as a function of time (in updates).\nFor small enough R and not too large ℓ (so that the\nproduct Rℓ is smaller than unity) we can approximate\nP ≈ Rℓ, and infinitesimal driving occurs in the limit\n⟨ǫ⟩Rℓ ≪ 1\nτ . (6)\nFurthermore\nτ ∼ L\nv (7)\nwith L the diameter of the system and v a typical Fisher\nvelocity. The fastest waves are those for which the latent\nheat is of the order of the new fitness, i.e., ∆ ǫ ∼ ǫ, in\nwhich case v ≈ ǫ (because D ∼ ǫ in Eq. (4), see Chu and\nAdami 1995) and a separation of time scales is assured\nwhenever\nRℓ ≫ L , (8)\nthat is, in the limit of vanishing mutation rate or small\npopulation sizes. For the L = 60 system used here, this\ncondition is obeyed (for the fastest waves) only for the\nsmallest mutation rate tested and sequence lengths of\nthe order of the ancestor.\nIn the following, we keep the population size constant\n(a 60 × 60 grid) and vary the mutation rate. From the\nprevious arguments, we expect true scale-free dynamics\nonly to appear in the limit of small mutation rates. As in\nthis limit avalanches occur less and less frequently, this\nis also the limit where data are increasingly difficult to\nobtain, and other finite size effects can come into play.\nWe shall try to isolate the scale-free regime by fitting the\ndistribution to a power law\nP (s) ∼ s− D(R) (9)\n\nand monitor the behavior of D from low to high mutation\nrates.\nIn Fig. 2, we display a typical history of ǫbest, i.e.,\nthe fitness of the dominant genotype 1. Note the “stair-\ncase” structure of the curve reflecting the “punctuated”\ndynamics, where each step reflects a new avalanche and\nconcurrently an extinction event. Staircases very much\nlike these are also observed in adapting populations of\nE. Coli (Lenski and Travisano 1994).\nAs touched upon earlier, the Avida world represents\nan environment replete with information, which we en-\ncode by providing bonuses for performing logical compu-\ntations on externally provided (random) numbers. The\ncomputations rewarded usually involve two inputs A and\nB, are finite in number and listed in Table 1. At the end\nof a typical run (such as Fig. 2) the population of pro-\ngrams is usually proficient in almost all tasks for which\nbonuses are given out, and the genome length has grown\nto several multiples of the initial size to accommodate\nthe acquired information.\nTable 1: Logical calculations on random inputs A and B\nrewarded, bonuses, and difficulty (in minimum number\nof nand instructions required). Bonuses bi increase the\nspeed of a CPU by a factor νi = 1 + 2 bi− 3.\nName Result Bonus bi Difficulty\nEcho I/O 1 –\nNot ¬A 2 1\nNand ¬(A ∧ B) 2 1\nNot Or ¬A ∨ B 3 2\nAnd A ∧ B 3 2\nOr A ∨ B 4 3\nAnd Not A ∧ ¬B 4 3\nNor ¬(A ∨ B) 5 4\nXor A xor B 6 4\nEquals ¬(A xor B) 6 4\nBecause the amount of information stored in the\nlandscape is finite, adaptation, and the associated\navalanches, must stop when the population has ex-\nhausted the landscape. However, we shall see that even a\n‘flat’ landscape (on which evolution is essentially neutral\nafter the sequence has optimized its replicative strategy )\ngives rise to a power law of genotype sizes, as long as the\nprograms do not harbor an excessive amount of “junk”\ninstructions2. A typical abundance distribution (for the\n1As the replication rate ǫ is exponential in the bonus ob-\ntained for a successful computation, ǫbest increases exponen-\ntially with time.\n2 “Junk” instructions do not code for any information,\nand do not affect the fitness of their bearer. Consequently,\nprograms with excessive amounts of junk code will give rise\nto many “degenerate” genotypes with no competitive advan-\ntage. In this regime, the genotype abundance distribution\nis exponential rather than of the power-law type, due to a\nrun depicted in Fig. 2) is shown in Fig. 3. As mentioned\nFigure 3: Distribution of genotypes sizes P (s) fitted to\na power law (solid line) at mutation rate R = 0.004.\nearlier, we can also turn off all bonuses listed in Tab. 1, in\nwhich case fitness is related to replicative abilities only.\nStill, avalanches occur (within the first 50,000 updates\nmonitored) due to minute improvements in fitness, but\nthe length of the genomes typically stays in the range of\nthe ancestor, a program of length 31 instructions. We\nexpect a change of dynamics once the “true” maximum\nof the local fitness landscape is reached, however, we did\nnot reach this regime in the experiments presented here.\nThe distribution of genotype sizes for the flat landscape\nis depicted in Fig. 4. Clearly then, even such landscapes\n(flat with respect to all other activities except replica-\ntion) are not neutral. Indeed, it is known that neutral\nviolation of condition (6).\nFigure 4: Distribution of genotypes sizes P (s) for a land-\nscape devoid of the bonuses listed in Tab. 1, at mutation\nrate R = 0.003.\n\nFigure 5: Fitted exponent of power law for 34 runs at\nmutation rates between R = 0 .0005 and R = 0 .01 copy\nerrors per instruction copied. The error bars reflect the\nstandard deviation across the sample of runs taken at\neach mutation rate. The solid line is to guide the eye\nonly.\nevolution, where the chance for a genotype to increase or\ndecrease in number is even, leads to a power law in the\nabundance distribution with exponent D = 1.5 (Adami,\nBrown, and Haggerty, 1995).\nIn order to test the dependence of the fitted expo-\nnent D(R) [Eq. (9)] on the mutation rate, we conduct a\nset of experiments at varying copy-mutation rates from\n0.5×10− 3 to 10 ×10− 3 and take data for 50,000 updates.\nAgain, a “best” genotype is not reached after this time,\nand we must assume that avalanches were still occurring\nat the end of these runs. Furthermore, in some runs we\nfind that a genotype comes to dominate the population\n(usually after most ‘genes’ have been discovered) which\ncarries an unusual amount of junk instructions. As men-\ntioned earlier, such species produce a distribution that\nis exponentially suppressed at large genotype sizes (data\nnot shown). To avoid contamination from such species,\nwe stop recording genotypes after a plateau of fitness\nwas reached, i.e., if the population had discovered most\nof the bonuses. Furthermore, in order to minimize finite\nsize effects on the determination of the critical exponent,\nwe excluded from this fit all genotype abundances larger\nthan 15, i.e., we only fitted the smallest abundances. In-\ndeed, at larger mutation rates the higher abundances are\ncontaminated by a pile-up effect due to the toroidal ge-\nometry, while at lower mutation rates a scale appears to\nenter which prevents scale-free behavior. We have not,\nas yet, been able to determine the origin of this scale.\nIn the results reported here, we show the dependence\nof the fitted exponent D as a function of the mutation\nrate R used in the run, which, however, is a good measure\nof the mutation probability P only at small R and if the\nsequence length is not excessive. As a consequence, data\npoints at large R, as well as runs where an excessive\nsequence length developed, carry a systematic error.\nFor the 34 runs that we obtained, the power D was\nmeasured for each run (for the low abundances), and an\naverage was calculated for all the runs at a particular\nmutation rate. This data is plotted in Fig. 5 and shows\na plateau in the fitted exponent only at intermediate\nmutation rates, with D = 2.0 ± 0.05. A fit of the middle\nabundances (10-100) produces a critical coefficient more\nor less independent of mutation rate, around D = 2 .0,\nbut with less accuracy (data not shown). At high R, we\nwitness a deviation from scale-free behavior (reflected in\nthe rising D for small abundances) which is most likely\ndue to pile-up, i.e., a finite toroidal lattice. This effect\nmay be avoided by using absorbing rather than periodic\nboundary conditions. We also see a violation of scaling\nat small R, which is due to the emergence of some other\nscale. While it is most likely a finite-size effect, the exact\norigin of this scale is as yet unclear. We comment on the\nsignificance of these results in Section V.\nStill, more control over the spread in exponents for\nfixed mutation rate would be desirable. This can obvi-\nously be achieved by plotting D versus P , rather than R,\nfor example, and by better keeping track of the coding\npercentage within a genotype, a variable that we know\nsignificantly affects the shape of the distribution. Such\nexperiments are planned for the near future.\nDistribution of Species Sizes\nIn Avida, it is possible to monitor groups of programs\nthat display the same “phenotype”, while differing in\ngenotype. Even though programs in this world are hap-\nloid (single-stranded) and do not reproduce sexually, it\nis convenient to label such groups taxonomically, i.e., we\nrefer to them as “species”. Strictly speaking, a species\nconsists out of all those genotypes that, when executed,\ngive rise to the same “chemistry”, i.e., such programs\ndiffer only in instructions that are either unexecuted,\nor else are neutral. Algorithmically, the determination\nwhether two genotypes belong to the same species is\ncomplicated by the fact that sequence length is not con-\nstant in these experiments. Thus, we need to be able to\ncompare strings with differing lengths, which is achieved\nby lining them up in such a manner that they are identi-\ncal in the maximum number of corresponding sites. Sub-\nsequently, a cross-over point is chosen randomly and the\ngenomes above and below this point are swapped. In\nother words, we construct a hybrid program from the two\ncandidates and test it for functionality, but without in-\ntroducing it in the population (see Adami 1998.) In the\nexperiments reported here, we actually test two cross-\nover points in order to rule out accidental matches. In\nretrospect, we find that almost all those strings classified\nas belonging to the same species by this method differ\nonly in “silent”, or at least inconsequential, instructions.\n\nFigure 6: Distribution of genotypes within species at\nR = 0.004, fitted to a power law with D = 2.44 ± 0.05.\nThe abundance distribution of genotypes within\nspecies more closely corresponds to the kind of geomet-\nric distributions investigated by Willis (1922) as well as\nBurlando (1990, 1993). Indeed, Burlando found, in an\nanalysis of distributions of subtaxa within taxa obtained\nfrom the fossil record as well as recorded flora and fauna,\nthat these distributions appear to be scale-free across\ntaxonomic hierarchies, with critical coefficients between\n2.0 < D < 2.5. This distribution can also be viewed as\na distribution of avalanches sizes, if avalanches are re-\ndefined as events that spawn different genotypes of the\nsame species. Indeed, in this manner it is possible to\ninvestigate hierarchies of avalanches, each higher level\npresumably sporting a higher critical coefficient.\nIn the experiments reported here, we found species\ncoefficients closer to D ≈ 2.5, but we also found viola-\ntions of power-law behavior which are most likely due\nto the contribution of species of different lengths to the\nabundance distribution. Indeed, the amount of “junk”\ninstructions in a species most likely governs the steep-\nness of the distribution, and several different such species\nmay give rise to a multifractal distribution rather than a\npure power law. In the future, we expect to disentangle\nsuch distributions by appealing to a an even higher level\nin taxonomy, reuniting all species of the same sequence\nlength within a genus. The latter taxonomic level could,\nfor example, be entirely phenotypic, by keeping track of\nwhich tasks a genus executes (irrespective of its geno-\ntype).\nStill, even though changing sequence lengths affect\nthe distribution of genotypes within species, those ex-\nperiments in which the sequence length does not change\nsignificantly can give rise to power laws with single ex-\nponents, as shown below in Fig. 6. The data for this\nexperiment were obtained from the same run as gave\nrise to Figs. 2 and 3.\nConclusions\nThe distribution of avalanche sizes in evolving systems,\nwhich is quite clearly related to the distribution of ex-\ntinction events, can reveal a fair amount of information\nabout the dynamics of the adapting agents. For exam-\nple, purely random systems in which there are no fit-\nness advantages, and where selection does not occur,\ncan still show power law behavior, as extinction events\nare governed by the return-to-zero probability of random\nwalks (Adami, Brown and Haggerty 1995). In Avida,\nwe observe a scaling exponent D = 2 .0 in an interme-\ndiate regime of mutation rates. While it is still unclear\nwhether the mixing of scales that we have observed at\nsmall and large mutation rates is due to the finite size\nof the lattice or the emergence of another scale, we can\nconclude with confidence that scale-free dynamics does\noccur. Scaling violations should be investigated by a\nthorough finite lattice-size analysis, and this is planned\nfor the future along with more refined methods for deal-\ning with explicit neutrality (i.e., “junk” code.)\nAn interesting hint at what the distribution might be\nlike in Nature comes from Raup’s analysis of a data set\nprepared by Sepkoski (Raup 1991): genera of marine in-\nvertebrates from the fossil record. Raup’s “kill-curve”\ncan be transformed into a distribution of sizes of extinc-\ntion evens (as shown by Newman 1996) governed by a\ncritical exponent close to D = 2 .0. This is tantalizingly\nclose to the coefficient we found in our genotype abun-\ndance distribution, but we must be careful in comparing\nthese distributions.\nThe avalanche-size distribution of genotypes gives us\na good indication of the strength of an evolutionary\nshock, but also about the length of time the particu-\nlar species dominates the dynamics, and therefore, of\nthe time between evolutionary transitions. Also, each\nevolutionary transition brings with it a wave of extinc-\ntion, as all previously extant genotypes and species of\nlower fitness must disappear on the heels of the new\n“discovery”. The size of extinction events proper, how-\never, is not measured by the “epoch-length” distribution\nreflected in the avalanche sizes, but rather by the abun-\ndance of genotypes within species (or any higher taxo-\nnomic abundance distribution) because each species ap-\npearing in this distribution must eventually go extinct,\nand thus this distribution must equal the distribution\nof extinction sizes. The latter distribution (measured in\nSection IV), appears to have a critical exponent around\nD ≈ 2.5, higher than the corresponding one from the\nfossil record. Furthermore, we must keep in mind the\nsimplicity of the model treated here when comparing to\nactual fossil data. As mentioned in the introduction, co-\nevolution does not play a role in the dynamics control-\nling the size of avalanches in this model, while we must\nassume that extinctions in Earth history have some co-\nevolutionary component. On the other hand, the abun-\n\ndance distribution of genotypes within species is consis-\ntent with those obtained by Burlando (1990, 1993), who\nargued that they represented evidence for a “fractal ge-\nometry of Nature”.\nFrom the present analysis, it is clear that there is\nas yet no reason to jump to conclusions from the evi-\ndence extracted either from the fossil record, theoreti-\ncal models of extinctions (Newman 1997), or else direct\nimplementation of the dynamics of adaptive avalanches\nas we have done here. We do, however, see clear evi-\ndence that avalanches not reigned in by any scale can\nand do develop in evolving and adapting systems with-\nout co-evolutionary pressures, via first-order transitions\nin populations occupying single ecological niches. Not\nonly do we find scale-free dynamics for the time between\ntransitions (as evidenced by the genotype abundance dis-\ntribution) but also for the strength of these transitions,\nmeasured by the distribution of species-sizes. It is left\nfor future experiments to determine how such dynamics,\ntaking place in interacting ecological niches, gives rise to\npower laws for co-evolutionary systems, and how the de-\nscription in terms of first-order transitions is ipso facto\ntransmutated into a second-order scenario.\nThis work was supported by NSF grant No. PHY-\n9723972.\nReferences\nAdami, C. 1995. Self-organized criticality in living sys-\ntems. Phys. Lett. A 203: 23.\nAdami, C. 1998. Introduction to Artificial Life . Santa\nClara: TELOS Springer-Verlag.\nAdami, C. and C. T. Brown. 1994. Evolutionary learning\nin the 2D Artificial Life system ‘Avida’. In Artificial\nLife IV , edited by R.A. Brooks and P. Maes. Cam-\nbridge, MA: MIT Press, p. 377.\nAdami, C., C. T. Brown and M. R. Haggerty. 1995.\nAbundance distributions in Artificial Life and stochas-\ntic models: ‘Age and Area’ revisited. Lect. Notes in\nArtif. Intell. 929: 503.\nBak, P. and M. Paczuski. 1996. In Physics of Biological\nSystems. Heidelberg: Springer-Verlag.\nBak, P. and K. Sneppen. 1993. Punctuated equilibrium\nand criticality in a simple model of evolution. Phys.\nRev. Lett. 71: 4083.\n[Bak, Tang, and Wiesenfeld, 1987] Bak, B., C. Tang, and\nK. Wiesenfeld. 1987. Self-organized criticality: An ex-\nplanation of 1/ f noise. Phys. Rev. Lett. 59: 381.\n[Bak, Tang, and Wiesenfeld, 1988] Bak, B., C. Tang, and\nK. Wiesenfeld. 1988. Self-organized criticality. Phys.\nRev. A 38: 364.\nBurlando, B. 1990. The fractal dimension of taxonomic\nsystems. J. Theor. Biol. 146: 99.\nBurlando, B. 1993. The fractal geometry of evolution. J.\nTheor. Biol. 163: 161.\nChu, J. and C. Adami. 1997. Propagation of informa-\ntion in populations of self-replicating code. In Proc.\nof Artificial Life V , edited by C. G. Langton and K.\nShimohara. Cambridge, MA: MIT Press, p. 462.\nElena, S. F., V. S. Cooper and R.E. Lenski. 1996. Punc-\ntuated evolution caused by selection of rare beneficial\nmutations. Science 272: 1802.\nGil, L. and D. Sornette. 1996. Landau-Ginzburg theory\nof self-organized criticality. Phys. Rev. Lett. 76: 3991.\nGould, S. J. and N. Eldredge. 1977. Punctuated equilib-\nria: The tempo and mode of evolution reconsidered.\nPaleobiology 3: 115.\nGould, S. J. and N. Eldredge. 1993. Punctuated equilib-\nrium comes of age. Nature 366: 223.\nLenski, R. and M. Travisano. 1994. Dynamics of adap-\ntation and diversification–A 10,000 generation experi-\nment with bacterial populations. Proc. Nat. Acad. Sci.\n91: 6808-6814.\nNewman, M. E. J. 1996. Self-organized criticality, evolu-\ntion, and the fossil extinction record. Proc. Roy. Soc.\nB 263: 1605–1610.\nNewman, M. E. J. 1997. A model of mass extinction.\nEprint adap-org/9702003.\nNewman, M. E. J., S. M. Fraser, K. Sneppen, and\nW.A. Tozier. 1997. Self-organized criticality in living\nsystems—Comment. Phys. Lett. A 228: 202.\nOfria, C., C. T. Brown and C. Adami. 1998. The Avida\nUser’s Manual. In Adami (1998). The Avida software\nis publicly available at ftp.krl.caltech.edu/pub/avida.\nRaup, D. M. 1991. A kill curve for phanerozoic marine\nspecies. Paleobiology 17: 37–48..\nSornette, D., A. Johansen, and I. Dornic. 1995. Mapping\nself-organized criticality to criticality. J. de Phys. I\n5: 325.\nSol´ e, R. V. and J. Bascompte. 1996. Are critical phe-\nnomena relevant to large-scale evolution? Proc. Roy.\nSoc. B 263: 161–168.\nWillis, J. C. 1922. Age and Area. Cambridge: Cambridge\nUniversity Press."}
{"id": "adap-org/9804003", "meta": {"categories": ["adap-org", "cond-mat", "nlin.AO"], "created": "1998-04-30", "extraction": {"body_chars": 39150, "cleaning": {"detected_repeated_margin_lines": ["COLLECTIVE BEHA VIOUR AND DIVERSITY 3"], "page_count": 13, "removed_boilerplate_lines": 7}, "method": "pypdf_no_ocr", "source_pdf_bytes": 133669, "text_chars": 40484}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9804003", "primary_category": "adap-org", "source": "arxiv", "title": "Collective Behaviour and Diversity in Economic Communities: Some Insights from an Evolutionary Game", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9804003"}, "text": "Collective Behaviour and Diversity in Economic Communities: Some Insights from an Evolutionary Game\n\nAbstract\nMany complex adaptive systems contain a large diversity of specialized components. The specialization at the level of the microscopic degrees of freedom, and diversity at the level of the system as a whole are phenomena that appear during the course of evolution of the system. We present a mathematical model to describe these evolutionary phenomena in economic communities. The model is a generalization of the replicator equation. The economic motivation for the model and its relationship with some other game theoretic models applied to ecology and sociobiology is discussed. Some results about the attractors of this dynamical system are described. We argue that while the microscopic variables -- the agents comprising the community -- act locally and independently, time evolution produces a collective behaviour in the system characterized by individual specialization of the agents as well as global diversity in the community. This occurs for generic values of the parameters and initial conditions provided the community is sufficiently large, and can be viewed as a kind of self-organization in the system. The context dependence of acceptable innovations in the community appears naturally in this framework.\n\narXiv:adap-org/9804003v1 30 Apr 1998\nCOLLECTIVE BEHA VIOUR AND DIVERSITY IN ECONOMIC COMMUNITIES:\nSOME INSIGHTS FROM AN EVOLUTIONAR Y GAME\nVIVEK S. BORKAR\nDepartment of Computer Science and Automation\nIndian Institute of Science\nBangalore 560 012, India\nSANJAY JAIN\nCentre for Theoretical Studies\nIndian Institute of Science\nBangalore 560 012, India\nAND\nGOVINDAN RANGARAJAN\nDepartment of Mathematics and Centre for Theoretical Studies\nIndian Institute of Science\nBangalore 560 012, India\nAbstract. Many complex adaptive systems contain a large diversity of s pecialized\ncomponents. The specialization at the level of the microsco pic degrees of freedom,\nand diversity at the level of the system as a whole are phenome na that appear\nduring the course of evolution of the system. We present a mat hematical model to\ndescribe these evolutionary phenomena in economic communi ties. The model is a\ngeneralization of the replicator equation. The economic mo tivation for the model\nand its relationship with some other game theoretic models a pplied to ecology and\nsociobiology is discussed. Some results about the attracto rs of this dynamical system\nare described. We argue that while the microscopic variable s – the agents comprising\nthe community – act locally and independently, time evoluti on produces a collective\nbehaviour in the system characterized by individual specia lization of the agents\nas well as global diversity in the community. This occurs for generic values of the\nparameters and initial conditions provided the community i s sufficiently large, and\ncan be viewed as a kind of self-organization in the system. Th e context dependence\nof acceptable innovations in the community appears natural ly in this framework.\nPreprint No. IISc-CTS-4/98\nTo appear in the proceedings of the Workshop on Econophysics held at Budapest,\nHungary, July 21-27, 1997.\n\n2 BORKAR, JAIN AND RANGARAJAN\n1. Introduction\nSeveral complex adaptive systems in the course of their evol ution exhibit the phe-\nnomenon that the individual components comprising the syst em evolve to perform\nhighly specialized tasks whereas the system as a whole evolv es towards greater diver-\nsity in terms of the kinds of components it contains or the tas ks that are performed\nin it. Here are some examples:\n1. Living systems are made of cells, which in turn are made of m olecules. Among\nthe various types of molecules are the proteins. Each type of protein molecule\nhas evolved to perform a very specific task, e.g., catalyse a s pecific reaction in the\ncell. At the same time, during the course of evolution, diver se kinds of protein\nmolecules have appeared – the range of specialized tasks bei ng performed by\nprotein molecules has increased.\n2. In an ecology, species with highly specialized traits app ear (e.g., butterflies with\na specific pattern of spots on their wings). Simultaneously, the ecology evolves\nto support a diverse variety of specialized species.\n3. Many early human societies (such as hunter-gatherer soci eties) were perhaps\ncharacterized by the fact that there were relatively few cho res (e.g., hunting,\ngathering, defending, raising shelter) to be performed, an d everyone in the com-\nmunity performed almost all the chores. These societies evo lved to have special-\nist hunters, tool makers, farmers, carpenters, etc. Indivi duals specialized, and\nsimultaneously a diverse set of specialists appeared.\n4. In an economic web, we find firms exploring and occupying inc reasingly spe-\ncialized niches, while the web as a whole supports an increas ingly diverse set of\nspecialists.\nIn the examples above the systems and their underlying dynam ics are quite differ-\nent. But they all share the twin evolutionary phenomena of in dividual specialization\nand global diversification. In all these systems, the nonlin ear interaction among the\ncomponents seems to play a crucial role in the manifestation of this type of be-\nhaviour. For example, in an ecology, the development of high ly specialized traits in\na species is a result of its interaction with and feedback fro m other species. In an\neconomic community, each agent’s choices depend upon feedb ack from exchanges\n(of goods, money, etc.) with other agents. Moreover, there i s no purposeful global\norganizing agency which directs the behaviour of individua l components and ordains\nthem to be specialists. The phenomenon happens ‘spontaneou sly’, arising from the\nlocal moves made at the level of individual components. Simi larly diversity also arises\nas individuals capitalize on innovations – mutations, tech nological innovations, etc.\n– which suit them in the existing context.\nIn this article, we describe a mathematical model which seem s to exhibit the\nabove twin evolutionary phenomena. The (idealized) behavi our of agents in eco-\nnomic communities provides the basic motivation of the mode l. The model consists\nof a set of coupled nonlinear ordinary differential equations describing the time evo-\nlution of the activities of individual agents. In the next se ction we motivate and\npresent the model and place it in the perspective of existing work. In section 3 we\ndefine more precisely the notions of specialization and dive rsity in the context of\nthe model and outline what type of behaviour we are looking fo r. Essentially we\n\nare seeking attractors of the dynamical model that have the p roperty of individual\nspecialization and global diversity. Section 4 states cert ain theorems and numerical\nresults for the attractors of the system and discusses their consequences. The results\nimply that under certain conditions that do not destroy gene ricity in parameter\nspace, the desired attractors (in which the system exhibits individual specialization\nand global diversity) exist and have basins of attraction th at cover the entire config-\nuration space. Thus the evolutionary phenomena mentioned a bove occur generically\nin the model. In this section we also discuss self-organizat ion and the emergence of\ninnovations in the model. Finally, section 5 contains a brie f summary.\n2. The model\nThe system is a community of N agents labeled by the index α = 1, 2, . . . , N . Each\nagent can perform s strategies or activities labelled by i ∈ S = {1, 2, . . . , s }. At time\nt, agent α performs strategy i with a probability pα\ni (t), ∑ s\ni=1 pα\ni (t) = 1. The vector\npα (t) = ( pα\n1 (t), p α\n2 (t), . . . , p α\ns (t)) is the mixed strategy profile of agent α at time t. In\nparticular, if pα\ni (t) = δij, then the agent α is said to pursue the pure strategy j or\nto have specialized in strategy j.\nThe vectors pα (t) constitute the basic dynamical variables of the model. The\nequation governing their evolution is taken to be\n˙pα\ni (t) = pα\ni (t)[\n∑\nβ ̸=α\ns∑\nj=1\naijpβ\nj (t) −\n∑\nβ ̸=α\ns∑\ni,j =1\npα\ni (t)aijpβ\nj (t)]. (1)\nHere aij denotes the ijth element of the s-dimensional payoff matrix A.\nThis dynamics is motivated as follows. Each agent is interac ting pairwise with\nall other agents and receiving a payoff at every interaction t hat depends on the\nstrategy pursued by each of the agents during that interacti on. Each agent updates\nher strategy profile based on the payoffs received, so as to incr ease her payoff at\nsubsequent interactions. In a time ∆ t, agent α has a total of m∆ t interactions with\nevery agent ( m assumed constant). If in a particular interaction with agen t β , agent\nα plays pure strategy k and β plays pure strategy j, then the payoff to α is akj\n(by the definition of payoff matrix elements). Since α plays the strategy k with\nprobability pα\nk and β plays the strategy j with probability pβ\nj , the average payoff to\nα from the m∆ t interactions with β is\nm∆ t\n∑\nk,j\npα\nk (t)akj pβ\nj (t).\nThe average payoff to α from the whole community is\nm∆ t\n∑\nβ ̸=α\n∑\nk,j\npα\nk (t)akj pβ\nj (t).\nThis is the second term in the [ ] in Eq. (1). We have assumed tha t ∆ t is large\nenough for there to be a statistically sufficient number of int eractions m∆ t so that\naverages make sense. Yet it is small enough compared to the ti me scale at which\nagents update their strategies so that pα\nk can be considered constant during ∆ t, i.e.,\n\n4 BORKAR, JAIN AND RANGARAJAN\nthere is a separation of time scales between the individual i nteractions of agents\n(which happen on a short time scale) and the time scale over wh ich agents update\ntheir strategy profile (a long time scale).\nIf agent α were to pursue not the mixed strategy profile pα during this interval\nbut instead the pure strategy i, then the payoff received during this period would\nhave been\nm∆ t\n∑\nβ ̸=α\n∑\nj\naijpβ\nj (t).\nThis is the first term in the [ ] in Eq. (1). This quantity depend s on i and for some i\nwill be greater than the average payoff and for some it will be l ess than the average\npayoff. At the end of period ∆ t, the agent α updates her strategy profile pα\ni to\npα\ni + ∆ pα\ni , adding a positive weight ∆ pα\ni to those strategies i that do better than\nthe average and a negative weight to those doing worse than th e average. ∆ pα\ni /p α\ni\nis chosen to be proportional to the amount by which the pure st rategy payoff differs\nfrom the average payoff:\n∆ pα\ni\npα\ni\n= cm∆ t[\n∑\nβ ̸=α\n∑\nj\naijpβ\nj (t) −\n∑\nβ ̸=α\n∑\ni,j\npα\ni (t)aijpβ\nj (t)]. (2)\nTaking the limit ∆ t → 0 and rescaling t by the factor cm, we recover Eq. (1).\nTherefore the equation embodies the statement that at all ti mes, all agents update\ntheir individual mixed strategy profiles so as to increase th eir own payoffs in the\ncurrent environment of the strategy profiles of other agents .\nThe reason why ∆ pα\ni /p α\ni and not just ∆ pα\ni appears in the l.h.s. of (2) is that the\ndynamics must respect the probability interpretation of pα\ni . If two pure strategies i\nand i′ provide the same payoff to agent α , she must increment them in proportion\nto their current strength in her profile. This is needed to ens ure that pα (t) remains\nnormalized at all times, ∑ s\ni=1 pα (t) = 1. If we start with normalized pα , the pro-\nportionality factor pα\ni on the r.h.s of (1) ensures that it remains normalized, since∑ s\ni=1 ˙pα\ni (t) = 0.\nThus, we have a community of N interacting agents, each responding to the rest\nof the environment by updating their own profile according to the above dynamical\nequation. This is a “non-cooperative game”. Agents act on th eir own (not in concert,\nper se ) and are selfish – their actions are designed to increase thei r own payoff,\nwithout consideration for others or the community as a whole . Agents also exhibit\n“bounded rationality” – they do not anticipate other agents ’ future strategies, but\nmerely respond to the aggregate of the other agents’ current strategies. There is\nno global organizing agency at work, the community evolves j ust through these\nindividual actions of the agents.\nNevertheless, we will argue that the community does exhibit a kind of global\norganization under certain circumstances. If the communit y starts with some arbi-\ntrary initial condition in which each pα at t = 0 is specified (each agent starts with\nsome mixed strategy profile which could be different for differen t agents) and evolves\naccording to Eq. (1), it will settle down to some attractor of the dynamics. The orga-\nnization referred to above is in the nature of the attractors . When the payoff matrix\nelements satisfy certain inequalities, and when the size of the community is larger\nthan a certain finite bound that depends on the payoff matrix (i .e., N is sufficiently\n\nlarge), then we find that these attractors are characterized by each individual agent\nhaving specialized to some pure strategy or the other, and at the same time the com-\nmunity as a whole retaining its full diversity of strategies , i.e., every pure strategy is\npursued by some agent or the other in the attractor configurat ion. Such attractors\nseem, generically, to be the only stable attractors of the sy stem under the above\nconditions. Most of the time, we will consider the system wit h a fixed set of pure\nstrategies. At the end, we will mention applications of our r esults for the innovation\nof new strategies. The instability of attractors in which th e community does not\nhave the full diversity of available strategies provides a m echanism by which new\npure strategies, or innovations, can invade the system.\nBefore proceeding further, we would like to place this model in the perspective of\nexisting work in the subject. Consider the “homogeneous sec tor”, where all agents\nhave the same (but in general mixed) strategy profile: pα = x ∀ α . Then Eq. (1)\nreduces to\n˙xi(t) = xi(t)[\n∑\nj\naijxj(t) −\n∑\nk,j\nxk(t)akj xj(t)](N − 1). (3)\nThe overall factor of N − 1 can be absorbed in a rescaling of time. This equa-\ntion is the well known replicator equation [1]. It has applic ations in diverse fields\nsuch as economics and sociobiology (where it models evoluti onary games) macro-\nmolecular evolution (describing evolution of autocatalyt ic networks, in particular the\nhypercyclic feedback), mathematical ecology (Eq. (3) maps onto the Lotka-Volterra\nequation) and population genetics (where it is the continuo us counterpart of the\ndiscrete selection equation). This system exhibits a great diversity of solutions in-\ncluding fixed points, limit cycles, heteroclinic cycles, et c. For more details, see [2]. In\nthese applications i labels strategies or species of molecules or organisms, dep ending\nupon the application. xi represents the fraction of individuals of type i in a large\npopulation, and Eq. (3) models the change of the composition of the population with\ntime.\nEq. (1) has been considered as a multi-population generaliz ation of the replicator\ndynamics and has been studied as such in the literature [2] [3 ]. The index α now labels\npopulations, e.g., α = 1 might correspond to a population of frogs, and α = 2 to a\npopulation of insects. The idea here is to model the co-evolu tion of the populations of\nfrogs and insects in interaction with each other. p1\ni now stands for the fraction of the\nfrog population with genotype i and p2\nj for the fraction of the insect population with\ngenotype j. The indices i and j run over values s1 and s2 respectively which need\nnot be equal, and now there are two payoff matrices, one for the frogs, A1 = ( a1\nij),\nwhose matrix element a1\nij equals the payoff to a frog of type i in an encounter with\nan insect of type j, and another for the insects, A2 = ( a2\nji) whose matrix element\na2\nji equals the payoff to the insect in the same encounter. The dyna mics for the two\npopulations is now given by\n˙p1\ni (t) = p1\ni (t)[\ns2∑\nj=1\na1\nijp2\nj (t) −\ns1∑\nk=1\ns2∑\nj=1\np1\nk(t)akj p2\nj (t)],\n˙p2\nj (t) = p2\nj (t)[\ns1∑\ni=1\na2\njip1\ni (t) −\ns2∑\nk=1\ns1∑\ni=1\np2\nk(t)akip1\ni (t)]. (4)\n\n6 BORKAR, JAIN AND RANGARAJAN\nThis is a so called ‘bimatrix game’ and reduces to Eq. (1) with N = 2, when\ns1 = s2 = s and A1 = A2 = A.\nOur interpretation of Eq. (1), presented earlier, is differen t from the “multi-\npopulation” interpretation. The index α labels individuals and not populations. In\nthe multipopulation interpretation, individuals in each p opulation are hard-wired to\nbe of some specific genotype, while the composition of the pop ulation is plastic and\nsubject to selection. For us, the composition of the mixed st rategy profile of each\nindividual is subject to selection. In the multipopulation interpretation there is no\nreason for A1 and A2 to be equal; frogs and insects are quite different. However a\nsingle payoff matrix A is natural in the present context if the community consists\nof N identical agents (identical in that the payoffs to agents in any interact ion de-\npends on the strategies played in that interaction and not on the identity of the\nagents). This allows us to study large communities (large N ) without the simulta-\nneous proliferation of parameters. To our knowledge, the in terpretation of Eq. (1)\nas modelling not N populations but a single community of N identical individuals\nis new. While we make use of existing mathematical results fo r Eq. (1), the new\ninterpretation prompts us to investigate certain other mat hematical properties of\nthe model which have not received attention. Since Eq. (1) is a generalization of\nthe replicator dynamics, we will refer to it as the generaliz ed replicator dynamics\n(GRD) whereas Eq. (3) will be referred to as pure replicator d ynamics (PRD).\nNote that in (4) frogs receive payoffs only from insects, not fr om other frogs,\nand insects only from frogs, not from other insects. This is b ecause the competition\namong the different genotypes of frogs happens not directly, b ut indirectly via their\ninteractions and competition with insects: the more succes sful genotypes among frogs\nmight be the ones (depending upon the payoff matrix) which do b etter at capturing\ninsects. Similarly insects do not compete with each other di rectly but only with frogs;\nthe insect population profile evolves because some insect ge notypes do better than\nothers at, say, evading frogs. A similar justification might be provided for agents\nin the present context. A single isolated agent has no compet ition and hence no\nmotivation to change her strategy profile. There is no direct competition among\nthe weights of different pure strategies within the strategy p rofile of a single agent;\nthis competition and consequent evolution arises indirect ly because of the external\npressure on the agent from the other agents. A firm that produc es a number of\ngoods in the economy need not change its production profile if there are no other\nproducers. But if other producers enter the fray, the firm may need to change (say,\nspecialize in the production of a few items), in order to comp ete effectively. This\nfeature is captured in the model by the exclusion of the β = α term on the r.h.s. of\n(1) – agents don’t compete with themselves but with other age nts. We will see later\nthat this property is important for the emergence of special ization in the model.\nA well known example is the “Hawk-dove game” [4], in which the re are two\npure strategies, “hawk” ( i = 1) and “dove” ( i = 2). The payoff matrix elements are\na11 = (g − c)/ 2, a12 = g, a21 = 0, and a22 = g/ 2, with (typically) c > g > 0. In this\ngame individuals interact pairwise and every interaction i s a competition for some\nresource. In an interaction, a hawk always escalates and figh ts, irrespective of what\nthe opponent does. A dove “displays”, but retreats if the opp onent escalates. Thus\nwhen hawk meets dove, the dove always retreats and gets zero p ayoff, while the hawk\n\ngains a payoff g from the resource. When dove meets dove, both have equal chan ce of\ngetting the resource or retreating, hence the average payoff to each party in such an\nencounter is g/ 2. When hawk meets hawk, there is a fight, and with equal probab ility\none wins without injury and gains g, while the other retreats with an injury resulting\nin a cost c. The average payoff in hawk-hawk encounters to each party is t herefore\n(g − c)/ 2. It is instructive to contrast the treatment of this game in PRD and GRD.\nIn the former, there is a large population of individuals, ea ch hardwired to be pure\nhawk or pure dove in every encounter. The fraction of the popu lation that is hawk,\nx1, and the fraction that is dove, x2 = 1 − x1, evolves according to (3) in response\nto selection pressure and birth/death processes. The point (x1, x 2) = ( g, c − g)/c is\na stable equilibrium of (3), and generically, the populatio n ends up in this attractor,\ni.e., with a ratio of hawks to doves being g/ (g−c). In GRD, one would have N agents,\neach allowed to play both hawk and dove strategy in an encount er with the respective\nprobabilities pα\n1 and pα\n2 . It is not obvious that each agent will end up specializing in\na pure hawk or pure dove strategy, but that is what does happen . A consequence\nof one of the theorems to be described later is that the only st able attractor of this\nGRD is a configuration where agents tend to distribute themse lves in a pure hawk\nor pure dove strategy roughly in the ratio g/ (c − g) for finite N , and exactly in\nthis ratio as N → ∞. Thus individual specialization, which was true by assumpt ion\nin PRD, is a dynamical outcome in GRD. Moreover, while indivi duals specialize in\ntheir self interest to some pure strategy or the other depend ing upon their initial\nconditions, collectively the community seems to obey some g lobal constraints.\n3. Definitions and Notation\nConsider\nJ = {x = (x1, x 2, . . . , x s) ∈ Rs|\ns∑\ni=1\nxi = 1, x i ≥ 0},\nwhich is the simplex of s-dimensional probability vectors. J is the full configuration\nspace of PRD dynamics and is invariant under it. The configura tion space for GRD\nis J N , the N -fold product. A generic point of J N is p = ( p1, p2, . . . , pN ), each pα\nbeing an s-dimensional probability vector pα (t) = ( pα\n1 (t), p α\n2 (t), . . . , p α\ns (t)) belonging\nto J (α ) (the latter being a copy of the simplex J corresponding to agent α ).\nA point x ∈ J such that xi = δij for some j is called the jth corner of J. If\nan agent α has specialized to the pure strategy j, then pα\ni = δij, i.e., pα has gone\nto the jth corner of J (α ). If every agent has specialized to some strategy or the\nother, the corresponding point in J N will be called a corner of J N and we say that\nthe community is fully specialized. Note that every corner of J N is an equilibrium\npoint of GRD, since the r.h.s. of (1) vanishes. Hence we refer to corners as corner\nequilibrium points (CEPs).\nA CEP can be characterized by an s-vector of non-negative integers n = (n1, n 2, . . . , n s)\nwhere ni denotes the number of agents pursuing the pure strategy i at CEP. There\ncan be many CEPs with the same n vector. These would differ only in the identity\nof the agents at various corners. In this article we will igno re the differences between\nsuch corners and characterize a CEP by its n-vector alone, since the agents are\nidentical and differ only in their strategy profile.\n\n8 BORKAR, JAIN AND RANGARAJAN\nConsider the following subset of J N : Fk ≡ { p ∈ J N |pα\nk = 0 ∀ α } for some fixed\nk ∈ S. By definition, at a point in Fk, every agent has opted out of strategy k. Fk\nis also invariant under (1), i.e., if pα\nk is zero at some time, it remains zero. At the\n“face” Fk, strategy k therefore becomes extinct from the population, and we say\nthat the full diversity of strategies is lost. As long as the s ystem is not in some Fk,\nwe say that the community exhibits the “full diversity” of strategies. Note that the\nword ‘diversity’, as used here, does not stand for variabili ty among agents, but to\nindicate that all strategies are supported. For example we c an have no variation but\nfull diversity at points p where pα = c ∀ α and none of the components of c are\nzero. This is a “homogeneous” point, since all agents are doi ng the same thing.\nIf a CEP is such that ni ̸= 0 for all i, i.e., each strategy is played by at least one\nagent at the CEP, we will refer to it as a fully diversified CEP or FDCEP. If one\nor more ni is zero, the full diversity of strategies is lost and such CEP s are called\nnon-FDCEPs.\nWe are interested in studying the circumstances under which FDCEPs are the\npreferred attractors of the dynamics, for then, individual specialization and global\ndiversity will arise dynamically in the community. If it hap pens that the FDCEPs\nare attractors and their basins of attraction cover most of J N (all of J N except a\nset of lower dimension), then for generic initial condition s the community is bound\nto end up in an FDCEP, which means that it will exhibit individ ual specialization\nand as well as global diversity.\n4. Results\nIn this section we discuss some results concerning attracto rs of GRD. The proofs of\nthe theorems are omitted here; these and further results can be found in [5, 6]. We\nwill discuss the significance of these results for specializ ation and diversity in GRD.\n4.1. INTERIOR EQUILIBRIUM POINTS\nAn equilibrium point of GRD is called an interior equilibriu m point (IEP) if none\nof the pα\ni is zero. We have the following theorem:\nTheorem 1: There is at most one isolated IEP. If there is one, it is homoge neous\nand is given by pα\ni = xi ∀ α, i where xi ≡ ui/ detB, ui is the cofactor of B0i, and B\nis the ( s + 1) × (s + 1) matrix (whose rows and columns are labelled by the indice s\n0, 1, 2, . . . , s )\nB ≡\n\n\n\n\n\n\n\n\n0 1 1 · · · 1\n−1\n−1 A\n.\n.\n.\n−1\n\n\n\n\n\n\n\n\n. (5)\nA necessary and sufficient condition for an isolated IEP to exi st is given by\nA1: ui ̸= 0 ∀ i, and all ui have the same sign.\nPRD also has an isolated IEP if condition A1 holds, which is then unique and\ngiven by the same xi as given above for GRD. Further, note that at the IEP in GRD,\nthe system exhibits full diversity since no strategy is opte d out of by any agent.\n\nHowever, there is no specialization. The above formula for t he IEP in particular\nyields the point ( x1, x 2) = ( g, c − g)/c for the hawk-dove game.\n4.2. SPECIALIZATION\nFor generic payoff matrices A, generic initial conditions, and sufficiently large N ,\nwe find that the system flows into a corner of J N . Thus specialization is a generic\noutcome of the dynamics. This observation is based on the fol lowing facts:\n1. Theorem 2: [2, 3] Any compact set in the interior of J N or the relative interior\nof any face cannot be asymptotically stable. An equilibrium point is asymptot-\nically stable if and only if it is a strict Nash equilibrium. ( A strict Nash equilib-\nrium is a point p such that at this point if any single agent unilaterally chan ges\nher strategy – unilaterally means that all other agents rema in where they are –\nthen her payoff strictly decreases.)\n2. Theorem 3: Every asymptotically stable attractor must contain at leas t one\ncorner equilibrium.\n3. Numerical W ork: The GRD equation for s = 3 was numerically integrated\nusing Runge-Kutta method of fourth order. We randomly gener ated ten 3 × 3\npayoff matrices and numerically integrated the GRD equation s for long times for\neach payoff matrix with ten randomly chosen initial conditio ns. When this was\ndone with N = 5, in 90 out of the 100 cases the dynamics converged to a corne r.\nThe remaining 10 cases (all corresponding to a single payoff m atrix) converged\nto a heteroclinic cycle. (In these 10 cases the system cycled between regions close\nto a few corners, moving rapidly between these regions, and a t every successive\ncycle spending increasing amounts of time near the corners a nd coming closer\nto them.) When N was increased to 10 for the same ten payoff matrices studied\nabove, all 100 cases converged to a corner. This suggests tha t the typically,\nthe stable attractors are corners or heteroclinic cycles, w ith corners becoming\noverwhelmingly more likely at larger N .\nWhy are corners the preferred attractors of this dynamics? W e give here an\nintuitive argument, which, though not rigorous or complete , provides some insight.\n(For rigorous arguments, refer to the proofs of the above men tioned theorems.) Recall\nthat according to the dynamics each agent updates her strate gy profile to increase\nher payoff in the current environment. Pick an agent α . Her payoff at any point is\nP α = ∑\nk pα\nk cα\nk where cα\nk ≡ ∑\nβ ̸=α akj pβ\nj . Given a set of s numbers cα\nk for a fixed\nα , generically one of them will be the largest. Let the largest one be cα\nl (for some\nparticular l). Then it is clear that since the payoff P α is linear in pα\nk , the choice\npα\nk = δkl will maximize it. Thus, as long as the index of the largest of t he cα\nk remains\nk = l, the agent α will move towards the pure strategy l. This argument can be\nmade for any agent. Thus every agent is, at any time, moving to wards some pure\nstrategy. In this argument it is crucial that cα\nk is independent of pα (which it is\nbecause of the exclusion of the β = α term in the payoff to α ). If it were not, then\nthe nonlinear dependence of P α on pα would have invalidated the argument (as is\nthe case in PRD, where the analogous quantity ∑\nk,j xkakj xj is quadratic in the xi,\nand corners are not the generic attractors).\n\n10 BORKAR, JAIN AND RANGARAJAN\nThis argument also sheds some light on why heteroclinic cycl es could be attrac-\ntors. The point is that cα\nk are not constants, but depend upon the strategy profiles\nof agents other than α . If the change in these profiles causes some other cα\nk (for some\nk = l′, different from l) to overtake cα\nl , then from that time onwards, agent α will\nhave to change track and move towards pure strategy l′ rather than l.\nIn particular the above theorems mean that the IEP is always u nstable in GRD.\n4.3. COLLECTIVE BEHA VIOUR\nWe have seen above that the GRD flows to corners generically. T he next step is to\ndetermine which corners the dynamics flows to. For the moment we restrict ours elves\nto FDCEPs. Characterizing an FDCEP by its n vector (described in the previous\nsection), we have the following result:\nTheorem 4: Let n and n′ be any two asymptotically stable FDCEPs with N ≥ s.\nIf condition A1 holds then all components of n − n′ are bounded by a function of\nthe payoff matrix A alone (and not of N ). Further,\nlim\nN →∞\nni\nN = xi.\nThus, out of a large number (of order N s−1) of FDCEPs all of which are equi-\nlibria for GRD, only a few are stable. Further, even though th e agents are acting\nindividually and selfishly and going to corners (specializi ng), the system as a whole\nretains a memory of the unique interior equilibrium point (w hich is guaranteed to\nexist under the conditions of the above theorem) and tunes th e ratios ni/N such\nthat they are close to the IEP xi values.\nOne can give a physical or “economic” interpretation of this collective behaviour.\nA stable equilibrium is by theorem 2 a strict Nash equilibriu m. Thus it cannot be\nadvantageous for any agent to switch her pure strategy unila terally. This means that\nall agents must receive more or less the same payoff. More preci sely, since a switch\nof strategy by a single agent causes changes of O(1) in the payoffs to other agents,\nat a strict Nash equilibrium it must be the case that difference s of payoffs among\nagents could not be larger than O(1), since otherwise it would be possible for some\nagent to make an advantageous switch without affecting others . Thus stability is\nachieved only at those equilibria at which differences in payo ff among agents are a\nvery small fraction of the total payoff to any agent, which is O(N ). This requirement\nof “near-equality” of payoffs narrows down the set of stable eq uilibria considerably.\nAs to why the ratio ni/N gets tuned to be close to the IEP, we remark that in both\nPRD and GRD, the IEP is characterized by exactly equal payoffs t o all strategies.\nThe above remarks also help explain some of our numerical res ults. When we\nincreased the number of agents from 5 to 10, we found that all c ases converged\nto corners. This is because as N increases, the ratios ni/N can reproduce the xi\nvalues corresponding to IEP more accurately and thereby ach ieve the near-equality\nof payoffs required for the existence of a strict Nash equilibr ium.\n4.4. DIVERSITY AND SELF-ORGANIZATION\nWe have seen above that among the FDCEPs, only a very small sub set can be\nasymptotically stable. Now we consider non-FDCEPs. It coul d happen that along\n\nwith an FDCEP, some non-FDCEPs are also stable. In that case, if the community\nstarts in the basin of attraction of a non-FDCEP, it would eve ntually lose its diver-\nsity. We would like to eliminate such attractors of the dynam ics. It turns out that\nthis can be achieved by imposing certain inequalities on the payoff matrix elements.\nConsider the following condition:\nA2: The payoff matrix is diagonally subdominant, i.e., aii < a ji∀j ̸= i.\nTheorem 5: For s = 2, if A2 holds, then all non-FDCEPs are unstable for\nN ≥ 2. For s = 3, if A1, A2 hold, then there exists a positive number N0 depending\non A, such that all non-FDCEPs are unstable for N > N 0.\nThe condition A2 means that each pure strategy gives more payoff to other\nstrategies than itself, clearly a tendency that would suppo rt diversity. It is interesting\nthat the model also has the desirable feature that larger com munities favour diversity.\nIt will be useful to have conditions for higher s also which make all the non-\nFDCEPs unstable. Partial results in this direction are cont ained in [6]. We expect\nthat for higher s the condition of sufficiently large N and further inequalities on the\npayoff matrix elements would ensure the instability of non-F DCEP.\nWe now discuss the behaviour of the system when such conditio ns hold. Notice\nthat since these conditions are inequalities on the payoff ma trix elements (and not\nequalities), the behaviour of the system is structurally st able or generic, i.e., is not\ndestroyed by a small perturbation of the parameters. From th e evidence presented\nin section 5.2, the system is expected to go to a corner with ge neric initial condi-\ntions. By theorem 5 (and its generalizations to higher s), this corner cannot be a\nnon-FDCEP. Hence it must be an FDCEP. But then theorem 4 appli es and tells us\nthat it must be a very specific corner. At this corner the numbe r of agents ni pursu-\ning the pure strategy i is fine tuned to a value close to N xi with xi determined by\nthe payoff matrix via Theorem 1, and all agents receive the sam e O(N ) payoff upto\ndifferences of O(1). The final state is fine-tuned but robust in that it arises w ithout\nfine-tuning the parameters or the initial state. In this sens e (of spontaneous dynam-\nical fine-tuning) the system exhibits self-organization, a lbeit without any obvious\ncritical behaviour. Furthermore, this self organization i s of the kind that we were\noriginally seeking, namely, in which the community exhibit s individual specialization\nand global diversity.\n4.5. INNOV ATIONS\nSo far we have considered strategy spaces of a fixed size s. However, the growth of\ndiversity in the systems mentioned in the introduction has t o do with the appearance\nof new strategies and disappearance of some old strategies. We now discuss how the\nabove considerations of the instability of non-FDCEPs are r elevant for the generation\nof innovations. As an example consider a community of N agents which has initially\nonly two strategies and a payoff matrix A satisfying condition A2. Then all non-\nFDCEPs are unstable and only those FDCEPs with n = (n1, n 2) such that the ratios\nni/N close to xi of the IEP are stable. Let us assume that the system has settle d\ninto such a state. Now, assume that a new, third strategy aris es which enlarges the\n2 × 2 matrix A into a 3 × 3 matrix A′ that contains A as a submatrix. The third\nrow and column of A′ represent the relationship of the new strategy with respect\nto the old – how much payoff it gives to them and receives from th em. At the time\n\n12 BORKAR, JAIN AND RANGARAJAN\nthis strategy arises, the system is in the state ( n1, n 2, 0) since the third strategy\n(being new) is as yet unpopulated. Note that this state is a no n-FDCEP for s = 3.\nNow if A′ is such that it satisfies the conditions A1, A2 in Theorem 5, and N is\nsufficiently large, then the above state is unstable. Therefo re, any small perturbation\nin which the agents start exploring the new strategy ever so s lightly will destroy the\nold state and take the system to a new stable state which must b e an FDCEP with\n3 strategies. By Theorem 4 this state will be ( n′\n1, n ′\n2, n ′\n3) with\nN (n′\n1, n ′\n2, n ′\n3) ≈ (x′\n1, x ′\n2, x ′\n3)\nwhere x′\ni are the components of the IEP corrsponding to the payoff matri x A′ and are\nall non-zero since A′ satisfies A1. Thus the innovation has destabilized the previous\nstate of the system and brought it to a new state where a finite f raction of the\npopulation has adopted strategy 3. In such a case, we say that the innovation has\nbeen “accepted” by the community. Note that the only require ments for this to\nhappen is that the elements of the new row and column in the pay off matrix satisfy\ncertain inequalities with respect to the existing matrix el ements (contained in the\nconditions A1, A2 to be satified by A′) and that the community be sufficiently\nlarge. This tells us what properties a new strategy should ha ve in the context of\nalready existing activities in order for it to be “accepted” by the community. Thus\nthe model suggests a natural mechanism for the emergence of c ontext dependent\ninnovations in the community.\n5. Conclusions\nTo summarize, Generalized Replicator Dynamics, eq. (1), is a nonlinear dynami-\ncal model of learning for a community of N mutually interacting agents with the\nfollowing features:\n1. Each agent is selfish and exhibits bounded rationality.\n2. This is a non-cooperative game and there is no global organ izing agency at\nwork. It is in general a non-hamiltonian system.\n3. Specialization of individual agents to pure strategies i s a generic outcome of the\ndynamics.\n4. Under certain generic conditions on the payoff matrix para meters the agents ex-\nhibit a collective behaviour, and for sufficiently large N , the community exhibits\ndiversity and self-organization.\n5. A ‘good’ innovation (one that satisfies conditions A1, A2 , etc., with respect to\nthe exisiting strategies) makes the society unstable and ev olve until the inno-\nvation is accepted.\nIt is noteworthy that in this dynamical system, order is gene rated at large N\n(unlike the systems where increasing the number of degrees o f freedom makes the\nsystem less orderly, in some sense). This order is not the usu al statistical mechanical\nkind of order, the order of appropriately defined macroscopi c variables, but an order\nin the original dynamical variables themselves. However, i n another sense, this order\nis also statistical since we do not know which pure strategy a n individual agent\nfollows. We only know about the fraction of agents pursuing a given strategy.\n\nAcknowledgement: SJ acknowledges support from the Jawaharlal Nehru Centre\nfor Advanced Scientific Research, Bangalore, and the Associ ateship of the Interna-\ntional Centre for Theoretical Physics, Trieste.\nEmail addresses :\nborkar@csa.iisc.ernet.in, jain@cts.iisc.ernet.in, ran garaj@math.iisc.ernet.in\nReferences\n1. Taylor, P. and Jonker, L. (1978) Evolutionarily stable st rategies and game dynamics, Math. Bio-\nsciences, 40, 145-156; Hofbauer, J., Schuster, P. and Sigmund, K. (1979) A note on evolutionarily\nstable strategies and game dynamics, J. Theor. Biol. , 81, 609-612; Gadgil, S., Nanjundiah, V.\nand Gadgil, M. (1980) On evolutionarily stable composition s of populations of interesting geno-\ntypes, J. Theor. Biol. , 84, 737-759; Schuster, P. and Sigmund, K. (1983) Replicator dy namics,\nJ. Theor. Biol. , 100, 533-538.\n2. Hofbauer, J. and Sigmund, K. (1988) The Theory of Evolution and Dynamical Systems . Cam-\nbridge University Press, Cambridge.\n3. Weibull, J. W. (1995) Evolutionary Game Theory . MIT Press, Cambridge.\n4. Maynard-Smith, J. (1972) Game theory and the evolution of fighting, in Maynard-Smith, J. On\nEvolution. Edinburgh University Press.\n5. Borkar, V. S., Jain, S. and Rangarajan, G. (1998) Dynamics of individual specialization and\nglobal diversification in communities, Complexity, Vol. 3, No. 3, 50-56.\n6. Borkar, V. S., Jain, S. and Rangarajan, G. (1997) Generali zed replicator dynamics as a model\nof specialization and diversity in societies, preprint num ber IISc-CTS-11/97."}
{"id": "adap-org/9804004", "meta": {"categories": ["adap-org", "cond-mat", "nlin.AO"], "created": "1998-04-30", "extraction": {"body_chars": 34984, "cleaning": {"detected_repeated_margin_lines": ["1"], "page_count": 13, "removed_boilerplate_lines": 13}, "method": "pypdf_no_ocr", "source_pdf_bytes": 154866, "text_chars": 35718}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9804004", "primary_category": "adap-org", "source": "arxiv", "title": "Dynamics of Individual Specialization and Global Diversification in Communities", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9804004"}, "text": "Dynamics of Individual Specialization and Global Diversification in Communities\n\nAbstract\nWe discuss a model of an economic community consisting of $N$ interacting agents. The state of each agent at any time is characterized, in general, by a mixed strategy profile drawn from a space of $s$ pure strategies. The community evolves as agents update their strategy profiles in response to payoffs received from other agents. The evolution equation is a generalization of the replicator equation. We argue that when $N$ is sufficiently large and the payoff matrix elements satisfy suitable inequalities, the community evolves to retain the full diversity of available strategies even as individual agents specialize to pure strategies.\n\narXiv:adap-org/9804004v1 30 Apr 1998\nDYNAMICS OF INDIVIDUAL SPECIALIZATION AND GLOBAL\nDIVERSIFICATION IN COMMUNITIES ∗\nVivek S. Borkar †\nDepartment of Computer Science and Automation,\nIndian Institute of Science, Bangalore 560 012, India\nSanjay Jain ‡\nCentre for Theoretical Studies,\nIndian Institute of Science, Bangalore 560 012, India\nGovindan Rangarajan §\nDepartment of Mathematics and Centre for Theoretical Studi es,\nIndian Institute of Science, Bangalore 560 012, India\nAbstract\nWe discuss a model of an economic community consisting of N interacting agents.\nThe state of each agent at any time is characterized, in general, by a mixed strategy\nprofile drawn from a space of s pure strategies. The community evolves as agents\nupdate their strategy profiles in response to payoffs received fro m other agents. The\nevolution equation is a generalization of the replicator equation. We a rgue that when\nN is sufficiently large and the payoff matrix elements satisfy suitable ineq ualities, the\ncommunity evolves to retain the full diversity of available strategies even as individual\nagents specialize to pure strategies.\n∗Journal Ref: Complexity, Vol. 3, No. 3, 50-56, (1998).\n†E-mail: borkar@csa.iisc.ernet.in\n‡Also at Jawaharlal Nehru Centre for Advanced Scientific Rese arch, Bangalore 560 064, and\nAssociate Member of ICTP, Trieste. E-mail: jain@cts.iisc.ernet.in\n§E-mail: rangaraj@math.iisc.ernet.in\n\nI. INTRODUCTION\nOne of the striking phenomena exhibited by a wide variety of complex a daptive systems\nis that individual agents or components of the system evolve to per form highly specialized\ntasks, and at the same time the system as a whole evolves towards a greater diversity in terms\nof the kinds of individual agents or components it contains or the ta sks that are performed in\nit. Some examples of this include living systems which have evolved incre asingly specialized\nand diverse kinds of interacting protein molecules, ecologies which de velop diverse species\nwith specialized traits, early human societies which evolve from a stat e where everyone\nshares in a small number of chores to a state with many more activitie s performed largely\nby specialists, and firms in an economic web that explore and occupy in creasingly specialized\nand diverse niches.\nIn this paper we study a mathematical model of economic communitie s that exhibits these\ntwin evolutionary phenomena of specialization and diversity. The sys tem is a community of\nN (say, human) agents. There are s strategies or activities each agent can perform labelled\nby i ∈ S ≡ {1, 2, . . . , s }, and at the time t the agent α (α = 1, . . . , N ) performs the activity\ni with a probability pα\ni (t) (thus\n∑ s\ni=1 pα\ni (t) = 1 ∀ α, t ). The vector pα (t) = ( pα\n1 (t), . . . , p α\ns (t))\nis called the mixed strategy profile of agent α at time t. If pα\ni (t) = δij for some j ∈ S then\nthe agent α is said to pursue the pure strategy j or to have ‘specialized’ in the strategy j at\ntime t. The set of vectors pα , α = 1 , . . . , N constitute the basic degrees of freedom of the\nmodel. The dynamics is defined by the equation\n˙pα\ni (t) = pα\ni (t)\n\n ∑\nβ ̸=α\n∑\nj\naijpβ\nj (t) −\n∑\nβ ̸=α\n∑\nk,j\npα\nk (t)akjpβ\nj (t)\n\n , 1 ≤ α ≤ N, 1 ≤ i ≤ s, (1.1)\nwhich determines the rate of change of an individual pα in terms of the current mixed\nstrategy profiles of all the agents and the payoff matrix A = [[aij]].\nWe motivate this model as follows: Agents interact with each other o n a short time scale,\nreceive payoffs based on each other’s activity, and update their ind ividual strategy profiles\non a longer time scale so as to increase their payoffs. As is usual in gam e theory, aij denotes\nthe payoff received by an agent pursuing a pure strategy i in a single interaction with an\nagent pursuing the pure strategy j. Then the average payoff received by the agent α from the\nrest of the community in the period t to t+ ∆ t is proportional to ∆ t ∑\nβ ̸=α\n∑\nk,j pα\nk (t)akjpβ\nj (t).\nThis assumes that every agent interacts equally often with all othe r agents and that there\nis a separation of time scales: ∆ t can be chosen long enough for there to be a statistically\nsufficient number of interactions during the period, yet short enou gh that the change in the\nstrategy profiles during this period can be ignored in the computatio n of the average payoff.\nIf α had played the pure strategy i in this period, she would have received an average payoff\nproportional to ∆ t\n∑\nβ ̸=α\n∑\nj aijpβ\nj (t). The agent α increments pα\ni by an amount proportional\nto pα\ni as well as to the difference between the average payoff she would ha ve got in this\ninterval if she had pursued a pure strategy i and the average payoff she actually received:\n∆ pα\ni = c∆ tpα\ni [\n∑\nβ ̸=α\n∑\nj aijpβ\nj (t) −\n∑\nβ ̸=α\n∑\nk,j pα\nk (t)akjpβ\nj (t)], where c is a constant. Equation\n(1.1) follows upon dividing by ∆ t, taking the limit, and rescaling time by a factor c. By\nconstruction, each agent makes a positive change in the weight of s trategy i in her own\nstrategy profile if she perceives that the pure strategy i would give a higher payoff in the\n\ncurrent environment than her current strategy profile, and a ne gative change if it were to\ngive a lower payoff.\nEq. (1.1) is nothing but the ‘multipopulation replicator equation’ discu ssed in [1] (and\nreferences therein). There each α represents a population, and pα\ni represents the fraction\nof individuals in the population α pursuing the strategy i. Since for us each α represents\nan individual and not a population, we refer to dynamics specified by ( 1.1) as simply the\ngeneralized replicator dynamics (GRD). By contrast the replicator dynamics (which we\nhereafter refer to as the ‘pure replicator dynamics’ (PRD)) is give n by (see [2])\n˙xi(t) = xi(t)[\n∑\nj\naijxj(t) −\n∑\nk,j\nxk(t)akj xj(t)], i = 1, . . . , s. (1.2)\nThis is a standard model in evolutionary biology describing the growth and decay of s species\nunder selection pressure with xi representing the fraction of the ith species in the population.\nPRD and its variants are also extensively studied in economics in game t heory as models\nfor dynamical selection of equilibria (see, e.g., [3] [4]). Its generalizat ions have also been\nstudied in the context of the emergence of organizations in complex adaptive systems (see\n[5] and references therein). For extensive accounts of more rec ent contributions to PRD and\nfurther references, see the recent books [6,7].\nWe view GRD as a model of learning in a community of N interacting agents. The\nagents are identical in that each is capable of pursuing the same set of strategies with the\nsame payoffs. This is a non-cooperative game in which the agents act selfishly (each is\nconcerned with increasing her own payoff without consideration of im pact on others or the\ncommunity), and exhibit bounded rationality (no anticipation of othe rs’ strategy, merely a\nresponse to the current aggregate behaviour of others). Ther e is no global organizing agency\nat work, individual actions alone are responsible for the evolution of the system.\nNevertheless, we shall argue that the community as a whole seems t o exhibit a kind of\nglobal organization under certain circumstances. Individual agen ts tend to specialize, while\nthe community as a whole retains its diversity, i.e., each pure strateg y is pursued by some\nagent or the other. We attempt to find conditions on the paramete rs of the model (the size\nN of the community and the s × s payoff matrix A) such that this behaviour occurs. While\nmost of the time we work with a strategy space of a fixed size (and re fer to diversity as the\nmaintenance of all strategies in this fixed size space) the results als o have bearing on the\nconditions under which new strategies can enter the community.\nSection 2 sets the notation and discusses some relationships betwe en PRD and GRD. In\nsection 3 we identify conditions under which attractors of GRD can e xhibit simultaneously\nspecialization and diversity, and characterize these attractors q uantitatively. Section 4 sum-\nmarizes the results, discusses their possible significance and outline s some open questions.\nDue to constraints on space, proofs for some of the results have not been included in this\npaper. These and other generalizations of our results are the sub ject of a detailed follow-up\npaper [8].\nII. RELA TIONSHIPS BETWEEN GRD AND PRD; INTERIOR EQUILIBRIA\n\nA. GRD preliminaries\nNotation, definition of specialization and diversity\nLet J denote the simplex of s-dimensional probability vectors:\nJ = {x = (x1, · · · , x s)T ∈ Rs|\ns∑\ni=1\nxi = 1, x i ≥ 0}. (2.1)\nJ is the full configuration space of PRD, and is invariant under it.\nThe configuration space of GRD will be denoted J N = Π N\nα =1J (α ) where J (α ) is a copy\nof J for the α th agent. A point of J N will be denoted p = ( p1, p2, . . . , pN ), where pα =\n(pα\n1 , p α\n2 , . . . , p α\ns ) ∈ J (α ). J N is invariant under GRD, as the norm of every pα is preserved\nunder (1.1).\nA point of J N at which every agent has specialized to some strategy or the other will\nbe referred to as a corner of J N , and at such a point we say that the community is ‘fully\nspecialized’. It is evident that a corner is an equilibrium point of (1.1) sin ce ˙pα\ni vanishes if pα\ni\ndoes and (1.1) preserves norm, hence we often refer to a corner as a ‘corner equilibrium point’\nor CEP. A CEP can be characterized by an s-vector of non-negative integers n = (n1, . . . , n s)\nwhere ni is the number of agents pursuing the pure strategy i at the CEP, 1 ≤ i ≤ s (thus∑\ni ni = N). Two CEPs with the same associated n vector are interchangeable, since they\ndiffer only in the identity of the agents, irrelevant for our purposes .\nThe set Fk ≡ { p ∈ J N |pα\nk = 0 ∀ α } for any k ∈ S is the subset of the boundary of J N\nwhere all agents have opted out of strategy k. At the ‘face’ Fk, strategy k becomes extinct\nfrom the population and the full diversity of strategies is lost. The c ommunity will be said\nto exhibit ‘diversity’ at all points that do not belong to some Fk. Note that we use the\nword ‘diversity’ not to signify the variation between individual agent s, but to indicate that\nall strategies are supported. Indeed we can have no variation but full diversity if all agents\npursue the same mixed strategy: for all α , pα = c ∈ J ◦. (The superscript ◦ for any set\ndenotes its relative interior.) When pα is independent of α , the community is completely\n‘homogeneous’ since all agents are doing the same thing. The commu nity can be fully\nspecialized and diversified at the same time: each agent chooses a pu re strategy and every\nstrategy is chosen by some agent or the other. This corresponds to CEP with n such that\neach ni is nonzero, which will be called a ‘fully diversified’ CEP or FDCEP. By cont rast,\nCEP where one or more strategies becomes extinct (some compone nts of n are zero) will be\ncalled non-FDCEP.\nIn this paper we are primarily interested in studying the circumstanc es in which FDCEP\nare the preferred attractors of the dynamics, since in that case individual specialization and\nglobal diversity will arise dynamically in the community.\nDifferences between PRD and GRD\nIf the initial point of a trajectory in GRD is homogeneous, the traje ctory remains homoge-\nneous for all time, and evolves according to (1.2) except that the t ime is speeded up by a\nfactor of N − 1. The sum ¯xi ≡ (1/N ) ∑ N\nα =1 pα\ni equals the probability that strategy i is being\nplayed in the entire community, and is therefore the analogue of xi in PRD. We can ask how\n¯xi evolves in GRD. It is easy to see that\n\n˙¯xi = N[¯xi\n∑\nj\naij ¯xj −\n∑\nk,j\nxikakj ¯xj − 1\nN\n∑\nj\nxijaij + 1\nN\n∑\nk,j\nxikjakj ], (2.2)\nwhere xik ≡ (1/N ) ∑\nα pα\ni pα\nk and xikj ≡ (1/N ) ∑\nα pα\ni pα\nk pα\nj . The r.h.s. of (2.2) is not propor-\ntional to the r.h.s. of (1.2), except for homogeneous trajectorie s in which case xik = ¯xi ¯xk,\nxikj = ¯xi ¯xk ¯xj. Thus in general ¯ xi does not follow the PRD. One might have hoped that\nwhen the number of agents N is large ¯xi follows PRD, but even that is not the case due to\nvariation among the agents. For example, at the corner n, the difference between xik and\n¯xi ¯xk is ni(N δij − nj)/N 2, which is comparable to the former two even for large N (except\nfor homogeneous corners).\nOne of our results in this paper is that even though variation among a gents, which\nis generic in GRD, causes the evolution of ¯ xi to be different from PRD, under suitable\nconditions ¯xi nevertheless converges to the interior equilibrium point of PRD.\nB. The interior equilibria of GRD\nConsider an interior equilibrium point (IEP) p of (1.1). By definition no pα\ni is zero in the\ninterior of J N . Therefore the bracket [ ] on the r.h.s. of (1.1) must vanish for all α, i . Define\nxα\n0 ≡\n∑\nβ ̸=α\n∑\ni,j pα\ni aijpβ\nj , and vα\ni ≡\n∑\nβ ̸=α pβ\ni . Then\n∑ s\ni=1 vα\ni = N − 1 ∀ α , and the interior\nequilibrium condition can be written as\nBXα = (N − 1)E0 ∀ α, (2.3)\nwhere Xα ≡ (xα\n0 , v α\n1 , v α\n2 , . . . , v α\ns )T , B is the s + 1-dimensional matrix\nB =\n\n\n\n\n\n\n\n\n\n0 1 1 · · · 1\n−1\n−1 A\n.\n.\n.\n−1\n\n\n\n\n\n\n\n\n\n, (2.4)\nand E0 is the s + 1-dimensional unit vector (1 , 0, 0, · · ·, 0)T .\nIt is not difficult to see (details in [8]) that (2.3) has an isolated solution if and only if\nthe following condition holds:\nA1: ui ̸= 0 ∀ i, and all ui have the same sign, where ui denotes the co-factor of B0i.\nFor the sake of notational simplicity, we have denoted the cofacto r of B0i by ui instead\nof u0i thereby suppressing the fixed first index. Under the above condit ion ( A1) det B =∑ s\ni=1 ui ̸= 0, and the solution is unique and given by vα\ni = (N − 1)xi ∀ α . Here,\nxi = ui/ detB (2.5)\nis nothing but the ith coordinate of the unique isolated interior equilibrium point of PRD.\n(Note that A1 is also the necessary and sufficient condition for PRD to have an isolat ed IEP,\nwhich, if it exists, is unique.) Since pα\ni − pβ\ni = vβ\ni − vα\ni = 0, it follows that the equilibrium\npoint is homogeneous and given by pα\ni = xi. Thus we have proved\nTheorem 2.1 There exists at most one isolated equilibrium in the interior of J N . It exists\nif and only if A1 is satisfied and then it is homogeneous (all agents pursue the same m ixed\nstrategy), and coincides with the isolated interior equilibrium point of PRD, pα\ni = xi ∀ α, i .\n\nIII. CORNER EQUILIBRIA OF GRD: DIVERSIFICA TION WITH\nSPECIALIZA TION\nA. Stability of corner equilibria\nThe IEP of GRD is always unstable to small perturbations. This is a con sequence of the\nfollowing theorem proved in [1]:\nTheorem 3.1 An equilibrium point of (1.1) is asymptotically stable if and only if it is\na strict Nash equilibrium. Further, any compact set in the relative int erior of a face cannot\nbe asymptotically stable.\nNote that strict Nash equilibria are perforce pure strategy Nash e quilibria and therefore\ncorrespond to CEP. As a consequence of this theorem, a traject ory either eternally moves\naround in the relative interior of some face or the interior of J N coming arbitrarily close to\nits boundaries and corners (the case of non-compact attractor ), or it converges to a corner of\nJ N . It is possible to construct payoff matrices for which there are no a symptotically stable\ncorners in J N , whereupon the former situation obtains. However, our numerica l work with\n3 × 3 payoff matrices suggests that this happens rarely (i.e., in a relative ly small region of\nR3×3); for most payoff matrices asymptotically stable corners do exist f or most values of\nN. Further, we randomly generated ten 3 × 3 payoff matrices and numerically integrated\nthe GRD equations for long times for each payoff matrix with ten rand omly chosen initial\nconditions. When this was done with N = 5, in 90 out of the 100 cases the dynamics\nconverged to a corner. With N = 10, all 100 cases converged to a corner. This suggests\nthat typically, at large N, not only do asymptotically stable corners exist, but also that\ntheir basins of attraction cover most of J N . Thus corners seem to be the most common\nattractors in GRD. These are numerical indications and need to be m ade more precise. In\nour interpretation of the model, a corner corresponds to a fully sp ecialized community. The\nabove theorem and numerical evidence therefore suggest that s pecialization of all the agents\nis the most common outcome in GRD.\nAt the CEP n, the payoff to an agent playing the jth pure strategy from the other N − 1\nagents is\nPj =\ns∑\nk̸=j\najknk + (nj − 1)ajj =\ns∑\nk=1\najknk − ajj = Pj − ajj , (3.1)\nwhere Pj ≡\n∑ s\nk=1 ajknk. If this agent were to suddenly switch to the ith pure strategy ( i ̸= j),\nall other agents remaining at their respective pure strategies, th en for this agent the payoff\nwould change to ∑ s\nk̸=j aiknk − aij(nj − 1) = Pi − aij. Thus the increase in payoff for an agent\nplaying the jth pure strategy at the FDCEP n (and this assumes nj ̸= 0) in switching to\nthe ith pure strategy is\nλ ij = Pi − Pj − hij, h ij ≡ aij − ajj . (3.2)\nTherefore, n is a strict Nash equilibrium if for every j such that nj ̸= 0, the conditions\nλ ij < 0 (3.3)\n\nare satisfied for all i ̸= j. At a FDCEP, all nj are nonzero and this is a set of s(s − 1)\nconditions. At a non-FDCEP where only s′ < s components of n are nonzero, the number\nof conditions is smaller, s′(s − 1).\nFrom Theorem 3.1, these are identical to the conditions for the asy mptotic stability of\nthe FDCEP’s associated with n. In fact, one can show that λ ij given by (3.2) are precisely\nthe eigenvalues of the Jacobian matrix of (1.1) linearized around a co rner of J N characterized\nby n.\nB. Stability of fully diversified corners\nTheorem 3.2 : Let PRD admit an isolated IEP x. That is, condition A1 holds [cf.\nSection II]. Let n, n′ be any pair of asymptotically stable FDCEP’s of GRD with N ≥ s.\nThen\n(i) all components of the difference n′ − n are bounded by a function of A alone, not of N,\nand\n(ii) lim N →∞\nni\nN = xi.\nThe proof is given in Appendix A.\nThe significance of this theorem is that it characterizes the FDCEP t hat are attractors\nof the dynamics. If the community is going to end up in a fully specialized and diversified\nconfiguration, the theorem quantifies the relative weights of all st rategies that will obtain in\nthat configuration: these relative weights are forced to be ‘close’ to the IEP configuration\ngiven by (2.5). The theorem does not guarantee the existence of a stable FDCEP. One can\nprove the existence of an infinite set of values of N at which stable FDCEP are guaranteed\nto exist under the conditions of the theorem. One can also identify s ufficient conditions for\nthe existence of stable FDCEP for any N ≥ s. These are presented in [8].\nC. Instability of non-fully diversified corners\nWe would like to define GRD as possessing diversity if all trajectories in the faces Fk\nbecome unstable at some time or the other with respect to perturb ations that take them away\nfrom these faces. With this in mind we now study corners at which one or more strategies\nbecome extinct and determine the conditions under which all such co rners become unstable.\nThen under small perturbations the population will dynamically flow ou t of such corners,\neliminating specialized configurations that do not carry the full diver sity of strategies. As\nremarked earlier, the number of conditions to be satisfied by a non- FDCEP to be stable is\nless than the number to be satisfied by a FDCEP. Thus a priori, things seem to be loaded\nagainst diversification. As we shall see, some further structure w ill need to be imposed on\nA in order to make the non-FDCEP unstable. At this point we do not hav e the general\nconditions for arbitrary s, but some insight gleaned from special cases s = 2, 3.\ns = 2, N arbitrary\nIn this case conditions (3.3) can be studied exhaustively. There are generically four cases.\n\nCase 1: a11 > a 21 and a22 > a 12: Both ( N, 0) and (0 , N ) are asymptotically stable, other\ncorners are not.\nCase 2: a11 > a 21 and a22 < a 12: ( N, 0) is the only asymptotically stable corner.\nCase 3: a11 < a 21 and a22 > a 12: (0 , N ) is the only asymptotically stable corner.\nCase 4: a11 < a 21 and a22 < a 12: The only asymptotically stable corners ( n1, n 2) (with\nn1 + n2 = N) are those for which n1 ̸= 0, n2 ̸= 0, and furthermore, (n2−1)\nn1\nh12 < h 21 <\nn2\n(n1−1)h12) if n2 < N − 1, and (n2−1)\nn1\nh12 < h 21 if n2 = N − 1.\nCases 2 and 3 correspond to dominated strategies. (The cases wit h one or more equalities\ninstead of inequalities have been disregarded as nongeneric. In any case, they are not difficult\nto handle.) The case of interest to us is the last one, which shows dive rsification. It is\nconvenient to introduce the\nDefinition : A is diagonally subdominant if aii < a ji ∀ j ̸= i, {i, j } ⊂ S.\nThat is, hij > 0 ∀ i ̸= j. From the above exhaustive list it follows that the condition\nA2: A is diagonally subdominant,\nis the necessary and sufficient condition for non-FDCEP to be unsta ble (for generic A). If\nA2 is satisfied, the only asymptotically stable CEP are the FDCEP, for wh ich Theorem 3.2\napplies. (Note that for s = 2, A2 implies A1, the IEP is given by pα = 1\nh12+h21\n(h12, h 21) for\nall α , and the inequalities involving n in Case 4 above are equivalent to the statement that\n(1/N )(n1, n 2) must be close to this IEP for arbitrary N and converge to it as N → ∞.)\nNote that GRD remains invariant under addition of an arbitrary cons tant to any column\nof the payoff matrix. Thus we may replace aij by hij, thus obtaining a matrix which under\nA2 has zero diagonal elements and nonnegative off-diagonal elements . It is interesting that\nthese conditions also arise in PRD in the context of population genetic s and ecological\nmodels [9] as well as in models of catalytic networks of chemically react ing molecules [10].\ns = 3, N arbitrary\nFor s = 3, A2 no longer implies A1; the latter is an independent condition. We now state\nTheorem 3.3 For s = 3, if both A1 and A2 hold, then there exists a positive number N0\ndepending on A such that for all N > N 0, all non-FDCEP are unstable.\nThe proof of this theorem can be found in Appendix B.\nWe remark that while N0 is finite, it may, depending upon A, be much larger than three.\nThe above result can be further generalized (with the imposition of a n additional condition)\nto prove that for s = 3 all points in Fk are unstable for sufficiently large N [8]. Note that\nour notion of diversity for GRD is related to the notions of ‘permanen ce’, ‘persistence’, etc.\nintroduced for PRD (see [2] and references therein). PRD is said to exhibit permanence if\nevery interior solution has components that remain bounded away f rom zero by a common\nconstant δ > 0. Strong persistence, in turn, is the weaker requirement where δ is trajectory\ndependent and persistence the even weaker requirement that ea ch component of an interior\ntrajectory not converge to zero. The biological implications are ob vious: the concept is\nclearly related to survival of species. The corresponding phenome non here is the survival of\npolicies. The conditions for, e.g., permanence in PRD (see [2]) may quite generally play a\nrole in discussions of diversity in GRD.\n\nIV. DISCUSSION AND CONCLUSIONS\nTo summarize:\n(i) We have considered the equation (1.1) as a model of evolution of a community of\nN agents, each agent being capable of performing any mix of a set of s strategies, and\nmodifying her mix depending upon the payoff received from other age nts. We have studied\nsome properties of the attractors of this system to gain insight on how the community is\nexpected to evolve.\n(ii) This model can exhibit specialization of the agents into pure strat egies. Evidence\nfor this comes from the previously known Theorem 3.1, supported w ith our numerical ob-\nservations. While individual specialization seems to be the most commo n outcome in this\nmodel, it would be interesting to characterize more precisely the circ umstances in which\nspecialization is guaranteed, i.e., when corners are the only attract ors, and when not.\n(iii) We have shown that under suitable conditions, while each agent sp ecializes to a single\npure strategy, it is guaranteed that the community as a whole pres erves the full diversity\nof strategies. These are that the community be sufficiently large ( N should be larger than\na number N0 that depends upon the payoff matrix), and the payoff matrix itself s hould\nsatisfy A1 (existence of an isolated interior equilibrium point) and A2 (diagonal entries\nof A be smaller than other entries in the same column). These guarantee (for upto three\nstrategies) that all corners where one or more strategy become s extinct are unstable to small\nperturbations (Theorem 3.3). To identify sufficient conditions for la rger s (and necessary\nand sufficient conditions for s ≥ 3) is a task for the future. The appearance of a lower limit\non the size of the community in this context (which could be much large r than the number\nof strategies) is interesting.\n(iv) Within the set of configurations where the community would exhib it full specializa-\ntion and diversity (the FDCEP), we have given a quantitative criterio n as to which ones\nwill be the attractors (Theorem 3.2). ni/N (where ni is the number of agents pursuing the\npure strategy i at the attractor) is forced to be close to xi and equal to it in the large N\nlimit, where xi is given by (2.5) and is the relative weight of the ith strategy at the interior\nequilibrium point of PRD. This constraint is a consequence of the fine b alance that exists\nfor every agent at a strict Nash equilibrium; any strategy switch fo r any agent reduces her\npayoff. This fine tuning, caused by the interaction of the agent with other agents, is a kind\nof organization exhibited by the system.\nThe conditions for the instability of non-FDCEP (Theorem 3.3) may als o be relevant to\nthe question: when does a society accept an innovation? For consid er a community of a\nlarge number of agents but with only two strategies, 1 and 2, at a st able corner where n1\nagents pursue the pure strategy 1 and n2 = N − n1 agents the pure strategy 2 (neither n1\nnor n2 is zero). Since this corner is assumed stable, the 2 × 2 matrix A satisfies condition\nA2 (diagonal subdominance). Now imagine that a new strategy 3 arises thereby enlarging\nthe payoff matrix to a 3 × 3 matrix A′ containing A as a 2 × 2 block. In the new context the\nearlier state of the community will be described by a three vector n = (n1, n 2, 0), which is in\nthe face F3. Now if the new payoff matrix satisfies A1,A2, and N is sufficiently large, then\nfrom Theorem 3.3, this configuration is unstable with respect to per turbations in which one\nof the agents begins to explore the new strategy. Thus if this agen t were to explore the new\nstrategy ever so slightly, her payoff would increase and a small pert urbation of the community\n\nwould grow until it settles down in another attractor. The new attr actor if described by\nTheorem 3.2 would have the property that a finite fraction of the po pulation pursues the\nnew strategy: the innovation has been accepted by the society. T hus the conditions A1,\nA2 of Theorem 3.3 indicate what the payoffs of a new strategy (innovat ion) should be with\nrespect to the existing ones, if the innovation is to be guaranteed a cceptance. (Conditions\nthat are both necessary and sufficient for diversity would consider ably strengthen the above\nremarks.)\nIt is worth mentioning that conditions A1, A2 are not equalities but inequalities. Thus\nthere is no fine tuning of parameters needed; the behaviour discus sed above emerges when-\never parameters cross certain thresholds.\nIt may be interesting to consider the ‘economic significance’ of cond itions which play an\nimportant role in preserving the full diversity of strategies. For ex ample, diagonal subdomi-\nnance, when translated as ‘each pure strategy gives more payoff t o other pure strategies than\nto itself’, carries a shade of an ‘altruism’ of sorts (at the level of st rategies, not individuals).\nNote that PRD with a payoff matrix in which diagonal entries are zero a nd offdiagonal ones\ngreater than or equal to zero is called a ‘catalytic network’ [2]. The g eneral message might\nbe that if the initial set of allowed strategies is chosen with the ‘right v ision’ (read ‘right\npayoffs’), then, even a community of identical and selfish individuals, if large enough, will\nexhibit diversity and accept only the ‘right’ innovations.\nAPPENDIX A\nThe proof of Theorem 3.2 in Section IIIB follows:\nProof: Note that Pi − Pj figures in both λ ij and λ ji. Therefore the s(s − 1) conditions\n(3.3) can be written in terms of s(s − 1)/ 2 “double-sided” inequalities\n− hji < P i − Pj < h ij. (4.1)\nDefine zi ≡ Pi − Pi+1 for i = 1 , . . . , s , with Ps+1 ≡ P1. Then zi = ∑ s\nj=1 cijnj with cij ≡\naij − ai+1,j , where it is again understood that as+1,j ≡ a1j. Now, since all the nj are not\nindependent, let us express zi in terms of only n1, . . . , n s−1 by eliminating ns = N −(n1+· · ·+\nns−1). This gives zi = yi + cisN where yi ≡ ∑ s−1\nj=1 dijnj and dij ≡ cij − cis, i, j = 1, . . . , s − 1.\nWith this notation, consider the subset of s − 1 inequalities obtained by setting j = i + 1 in\n(4.1), with i = 1, . . . , s − 1. These involve zi and take the form\n− hi+1,i − cisN < y i < h i,i +1 − cisN, i = 1, . . . , s − 1. (4.2)\nThese inequalities mean that for any stable FDCEP n, the yi, which are linear combinations\nof n1, . . . , n s−1, are constrained to be in an open interval of the real line. While the lo cation\nof this interval is N dependent, it follows from (4.2) that the size of this interval is finite ,\nindependent of N, and depends only on the payoff matrix (for yi the size of the interval is\nhi+1,i + hi,i +1).\nIf the s − 1 dimensional matrix D = (dij) has an inverse, we can invert yi ≡ ∑ s−1\nj=1 dijnj to\nexpress the nj in terms of yi. Then, (4.2) will get converted into inequalities for n1, . . . , n s−1.\nSince the vector ˜y = (y1, . . . , y s−1) in the s−1 dimensional cartesian space whose axes are the\nyi is constrained by (4.2) to lie in a (rectangular) parallelepiped, the vec tor ˜n = (n1, . . . , n s−1)\n\nin the s − 1 dimensional cartesian space whose axes are the ni will also lie in a (in general\noblique) parallelepiped which is the image, under D−1, of the rectangular parallelepiped\nin y-space defined by (4.2). Again, while the location of the parallelepiped in n1, . . . , n s−1\nspace will depend upon N, its size, i.e., its extent along any of the coordinate axes, will be\nindependent of N. This is because the matrix D−1, if it exists, depends only on the payoff\nmatrix and not on N. Therefore, if D−1 exists, the differences in ni, i = 1 , . . . , s − 1 for\nall FDCEP are bounded by some function of A alone, not of N. The same is true for ns\nalso since\n∑ s\ni=1 ni = N. One can show [8] that det D = det B. The existence of D−1 is thus\nguaranteed by condition A1, completing the proof of the first part of Theorem 3.2.\nTo prove the second part of Theorem 3.2, divide all sides of (4.1) by N and take the\nlimit N → ∞ . This yields lim N →∞[ Pi\nN − Pj\nN ] = 0. Defining x′\ni ≡ limN →∞(ni/N ) along an\nappropriate subsequence independent of i, this is equivalent to the statement that ∑ s\nk=1 aikx′\nk\nis independent of i, which implies that x′ is the same as x, the IEP of PRD. ✷\nAPPENDIX B\nThe proof of Theorem 3.3 in Section IIIC follows:\nProof: Any non-FDCEP n must belong to some Fk, in this case to F1, F2 or F3. For\nevery i such that the component ni of n ∈ Fk is nonzero, consider the eigenvalue\nλ ki = Pk − Pi − hki. (4.3)\nFrom the discussion of Eq. (3.2) it follows that if any one (or the large st) of the λ ki at n\nis greater than zero, then the CEP n is unstable against perturbations in which an agent\npursuing the pure strategy i moves towards strategy k (i.e., the perturbations which restore\nthe extinct strategy k will then grow).\nFor concreteness consider F3. Corners of F3 are of two types.\nCase 1: Only one strategy survives at the corner. Then n = ( N, 0, 0) or (0 , N, 0). In the\nformer case (4.3) implies λ 31 = (N − 1)h31 and in the latter case λ 32 = (N − 1)h32. By A2\nboth corners are unstable.\nCase 2: Both strategies 1 and 2 survive at the corner of F3. Then n = (n1, n 2, 0) with both\nn1 and n2 positive integers and n1 + n2 = N. There are then two eigenvalues from (4.3),\nλ 31 = h31n1 + h32n2 − h12n2 − h31, and λ 32 = h31n1 + h32n2 − h21n1 − h32. Let us assume\nthat this corner is stable, hence both λ 31, λ 32 are negative. The condition λ 31 < 0 (upon\neliminating n1 = N − n2) reduces to ( h12 + h31 − h32)n2 > (N − 1)h31. Since n2, N − 1, h 31\nare all positive this means that the combination h12 + h31 − h32 is also positive, and\n(N − 1)h31\nh12 + h31 − h32\n< n 2. (4.4)\nSimilarly λ 32 < 0 implies that h21 + h32 − h31 is positive (as can be seen by eliminating n2)\nand further,\nn2 < (N − 1)(h21 − h31)\nh21 + h32 − h31\n+ 1. (4.5)\nCombining the two, we get\n\n(N − 1)h31\nh12 + h31 − h32\n< (N − 1)(h21 − h31)\nh21 + h32 − h31\n+ 1, (4.6)\nwhich can be rearranged into the form\n(N − 1)[−h21h12 + h21h32 + h31h12] < (h12 + h31 − h32)(h21 + h32 − h31). (4.7)\nBut the quantity in [ ] on the l.h.s. of this inequality is just u3 (as evaluated from the\ndefinition given in A1), which is positive. (The positivity of det B and hence u1, u 2, u 3 also\nfollows from A1 and A2.) Thus we have\nN < (h12 + h31 − h32)(h21 + h32 − h31)\nu3\n+ 1. (4.8)\nNote that the r.h.s. is a function of A alone and is finite, say N0(A). If N is chosen larger\nthan N0(A), this inequality is violated. That is, for N > N 0(A), the corner of F3 under\nconsideration cannot be stable. We have thus proved that under A1,A2, all corners of F3\nare unstable for N > N 0(A). Similarly one may consider F1, F 2, which will yield the same\nresult but with different finite bounds in place of N0(A). We can henceforth use N0 for the\nlargest of the three. The claim follows. ✷\n\nREFERENCES\n[1] J. W. Weibull. Evolutionary game theory, MIT Press, Cambridge, 1 995.\n[2] J. Hofbauer and K. Sigmund. The theory of evolution and dynamic al systems, Cam-\nbridge University Press, Cambridge, 1988.\n[3] G. Mailath: Introduction: Symposium on evolutionary game theor y. J. Economic The-\nory 57: 1992, pp. 259-277.\n[4] D. Fudenberg and D. Levine: Learning and evolution in games. Plen ary talk at the\nSummer meeting of the Econometric Society: 1996.\n[5] P. F. Stadler, W. Fontana and J. H. Miller: Random catalytic react ion networks. Physica\nD63: 1993, pp. 378-392.\n[6] L. Samuelson. Evolutionary games and equilibrium selection, MIT Pr ess, Cambridge,\n1996.\n[7] F. Vega-Redondo. Evolution, games and economic behaviour, Ox ford University Press,\nNew York, 1996.\n[8] V. S. Borkar, S. Jain, and G. Rangarajan: Generalized replicato r dynamics as a model\nof specialization and diversity in societies. Preprint No. IISc-CTS-1 1/97.\n[9] S. Gadgil, V. Nanjundaiah, and M. Gadgil: On evolutionarily stable co mpositions of\npopulations of interesting genotypes. J. Theor. Biol. 84: 1980, pp . 737-759.\n[10] J. Hofbauer, P. Schuster, K. Sigmund, and R. Wolff: Dynamical systems under constant\norganization II: Homogeneous growth functions of degree 2. SIA M J. Appl. Math. 38:\n1980, pp 282-304."}
{"id": "adap-org/9806001", "meta": {"categories": ["adap-org", "nlin.AO", "q-bio.NC"], "created": "1998-06-03", "extraction": {"body_chars": 78008, "cleaning": {"detected_repeated_margin_lines": ["2"], "page_count": 52, "removed_boilerplate_lines": 133}, "method": "pypdf_no_ocr", "source_pdf_bytes": 465682, "text_chars": 79075}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9806001", "primary_category": "adap-org", "source": "arxiv", "title": "Neural network design for J function approximation in dynamic programming", "updated": null, "url": "https://arxiv.org/abs/adap-org/9806001"}, "text": "Neural network design for J function approximation in dynamic programming\n\nAbstract\nThis paper shows that a new type of artificial neural network (ANN) -- the Simultaneous Recurrent Network (SRN) -- can, if properly trained, solve a difficult function approximation problem which conventional ANNs -- either feedforward or Hebbian -- cannot. This problem, the problem of generalized maze navigation, is typical of problems which arise in building true intelligent control systems using neural networks. (Such systems are discussed in the chapter by Werbos in K.Pribram, Brain and Values, Erlbaum 1998.) The paper provides a general review of other types of recurrent networks and alternative training techniques, including a flowchart of the Error Critic training design, arguable the only plausible approach to explain how the brain adapts time-lagged recurrent systems in real-time. The C code of the test is appended. As in the first tests of backprop, the training here was slow, but there are ways to do better after more experience using this type of network.\n\narXiv:adap-org/9806001v1 3 Jun 1998\nNeural Network Design for J Function\nApproximation in Dynamic Programming\nXiaozhong Pang Paul J. Werbos\nAbstract\nThis paper will show that a new neural network design can solv e an\nexample of difficult function approximation problems which a re crucial to\nthe field of approximate dynamic programming(ADP). Althoug h conven-\ntional neural networks have been proven to approximate smoo th functions\nvery well, the use of ADP for problems of intelligent control or planning\nrequires the approximation of functions which are not so smo oth. As an\nexample, this paper studies the problem of approximating th e J function\nof dynamic programming applied to the task of navigating maz es in gen-\neral without the need to learn each individual maze. Convent ional neural\nnetworks, like multi-layer perceptrons(MLPs), cannot lea rn this task. But\na new type of neural networks, simultaneous recurrent netwo rks(SRNs),\ncan do so as demonstrated by successful initial tests. The pa per also ex-\namines the ability of recurrent neural networks to approxim ate MLPs and\nvice versa.\nKeywords: Simultaneous recurrent networks(SRNs), multi-layer percep-\ntrons(MLPs), approximate dynamic programming, maze navigation , neural net-\nworks.\n1 Introduction\n1.1 Purpose\nThis paper has three goals:\nFirst, to demonstrate the value of a new class of neural network w hich pro-\nvides a crucial component needed for brain-like intelligent control s ystems for\nthe future.\nSecond, to demonstrate that this new kind of neural network pro vides better\nfunction approximate ability for use in more ordinary kinds of neural network\napplications for supervised learning.\nThird, to demonstrate some practical implementation techniques n ecessary\nto make this kind of network actually work in practice.\n\nX(t) Y(t)Supervised Learning\nSystem\nActual Y(t)\n^\nFigure 1: What is supervised learning?\n1.2 Background\nAt present, in the neural network field perhaps 90% of neural net work applica-\ntions involve the use of neural networks designed to performance a task called\nsupervised learning(Figure 1). Supervised learning is the task of lea rning a non-\nlinear function which may have several inputs and several outputs based on some\nexamples of the function. For example, in character recognition, t he inputs may\nbe an array of pixels seen from a camera. The desired outputs of th e network\nmay be a classification of character being seen. Another example wo uld be for\nintelligent sensing in the chemical industry where the inputs might be s pectral\ndata from observing a batch of chemicals, and the desired outputs would be\nthe concentrations of the different chemicals in the batch. The pur pose of this\napplication is to predict or estimate what is in the batch without the ne ed for\nexpensive analytical tests.\nThe work in this paper will focus totally on certain tasks in supervised\nlearning. Even though existing neural networks can be used in supe rvised learn-\ning, there can be performance problems depending on what kind of f unction is\nlearned. Many people have proven many theorems to show that neu ral networks,\nfuzzy logic, Taylor theories and other function approximation have a universal\nability to approximate functions on the condition that the functions have certain\nproperties and that there is no limit on the complexity of the approxim ation.\nIn practice, many approximation schemes become useless when the re are many\ninput variables because the required complexity grows at an expone ntial rate.\nFor example, one way to approximate a function would be to constru ct a\ntable of the values of the function at certain points in the space of p ossible\ninputs. Suppose there are 30 input variables and we consider 10 pos sible values\nof each input. In that case, the table must have 10 30 numbers in it. This\nis not useful in practice for many reasons. Actually, however, man y popular\n\napproximation methods like radial basis function(RBF) are similar in sp irit to\na table of values.\nIn the field of supervised learning, Andrew Barron[30] has proved s ome func-\ntion approximation theorems which are much more useful in practice . He has\nproven that the most popular form of neural networks, the multi- layer percep-\ntron(MLP), can approximate any smooth function. Unlike the case with the\nlinear basis functions (like RBF and Taylor series), the complexity of t he net-\nwork does not grow rapidly as the number of input variables grows.\nUnfortunately there are many practical applications where the fu nctions to\nbe approximated are not smooth. In some cases, it is good enough j ust to add\nextra layers to an MLP[1] or to use a generalized MLP[2]. However, th ere are\nsome difficult problems which arise in fields like intelligent control or image\nprocessing or even stochastic search where feed-forward netw orks do not appear\npowerful enough.\n1.3 Summary and Organization of This Paper\nThe main goal of this paper is to demonstrate the capability of a differ ent\nkind of supervised learning system based on a kind of recurrent net work called\nsimultaneous recurrent network(SRN). In the next chapter we w ill explain why\nthis kind of improved supervised learning system will be very importan t to\nintelligent control and to approximate dynamic programming. In effe ct this\nwork on supervised learning is the first step in a multi-step effort to b uild more\nbrain-like intelligent systems. The next step would be to apply the SRN to static\noptimization problems, and then to integrate the SRNs into large sys tems for\nADP.\nEven though intelligent control is the main motivation for this work, t he\nwork may be useful for other areas as well. For example, in zip code r ecog-\nnition, AT &T [3] has demonstrated that feed-forward networks can achieve a\nhigh level of accuracy in classifying individual digits. However, AT &T and the\nothers still have difficulty in segmenting the total zip codes into individ ual dig-\nits. Research on human vision by von der Malsburg[4] and others has suggested\nthat some kinds of recurrency in neural networks are crucial to t heir abilities\nin image segmentation and binocular vision. Furthermore, research ers in image\nprocessing like Laveen Kanal have showed that iterative relaxation algorithms\nare necessary even to achieve moderate success in such image pro cessing tasks.\nConceptually the SRN can learn an optimal iterative algorithm, but th e MLP\ncannot represent any iterative algorithms. In summary, though w e are most\ninterested in brain-like intelligent control, the development of SRNs c ould lead\nto very important applications in areas such as image processing in th e future.\nThe network described in this paper is unique in several respects. H owever,\nit is certainly not the first serious use of a recurrent neural netwo rk. Chapter\n3 of this paper will describe the existing literature on recurrent net works. It\nwill describe the relationship between this new design and other desig ns in the\n\nliterature. Roughly speaking, the vast bulk of research in recurre nt networks\nhas been academic research using designs based on ordinary differe ntial equa-\ntions(ODE) to perform some tasks very different from supervised learning —\ntasks like clustering, associative memory and feature extraction. The simple\nHebbian learning methods[13] used for those tasks do not lead to th e best per-\nformance in supervised learning. Many engineers have used anothe r type of\nrecurrent network , the time lagged recurrent network(TLRN), where the recur-\nrency is used to provide memory of past time periods for use in forec asting the\nfuture. However, that kind of recurrency cannot provide the ite rative analysis\ncapability mentioned above. Very few researchers have written ab out SRNs,\na type of recurrent network designed to minimize error and learn an optimal\niterative approximation to a function. This is certainly the first use o f SRNs to\nlearn a J function from dynamic programming which will be explained more in\nchapter 2. This may also be the first empirical demonstration of the need for\nadvanced training methods to permit SRNs to learn difficult functions .\nChapter 4 will explain in more detail the two test problems we have use d\nfor the SRN and the MLP, as well as the details of architecture and le arning\nprocedure.\nThe first test problem was used mainly as an initial test of a simple form of\nSRNs. In this problem, we tried to test the hypothesis that an SRN c an always\nlearn to approximate a randomly chosen MLP, but not vice versa. Alt hough\nour results are consistent with that hypothesis, there is room for more extensive\nwork in the future, such as experiments with different sizes of neur al networks\nand more complex statistical analysis.\nThe main test problem in this work was the problem of learning the J\nfunction of dynamic programming. For a maze navigation problem, ma ny neural\nnetwork researchers have written about neural networks which learn an optimal\npolicy of action for one particular maze[5]. This paper will address the more\ndifficult problem of training a neural network to input a picture of a ma ze and\noutput the J function for this maze. When the J function is known, it is a trivial\nlocal calculation to find the best direction of movement. This kind of n eural\nnetwork should not require retraining whenever a new maze is encou ntered.\nInstead it should be able to look at the maze and immediately ”see” the optimal\nstrategy. Training such a network is a very difficult problem which has never\nbeen solved in the past with any kind of neural network. Also it is typic al of the\nchallenges one encounters in true intelligent control and planning. T his paper\nhas demonstrated a working solution to this problem for the first tim e. Now\nthat a system is working on a very simple form for this problem, it would be\npossible in the future to perform many tests of the ability of this sys tem to\ngeneralize its success to many mazes.\nIn order to solve the maze problem, it was not sufficient only to use an SRN.\nThere are many choices to make when implementing the general idea o f SRNs\nor MLPs. Chapter 5 will describe in detail how these choices were mad e in this\nwork. The most important choices were:\n\n1. Both for the MLP and for the feed-forward core of the SRN we u sed the\ngeneralized MLP design[2] which eliminates the need to decide on the nu mber\nof layers.\n2. For the maze problem, we used a cellular or weight-sharing archite cture\nwhich exploits the spatial symmetry of the problem and reduces dra matically\nthe number of weights. In effect we solved the maze problem using on ly five\ndistinct neurons. There are interesting parallels between this netw ork and the\nhippocampus of the human brain.\n3. For the maze problem, an adaptive learning rate(ALR) procedur e was\nused to prevent oscillation and ensure convergence.\n4. Initial values for the weights and the initial input vector for the S RN were\nchosen essentially at random, by hand. In the future, more syste matic methods\nare available. But this was sufficient for success in this case.\nFinally Chapter 6 will discuss the simulation results in more detail, give th e\nconclusions of this paper and mention some possibilities for future wo rk.\n2 Motivation\nIn this chapter we will explain the importance of this work. As discuss ed above,\nthe paper shows how to use a new type of neural network in order t o achieve\nbetter function approximation than what is available from the types of neu-\nral networks which are popular today. This chapter will try to expla in why\nbetter function approximation is important to approximate dynamic program-\nming(ADP), intelligent control and understanding the brain. Image processing\nand other applications have already been discussed in the Introduc tion. These\nthree topics — ADP, intelligent control and understanding the brain — are all\nclosely related to each other and provide the original motivation for the work\nof this paper.\nThe purpose of this paper is to make a core contribution to developin g the\nmost powerful possible system for intelligent control.\nIn order to build the best intelligent control systems, we need to co mbine\nthe most suitable mathematics together with some understanding o f natural\nintelligence in the brain. There is a lot of interest in intelligent control in the\nworld. Some control systems which are called intelligent are actually v ery quick\nand easy things. There are many people who try to move step by ste p to add\nintelligence into control , but a step-by-step approach may not be enough by\nitself.\nSometimes to achieve a complex difficult goal, it is necessary to have a p lan,\nthus some parts of the intelligent control community have develope d a more\nsystematic vision or plan for how it could be possible to achieve real int elligent\ncontrol. First, one must think about the question of what is intelligen t control.\nThen, instead of trying to answer this question in one step, we try t o develop a\nplan to reach the design. Actually there are two questions:\n\n1. How could we build an artificial system which replicates the main capa -\nbilities of brain-like intelligence, somehow unified together as they are unified\ntogether in the brain?\n2. How can we understand what are the capabilities in the brain and ho w\nthey are organized in a functional engineering view? i.e. how are thos e circuits\nin the human brain arranged to learn how to perform different tasks ?\nIt would be best to understand how the human brain works before b uilding\nan artificial system. However, at the present time, our understa nding of the\nbrain is limited. But at least we know that local recurrency plays critic al rule\nin the higher part of the human brain[6][7][8][4].\nAnother reason to use SRNs is that SRNs can be very useful in ADP m ath-\nematically. Now we will discuss what ADP can do for intelligent control a nd\nunderstanding the brain.\nThe remainder of this chapter will address three questions in order :\n1. What is ADP?\n2. What is the importance of ADP to intelligent control and understa nding the\nbrain?\n3. What is the importance of SRNs to ADP?\n2.1 What is ADP and J Function?\nTo explain what is ADP, let us consider the original Bellman equation[9]:\nJ(R(t)) = max\nu(t)\n(U (R(t), u (t))+ < J (R(t + 1)) >)/ (1 + r) − U0 (1)\nwhere r and u0 are constants that are used only in infinite-time-horizon problems\nand then only sometimes, and where the angle brackets refer to ex pectation\nvalue. In this paper we actually use:\nJ(R(t)) = max\nu(t)\n(U (R(t), u (t))+ < J (R(t + 1)) >) (2)\nsince the maze problem do not involve an infinite time-horizon.\nInstead of solving for the value of J in every possible state, R(t), we can use\na function approximation method like neural networks to approxima te the J\nfunction. This is called approximate dynamic programming(ADP). This paper\nis not doing true ADP because in true ADP we do not know what the J function\nis and must therefore use indirect methods to approximate it. Howe ver, before\nwe try to use SRNs as a component of an ADP system, it makes sense to first\ntest the ability of an SRN to approximate a J function, in principle.\nNow we will try to explain what is the intuitive meaning of the Bellman\nequation(Equation(1)) and the J function according to the treatment taken\nfrom[2].\nTo understand ADP, one must first review the basics of classical dy namic\nprogramming, especially the versions developed by Howard[28] and B ertsekas.\n\nClassical dynamic programming is the only exact and efficient method t o com-\npute the optimal control policy over time, in a general nonlinear sto chastic envi-\nronment. The only reason to approximate it is to reduce computatio nal cost, so\nas to make the method affordable (feasible) across a wide range of a pplications.\nIn dynamic programming, the user supplies a utility function which may take\nthe form U (R(t), u (t)) — where the vector R is a Representation or estimate of\nthe state of the environment (i.e. the state vector) — and a stoch astic model of\nthe plant or environment. Then ”dynamic programming” (i.e. solution of the\nBellman equation) gives us back a secondary or strategic utility func tion J(R).\nThe basic theorem is that maximizing U (R(t), u (t)) + J(R(t + 1)) yields the\noptimal strategy, the policy which will maximize the expected value of U added\nup over all future time. Thus dynamic programming converts a difficu lt prob-\nlem in optimizing over many time intervals into a straightforward proble m in\nshort-term maximization. In classical dynamic programming, we find the exact\nfunction J which exactly solves the Bellman equation. In ADP, we learn a kind\nof ”model” of the function J; this ”model” is called a ”Critic.” (Alternatively,\nsome methods learn a model of the derivatives of J with respect to the variables\nRi ; these correspond to Lagrange multipliers, λ i , and to the ”price variables”\nof microeconomic theory. Some methods learn a function related to J, as in the\nAction-Dependent Adaptive Critic (ADAC)[29].\n2.2 Intelligent Control and Robust Control\nTo understand the human brain scientifically, we must have some suit able math-\nematical concepts. Since the human brain makes decisions like a cont rol system,\nit is an example of an intelligent control system. Neuroscientists do n ot yet un-\nderstand the general ability of the human brain to learn to perform new tasks\nand solve new problems even though they have studied the brain for decades.\nSome people compare the past research in this field to what would hap pen\nif we spent years to study radios without knowing the mathematics o f signal\nprocessing.\nWe first need some mathematical ideas of how it is possible for a compu ting\nsystem to have this kind of capability based on distributed parallel co mputation.\nThen we must ask what are the most important abilities of the human b rain\nwhich unify all of its more specific abilities in specific tasks. It would be s een\nthat the most important ability of brain is the ability to learn over time h ow to\nmake better decisions in order to better maximize the goals of the or ganism. The\nnatural way to imitate this capability in engineering systems is to build s ystems\nwhich learn over time how to make decisions which maximize some measur e\nof success or utility over future time. In this context, dynamic pro gramming\nis important because it is the only exact and efficient method for maxim izing\nutility over future time. In the general situation, where random dis turbances\nand nonlinearity are expected, ADP is important because it provides both the\nlearning capability and the possibility of reducing computational cost to an\n\naffordable level. For this reason, ADP is the only approach we have to imitating\nthis kind of ability of the brain.\nThe similarity between some ADP designs and the circuitry of the brain has\nbeen discussed at length in [10] and [11]. For example, there is an impor tant\nstructure in the brain called the limbic system which performs some kin ds of\nevaluation or reinforcement functions, very similar to the function s of the neural\nnetworks that must approximate the J function of dynamic programming. The\nlargest part of the limbic system, called the hippocampus, is known to possess\na higher degree of local recurrency[8].\nIn general, there are two ways to make classical controllers stable despite\ngreat uncertainty about parameters of the plant to be controlled . For example,\nin controlling a high speed aircraft, the location of the center of the gravity\nis not known. The center of gravity is not known exactly because it d epends\non the cargo of the air plane and the location of the passengers. On e way to\naccount for such uncertainties is to use adaptive control method s. We can get\nsimilar results, but more assurance of stability in most cases[16] by u sing related\nneural network methods, such as adaptive critics with recurrent networks. It is\nlike adaptive control but more general. There is another approach called robust\ncontrol or H∞ control, which trys to design a fixed controller which remains\nstable over a large range in parameter space. Baras and Patel[31] h ave for\nthe first time solved the general problem of H∞ control for general partially\nobserved nonlinear plants. They have shown that this problem redu ces to a\nproblem in nonlinear, stochastic optimization. Adaptive dynamic prog ramming\nmakes it possible to solve large scale problems of this type.\n2.3 Importance of the SRN to ADP\nADP systems already exist which perform relatively simple control ta sks like\nstabilizing an aircraft as it lands under windy conditions [12]. However t his\nkind of task does not really represent the highest level of intelligenc e or planning.\nTrue intelligent control requires the ability to make decisions when fu ture time\nperiods will follow a complicated, unknown path starting from the initia l state.\nOne example of a challenge for intelligent control is the problem of nav igating a\nmaze which we will discuss in chapter 4. A true intelligent control syst em should\nbe able to learn this kind of task. However, the ADP systems in use to day\ncould never learn this kind of task. They use conventional neural n etworks to\napproximate the J function. Because the conventional MLP cannot approximate\nsuch a J function, we may deduce that ADP system constructed only from\nMLPs will never be able to display this kind of intelligent control. Theref ore,\nit is essential that we can find a kind of neural network which can per form\nthis kind of task. As we will show, the SRN can fill this crucial gap. The re\nare additional reasons for believing that the SRN may be crucial to in telligent\ncontrol as discussed in chapter 13 of [9].\n\n3 Alternative Forms of Recurrent Networks\n3.1 Recurrent Networks in General\nThere is a huge literature on recurrent networks. Biologists have u sed many\nrecurrent models because the existence of recurrency in the bra in is obvious.\nHowever, most of the recurrent networks implemented so far hav e been clas-\nsic style recurrent networks, as shown on the left hand of Figure 2 . Most of\nthese networks are formulated from ordinary differential equatio n(ODE) sys-\ntems. Usually their learning is based on a restricted concept of Hebb ian learn-\ning. Originally in the neural network field, the most popular neural ne tworks\nwere recurrent networks like those which Hopfield[14] and Grossbe rg[15] used\nto provide associative memory.\nFEATURE EXTRACTION\nART,SOM, ...\nMINIMIZATION\nHOPFIELD, CAUCHY\nCLASSICAL RECURRENT NETWORKS\nRECURRENT NETWORKS\nTLRN\nSRN\n(Dynamic Systems,\nPrediction)\n(Better function\napproximation)\nASSOCIATIVE MEMORY\nSTATIC FUNCTION\nCLUSTERING\nHOPFIELD, HASSOUN\nFigure 2: Recurrent networks\nAssociative memory networks can actually be applied to supervised le arning.\nBut in actuality their capabilities are very similar to those of look-up ta bles and\nradial basis functions. They make predictions based on similarity to p revious\n\nexamples or prototypes. They do not really try to estimate genera l functional\nrelationships. As a result these methods have become unpopular in p ractical\napplications of supervised learning. The theorems of Barron discus sed in the\nIntroduction show that MLPs do provide better function approxim ation than\ndo simple methods based on similarity.\nThere has been substantial progress in the past few years in deve loping new\nassociative memory designs. Nevertheless, the MLP is still better f or the specific\ntask of function approximation which is the focus of this paper.\nIn a similar way, classic recurrent networks have been used for tas ks like\nclustering, feature extraction and static function optimization. B ut these are\ndifferent problems from what we are trying to solve here.\nActually the problem of static optimization will be considered in future\nstages of this research. We hope that the SRN can be useful in suc h appli-\ncations after we have used it for supervised learning. When people u se the\nclassic Hopfield networks for static optimization, they specify all th e weights\nand connections in advance[14]. This has limited the success of this kin d of\nnetwork for large scale problems where it is difficult to guess the weigh ts. With\nthe SRN we have methods to train the weights in that kind of structu re. Thus\nthe guessing is no longer needed. However, to use SRNs in that applic ation\nrequires refinement beyond the scope of this paper.\nThere have also been researchers using ODE neural networks who have tried\nto use training schemes based on a minimization of error instead of He bbian\napproaches. However, in practical applications of such networks , it is important\nto consider the clock rates of computation and data sampling. For t hat reason,\nit is both easier and better to use error minimizing designs based on dis crete\ntime rather than ODE.\n3.2 Structure of Discrete-Time Recurrent Networks\nIf the importance of neural networks is measured by the number o f words pub-\nlished, then the classic networks dominate the field of recurrent ne tworks. How-\never, if the value is measured based on economic value of practical a pplication,\nthen the field is dominated by time-lagged recurrent networks(TLR Ns). The\npurpose of the TLRN is to predict or classify time-varying systems u sing re-\ncurrency as a way to provide memory of the past. The SRN has some relation\nwith the TLRN but it is designed to perform a fundamentally different t ask.\nThe SRN uses recurrency to represent more complex relationships between one\ninput vector X(t) and one output Y (t) without consideration of the other times\nt. Figure 3 and Figure 4 show us more details about the TLRN and the SR N.\nIn control applications, u(t) represents the control variables which we use to\ncontrol the plant. For example, if we design a controller for a car en gine, the X(t)\nvariables are the data we get from our sensors. The u(t) variables would include\nthe valve settings which we use to try to control the process of co mbustion. The\nR(t) variables provide a way for the neural networks to remember pas t time\n\nZ -1\nX(t)\nR(t)\nX(t+1)\nu(t)\nTLRN\nFigure 3: Time lagged recurrent network(TLRN)\nX\nf\ny\nFigure 4: Simultaneous recurrent network(SRN)\n\ncycles, and to implicitly estimate important variables which cannot be o bserved\ndirectly. In fact, the application of TLRNs to automobile control is t he most\nvaluable application of recurrent networks ever developed so far.\nA simultaneous recurrent network(Figure 4) is defined as a mapping :\nˆY (t) = F (X(t), W ) (3)\nwhich is computed by iterating over the following equation:\ny(n+1)(t) = f (y(n)(t), X (t), W ) (4)\nwhere f is some sort of feed-forward network or system, and ˆY is defined as:\nˆY (t) = lim\nn→∞\ny(n)(t) (5)\nWhen we use ˆY in this paper, we use n = 20 instead of ∞ here.\nIn Figure 4, the outputs of the neural network come back again as inputs\nto the same network. However, in concept there is no time delay. Th e inputs\nand outputs should be simultaneous. That is why it is called a simultaneo us\nrecurrent network(SRN). In practice, of course, there will alwa ys be some phys-\nical time delay between the outputs and the inputs. However if the S RN is\nimplemented in fast computers, this time delay may be very small comp ared to\nthe delay between different frames of input data.\nIn Figure 4, X refers to the input data at the current time frame t. The vector\ny represents the temporary output of the network, which is then r ecycled as an\nadditional set of inputs to the network. At the center of the SRN is actually the\nfeed-forward network which implements the function f . (In designing an SRN,\nyou can choose any feed-forward network or system as you like. T he function f\nsimply describes which network you use). The output of the SRN at a ny time\nt is simply the limit of the temporary output y.\nIn Equation (3) and (4), notice that there are two integers — n and t —\nwhich could both represent some kind of time. The integer t represents a slower\nkind of time cycle, like the delay between frames of incoming data. The integer\nn represents a faster kind of time, like the computing cycle of a fast e lectronic\nchip. For example, if we build a computer to analyze images coming from a\nmovie camera, ” t” and ” t + 1” represent two successive incoming pictures with\na movie camera. There are usually only 32 frames per second. (In th e human\nbrain, it seems that there are only about 10 frames per second com ing into the\nneocortex.) But if we use a fast neural network chip, the computa tional cycle\n— the time between ” n” and ” n + 1” — could be as small as a microsecond.\nIn actuality, it is not necessary to choose between time-lagged rec urrency\n(from t to t + 1) and simultaneous recurrency (from n to n + 1). It is possible to\nbuild a hybrid system which contains both types of recurrency. This could be\nvery useful in analyzing data like movie pictures, where we need both memory\nand some ability to segment the images. [9] discusses how to build such a hybrid.\n\nHowever, before building such a hybrid, we must first learn to make S RNs work\nby themselves.\nFinally, please note that the TLRN is not the only kind of neural netwo rk\nused in predicting dynamical systems. Even more popular is the time d elayed\nneural network(TDNN), shown in Figure 5. The TDNN is popular beca use it\nis easy to use. However, it has less capability, in principle, because it h as no\nability to estimate unknown variables. It is especially weak when some o f these\nvariables change slowly over time and require memory which persists o ver long\ntime periods. In addition, the TLRN fits the requirements of ADP dire ctly,\nwhile the TDNN does not[9][16].\nX(t)\nX(t-1)\nX(t-k)\nTDNN\nX(t+1)\nu(t-k)\nu(t-1)\nu(t)\nFigure 5: Time delayed neural network(TDNN)\n3.3 Training of SRNs and TLRNs\nThere are many types of training that have been used for recurre nt networks.\nDifferent types of training give rise to different kinds of capabilities fo r different\ntasks. For the tasks which we have described for the SRN and the T LRN, the\nproper forms of training all involve some calculation of the derivative s of error\nwith respects to the weights. Usually after these derivatives are k nown, the\nweights are adapted according to a simple formula as follows:\nnewWi,j = oldWi,j − LR ∗ ∂Error\n∂W i,j\n(6)\nwhere LR is called the learning rate.\nThere are five main ways to train SRNs, all based on different method s for\ncalculating or approximating the derivatives. Four of these method s can also\nbe used with TLRNs. Some can be used for control applications. But the\ndetails of those applications are beyond the scope of this paper. Th ese five\n\ntypes of training are listed in Figure 6. For this paper, we have used t wo of\nthese methods: Backpropagation through time(BTT) and Trunca tion.\nTypes of SRN\nSimultaneous Backpropagation\nBackpropagation Through Time\nForward Propagation\nTraining\nTruncation\nError Critics\nFigure 6: Types of SRN Training\nThe five methods are:\n1. Backpropagation through time(BTT). This method and forward propa-\ngation are the two methods which calculate the derivatives exactly. BTT is also\nless expensive than forward propagation.\n2. Truncation. This is the simplest and least expensive method. It us es only\none simple pass of backpropagation through the last iteration of th e model.\nTruncation is probably the most popular method used to adapt SRNs even\nthough the people who use it mostly just call it ordinary backpropag ation.\n3. Simultaneous backpropagation. This is more complex than trunca tion,\nbut it still can be used in real time learning. It calculates derivatives w hich are\nexact in the neighborhood of equilibrium but it does not account for t he details\nof the network before it reaches the neighborhood of equilibrium.\n4. Error critics(shown in Figure 7). This provides a general approx imation\nto BTT which is suitable for use in real-time learning[9].\n\nTLRN\nTLRN\nCriticError\nCriticError\nError\nError\nR(t)\nX(t)\nu(t)\nX(t) u(t)\nX(t+1) X(t+1) X(t=1)\nu(t+1)\nR(t+1)\nλ^ (t)\nλ^\nλ (t)\n(t+1)\nFigure 7: Error Critics\n\n5. Forward propagation. This, like BTT, calculates exact derivative s. It\nis often considered suitable for real-time learning because the calcu lations go\nforward in time. However, when there are n neurons and m connections, the\ncost of this method per unit of time is proportional to n ∗ m. Because of this\nhigh cost, forward propagation is not really brain-like any more than BTT.\n3.3.1 Backpropagation through time(BTT)\nBTT is a general method for calculating all the derivative of any outc ome or\nresult of a process which involves repeated calls to the same networ k or net-\nworks used to help calculate some kind of final outcome variable or re sult E.\nIn some applications, E could represent utility, performance, cost or other such\nvariables. But in this paper, E will be used to represent error. BTT was first\nproposed and implemented in [17]. The general form of BTT is as follows :\nfor k = 1 to T do forward\ncalculation(k);\ncalculate result E;\ncalculate direct derivatives of E with respect to outputs of forward calculations;\nfor k = T to 1 backpropagate through forwards calculation(k), calculating run-\nning totals where appropriate.\nThese steps are illustrated in Figure 8. Notice that this algorithm can be\napplied to all kinds of calculations. Thus we can apply it to cases where k\nrepresents data frames t as in the TLRNs, or to cases where k represents internal\niterations n as in the SRNs. Also note that each box of calculation receives input\nfrom some dashed lines which represent the derivatives of E with respect to the\noutput of the box. In order to calculate the derivatives coming out of each\ncalculation box, one simply uses backpropagation through the calcu lation of\nthat box starting out from the incoming derivatives. We will explain in m ore\ndetail how this works in the SRN case and the TLRN case.\nSo far as we know BTT has been applied in published working systems fo r\nTLRNs and for control, but not yet for SRNs until now. However, R umelhart,\nHinton and Williams[18] did suggest that someone should try this.\nThe application of BTT for TLRNs is described at length in [2] and [9]. The\nprocedure is illustrated in Figure 9. In this example the total error is actually\nthe sum of error over each time t where t goes from 1 to T . Therefore the\noutputs of the TLRN at each time t (t < T ) have two ways of changing total\nerrors:\n(1)A direct way when the current predictions ˆY (t) are different from the\ncurrent targets Y (t);\n(2)An indirect way based on the impact of R(t) on errors in later time\nperiods.\nTherefore the derivative feedback coming into the TLRN is actually t he sum\nof two feedbacks from two different sources. As a technical deta il, note that\nR(0) needs to be specified somehow. However, we will not discuss this point\nhere because the focus of this paper is on SRNs.\n\nk=T\nk=1\nk=2\nResult\n(Error, Utility\nCalculation\nCalculation\nError\nError\nError\nCalculation\n)\nFigure 8: Backpropagation through time(BTT)\n\nX(T) Y(T) Y(T)\nY(1)Y(1)X(1)\nX(2) Y(2) Y(2)\nR(0)\nR(T-1)\nR(2)\nR(1)\nTLRN\nTLRN\nTLRN\n^\n^\n^\nFigure 9: BTT for TLRN\n\ny(1)\ny(0)X(t)\nF_y(2)\nF_y(1)\nF_y(N)\nError\nY\nY\n= y(n)^\nFigure 10: BTT for SRN\n\nFigure 10 shows the application of BTT to training an SRN. This figure\nalso provides some explanation of our computer code in the appendix . In this\nfigure, the left-hand side(the solid arrows) represents the neur al network which\npredicts our desired output Y . (In our example, Y represents the true values of\nthe J function across all points in the maze). Each box on the left repres ents a\ncall to a feed-forward system. The vector X(t) represents the external inputs to\nthe entire system. In our case, X(t) consists of two variables, indicating which\nsquares in the maze contain obstacles and which contains the goal r espectively.\nFor simplicity, we selected the initial vector y(0) as a constant vector as we\nwill describe below. Each call to the feed-forward system includes c alls to a\nsubroutine which implements the generalized MLP.\nOn the right-hand side of Figure 10, we illustrate the backpropagat ion cal-\nculation used to calculate the derivatives. For the SRN, unlike the TL RN, the\nfinal error depends directly only on the output of the last iteration . Therefore\nthe last iteration receives feedback only from the final error but t he other iter-\nations receive feedback only from the iterations just after them. Each box on\nthe right-hand side represents a backpropagation calculation thr ough the feed-\nforward system on its left. The actual backpropagation calculatio n involves\nmultiple calls to the dual subroutine F net2, which is similar to a subroutine in\nchapter 8 of [2].\nNotice that the derivative calculation here costs about the same am ount as\nthe forward calculation on the left-hand side. Thus BTT is very inexp ensive in\nterms of computer time. However, the backpropagation calculatio ns do require\nthe storage of many intermediate results. Also we know that the hu man brain\ndoes not perform such extended calculations backward through t ime. Therefore\nBTT is not a plausible model of true brain-like intelligence. We use it here\nbecause it is exact and therefore has the best chance to solve this difficult\nproblem never before solved. In future research, we may try to s ee whether this\nproblem can also be solved in a more brain-like fashion.\n3.3.2 T runcation for SRNs\nTruncation is probably the most popular method to train SRNs even t hough the\nterm truncation is not often used. For example, the ”simple recurr ent networks”\nused in psychology are typically just SRNs adapted by truncation[19 ].\nStrictly speaking there are two kinds of truncation — ordinary one- step trun-\ncation(Figure 11) and multi-step truncation which is actually a form o f BTT.\nOrdinary truncation is by far the most popular. In the derivative ca lculation\nof ordinary truncation, the memory inputs to the last iteration are treated as if\nthey were fixed external inputs to the network. In truncation th ere is only one\npass of ordinary backpropagation involving only the last iteration of the net-\nwork. Many people have adapted recurrent networks in this simple w ay because\nit seems so obvious. However, the derivatives calculated in this way a re not\nexactly the same because they do not totally represent the impact of changing\n\nthe weights on the final error. The reason for this is that changing the weights\nwill change the inputs to the final iteration. It is not right to treat t hese inputs\nas constants because they are changed when the weights are cha nged.\ny(1)\ny(0)X(t)\nF_y(N)\nError\nY\nY\n= y(n)^\nFigure 11: Truncation\nThe difference between truncation and BTT can be seen even in a simp le\nscalar example, where n=2 and the feed-forward calculation is linear . In this\ncase, the feed-forward calculation is:\ny(1) = A ∗ y(0) + B ∗ X (7)\ny(2) = A ∗ y(1) + B ∗ X (8)\nIn additon,\nE = Error = 1\n2 (Y − y(2))2 (9)\n\n∂E\n∂y (2) = y(2) − Y (10)\nIn truncation, we use Equation (8) and deduce:\n∂E\n∂B = ∂E\n∂y (2) ∗ ∂y (2)\n∂B = (y(2) − Y ) ∗ X (11)\nBut for a complete calculation, we substitute (7) into (8), deriving:\ny(2) = A2 ∗ y(0) + A ∗ B ∗ X + B ∗ X (12)\nwhich yields:\n∂E\n∂B = (y(2) − Y ) ∗ (A ∗ X + X) (13)\nThe result in Equation(11) is usually different from the result in Equa-\ntion(13), which is the true result, and comes from BTT. Depending o n the\nvalue of A, these results could even have opposite signs.\nIn this paper, we have tried to used truncation because it is so easy and so\npopular. If truncation had worked, it would be the easiest way to so lve this\nproblem. However, it did not work.\n3.3.3 Simultaneous Backpropagation\nSimultaneous backpropagation is a method developed independently in differ-\nent forms by Werbos, Almeida and Pineda[20][21][22]. The most general form\nof this method for SRNs can be found in chapter 3 of[9] and in [23]. This\nmethod is guaranteed to converge to the exact derivatives for th e neighborhood\nof the equilibrium y(∞ ) in the case where the forward calculations have reached\nequilibrium[20].\nAs with BTT, the derivative calculations are not expensive. Unlike BTT\nthere is no need for intermediate storage or for calculation backwa rd through\ntime. Therefore simultaneous backpropagation could be plausible as a model\nof learning in the brain. On the other hand, these derivative calculat ions do\nnot account for the details of what happened in the early iterations . Unlike\nBTT, they are not guaranteed to be exact in the case where the fin al y(n) is\nnot an exact equilibrium. Even in modeling the brain there may be some n eed\nto train SRNs so as to improve the calculation in early iterations. In su mmary,\nthough simultaneous backpropagation may be powerful enough to solve this\nproblem, there was sufficient doubt that we decided to wait until late r before\nexperimenting with this method.\n\n3.3.4 Error Critic\nThe Error Critic, like simultaneous backpropagation, provides appr oximate deriva-\ntives. Unlike simultaneous backpropagation, it has no guarantee of yielding ex-\nact results in equilibrium. On the other hand, because it approximate s BTT\ndirectly in a statistically consistent manner, it can account for the e arly itera-\ntions. Chapter 13 of [9] has argued that the Error Critic is the only p lausible\nmodel for how the human brain adapts the TLRNs in the neocortex. It would\nbe straightforward in principle to apply the Error Critic to training SR Ns as\nwell.\nFigure 7 shows the idea of an Error Critic for TLRNs. This figure shou ld be\ncompared with Figure 10. The dashed input coming into the TLRN in Figu re\n7 is intended to be an approximation of the same dashed line coming into the\nTLRN in the Figure 9. In effect, the Error Critic is simply a neural netw ork\ntrained to approximate the complex calculations which lead up to that dashed\nline in the Figure 8. The line which ends as the dashed line in Figure 7 begins\nas a solid line because those derivatives are estimated as the ordinar y output of\na neural network, the Error Critic. In order to train the Error Cr itic to output\nsuch approximations, we use the error calculation illustrated on the lower right\nof Figure 7. In this case, the output of the Error Critic from the pr evious\ntime period is compared against a set of targets coming from the TLR N. These\ntargets are simply the derivatives which come out of the TLRN after one pass of\nbackpropagation starting from these estimated derivatives from the later time\nperiod. This kind of training may seem a bit circular but in fact it has an e xact\nparallel to the kind of bootstrapping used in the well known designs f or adaptive\ncritics or ADP.\nAs with simultaneous backpropagation, we intend to explore this kind of\ndesign in the future, now that we have shown how SRNs can in fact so lve the\nmaze problem.\n3.3.5 F orward Propagation\nThe major characteristics of this method have been described abo ve. This\nmethod has been independently rediscovered many times with minor v ariations.\nFor example, in 1981 Werbos called it conventional perturbation[2]. W illiams\nhas called it the Williams – Zipser method[5]. Narendra has called it dynamic\nbackpropagation.\nNevertheless, because this method is more expensive than BTT, ha s no per-\nformance advantage over BTT, and is not plausible as a model of lear ning in\nthe brain, we see no reason to use this method.\n\n4 Two Test Problems\nIn this paper we use two examples to show that the SRN design has mo re general\nfunction approximation capabilities than does the MLP. Our primary f ocus was\non the maze problem because of its relation to intelligent control as d iscussed\nin Chapters 1 and 2. However, before studying this more specialized example,\nwe performed a few experiments on a more general problem which we call Net\nA/Net B. This chapter will discuss these two problems in more detail.\n4.1 Net A/Net B\nIn the Net A/Net B problem, our fundamental goal is to explore the idea that\nthe functions that an MLP can approximate are a subset of what an SRN can.\nIn other words, we hypothesize that an SRN can learn to approxima te any\nfunctions which an MLP can represent without adding too much comp lexity,\nbut not vice versa. To consider the functions which an MLP can repr esent, we\ncan simply sample a set of randomly selected MLPs, and then test the ability\nof SRNs to learn those functions. Similarly we can generate SRNs at r andom\nand test the ability of MLPs to learn to approximate the SRNs.\nIn order to implement this idea, we used the approach shown in Figure 12.\nThe first step in the process was to pick Net A at random. In some ex periments,\nNet A was an SRN, while in the other experiments, it was an MLP. In all t hese\nexperiments, Net B was chosen to be the opposite kind of network f rom Net A.\nIn picking Net A, we always used the same feed-forward structure . But we used\na random number generator to set the weights. After Net A was ch osen and\nNet B was initialized, we generated a stream of random inputs betwee n -1 and\n+1 following a uniform distribution. For each set of inputs, we trained Net B\nto try to imitate the output of Net A. Of course Net A was fixed. The results\ngave an indication of the ability of Net B to approximate Net A.\nNet B\nNet A\nRandom Inputs\nFigure 12: Net A/Net B\n\nOur preliminary experiments did show that the SRNs have some advan tage\nover the MLPs. However, in all of these experiments, the SRN was t rained\nwith truncation, not BTT. To fully explore all the theoretical issues would\nrequire a much larger set of computer runs. Still, these initial exper iments were\nvery useful in testing some general computer code in order to pre pare for the\ncomplexities of the maze problem.\n4.2 The Maze Problem\nIn the classic form of the maze problem, a little robot is asked to find t he\nshortest path from the starting position to a goal position on a two -dimensional\nsurface where there are some obstacles. For simplicity, this surfa ce is usually\nrepresented as a kind of chess board or grid of squares in which eve ry square is\neither clear or blocked by an obstacle. In formal terms, this means that we can\ndescribe the state of the maze by providing three pieces of informa tion:\n(1) An array A[ix][iy] which has the value 0 when the square is clear and 1 when\nit is covered by an obstacle;\n(2) The coordinates of the goal;\n(3) The coordinates of the starting square.\nIn our case, we used a large number to represent the obstacles.\nAs discussed in the introduction, many researchers have trained n eural net-\nworks to learn an individual maze[5]. Our goal was to train a network t o input\nthe array A and to output J[ix][iy] for all the clear squares. According to dy-\nnamic programming, the best strategy of motion for a robot is simply to move\nto that neighboring square which has the smallest J.\nThis more general problem has not been solved before with neural n etworks.\nFor example, Houillon etc[24] initially attempted to solve this problem wit h\nMLPs, but were unsuccessful. Widrow in several plenary talks has r eported\nthat his neural truck backer upper has some ability to see and avoid obstacles.\nHowever, this ability was based on an externally developed potential function\nwhich was not itself learned by neural networks. Such potential fu nctions are\nanalogous to the J function which we are trying to learn.\nIn fact, this maze problem can always be solved directly and economic ally\nby dynamic programming. Why then do we bother to use a neural net work on\nthis problem? The reason for using this test is not because this simple maze\nis important for its own sake, but because this is a very difficult proble m for\na learning system, and because the availability of the correct solutio n is very\nuseful for testing. It is one thing for a human being to know the ans wer to a\nproblem. It is a different thing to build a learning system which can figur e out\nthe answer for itself. Once the simple maze problem is fully conquered , we can\nthen move on to solve more difficult navigation problems which are too c omplex\nfor exact dynamic programming.\nIn order to represent the maze problem as a problem for supervise d learning,\n\nwe need to generate both the inputs to the network(the array A) and the desired\noutputs(the array B)(Refer to the Appendix). For this basic experiment, we\nchose to study the example maze shown in Figure 13. In this figure, G represents\nthe goal position, which is assigned a value of ”1”; the other number s represent\nthe true values of the J function as calculated by dynamic programming (sub-\nroutine ”Synthesis” in the attached code in the appendix). Intuitiv ely each J\nvalue represents the length of the shortest path from that squa re to the goal.\n10 2 3 4 5 6\n4 35\n7 8\nG 2\n2 3\nFigure 13: Desired J function of a maze\nInitially we chose to study this particular maze because it poses some very\nunique difficulties. In particular there are four equally good direction s starting\nfrom one of these squares in this maze — a feature which can be very confusing\nto neural networks, even human. If we had used a fully connected conventional\nneural network, then the use of a single test maze would have led to over-training\nand meaningless results. However, as we will discuss later in this chap ter, we\nconstrained all of our networks to prevent this problem. Neverth eless, a major\ngoal of our future research will be to test the ability of SRNs to pre dict new\nmazes after training on different mazes.\nThis problem of maze navigation has some similarity to the problem of co n-\nnectedness described by Minsky[25]. Logically we know that the desir ed output\nin any square can depend on the situation in any other square. Ther efore, it\nis hard to believe that a simple feed-forward calculation can solve this kind of\nproblem. On the other hand, the Bellman equation(Equation(1)) its elf is a sim-\nple recurrent equation based on relationships between ”neighborin g”(successive)\n\nstates. Therefore it is natural to expect that a recurrent stru cture could approx-\nimate a J function. The empirical results in this paper confirm these expectio ns.\n5 Details of Architecture and Learning Proce-\ndure\nThe architecture and learning used for the net A/Net B problem wer e all very\nstandard. They will be discussed briefly in section 5.1. The bulk of this chapter\nwill then describe the two special feature – cellular architecture an d adaptive\nlearning rate(ALR) used for the maze problem.\n5.1 Details for the Net A/Net B Problem\nIn all these experiments, the MLP network and the feed-forward network f in\nthe SRN was a standard MLP with two hidden layers. The input vector X\nconsisted of six numbers between -1 and +1. The two hidden layers a nd the\noutput layers all had three neurons. The initial weights were chose n at random\naccording to a uniform distribution between -1 and +1. Training was d one by\nstandard backpropagation with a learning rate of 0 . 1.\n5.2 Weight-sharing and Cellular Architecture\n5.2.1 What is Weight-sharing?\nIn theoretical terms, weight-sharing is a generalized technique fo r exploiting\nprior knowledge about some symmetry in the function to be approxim ated.\nWeight-sharing has sometimes been called ”windowing” or ”Lie Group” tech-\nniques.\nWeight-sharing has been used almost exclusively for applications like c har-\nacter recognition or image processing where the inputs form a two- dimensional\narray of pixels[3][26]. In our maze problem the inputs and outputs also form\narrays of pixels. Weight-sharing leads to a reduction in the number o f weights.\nFewer weights lead in turn to better generalization and easier learnin g.\nAs an example, suppose that we have an array of hidden neurons wit h volt-\nages net[ix][iy], while the input pixels form an array X[ix][iy]. In that case, the\nvoltages for a conventional MLP would be determined by the equatio n:\nnet[i][j] =\n∑\nix,iy\nW (i, j, ix, iy ) ∗ X(ix, iy ) (14)\nThus if each array has a size 20 ∗ 20, the weights form an array of size\n20 ∗ 20 ∗ 20 ∗ 20. This means 160,000 weights — a very big problem. In basic\nweight-sharing, this equation would be replaced by:\n\nnet[i][j] =\n∑\nd1,d2\nW (d1, d 2) ∗ X(i + d1, i + d2) (15)\nFurthermore, if d1 and d2 are limited to an range like [ − 5, 5], then the\nnumber of weights can be reduced to just over 100. Actually this wo uld make\nit possible to add two or three additional types of hidden neurons wit hout\nexceeding 1,000 weights. This trick was used by Guyon etc[3]. They us ed it to\ndevelop the most successful zip code digit recognizer in existence.\nIntuitively AT &T justified this idea by arguing that similar patterns in dif-\nferent locations have similar meanings. However, there is a more rigo rous math-\nematical justification for this procedure as we will see.\n5.2.2 Lie Group Symmetry and Weight-sharing\nThe technique of weight-sharing in neural networks is really just a s pecial case\nof the Lie-group method pioneered much earlier by Laveen Kanal an d others\nin image processing. Formally speaking, if we know that the function F to be\napproximated must obey a certain symmetry requirement then we c an impose\nthe same symmetry on the neural network which we use to approxim ate F .\nMore preciously, if Y = F (x) always implies that M Y = F (M x), where M is\nsome kind of simple linear transformation, then we can require that t he neural\nnetwork possess the same symmetry.\nBoth in image processing and in the maze problem, we can use the symm etry\nwith respect to those transformations M which move all the pixels by the same\ndistance to the left, to the right or up and down. In the language of physics,\nthese are called spatial translations.\nBecause we know that the best form of the neural network must a lso obey\nthis symmetry, we have nothing to lose by restricting our weights as required\nby the symmetry.\n5.2.3 How We implemented Weight-sharing\nIn order to exploit Lie group symmetry in a rigorous way, we first ref ormulated\nthe task to be solved so as to ensure exact Lie group symmetry. To do this, we\ndesigned our neural network to solve the problem of maze defined o ver a torus.\nFor our purposes, a torus was simply an N by N square where the right-hand\nneighbor of [ i, N ] is the point [ i, 0], and likewise for the other edges. This system\ncan still solve an ordinary maze as in Figure 13, where the maze is surr ounded\nby walls of obstacles.\nNext we used a cellular structure for our neural network including b oth the\nMLPs and SRNs. A cellular structure means that the network is made up of a\nset of cells each made up of a set of neurons. There is one cell for ea ch square\nin the maze. The neurons and the weights in each cell are the same as those\n\nin any other cell. Only the inputs and outputs are different because t hey come\nfrom different locations.\nThe general idea of our design is shown in Figure 14. Notice that each cell is\nmade up of two parts: a connector part and a local memory part wh ich includes\n4 neighbors and the memory from itself in Fingure 15. The connector part\nreceives the inputs to the cell and transmits its output to all four n eighboring\ncells. In addition, the local memory part sends all its outputs back a s inputs,\nbut only to the same cell. Finally the forecast of J is based on the output of\nthe local memory part.\nLocalLocalLocalLocal\nY(ix,iy)^\ngoal\nCONNECTOR\nobstacle\nGeneral Idea of Cellular Network\nneigbourneigbour\nConnector Connector\ngoal obstacle\nFigure 14: General Idea of the Cellular Network\nThe exact structure which we used is shown completely in Figure 15. I n this\nfigure it can be seen that each cell receives 11 inputs on each iterat ion. Two of\nthese inputs represent the goal and obstacle variables, A[ix][iy] and B[ix][iy], for\nthe current pixel. The next four inputs represent the outputs of the connector\nneuron from the four neighboring cells from the previous iteration. The final\nfive inputs are simply the outputs of the same cell from the previous iteration.\nThen after the inputs, there are only five actual neurons. The co nnector part\nis only one neuron in our case. The local memory part is four neurons . The\nprediction of J[ix][iy] results from multiplying the output of the last neuron by\nW s, a weight used to rescale the output.\nTo complete this description, we must specify how the five active neu rons\nwork. In this case, each neuron takes inputs from all of the neuro ns to its left,\nas in the generalized MLP design[2]. Except for ˆJ, all of the inputs and outputs\n\nmemory\nfrom itself\nmemory\n4 neighbors\nInputs\nWs\ngoal obstacle\nY^Figure 15: Inputs, Outputs and Memory of Each Cell\nrange between -1 and 1, and the tanh function is used in place of the usual\nsigmoid function.\nTo initialize the SRN on iteration zero, we simply picked a reasonable look ing\nconstant vector for the first four neurons out of the five. We se t the initial\nstarting value to -1. For the last neuron, we set it to 0. In future w ork, we shall\nprobably experiment with the adaptation of the starting vector y(0).\nIn order to backpropagate through this entire cellular structure , we simply\napplied the chain rule for ordered derivatives as described in [2].\n5.3 Adaptive Learning Rate\nIn our initial experiments with this structure, we used ordinary dyn amic pro-\ngramming with only one special trick. The trick was that we set the nu mber of\niterations for SRN to only 1 on the first 20 trials, and then to 2 for th e next 20\ntrials... and so on up until there were 20 iterations.\nWe found that ordinary weight adjustment led to extremely slow lear ning\ndue to oscillation. This was not totally unexpected because slow learn ing and\noscillation are a common result of simple steepest descent methods. There\nare many methods available to accelerate the learning. Some of thes e like the\nDEKF method developed by Ford Motor Company are similar to quasi-N ewton\nmethods[27] which are very powerful but also somewhat expensive . For this work\nwe chose to use a method called the adaptive learning rate(ALR) as d escribed\nin chapter 3 of [9]. This method is relatively simple and cheap, but far mo re\nflexible and powerful than other simple alternatives.\nIn this method, we maintain a single adapted learning rate for each gr oup\nof weights. In this case, we chose three groups of weights:\n1. The weight W s used for rescaling of the output;\n2. The constant or bias weights ww;\n\n3. All the other weights W .\nFor each group of weights the learning rate is updated on each trial according\nto the following formula:\nLR(t + 1) = LR(t) ∗ (0. 9 + 0. 2 ∗\n∑\nk Wk(t) ∗ Wk(t − 1)∑\nk Wk(t − 1) ∗ Wk(t − 1) ) (16)\nwhere the sum over k actually refers to the sum over all weights in the same\ngroup. In addition, to prevent overshoot, we would reset the lear ning rate to:\nLR ∗ E\n∑\nk ( ∂E\n∂W k\n)\n2 (17)\nwhere the sum is taken over all weights. In this special case where t he error on\nthe next iteration would be predicted to be less than zero, i.e.:\nE −\n∑\nk\n(Wk(t + 1) − Wk(t)) ∗ ∂E\n∂W k\n(t)\n= E −\n∑\nk\n(LR ∗ ∂E\n∂W k\n(t)) ∗ ∂E\n∂W k\n(t)\n= E − LR ∗\n∑\nk\n( ∂E\n∂W k\n(t))\n< 0 (18)\nwhere Wk(t + 1) is the new value for the weights which would be used if the\nlearning rates were not reset. In our case, we modified this proced ure slightly\nto apply it separately to each group of weights.\nAfter the adaptive learning rates were installed the process of lear ning be-\ncame far more reliable. Nevertheless, because of the complex natu re of the\nfunction J, there was still some degree of local minimum problem. For our pur-\nposes, it was good enough to simply try out a handful of initial values which we\nguessed at random. However, in future research, we would like to e xplore the\nconcept of shaping as described in [9].\n6 Simulation Results and Conclusions\nIn this chapter, we will see some simulation results for the two test p roblems\ndiscussed before. From analyzing the results, we can conclude tha t compared to\nthe MLPs, the SRNs are more powerful in nonsmooth function appr oximation.\nIn addition, our new design — the cellular structure — can really solve t he maze\nproblem.\n\n6.1 Results for the Net A/Net B Problem\nFrom Figure 16 to Figure 19 we can see that the SRN using the same th ree-\nlayered neural network structure(9 inputs, 3 outputs, and 3 ne urons for each\nhidden layer) as the MLP can achieve better simulation result. The SR N not\nonly converged more rapidly than the MLP(Figure 16 and Figure 17, b ut also\nreached a smaller error(Figure 18 and Figure 19), about 1 . 25 ∗ 10− 4, while the\nMLP reached 5 ∗ 10− 4. Thus we can say that, in this typical case, an SRN has\nbetter ability to learn an MLP than an MLP to learn an SRN.\n0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000\n0.002\n0.004\n0.006\n0.008\n0.01\n0.012\nTrials\nError\nFigure 16: The MLP learned the SRN\n6.2 Results for the Maze Problem\nThere are two parts of the results for the maze problem.\nFirst, we compare the J function in each pixels of the same maze as predicted\nby an SRN trained by BTT and an SRN trained by truncation respectiv ely with\nthe actual J function for the maze. Figure 20 and Figure 21 show th at the SRN\ntrained by BTT can really approximate the J function, but the SRN trained\nby truncation cannot. Moreover, the SRN trained by BTT can learn the ability\nto find the optimal path from the start to the goal as calculated by dynamic\nprogramming. Although there is some error in the approximation of J by the\nSRN trained by BTT, the errors are small enough that a system gov erned by\nthe approximation of J would always move in an optimal direction.\n\n0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000\n0.2\n0.4\n0.6\n0.8\n1.2\n1.4 x 10\n−3\nTrials\nError\nFigure 17: The SRN learned the MLP\n0 200 400 600 800 1000 1200\n10 x 10\n−4\nTrials\nError\nFigure 18: The last 1000 trials of figure 16\n\n0 200 400 600 800 1000 1200\n0.5\n1.5\n2.5\n3 x 10\n−4\nTrials\nError\nFigure 19: The last 1000 trials of figure 17\n10 2 3 4 5 6\n12.1 2.1\n2.92.13.43.9\n6.18.49.1\n8.4\n7.1\n9.1 7.9\n7.1\n7.9\n6.1\n4.6\n3.42.1\n3.9\n4.6\nFigure 20: J function as predicted by SRN-BTT(I)\n\n10 2 3 4 5 6\n3.03.0\n3.7 4.4\n2.9\n2.5 2.5\n2.54.23.92.9\n3.5 4.3\n4.6\n4.5 4.6 3.5 3.0\n3.0\n2.5\n2.6\n3.9\nFigure 21: J function as predicted by SRN-Truncation(I)\nSecond, we show some error curves from Figure 22 to Figure 27. Fr om the\nfigures we can see the error curve of SRN trained by BTT not only co nverged\nmore rapidly than the curve of the SRN trained by truncation, but a lso reached\na much smaller level of error. The errors with the MLP did not improve at all\nafter about 80 trials(Figure 26 and Figure 27).\n6.3 Conclusions\nIn this paper, we have described a new neural network design for J function\napproximation in dynamic programming. We have tested this design in t wo test\nproblems: Net A/Net B and the maze problem. In the Net A/ Net B pro blem,\nwe showed that SRNs can learn to approximate MLPs better than ML Ps can\nlearn SRNs.\nIn the maze problem, a much more complex problem, we showed that w e\ncan achieve good results only by training an SRN with a combination of B TT\nand adaptive learning rates. In addition, we needed to use a special design — a\ncellular structure — to solve this problem. On the other hand, neithe r an MLP\nnor an SRN trained by truncation could solve this problem.\nNow that it has been proven that neural networks can solve these kinds\nof problems, the next step in research is to consider many variation s of these\nproblems in order to demonstrate generalization ability and the ability to solve\noptimization problems while the J function is not known.\n\n0 0.5 1 1.5 2 2.5 3\nx 10\nFigure 22: Error curve of J function as predicted by SRN-BTT(II)\n1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3\nx 10\n0.85\n0.9\n0.95\n1.05\n1.1\n1.15\n1.2\n1.25\n1.3\n1.35\nTrials\nError\nFigure 23: Error curve of J function as predicted by SRN-BTT(III )\n\n0 0.5 1 1.5 2 2.5 3\nx 10\n0.5\n1.5\n2.5\n3.5 x 10\nTrials\nError\nFigure 24: Error curve of J function as predicted by SRN-Truncat ion(II)\n1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3\nx 10\nTrials\nError\nFigure 25: Error curve of J function as predicted by SRN-Truncat ion(III)\n\n0 10 20 30 40 50 60 70 80 90 100\n0.5\n1.5\n2.5 x 10\nTrials\nError\nFigure 26: Error curve of J function as predicted by MLP(I)\n0 100 200 300 400 500 600 700 800 900 1000\n0.5\n1.5\n2.5 x 10\nTrials\nError\nFigure 27: Error curve J function as predicted by MLP(II)\n\nAcknowledgments\nThe work described herein was truly collaborative work. More than h alf\nof it was performed when both authors worked together at the Un iversity of\nMaryland, College Park, or a nearby location. Neither author receiv ed any fi-\nnancial support covering the time when this paper was written; how ever, this\nwork would have been impossible without prior support and ongoing co nnec-\ntions involving: the University of Maryland (Prof. John S. Baras and Robert W.\nNewcomb); the National University of Singapore; the IEEE Singapo re Section;\nScientific Cybernetics Inc.; the International Joint Conference o n Neural Net-\nworks; the National Science Foundation of the U.S. and the Nationa l Natural\nScience Foundation of China.\nReferences\n[1] E. D. Sontag, “Feedback stabilization using two-hidden-layer ne ts”, IEEE\nTrans. Neural Networks , Vol. 3, No.6, 1992.\n[2] P. Werbos, The Roots of Backpropagation: From Ordered Derivatives to\nNeural Networks and Political Forecasting , Wiley, 1994.\n[3] I. Guyon, I. Poujaud, L. Personnaz, G. Dreyfus, J. 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Chew, Symmetric rank-one update and quasi-\nNewton methods, Optimization Techniques and Applications, Proceedings of\nthe International Conference on Optimization Techniques a nd Applications,\nK.H. Phua et al., eds., World Scientific, 1992, Singapore, pp. 52–63.\n[28] R. Howard, Dynamic Programming and Markhov Processes , MIT press,\nCambridge, MA, 1960.\n[29] P. Werbos, Neural networks for control and system identific ation, IEEE\nConference on Decision and Control (Florida), IEEE, New York, 1989.\n[30] A. Barron, “Asymptotically optimal functional estimation by min imum\ncomplexity criteria”, Proceedings of 1994 IEEE International Symposium\non Information Theory , IEEE, New York, 1994.\n[31] J. S. Baras and N. S. Patel, “Information state for robust co ntrol of set-\nvalued discrete time systems”, Proceedings of 34th Conference on Decision\nand Control , IEEE, 1995, p. 2302.\nA Appendix: The program of the maze problem\nusing SRN trained by BTT\n/* The program of the maze problem using SRN trained by BTT*/\n/* Using the SRN trained by BTT to learn the optimal path of a 5* 5 maze: */\n/* Learning Rate Adaptive --- Lr_Ws,Lr_W,Lr_ww */\n/* When change line 136 and 137 into p=0 then the program will b e MLP*/\n/* When change line 243 into F_x[N+i]=0 then the program will be the\nSRN trined by Truncation*/\n#include <stdio.h>\n#include <stdlib.h>\n#include <math.h>\n#include <time.h>\nvoid F_NET2(double F_Yhat, double W[30][30],double x[30] ,int n,int m,int N,\ndouble F_W[30][30], double F_net[30],double F_Ws[30],do uble Ws,double F_x[30]);\nvoid NET(double W[30][30],double x[30],double ww[30],\nint n, int m, int N, double Yhat[30]);\n\nvoid synthesis(int B[30][30],int A[30][30],int n1,int n2 );\nvoid pweight(double Ws,double F_Ws_T,double ww[30],doub le F_net_T[30],\ndouble W[30][30], double F_W_T[30][30],int n,int N,int m) ;\nint minimum(int s,int t,int u,int v);\nint min(int k,int l);\ndouble f(double x);\nvoid main()\n{\nint i,j,it,iz,ix,iy,lt,m,maxt,n,n1,n2,nn,nm,N,p,q,po ,t,TT;\nint A[30][30],B[30][30];\ndouble a,b,dot,e,e1,e2,es,mu,s,sum,F_Ws_T,Ws,F_Yhat, wi;\ndouble W[30][30],x[30],F_net_T[30],F_Ws[30],F_W_O[30 ][30],F_W[30][30];\ndouble F_W_T[30][30],F_net[30],ww[30],yy[21][12][8][ 8];\ndouble Yhat[30],F_y[21][12][8][8],F_x[30],F_Jhat[30] [30];\ndouble S_F_W1,S_F_W2,Lr_W,S_F_net1,S_F_net2,Lr_ww,Lr _Ws,F_Ws_O;\ndouble y[50][50],F_net_O[50], F_Ws1, F_Ws2,W_O[50][50] ,ww_O[50],Ws_O;\nFILE *f;\n/* Number of inputs,neurons and output:7,3,1 */\n/* ’n’ is the number of the active neurons */\n/* ’m’ and ’N’ both are the number of inputs */\n/* ’nm’ is the number of memory is: 5 */\n/* ’nn+1’*’nn+1’ is the size of the maze’ */\n/* ’TT’ is the number of trials */\n/* ’lt’ is the number of the interval time */\n/* ’maxt’ is the max number for T in figure[8] */\n/* Lr-Ws,Lr_ww and Lr_W are the learning rates for Ws,ww and W */\na=0.9; b=0.2;\nn=5;m=11;N=11;nn=6;nm=5;TT=30000;lt=50;maxt=20;wi=25;Ws=40;\ne=0;po=pow(2,31) -1;\n/* Initial values of Old */\nF_Ws_O=1;\nfor(i=m+1;i<N+n+1;i++)\n{\nfor(j=1;j<i;j++)\nF_W_O[i][j]=1;\nF_net_O[i]=1;\n}\n\nLr_W=Lr_ww=Lr_Ws=10;\n/* Initial values of weights */\nfor (i=1;i<N+n+1;i++)\nx[i]=0;\nfor (i=m+1;i<N+n+1;i++)\nfor (j=0;j<i;j++)\n{\nsrand(rand());\nW[i][j]=rand();\nW[i][j]=0.2091;\n}\nfor (i=m+1;i<N+n+1;i++)\n{\nsrand(rand());\nww[i]=rand();\nww[i]=0.00678;\n}\n/* Input Maze */\nn2=5*5;\nn1=5*5-1;\nfor (i=0;i<7;i++)\nfor(j=0;j<7;j++)\n{\nif ((i==0)||(j==0)||(i==6)||(j==6))\nB[i][j]=n2;\nelse\nB[i][j]=n1;\n}\n/* Generate Obstacle */\nB[2][2]=B[3][3]=B[4][4]=n2;\n/* Generate Start */\nB[2][4]=1;\nfor (i=0;i<7;i++)\nfor(j=0;j<7;j++)\n{\nA[i][j]=0;\n\n}\n/* Desired outputs */\nsynthesis(B,A,n1,n2);\nif((f=fopen(\"results5\",\"w\"))==NULL) {\nprintf(\"Cannot open file\");\nexit(1);\n}\n/* Learning Pattern */\nfor(t=0;t<TT;t++)\n{\nfor(i=m+1;i<N+n+1;i++)\n{\nfor(j=1;j<i;j++)\n{\nF_W_T[i][j]=0;\n}\nF_net_T[i]=0;\n}\nfor (i=1;i<n;i++)\nfor(ix=0;ix<nn+1;ix++)\nfor(iy=0;iy<nn+1;iy++)\nyy[0][i][ix][iy]=-1;\nfor(ix=0;ix<nn+1;ix++)\nfor(iy=0;iy<nn+1;iy++)\nyy[0][n][ix][iy]=0;\ne=F_Ws_T=s=0;\n/* If the next two lines are changed into p=0 then it is MLP */\np=(t/lt)+1;\np=(p<maxt ? p:maxt);\nfor(q=0;q<p+1;q++)\n{\ne=0;\nfor(ix=0;ix<nn+1;ix++)\nfor(iy=0;iy<nn+1;iy++)\n{\nif (B[ix][iy]==25)\nx[1]=B[ix][iy];\n\nelse if (B[ix][iy]!=1)\nx[1]=0;\nx[2]=1;\nif (ix!=0)\nx[3]=yy[q][1][ix-1][iy];\nelse\nx[3]=yy[q][1][nn][iy];\nif (iy!=0)\nx[4]=yy[q][1][ix][iy-1];\nelse\nx[4]=yy[q][1][ix][nn];\nif (ix!=nn)\nx[5]=yy[q][1][ix+1][iy];\nelse\nx[5]=yy[q][1][0][iy];\nif (iy!=nn)\nx[6]=yy[q][1][ix][iy+1];\nelse\nx[6]=yy[q][1][ix][0];\nfor (i=1;i<n+1;i++)\nx[6+i]=yy[q][i][ix][iy];\nNET(W,x,ww,n,m,N,Yhat);\nfor (i=1;i<n+1;i++)\nyy[q+1][i][ix][iy]=Yhat[i];\n}\n}\ne=0;\nfor (ix=0;ix<nn+1;ix++)\nfor(iy=0;iy<nn+1;iy++)\n{\nif (t==(TT-1))\ny[ix][iy]=yy[p+1][n][ix][iy];\nif (B[ix][iy]!=25)\nF_Jhat[ix][iy]=Ws*yy[p+1][n][ix][iy]\n-A[ix][iy];\nelse\nF_Jhat[ix][iy]=0;\ne+=F_Jhat[ix][iy]*F_Jhat[ix][iy];\n}\nprintf(\"\\n t e %d %e\",t,e);\nfprintf(f,\"\\n%d %e\",t,e);\n/* Initialize F_y */\n\nfor(q=1;q<21;q++)\nfor(ix=0;ix<nn+1;ix++)\nfor(iy=0;iy<nn+1;iy++)\nfor(i=1;i<n+1;i++)\nF_y[q][i][ix][iy]=0;\nfor(q=p;q>-1;q--)\n{\nfor (ix=0;ix<nn+1;ix++)\nfor(iy=0;iy<nn+1;iy++)\n{\nif (B[ix][iy]==25)\nx[1]=B[ix][iy];\nelse if (B[ix][iy]!=1)\nx[1]=0;\nx[2]=1;\nif (ix!=0)\nx[3]=yy[q][1][ix-1][iy];\nelse\nx[3]=yy[q][1][nn][iy];\nif (iy!=0)\nx[4]=yy[q][1][ix][iy-1];\nelse\nx[4]=yy[q][1][ix][nn];\nif (ix!=nn)\nx[5]=yy[q][1][ix+1][iy];\nelse\nx[5]=yy[q][1][0][iy];\nif (iy!=nn)\nx[6]=yy[q][1][ix][iy+1];\nelse\nx[6]=yy[q][1][ix][0];\nfor (i=1;i<n+1;i++)\nx[6+i]=yy[q][i][ix][iy];\nNET(W,x,ww,n,m,N,Yhat);\nif (q==p)\n{\nF_Yhat=F_Jhat[ix][iy];\nfor(i=1;i<n+1;i++)\nF_x[N+i]=0;\n}\nelse\n{\nF_Yhat=0;\nfor(i=1;i<n+1;i++)\n\nF_x[N+i]=F_y[q+1][i][ix][iy];\n}\nF_NET2(F_Yhat,W,x,n,m,N,F_W,F_net,\nF_Ws,Ws,F_x);\nif (ix!=0)\nF_y[q][1][ix-1][iy]+=F_x[3];\nelse\nF_y[q][1][nn][iy]+=F_x[3];\nif (iy!=0)\nF_y[q][1][ix][iy-1]+=F_x[4];\nelse\nF_y[q][1][ix][nn]+=F_x[4];\nif (ix!=nn)\nF_y[q][1][ix+1][iy]+=F_x[5];\nelse\nF_y[q][1][0][iy]+=F_x[5];\nif (iy!=nn)\nF_y[q][1][ix][iy+1]+=F_x[6];\nelse\nF_y[q][1][ix][0]+=F_x[6];\nfor(i=1;i<n+1;i++)\nF_y[q][i][ix][iy]+=F_x[6+i];\nif (q==p) F_Ws_T+=F_Ws[1];\nfor(i=m+1;i<N+n+1;i++)\n{\nfor(j=1;j<i;j++)\n{\nF_W_T[i][j]+=F_W[i][j];\n}\nF_net_T[i]+=F_net[i];\n}\n}\n}\ndot=0;\nfor(i=m+1;i<N+n+1;i++)\nfor(j=1;j<i;j++)\n{\ndot+=F_W_O[i][j]*F_W_T[i][j];\n}\nS_F_W1=S_F_W2=0;\n\nfor(i=m+1;i<N+n+1;i++)\nfor(j=1;j<i;j++)\n{\nS_F_W1 += F_W_O[i][j] * F_W_T[i][j];\nS_F_W2 += F_W_O[i][j] * F_W_O[i][j];\ns+=F_W_T[i][j]*F_W_T[i][j];\n}\nif ((S_F_W1>S_F_W2) || (S_F_W1==S_F_W2))\nLr_W=Lr_W*(a+b);\nelse if (S_F_W1<(-2)*S_F_W2)\nLr_W=Lr_W*(a-2*b);\nelse\nLr_W=Lr_W*(a+b*(S_F_W1/S_F_W2));\nS_F_net1=S_F_net2=0;\nfor(i=m+1;i<N+n+1;i++)\n{\ns+=F_net_T[i]*F_net_T[i];\nS_F_net1 +=F_net_O[i] *F_net_T[i];\nS_F_net2 +=F_net_O[i] *F_net_O[i];\n}\nif ((S_F_net1>S_F_net2) || (S_F_net1==S_F_net2))\nLr_ww=Lr_ww*(a+b);\nelse if (S_F_net1<(-2)*S_F_net2)\nLr_ww=Lr_ww*(a-2*b);\nelse\nLr_ww=Lr_ww*(a+b*(S_F_net1/S_F_net2));\nF_Ws1=F_Ws_O*F_Ws_T;\nF_Ws2=F_Ws_O*F_Ws_O;\nif ((F_Ws1>F_Ws2) || (F_Ws1==F_Ws2))\nLr_Ws=Lr_Ws*(a+b);\nelse if (F_Ws1<(-2)*F_Ws2)\nLr_Ws=Lr_Ws*(a-2*b);\nelse\nLr_Ws=Lr_Ws*(a+b*(F_Ws1/F_Ws2));\ns+=F_Ws_T*F_Ws_T;\nes=e/s;\nif ((e-Lr_W*s)<0)\nLr_W=Lr_W*es;\nif ((e-Lr_ww*s)<0)\nLr_ww=Lr_ww*es;\nfor(i=m+1;i<N+n+1;i++)\n{\nfor(j=1;j<i;j++)\n\n{\nW_O[i][j]=W[i][j];\nW[i][j]-=Lr_W*F_W_T[i][j];\n}\nww_O[i]=ww[i];\nww[i]-=Lr_ww*F_net_T[i];\n}\nif ((e-Lr_Ws*s)<0)\nLr_Ws=Lr_Ws*es;\nWs_O=Ws;\nWs-=Lr_Ws*F_Ws_T;\nsum=0;\nfor(i=m+1;i<N+n+1;i++)\n{\nfor(j=1;j<i;j++)\n{\nF_W_O[i][j]=F_W_T[i][j];\n}\nF_net_O[i]=F_net_T[i];\n}\nF_Ws_O=F_Ws_T;\n}\nfclose(f);\n}\nvoid synthesis(int B[30][30],int A[30][30],int n1,int n2 )\n{\nint k,mini,no,i,j;\n/* Initialization */\nk=0;\nfor (i=0;i<7;i++)\nfor(j=0;j<7;j++)\n{\nA[i][j]=B[i][j];\n}\n/* Searching the optimal path */\n/* Calculating the Utility */\nno = n2-3-1;\nwhile (k!=no)\n\n{\nk=0;\nfor(i=1;i<6;i++)\nfor(j=1;j<6;j++)\n{\nmini = 1 + minimum(A[i-1][j],A[i][j-1],A[i+1][j],A[i][j +1]);\nif ((A[i][j]!=n2) && (A[i][j]!=1))\n{\nif ((A[i][j]==mini) && (A[i][j]!=n1))\nk++;\nelse\nif (mini!=n2)\nA[i][j]=mini;\n}\n}\n}\n}\n/* minimum: return the minimum value */\nint minimum(int s,int t,int u,int v)\n{\nint mini;\nmini=0;\nmini=min(min(min(s,t),u),v);\nreturn mini;\n}\nvoid NET(double W[30][30],double x[30],double ww[30],\nint n, int m, int N, double Yhat[30])\n{\nint i,j;\ndouble net;\nfor (i=m+1;i<N+n+1;i++)\n{\nnet=0;\nfor (j=1;j<i;j++)\n{\nnet += W[i][j] * x[j];\n}\nx[i]=f(net+ww[i]);\n\n}\nfor (i=1;i<n+1;i++)\n{\nYhat[i]=x[i+N];\n}\n}\ndouble f(double x)\n{\ndouble z;\nz=(1-exp(-x))/(1+exp(-x));\nreturn z;\n}\nvoid F_NET2(double F_Yhat,double W[30][30],double x[30] ,int n,int m,int\nN,double F_W[30][30],double F_net[30],double F_Ws[30], double Ws,\ndouble F_x[30])\n/* This subroutine calculates the F_W terms needed to adapt a fully connected */\n/* generalized MLP. It does not backpropagate through the ne twork and it */\n/* does not permit the switching off of weights. */\n{\nint i,j;\nfor (i=1;i<N+1;i++)\nF_x[i]=0;\nF_x[N+n]+=F_Yhat*Ws;\nF_Ws[1]=F_Yhat*x[N+n];\nF_net[N+n]=F_x[N+n]*(1-x[N+n]*x[N+n])*0.5;\nfor (j=1;j<N+n;j++)\nF_W[N+n][j]=F_net[N+n]*x[j];\nfor (i=N+n-1;i>m;i--)\n{\nfor (j=i+1;j<N+n+1;j++)\n{\nF_x[i]+=W[j][i]*F_net[j];\n}\nF_net[i]=F_x[i]*(1-x[i]*x[i])*0.5;\nfor (j=1;j<i;j++)\n{\nF_W[i][j]=F_net[i]*x[j];\n}\n}\nfor(i=m;i>0;i--)\nfor(j=m+1;j<N+n+1;j++)\nF_x[i]+=W[j][i]*F_net[j];\n\n}\nint min(int k, int l)\n{\nint r;\nif (k>l) r=l;\nelse r=k;\nreturn r;\n}\nvoid pweight(double Ws,double F_Ws_T,double ww[30],doub le F_net_T[30],\ndouble W[30][30], double F_W_T[30][30],int n,int N,int m)\n{\nint i,j;\nfor(i=m+1;i<N+n+1;i++)\n{\nfor(j=1;j<i;j++)\n{\nprintf(\"\\n W[i][j] F_W_T %e %e\",W[i][j],F_W_T[i][j]);\n}\nprintf(\"\\n ww F_net_T %e %e\",ww[i],F_net_T[i]);\n}\nprintf(\"\\n Ws F_Ws_T %e %e\",Ws,F_Ws_T);\n}"}
{"id": "adap-org/9806002", "meta": {"categories": ["adap-org", "nlin.AO"], "created": "1998-06-05", "extraction": {"body_chars": 11192, "cleaning": {"detected_repeated_margin_lines": ["1"], "page_count": 4, "removed_boilerplate_lines": 6}, "method": "pypdf_no_ocr", "source_pdf_bytes": 126349, "text_chars": 11676}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9806002", "primary_category": "adap-org", "source": "arxiv", "title": "On the 1/f fluctuations in the nonlinear systems affected by noise", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9806002"}, "text": "On the 1/f fluctuations in the nonlinear systems affected by noise\n\nAbstract\nWe investigate a problem of the necessary and sufficient conditions for appearance of the 1/f fluctuations in the simple systems affected by the external random perturbations, i.e. the power spectral density of the flux of particles moving in some contours and perturbed by the external forces. In some cases we observe the 1/f behavior but only in some range of frequencies and parameters of the systems.\n\narXiv:adap-org/9806002v1 5 Jun 1998\nON THE 1/f FLUCTUATIONS IN THE NONLINEAR SYSTEMS\nAFFECTED BY NOISE\nB. Kaulakys\nInstitute of Theoretical Physics and Astronomy, A. Goˇ stauto 12, 2600 Vilnius, Lithuania,\nT. Meˇ skauskas\nDepartment of Mathematics, Vilnius University, Naugarduk o 24, 2006 Vilnius, Lithuania\nABSTRACT\nWe investigate a problem of the necessary and sufficient conditions f or ap-\npearance of the 1 /f fluctuations in the simple systems affected by the external\nrandom perturbations, i.e. the power spectral density of the flux of particles moving\nin some contours and perturbed by the external forces. In some cases we observe\nthe 1 /f δ (δ ≃ 1) behavior but only in same range of frequencies and parameters o f\nthe systems.\nINTRODUCTION\nMany processes of the greatest diversity exhibit temporal evolut ion charac-\nterized by a power spectral density S(f ) which behaves like 1 /f δ (δ ≃ 1) at low\nfrequencies. Despite considerable efforts, the generally accepte d theory of the 1 /f\nnoise is still lacking. Possibly, there are several physical mechanism s giving rise to\n1/f noise.\nUsually a study of 1 /f noise in materials starts from the approximate em-\npirical relation given in 1969 by Hooge which for the current I may be written as\n[1]\nS(f ) = ¯I\n2 α\nN f . (1)\nHere ¯I\nis the squared average current, α is a dimensionless constant which charac-\nterizes the noise intensity and N is the total number of the free charge carriers or\nfluctuating particles. Since the current is proportional to the num ber of particles, the\npower spectral density is proportional to the number of particles N too. Therefore,\nit is likely that every particle influences on the spectral power additiv ely and it is\nlogical to search of the 1 /f fluctuations in the few-particle systems affected by some\nperturbations.\nMODELS AND RESULTS\nOne of the relatively universal and approved mechanisms which gene rates 1/f\nnoise is the intermittency [2]. Intermittent chaos is often observed phenomenon in\nnonlinear dynamical systems. Investigating a transition from chao tic to nonchaotic\nbehavior in the randomly driven systems [3] we have observed the int ermittency\nroute to chaos too [4]. Therefore, it is interesting to analyze the po wer spectral\n\ndensity of the flux of small number of particles, or even one particle , moving in\nsome contours and perturbed by external random forces.\nThe intensity of one-dimensional flux of small particles in some space point\nxs may be expressed as\nI(t) =\n∑\nk\nsign(vk) δ(t − tk), (2)\nwhere tk is a sequence of transit times when one of the particles crosses the point\nxs and sign( vk) is the sign of velocity in this point. The power spectral density of\nthe flux (2) is\nS(f ) = 2\nT\n⟨ ⏐\n⏐\n⏐\n⏐\n⏐\n∑\nk\nsign(vk)e−i2πf tk\n⏐\n⏐\n⏐\n⏐\n⏐\n2⟩\n, (3)\nwhere T is the whole observation time and the brackets ⟨... ⟩ denote averaging over\nthe realizations. When the averaged flux ¯I is nonzero the dominant term in the\npower spectral density at low frequencies regardless of the deta iled structure of the\nflux is Sdc(f ) = ¯I\n/T (πf )2. To eliminate this term we divide the whole observation\ntime T into pieces of equal length and calculate the averaged power spect ral density\nof the difference of two pieces fluxes [6].\nWe analyze the power spectral density of particles moving in some po tential\nand perturbed by the random noise modelled like in papers [3, 4] by the resets\nof velocity of all particles after every time interval τ according to the replacement\nvnew(kτ) = αv old(kτ) + βv ran\nk , k = 1 , 2, . . . .Here α and β are positive parameters\nand vran\nk are random velocities from the Gaussian distribution of variance σ = 1.\nFirst of all we have calculated the power spectral density of the flu x of particles\nbounded in the Duffing potential V (x) = x4−x2. Under some conditions a transition\nfrom chaotic to nonchaotic behavior and synchronization of partic les in such system\nmay be observed [3-5]. The calculated spectral density of the curr ent of particles in\nsuch systems usually was very close to the white noise. Only for α = 1 we observe\nthe 1 /f 2 noise, while for α close to 1 the noise dependence on the frequency is\nbetween 1 /f 2 and 1 /f (see Fig. 1(a)). A very small component of 1 /f noise in the\nflux fluctuations of such system may be explained on the basis of Eq. (1) according to\nwhich the intensity of 1 /f noise component is proportional to the squared averaged\ncurrent ¯I\n. But in such model the averaged current equals zero.\nTherefore further we have analyzed the power spectral density of the current\nof particles moving in the closed contour and perturbed by the comm on for all\nparticles noise. The simplest equations of motion for such model are of the form [5]\n˙v = F (x) − γv, ˙x = v (4)\nHere F (x) = F (x + L) is the (regular) periodic in the space force with L being the\nlength of the contour and γ is the friction coefficient. Because of the directed drift the\naveraged current in this model is nonzero. We have investigated th e dependence of\nthe power spectral density on the nonlinearity of equations of mot ion and on intensity\nof the perturbation. Moreover, we have analyzed the ensemble of particles moving\nwith the friction in the periodically time-dependent force F (x + L, t + Tp) = F (x, t )\nand affected of the identical for all particles noise. We have observ ed the relatively\nweak sensitivity of the spectral density dependence on the frequ ency to the nonlinear\n\nand periodically time-dependent terms in the equation for the velocit y. Moreover,\nthe spectral density is not sensitive to the number of the particles N moving in the\ncontour too. Therefore, we show results of calculations for the s implest versions\nof the model (4), i.e. for the motion of one particle affected by the c onstant force\nand for the free motion with perturbations modelled by the velocity r eplacements\nevery time interval τ (see Figs. 1 (b) and (d)). For some parameters τ when α ̸= 1\nwe observe the S(f ) dependence on the frequency f (in same range of frequencies)\nclose to the 1 /f -dependence. Although, when we analyze the current power spec tral\ndensity in the large range of frequency we can, as a rule, discover t he Lorentzian-\ntype curves. However the increase of the S(f ) with the decrease of the frequency\nusually is not like 1 /f 2 but often as 1 /f .\nFrom the analyzes above it appears that the current power spect ral density in\nthe most degree depends not on the type of the potential (and fo rce F (x, t )) in which\nthe particles move but much more on the type and intensity of the pe rturbation.\nTherefore, we have also analyzed the simplest model for the pertu rbed flux: the\nfree motion of the particles with the Brownian-type additional fluct uations of the\ntransition times. For such model the transit times may be calculated from the\niterative sequence tk = tk− 1 + 1 + βδt ran\nk . Here the period of the free motion in\nthe closed contour is chosen to be equal 1 and δtran\nk are random time increments\nfrom the Gaussian distribution of variance σ = 1. Results of calculations of S(f )\nfor different values of intensity of perturbation β are presented in Fig. 1 (c). With\nincrease of the perturbation intensity we observe transition from the white noise at\nβ ≤ 1 to the Lorentzian-type, 1 /f - and 1 /f 1. 3-type noise at β ≥ 2.\nSummarizing, we have analyzed the problem of necessary and sufficie nt\nconditions for appearance of the 1 /f -type fluctuations in the current of the simple\nsystems of particles moving in some contour. Even in such simplest mo dels consisting\nof few or even one particle moving in the closed contours and pertur bed by the\nappropriate external noise we may observe the power spectral d ensity S(f ), which\nbehaves like 1 /f δ (δ ≃ 1) in same range of frequencies.\nACKNOWLEDGMENTS\nThe research described in this publication was supported in part by t he\nLithuanian State Science and Studies Foundation.\nREFERENCES\n1. F. N. Hooge, T. G. M. Kleinpenning and L. K. J. Vandamme. Rep. Prog. Phys.\n44, 479 (1981).\n2. H. G. Schuster. Deterministic chaos: an introduction. (VCH, Weinheim, 1989).\n3. B. Kaulakys and G. Vektaris. Phys. Rev. E 52, 2091 (1995).\n4. B. Kaulakys and G. Vektaris. In: Proc. 13th Int. Conf. on Noise in Physical\nSystems and 1/f Fluctuations. Eds V. Bareikis and R. Katilius. (World Scientific:\nSingapore, 1995) p.677.\n5. B. Kaulakys, F. Ivanauskas and T. Meˇ skauskas. In: Proc. Intern. Conf. on\nNonlinearity, Bifurcation and Chaos: the Doors to the Futur e. (Lodz-Dobieszkow,\nPoland, September 16-18, 1996) p.145.\n6. T. Musha and H. Higuchi. Jap. J. Appl. Phys. 15, 1271 (1976).\n\n/G14/G13/G10/G17 /G14/G13/G10/G16 /G14/G13/G10/G15 /G14/G13/G10/G14/G14/G13/G10/G16\n/G14/G13/G10/G15\n/G14/G13/G10/G14\n/G14/G13/G13\n/G14/G13/G14\n/G14/G13/G15\n/G14/G13/G16\n/G0B/G44/G0C\n/G49\n/G36/G0B/G49/G0C\nα /G03/G20/G03/G14\nα /G03/G20/G03/G13/G11/G1C/G18\nα /G03/G20/G03/G13/G11/G18\n/G14/G12/G49\n/G14/G12/G49/G15\n/G14/G13/G10/G17 /G14/G13/G10/G16 /G14/G13/G10/G15 /G14/G13/G10/G14\n/G14/G13/G10/G14\n/G14/G13/G13\n/G14/G13/G14\n/G14/G13/G15\n/G14/G13/G16\n/G0B/G45/G0C\n/G49\n/G36/G0B/G49/G0C\nτ/G03/G20/G03/G13/G11/G14\nτ/G03/G20/G03/G14/G11/G17\nα /G03/G20/G03/G13/G11/G18\nα /G03/G20/G03/G14/G0F/G03τ/G03/G20/G03/G13/G11/G14\n/G14/G12/G49\n/G14/G12/G49/G15\n/G14/G13/G10/G17 /G14/G13/G10/G16 /G14/G13/G10/G15 /G14/G13/G10/G14\n/G14/G13/G10/G14\n/G14/G13/G13\n/G14/G13/G14\n/G14/G13/G15\n/G0B/G46/G0C\n/G49\n/G36/G0B/G49/G0C\nβ/G03/G20/G03/G1B\nβ/G03/G20/G03/G17\nβ/G03/G20/G03/G15\nβ/G03/G20/G03/G13/G11/G18\n/G14/G12/G49\n/G14/G12/G49/G14/G11/G16\n/G14/G12/G49/G15\n/G29/G4C/G4A/G11/G03/G14/G11/G03/G03/G33/G52/G5A/G48/G55/G03/G56/G53/G48/G46/G57/G55/G44/G4F/G03/G47/G48/G51/G56/G4C/G57/G4C/G48/G56/G03/G44/G56/G03/G49/G58/G51/G46/G57/G4C/G52/G51/G56/G03/G52/G49/G03/G49/G55/G48/G54/G58/G48/G51/G46/G5C/G03/G49/G52/G55/G03/G49/G4F/G58/G5B/G03/G52/G49/G03/G53/G44/G55/G57/G4C/G46/G4F/G48/G1D/G03/G0B/G44/G0C\n/G4C/G51/G03/G57/G4B/G48/G03/G27/G58/G49/G49/G4C/G51/G4A/G03/G53/G52/G57/G48/G51/G57/G4C/G44/G4F/G03/G5A/G4C/G57/G4B/G03τ/G03/G20/G03/G16/G0F/G03β/G03/G20/G03/G14/G03/G44/G51/G47/G03/G47/G4C/G49/G49/G48/G55/G48/G51/G57/G03α /G1E/G03/G49/G52/G55/G03/G50/G52/G57/G4C/G52/G51/G03/G4C/G51/G03/G46/G4F/G52/G56/G48/G47\n/G46/G52/G51/G57/G52/G58/G55/G03/G44/G46/G46/G52/G55/G47/G4C/G51/G4A/G03/G57/G52/G03/G28/G54/G11/G0B/G17/G0C/G1D/G03/G0B/G45/G0C/G03/G5A/G4C/G57/G4B/G03/G29/G0B/G5B/G0C/G03/G20/G03/G14/G0F/G03γ/G03/G20/G03/G13/G11/G14/G0F/G03/G2F/G03/G20/G03/G15/G0F/G03β/G03/G20/G03/G14/G1E/G03/G0B/G47/G0C/G03/G5A/G4C/G57/G4B\n/G29/G0B/G5B/G0C/G03/G20/G03γ/G03/G20/G03/G13/G0F/G03/G2F/G03/G20/G03/G14/G0F/G03β/G03/G20/G03/G14/G1E/G03/G44/G51/G47/G03/G0B/G46/G0C/G03/G49/G52/G55/G03/G57/G4B/G48/G03/G4C/G57/G48/G55/G44/G57/G4C/G59/G48/G03/G56/G48/G54/G58/G48/G51/G46/G48/G03/G52/G49/G03/G57/G55/G44/G51/G56/G4C/G57/G03/G57/G4C/G50/G48/G56/G11\n/G14/G13/G10/G15 /G14/G13/G10/G14\n/G14/G13/G10/G14\n/G14/G13/G13\n/G14/G13/G14\n/G14/G13/G15\n/G14/G13/G16\n/G49\n/G36/G0B/G49/G0C\n/G0B/G47/G0C\nα /G03/G20/G03/G13/G11/G18/G0F/G03τ/G03/G20/G03/G14\nα /G03/G20/G03/G13/G11/G1C/G0F/G03τ/G03/G20/G03/G13/G11/G16\nα /G03/G20/G03/G13/G11/G1C/G0F/G03τ/G03/G20/G03/G14\nα /G03/G20/G03/G14/G0F/G03τ/G03/G20/G03/G14\n/G14/G12/G49/G14/G11/G18\n/G14/G12/G49\n/G14/G12/G49/G15"}
{"id": "adap-org/9806003", "meta": {"categories": ["adap-org", "nlin.AO"], "created": "1998-06-05", "extraction": {"body_chars": 17442, "cleaning": {"detected_repeated_margin_lines": ["1"], "page_count": 11, "removed_boilerplate_lines": 11}, "method": "pypdf_no_ocr", "source_pdf_bytes": 297212, "text_chars": 18167}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9806003", "primary_category": "adap-org", "source": "arxiv", "title": "How fast do structures emerge in hypercycle-systems?", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9806003"}, "text": "How fast do structures emerge in hypercycle-systems?\n\nAbstract\nA general framework for the simulation of reaction-diffusion systems with probabilistic cellular automata is presented. The basic reaction probabilities of the chemical model translate directly into the transition rules of the automaton, thus allowing a clear comparison between simulation results and analytic calculations. This framework is then applied to simulations of hypercycle-systems in up to three dimensions. Furthermore, a new measurement quantity is introduced and applied to the hypercycle-systems in two and three dimensions. It can be shown that this quantity can be interpreted as a measure for the macroscopic order of the hypercycle systems.\n\narXiv:adap-org/9806003v1 5 Jun 1998\nHow fast do structures emerge in\nhypercycle-systems?\nStephan Altmeyer, Claus Wilke, and Thomas Martinetz\nInstitut f¨ ur Neuroinformatik\nRuhr-Universit¨ at Bochum\nD-44780 Bochum, Germany\nStephan.Altmeyer@neuroinformatik.ruhr-uni-bochum.de\nNovember 19, 2018\nAbstract\nA general framework for the simulation of reaction-diffusion sys-\ntems with probabilistic cellular automata is presented. The basic re-\naction probabilities of the chemical model translate directly into the\ntransition rules of the automaton, thus allowing a clear comparison\nbetween simulation results and analytic calculations. This framework\nis then applied to simulations of hypercycle-systems in up to three di-\nmensions.\nFurthermore, a new measurement quantity is introduced and applie d to\nthe hypercycle-systems in two and three dimensions. It can be sho wn\nthat this quantity can be interpreted as a measure for the macros copic\norder of the hypercycle systems.\n1 The automaton\nIn the past a lot of work was done in the field of algorithmic che mistry [1, 2]\nand chemistry on cellular automata (CA) [3, 4]. Our goal is to give a com-\nplete framework which begins with the determination of the t ransition rules\nfrom basic reaction probabilities and ends with the compari son between the\nemergence of spatially ordered structures and a macroscopi c measurement\nquantity in hypercycle-systems.\nIn this paper we can only give a brief introduction of our prob abilistic CA\nwith asynchronous updating. Our aim was to establish a one-t o-one corre-\nspondence between real chemical reactions (defined by their reaction prob-\nabilities) and the simulations on a CA (defined by the transit ion rules).\nTherefore, we explain the calculation of these transition r ules for a simple\nparticle production process in detail and then present the t ransition rules\nfor more complex reaction types without derivation.\n\n1.1 Particle production processes\nIn our system, particles can be produced through spontaneou s or induced\nreactions. The result of both reactions is the change of an em pty cell to an\noccupied one. Since we evaluate production processes for em pty cells, we\nwill usually have the situation that several different reacti ons are possible,\nresulting in a different final state for the previously empty ce ll. If we think,\ne.g. of spontaneous self-replication, all the nearest neig hbors of the empty\ncell are in principle equally likely to reproduce into this c ell. In chemistry\nsuch symmetry breaking phenomena are induced by statistica l fluctuations,\ne.g. temperature gradients.\nLet pi, i ∈ A , be the probability for a single species i to replicate into the\nempty cell by a special reaction, e.g. self-replication. He re A denotes an\nindex set containing all indices which are necessary to enum erate all species\nin the given context and Ni will denote the number of molecules of species\ni.\nIn the case there are several nearest neighbor cells occupie d, we first have\nto calculate the total probability ptot that at least one arbitrary nearest-\nneighbor-molecule will occupy the empty cell. We get\nptot = 1 −\n∏\ni∈A\n(1 − pi)Ni. (1)\nFrom ptot we can calculate the effective probabilities for the nearest n eighbor\nmolecules to replicate in the given arrangement of species. We will denote\nthese probabilities with peff\ni . In this calculation we have to make sure that\nthe ratio pi/p j is equal to peff\ni /p eff\nj , ∀i, j ∈ A . Since the sum over all pi is\nnot necessarily 1, we introduce a normalization factor ∑\ni∈A Nipi.1 We get\npeff\ni = pi\nptot\n∑\ni∈A Nipi\n. (2)\nNote that ptot is the sum of Nipeff\ni over all i.\n1.1.1 Type I reactions\nReactions of type I,\nMi\npi,m,n\n− − − →Mm + Mn,\nrepresent the group of spontaneous reactions with two react ion products.\nAfter some work and with the calculations done above we get:\npeff\ni,m,n= pi,m,n\n1 − ∏\ni∈A(1 − pi,m,n)Ni\n∑\ni∈A Nipi,m,n\n. (3)\n1The normalization factor is introduced to ensure that the su m over all peff\ni and 1 − ptot\nis 1.\n\nThe corresponding transition rule then reads:\nA cell with state zero will change to state n and a nearest-neighbor cell with\nstate i will change to m with the probability peff\ni,m,n.\n1.1.2 Type II reactions\nReactions of type II,\nMi + Mj\npi,j,m,n,o\n− − − − − →Mm + Mn + Mo,\ndescribe a group of two-body reactions with three reaction p roducts. Obvi-\nously, such reaction processes are more difficult to handle si nce we have to\nimplement the determination of the correct reaction partne r. The related\nquantities are primed.\nAfter some calculations we get:\npeff\ni,j,m,n,o= p′\ni,j,m,n,o\n1 − ∏\ni∈A(1 − p′\ni,j,m,n,o)N ′\ni\n∑\ni∈A Nip′\ni,j,m,n,o\n. (4)\nwith\np′\ni,j,m,n,o= p(N ′ ̸= 0) N ′\nj\nN ′ pi,j,m,n,o. (5)\np(N ′ ̸= 0) denotes the probability that there is at least one molecu le to\nreact with. For the derivation of p′\ni,j,m,n,owe have implicitly used the idea\nof an attractive force between the molecules which means tha t the reactive\nmolecule therefore ”jumps” to a reaction partner. On the oth er hand, if we\ndo not assume such an attractive force, which means a molecul e can also\n”jump” into a hole, we have to substitute p′\ni,j,m,n,owith N ′\nj/N ′\nmax.\nThen we can write down the transition rule:\nThe probability that a cell in state zero, with a nearest-neig hbor cell in state\ni and with a ”reaction” cell in state j will change to o, while the nearest-\nneighbor cell changes to m and the ”reaction” cell changes to n, is peff\ni,j,m,n,o.\n1.1.3 Particle decay\nThe decay process,\nMi\npdec\ni\n− − →0,\ncan change an occupied cell into an empty one. The transition rule descrip-\ntion is rather easy:\nThe probability that a cell in state i will change to a cell in st ate zero is pdec\ni .\n\n1.2 Diffusion\nThe diffusion is an additional process. Since we use an asynchr onous au-\ntomaton, an often used diffusion algorithm for CA [5] is not nec essary.\nInstead, we use a more natural algorithm that can be divided i nto three\nsubprocesses:\n1. An arbitrary cell of the CA is chosen.\n2. One of it’s nearest neighbor cells is determined at random .\n3. The states of the two cells are interchanged.\n1.3 Automaton update\nWe use asynchronous updating, since this seems to be the more natural way\n[6]. The basic processes are then executed in the following w ay:\nChoose an arbitrary cell at random. If the cell is occupied we use the decay-\ntransition rule. Otherwise the transition rule for reactio ns of type I or of\ntype II or a combination of both are used.\nThe reaction processes and the diffusion process are then comb ined to the\nfull simulation in the following way. Let k be the total number of cells in\na CA and d the ratio of the number of basic processes and the number of\ndiffusion steps. Then a CA-time-step t consists of k basic processes and dk\ndiffusion steps. It is important that the basic processes and t he diffusion\nsteps are mixed up very well.\nNote that t is independent of the system’s size, since we scale the numbe r\ninternal processes with it.\n2 Simulation\nWe have simulated systems with (symmetrical) hypercycles [ 7, 8], which\nmeans we set\npi,j,k=\n{\npself if i = j = k,\n0 else (6)\nand\npi,j,m,n,o=\n{\npcat if i = m = o, j = n and i = j + 1 (cyclic) ,\n0 else (7)\n\n2.1 Stable patterns\nIn this part of the paper we will show the well-known spiral wa ves [4] in\ntwo dimensions and the emergence of a stable scroll wave stru ctures [9] in\nthree dimensions. In the latter case we have used a two-dimen sional spiral\nas initial condition. We will see the system going through di fferent unstable\npatterns. For reasons of clear visualization, in the pictur es we show only\none of the six species present in the simulation.\nThe three-dimensional analog of a spiral wave is a scroll wav e [9]. As we\ncan see in Fig. 3, even in the two dimensional case, structure s can be very\ncomplex. Since there are more degrees of freedom in three spa tial dimen-\nsions, the structures arising from non-ordered initial con ditions are much\nmore complex. Therefore, we use a 2d spiral as a symmetry-bre aking initial\ncondition for our simulations to simplify the emergent stru ctures.\nWe can see the spiral’s growth in the z-direction (see Fig. 1, 2). As a matter\nof fact, the reproduction activity of a species becomes high if it is adjacent to\nit’s ”supporter”. Therefore, in a layer above, molecules ar e first created over\nthe small region where their own and their ”supporter” are pr esent. Since\nthe spirals rotate in the ”supporter” direction, we can see t he phenomenon\nthat the new spirals, growing on top of the old ones, are phase shifted in\nthe rotation direction. This continues until the space in z- direction is oc-\ncupied. The resulting structure looks like a rotating screw . When all the\navailable space is occupied, the molecules cannot replicat e freely any more.\nSince space is rare than, the molecules have to compete for em pty space to\nreplicate. For this reason, the molecules begin to arrange t hemselves in a\ndifferent way so that the ”free” screw begins to change to a cone structure.\nThis cone structure is stable so that a real scroll wave canno t be seen.\nThe whole re-arrangement process takes much more time than t he occupa-\ntion at the beginning. The pictures show the emergence of the cone struc-\nture. At the top of the cone the emergence of other small cones can be seen\nso that the whole structure becomes more complex.\n2.2 A microscopic measurement quantity\nNow we will introduce a measurement quantity with which we ca n measure\nthe arrangement of the molecules in our automaton. The defini tion of this\norder-quantity, called α k, is rather easy:\nFor every molecule, we take the nearest neighbor molecules. If an adjacent\nmolecule is a ”supporter”, the variable S is incremented by one. Otherwise\nthe variable N is incremented by one. If a cell is not occupied, nothing is\ndone. With these microscopic values we can define the macrosc opic quantity\n\nFigure 1: The first two pictures show the system’s behavior sh ortly after the\n”injection” of the 2d spiral. We can see the emergence of a rot ating screw.\nThe left picture was taken at t = 150 and the right one at t = 250.\nFigure 2: If the space is nearly completely occupied the mole cules cannot\nreplicate freely and have to compete for empty space. Theref ore the struc-\nture changes from a screw to a cone-like pattern.\n\nFigure 3: The three pictures show the time dependent behavio r of the sys-\ntem. The times when the pictures have been taken are (from lef t to right):\nt = 0, t = 120 and t = 9000. See also Fig. 4 and text for further information.\nα k:\nα k = kS\nN + kS , (8)\nwhere k ≥ 0 is a tuning constant. The larger k, the more sensitive is α k on\nS.\nSince both N, S ≥ 0, the quantity α k ranges between 0 and 1. We will later\nsee that α k is even constant for large t. (Obviously, if α k is constant, α k′,\nwith another k′ ̸= k, is also constant.)\nMoreover, since we do not take into account empty cells, α k is independent\nof the system’s concentration and therefore independent of pdec.\nFor our simulations we have used k = 10 and applied α 10 to both the two\nand three dimensional case.\nNote that there are a lot of possible microscopic/macroscop ic quantities\nwhich all can describe the system from different points of view .\n2.3 Two dimensions\nFor the two dimensional case we have used two different initial conditions.\nIn both cases the parameter set is pdec = 0 . 1, pself = 0 . 05, pcat = 0 . 9 and\nd = 1.\nFirst, the hypercycle-system was initialized at random, se e Fig. 3. We\nhave seen α 10 converging rapidly to a constant value at the very beginning .\nAfter t = 120, however, α 10 changes only in very small amounts. In compar-\nison, the global structure of the system also changes very sl owly, see Fig. 4.\nIt seems that the system’s rough structure is determined at t he very begin-\nning and will then only be refined, later on. This is very inter esting, since\nthere are a lot of possible stable patterns. However, the dec ision is made\nvery early.\nFor the asymptotic value of α 10 we have found α 10 ≈ 0. 72, independent\n\nFigure 4: The picture shows the concentrations of the six hyp ercycle species\n(solid lines) and the value of α 10 versus time for a system with random initial\ncondition.\nFigure 5: The three pictures show the time dependent behavio r of the sys-\ntem. The times when the pictures have been taken are (from lef t to right):\nt = 0, t = 140 and t = 550. See also Fig. 6 and text for further information.\nof the number of species in the hypercycle and of the system’s size. Fur-\nthermore, α k depends only weakly on d for d < 1. The next step was to\nexamine a system where only one stable final pattern exits. Th erefore, we\nhave used a special initial condition which is displayed in F ig. 5. The final\nstructure will be of course a single rotating spiral.\nWhat we have seen was α 10 changing comparably to the transformation of\nthe triangular structure to the spiral structure. Since the major part of this\ntransformation is done at the beginning, α also increases fast during the first\nsimulation steps and then increases slower and slower. The s ystem behaves\nsimilar to the case with random initialization, although th e convergence of\nα k to it’s asymptotic value is significantly slower. But what is more surpris-\ning, the asymptotic value of α 10 is again the same, α 10 ≈ 0. 72.\n\nFigure 6: The picture shows the concentrations of the six hyp ercycle species\n(solid lines) and the value of α 10 versus time for a system with ordered\ntriangles as initial condition.\nWe will now go a step beyond and apply α 10 on our three dimensional sim-\nulator.\n2.4 Three dimensions\nAs we have seen above (Fig. 1, 2) in three dimensions there are several\ndifferent unstable structures, beginning with a single two di mensional spiral\nas initial condition and ending with a stable cone-structur e.\nIn Fig. 7 we can see α 10 starting with α 10 ≈ 0. 72. But then we see α k\nincreasing in the time interval when the spiral occupies the empty space by\ngrowing in the z-direction. After that, when the available s pace is occupied,\nnew boundaries are present and α 10 decreases to the former value.\nThis is a bit surprising since the cone structures are quite d ifferent and at the\nfirst glance there seems to be no reason why the macroscopic or der should\nbe similar.\n3 Conclusions\nIn this paper we have shown pictures of the transformation fr om two di-\nmensional patterns to three dimensional ones and have there fore seen the\nsystem walking through different states of arrangement. But w hat is more\ninteresting is the fact that our macroscopic quantity α k (here: α 10) is quite\nconstant after a short time and has even the same asymptotic v alue in dif-\nferent settings.\nIn all our simulations, especially in two dimensions, we hav e seen α 10 con-\n\nFigure 7: The picture shows the concentrations of the six hyp ercycle species\n(solid lines) and the value of α 10 versus time for a system in three spatial\ndimensions. Note that a single two dimensional spiral was us ed as initial\ncondition.\nverging rapidly to the fixed value, α 10 ≈ 0. 72. Thereof we can deduce that\nsuch catalytic systems have the urge to determine their final structures at\nthe very beginning. Later on structures are only refined.\nAlthough the hypercycles have been studied very well in the p ast years some\nnew features can still be found. We think that the definition o f microscopic\nquantities and their interpretation in a macroscopic sense is one of the right\nways to learn more about catalytic systems which can be even m ore complex\nthan such ”simple” symmetric hypercycles.\n4 Outlook\nIt seems to be interesting for future work to study whether st ability against\nparasites is possible in three dimensions, similar to the tw o-dimensional case\n[4, 10]. Furthermore, the behavior of α k in such parasite-systems might be\ninteresting.\nReferences\n[1] R. J. Bagley and J. D. Farmer. Spontaneous emergence of a m etabolism.\nIn C. Langton, C. Taylor, J. Farmer, and S. Rasmussen, editor s, Arti-\nficial Life II . Santa Fe Institut, Addison-Wesley, 1992.\n\n[2] R. J. Bagley, J. D. Farmer, and W. Fontana. Evolution of a m etabolism.\nIn C. Langton, C. Taylor, J. Farmer, and S. Rasmussen, editor s, Arti-\nficial Life II . Santa Fe Institut, Addison-Wesley, 1992.\n[3] M. Gerhardt, H. Schuster, and J. J. Tyson. A cellular auto maton de-\nscribing the formation of spatially ordered structures in c hemical sys-\ntems. Physica D , 36, 1989.\n[4] M. Boerlijst and P. Hogeweg. Self-structuring and selec tion: Spiral\nwaves as a substrate for prebiotic evolution. In C. Langton, C. Tay-\nlor, J. Farmer, and S. Rasmussen, editors, Artificial Life II . Santa Fe\nInstitut, Addison-Wesley, 1992.\n[5] T. Toffoli and N. Margolus. Cellular Automata Machines . MIT Press,\n1987.\n[6] I. Harvey and T. Bossomaier. Time out of joint: Attractor s in ran-\ndom boolean networks. In P. Husbands and I. Harvey, editors, Fourth\nEuropean Conference on Artificial Life . MIT Press, 1997.\n[7] M. Eigen and P. Schuster. The Hypercycle: A Principle of Natural\nSelf-Organisation. Springer, 1979.\n[8] J. Hofbauer and K. Sigmund. The Theory of Evolution and Dynamical\nSystems. Cambridge University Press, 1988.\n[9] J. J. Tyson and J. P. Keener. Singular pertubation theory of travelling\nwaves in excitable media. Physica D , 32:327–361, 1988.\n[10] M. Cronhjort and C. Blomberg. Chasing: A mechanism for r esistance\nagainst parasites in self-replicating systems. In C. Langt on and K. Shi-\nmohara, editors, Artificial Life V . MIT Press, 1996."}
{"id": "adap-org/9807001", "meta": {"categories": ["adap-org", "nlin.AO"], "created": "1999-07-21", "extraction": {"body_chars": 19924, "cleaning": {"detected_repeated_margin_lines": ["/G57/G0F/G03/G3E/G56/G40", "0.0 10 .0 20 .0 30 .0 40 .0 50 .0", "0.00", "dashed curve denoted as MFPT (mean First Passage Time) repre sents exponential approximation with MFPT substituted int o", "kT = 0.5", "kT = 3", "M FP T , kT = 1", "1", "the factor of exponent."], "page_count": 9, "removed_boilerplate_lines": 43}, "method": "pypdf_no_ocr", "source_pdf_bytes": 169062, "text_chars": 20473}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9807001", "primary_category": "adap-org", "source": "arxiv", "title": "Nondecay probability of a metastable state: almost exact analytical description in wide range of noise intensity", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9807001"}, "text": "Nondecay probability of a metastable state: almost exact analytical description in wide range of noise intensity\n\nAbstract\nThis paper presents a complete description of noise-induced decay of a metastable state in a wide range of noise intensity. Recurrent formulas of exact moments of decay time valid for arbitrary noise intensity have been obtained. The nondecay probability of a metastable state was found to be really close to the exponent even for the case when the potential barrier height is comparable or smaller than the noise intensity.\n\narXiv:adap-org/9807001v2 21 Jul 1999\nNondecay probability of a metastable state: almost exact an alytical description\nin wide range of noise intensity\nSvetlana P. Nikitenkova\nNizhny Novgorod State Technical University, Applied Mathe matics Dpt.,\n24 Minin str., Nizhny Novgorod, 603155, Russia. E-mail: spn @waise.nntu.sci-nnov.ru\nAndrey L. Pankratov\nInstitute for Physics of Microstructures of RAS, GSP 105,\nNizhny Novgorod, 603600, RUSSIA. E-mail: alp@ipm.sci-nno v.ru\nThis paper presents a complete description of noise-induce d decay of a metastable state in a\nwide range of noise intensity. Recurrent formulas of exact m oments of decay time valid for arbitrary\nnoise intensity have been obtained. The nondecay probabili ty of a metastable state was found to\nbe really close to the exponent even for the case when the pote ntial barrier height is comparable or\nsmaller than the noise intensity.\nPACS numbers: 05.40.+j\nI. INTRODUCTION\nOverdamped Brownian motion in a field of force (Markov process) is a model widely used for description of noise-\ninduced transitions in different polystable systems. In many practic al tasks (e.g. tasks of Josephson electronics,\nkinetics of chemical reactions and so on) it is enough to know the pro bability of transition or even only a time scale\nof transition. When the transition occurs over a potential barrier high enough in comparison with noise intensity,\nthe probability of transition is a simple exponent ∼ exp(−t/τ ) [1], where τ is the mean transition time. In this case\nthe mean transition time gives complete information about the proba bility evolution. The boundaries of validity of\nexponential approximation of the probability were previously studie d in [2], [3]. In [2] authors extended the Mean\nFirst Passage Time to the case of ”radiation” boundary condition an d for two barrierless examples demonstrated good\ncoincidence between exponential approximation and numerically obt ained probability. In a more general case the\nexponential behavior of observables was demonstrated in [3] for r elaxation processes in systems having steady states.\nUsing the approach of ”generalized moment approximation” the aut hors of [3] obtained the exact mean relaxation time\nto steady state and for particular example of a rectangular poten tial well demonstrated good coincidence of exponential\napproximation with numerically obtained observables. The considere d in [3] example of the rectangular well does not\nhave a potential barrier, and the authors of that paper suppose d that their approach (and the corresponding formulas)\nshould also give good approximation in tasks with diffusive barrier cros sing for a wide range of noise intensity.\nIn the frame of this paper we consider a different case than in [2], [3]: w e present investigation of nondecay\nprobability of a metastable state. We treat the decay as a transitio n of Markov process trajectory outside the region\nof a metastable state. We consider namely the probability to find a re lization of Markov process in a given interval,\nbut not the probability to pass the boundary for the first time (Firs t Passage Time formalizm), since usually in\nexperiment we can measure only the first one. In most of applied tas ks no absorbing boundaries can be introduced\nand the precision of the used equipment allows to register the only fa ct of leaving the considered domain, but the\nboundary may be crossed many times during the transition (infinite n umber of times for Markov process, since this\nprocess is not differentiable).\nUsing the approach proposed by Malakhov [4], [5], that requires only k nowledge of the behavior of a potential at\n±∞, we have decomposed the nondecay probability into a set of moment s (cumulants), obtained recurrent formulas for\nthese moments and approximately summarized them into the require d probability. The obtained nondecay probability\ndemonstrates exponential behavior with a good precision even in th e case of a small potential barrier in comparison\nwith noise intensity.\nII. MAIN EQUATIONS AND SET UP OF THE PROBLEM\nConsider a process of Brownian diffusion in a potential profile Φ (x). Let a coordinate x(t) of the Brownian particle\ndescribed by the probability density W (x, t ) at initial instant of time has a fixed value x(0) = x0 within the interval\n(c, d ), i.e. the initial probability density is the delta function: W (x, 0) = δ(x − x0), x0 ∈ (c, d ).\n\nIn this case the one-dimensional probability density W (x, t ) is the transition probability density from the point x0\nto the point x: W (x, t ) = W (x, t ; x0, 0). It is known that the probability density W (x, t ) of the Brownian particle in\nthe overdamped limit satisfies to the Fokker–Planck equation (FPE) :\n∂W (x, t )\n∂t = − ∂G(x, t )\n∂x = 1\nB\n{ ∂\n∂x\n[ dϕ (x)\ndx W (x, t )\n]\n+ ∂ 2W (x, t )\n∂x 2\n}\n. (1)\nwith the delta-shaped initial distribution. Here B = h/kT , G(x, t ) is the probability current, h is the viscosity (in\ncomputer simulations we put h = 1), T is the temperature, k is the Boltzmann constant and ϕ (x) = Φ (x)/kT is\nthe dimensionless potential profile. In this paper we restrict ourse lves by the case of metastable potentials, i.e. we\nconsider an overdamped Brownian motion in a potential field ϕ (x) in systems, having metastable states, such that\nϕ (−∞) = + ∞ and ϕ (+∞) = −∞. This leads to the following boundary conditions: G(−∞, t ) = W (+∞, t ) = 0.\nNote, that the results obtained may be generalized for potentials o f arbitrary types, e.g. for such that ϕ (±∞) = ∞.\nIt is necessary to find the probability P (x0, t ) of a Brownian particle, located at the point x0 (t = 0) within the\ninterval (c, d ) to be at the time t > 0 inside the considered interval: P (x0, t ) =\nd∫\nc\nW (x, t )dx. Further we for simplicity\nwill call the probability P (x0, t ) as nondecay probability. We suppose, that c and d are arbitrary chosen points of an\narbitrary potential profile ϕ (x) and boundary conditions at these points may be arbitrary: W (c, t ) ≥ 0, W (d, t ) ≥ 0.\nIn this case there is the possibility for a Brownian particle to come bac k in the interval ( c, d ) after crossing boundary\npoints.\nIII. MOMENTS OF DECAY TIME\nConsider the nondecay probability P (x0, t ). We can decompose this probability to the set of moments. On the\nother hand, if we know all moments, we can in some cases construct a probability as the set of moments. Thus,\nanalogically to moments of the First Passage Time [6]- [8] we can introdu ce moments of decay time τn(c, x 0, d ) (or,\ngenerally, moments of transition time, see [9], where it was performe d for the probability Q(x0, t ) = 1 − P (x0, t )):\nτn(c, x 0, d ) =< t n >=\n∞∫\ntn ∂P (x0,t )\n∂t dt\nP (x0, ∞) − P (x0, 0) . (2)\nHere we can formally denote the derivative of the probability divided b y the normalization factor as w(x0, t ) and\nthus introduce the probability density of decay time w(x0, t ) in the following way [9]:\nw(x0, t ) = ∂P (x0, t )\n∂t\n[P (x0, ∞) − P (x0, 0)] . (3)\nIt is important to mention that the moments of decay (transition) t ime (2) is a generalization of the well-known\nmoments of the First Passage Time for the case of arbitrary bound ary conditions (see discussion in [9]). For example,\nin the considered case of the potential ϕ (x) (such that ϕ (−∞) = + ∞ and ϕ (+∞) = −∞) the moments of decay time\ncoincide with the corresponding moments of the First Passage Time, if a reflecting boundary at the point c and an\nabsorbing boundary at the point d are introduced. On the other hand, if we consider the decay of met astable state\nas transition over a barrier top, and compare mean decay time obta ined via approach discussed in the present paper\n(case of a smooth potential without absorbing boundary) and the mean First Passage Time (MFPT) of the absorbing\nboundary located at the barrier top, we get two times difference be tween these time characteristics even in the case of\na high potential barrier in comparison with the noise intensity. This is d ue to the fact, that the MFPT does not take\ninto account the backward probability current and therefore is se nsitive to the location of an absorbing boundary. For\nthe considered situation, if we will move the boundary point down fro m the barrier top, the MFPT will increase up to\ntwo times and tend to reach value of the corresponding mean decay time, which is less sensitive to the location of the\nboundary point over a barrier top. Such weak dependence of the m ean decay time from the location of the boundary\npoint at the barrier top or further is intuitively obvious: much more t ime should be spent to reach the barrier top\n(activated escape) than to move down from the barrier top (dyna mic motion).\nThe required moments of decay time may be obtained via the approac h proposed by Malakhov [4], [5]. This\napproach is based on the Laplace transformation method of the FP E (1). Following this approach, one can introduce\nthe function H(x, s ) ≡ s ˆG(x, s ), where ˆG(x, s ) =\n∞∫\nG(x, t )e− stdt is the Laplace transformation of the probability\ncurrent, and expand it in the power series in s:\n\nH(x, s ) ≡ s ˆG(x, s ) = H0(x) + sH1(x) + s2H2(x) + . . . (4)\nIt is possible to find the differential equations for Hn(x) (see [4], [5]; dH0(x)/dx = 0):\ndH1(x)\ndx = δ(x − x0),\nd2Hn(x)\ndx2 + dϕ (x)\ndx\ndHn(x)\ndx = BHn− 1(x), n = 2, 3, 4, . . .\n(5)\nUsing the boundary conditions W (+∞, t ) = 0 and G(−∞, t ) = 0, one can obtain from (5) H1(x) = 1( x − x0) and\nH2(x) = −B\nx∫\n−∞\ne− ϕ (v)\n∞∫\nv\neϕ (y)1(y − x0)dydv,\nHn(x) = −B\nx∫\n−∞\ne− ϕ (v)\n∞∫\nv\neϕ (y)Hn− 1(y)dydv, n = 3, 4, 5, . . .\n(6)\nWhy did we calculate this recurrent formula for the functions Hn(x)? The matter is, that from formula (2) (taking\nthe integral by parts and Laplace transforming it using the proper ty P (x0, 0) − s ˆP (x0, s ) = ˆG(d, s ) − ˆG(c, s ) together\nwith the expansion (4)) one can get the following expressions for mo ments of decay time:\nτ1(c, x 0, d ) = −(H2(d) − H2(c)),\nτ2(c, x 0, d ) = 2( H3(d) − H3(c)),\nτ3(c, x 0, d ) = −2 · 3(H4(d) − H4(c)), . . .\nτn(c, x 0, d ) = ( −1)nn!(Hn+1(d) − Hn+1(c)).\n(7)\nOne can represent the n-th moment in the following form:\nτn(c, x 0, d ) = n!τn\n1 (c, x 0, d ) + rn(c, x 0, d ). (8)\nThis is a natural representation of τn(c, x 0, d ) due to the structure of recurrent formulas (6), which is seen fr om the\nparticular form of the first and the second moments for the case c = −∞ (c < x 0 < d). From the recurrent formulas\n(6), (7) one can obtain:\nτ1(−∞, x 0, d ) = B\n\n\n\nd∫\n−∞\ne− ϕ (x)dx ·\n∞∫\nx0\neϕ (v)dv −\nd∫\nx0\ne− ϕ (x)\nx∫\nx0\neϕ (v)dvdx\n\n\n . (9)\nτ2(−∞, x 0, d ) = 2 B2\n{\n[τ1(−∞, x 0, d )]2 +\n+\nd∫\n−∞\ne− ϕ (x)dx ·\n∞∫\nx0\neϕ (v)\nv∫\nd\ne− ϕ (u)\n∞∫\nu\neϕ (z)dzdudv − (10)\n−\nd∫\nx0\ne− ϕ (x)\nx∫\nx0\neϕ (v)\nv∫\nd\ne− ϕ (u)\n∞∫\nu\neϕ (z)dzdudvdx\n\n\n .\nUsing the approach, applied in the paper by Shenoy and Agarwal [10] for analysis of moments of the First Passage\nTime, it can be demonstrated, that in the limit of a high barrier ∆ ϕ ≫ 1 (∆ ϕ = ∆ Φ/kT is the dimensionless barrier\nheight) the remainders rn(c, x 0, d ) in formula (8) may be neglected. For ∆ ϕ ≈ 1, however, a rigorous analysis should\nbe performed for estimation of rn(c, x 0, d ). Let us suppose, that the remainders rn(c, x 0, d ) may be neglected in wide\nrange of parameters and further we will check numerically when our assumption is valid.\nThe cumulants of decay time æ n [11], [8] are much more useful for our purpose to construct the pr obability\nP (x0, t ), that is the integral transformation of the introduced probabilit y density of decay time w(x0, t ) (3). Unlike\nthe representation via moments, the Fourier transformation of t he probability density (3) - the characteristic function\n- decomposed into the set of cumulants may be inversely transform ed into the probability density.\nAnalogically to representation for moments (8), similar representa tion can be obtained for cumulants æ n:\næ n(c, x 0, d ) = ( n − 1)!æ n\n1 (c, x 0, d ) + Rn(c, x 0, d ). (11)\n\nIt is known that the characteristic function Θ( x0, ω ) =\n∞∫\nw(x0, t )ejωt dt (j = √−1) can be represented as the set of\ncumulants (w(x0, t ) = 0 for t < 0):\nΘ( x0, ω ) = exp\n[ ∞∑\nn=1\næ n(c, x 0, d )\nn! (jω )n\n]\n. (12)\nIn the case, when the remainders Rn(c, x 0, d ) in (11) (or rn(c, x 0, d ) in (8)) may be neglected, the set (12) may be\nsummarized and inverse Fourier transformed:\nw(x0, t ) = e− t/τ\nτ , (13)\nwhere τ is the mean decay time [4], [5] ( τ(c, x 0, d ) ≡ τ1 ≡ æ 1):\nτ(c, x 0, d ) = B\n\n\n\nd∫\nx0\neϕ (x)\nx∫\nc\ne− ϕ (v)dvdx +\n∞∫\nd\neϕ (x)dx\nd∫\nc\ne− ϕ (v)dv\n\n\n . (14)\nThis expression is a direct transformation of formula (9), where c is arbitrary, such that c < x 0 < d.\nProbably, similar procedure was previously used (see [7], [10], [12], [13]) for summation of the set of moments of the\nFirst Passage Time, when exponential distribution of the First Pass age Time probability density was demonstrated\nfor the case of a high potential barrier in comparison with noise inten sity.\nIV. NONDECAY PROBABILITY EVOLUTION\nIntegrating probability density (13), taking into account definition (3), we get the following expression for the\nnondecay probability P (x0, t ) ( P (x0, 0) = 1, P (x0, ∞) = 0):\nP (x0, t ) = exp( −t/τ ), (15)\nwhere mean decay time τ is expressed by (14). Probability (15) represents a well-known exp onential decay of a\nmetastable state with a high potential barrier [1]. Where is the bound ary of validity of formula (15) and when can we\nneglect by reminders rn and Rn in formulas (8),(11)? To answer this question we have considered th ree examples of\npotentials having metastable states and compared numerically obta ined nondecay probability P (x0, t ) =\nd∫\nc\nW (x, t )dx\nwith its exponential approximation (15). We used the usual explicit d ifference scheme to solve the FPE (2), supposing\nthe reflecting boundary condition G(cb, t ) = 0 ( cb < c ) far above the potential minimum and the absorbing one\nW (db, t ) = 0 ( db > d ) far below the potential maximum, instead of boundary conditions a t ±∞, such that the\ninfluence of phantom boundaries at cb and db on the process of diffusion was negligible.\nThe first considered system is described by the potential Φ (x) = ax2 − bx3. We have taken the following particular\nparameters: a = 2, b = 1 that leads to the barrier height ∆ Φ ≈ 1. 2, c = −2, d = 2 a/ 3b, and kT = 0 . 5; 1; 3.\nThe corresponding curves of the numerically simulated probability an d its exponential approximation are presented\nin Fig.1. In the worse case when kT = 1 the maximal difference between the corresponding curves is 3 . 2%. For\ncomparison, there is also presented a curve of exponential appro ximation with the mean First Passage Time (MFPT)\nof the point d for kT = 1 (dashed line). One can see, that in the latter case the error is sig nificantly larger.\nThe second considered system is described by the potential Φ (x) = ax4 −bx5. We have taken the following particular\nparameters: a = 1, b = 0 . 5 that leads to the barrier height ∆ Φ ≈ 1. 3, c = −1. 5, d = 4 a/ 5b, and kT = 0 . 5; 1; 3. The\ncorresponding curves of the numerically simulated probability and its exponential approximation are presented in\nFig.2. In the worse case ( kT = 1) the maximal difference between the corresponding curves is 3 . 4%.\nThe third considered system is described by the potential Φ (x) = 1 − cos(x) − ax. This potential is multistable. We\nhave considered it in the interval [ −10, 10], taking into account three neighboring minima. We have taken a = 0 . 85\nthat leads to the barrier height ∆ Φ ≈ 0. 1, c = −π − arcsin(a), d = π − arcsin(a), x0 = arcsin(a), and kT = 0. 1; 0. 3; 1.\nThe corresponding curves of the numerically simulated probability an d its exponential approximation are presented\nin Fig.3. In difference with two previous examples, this potential was c onsidered in essentially longer interval and\nwith smaller barrier. The difference between curves of the numerica lly simulated probability and its exponential\napproximation is larger. Nevertheless, the qualitative coincidence is good enough.\n\nFinally, we have considered an example of metastable state without p otential barrier: Φ (x) = −bx3, where b = 1,\nx0 = −1, d = 0, c = −3 and kT = 0 . 1; 1; 5. By dashed curve an exponential approximation with the MFP T of the\npoint d for kT = 1 is presented. It is seen, that even for such example the expone ntial approximation (with the mean\ndecay time (14)) gives an adequate description of the probability ev olution and that this approximation works better\nfor larger noise intensity.\nV. CONCLUSION\nIn the present paper the decay of metastable states, described by the model of Markov process, has been considered.\nRecurrent formulas of exact moments of decay time, valid for arbit rary noise intensity, have been obtained. Some\nconcrete examples of metastable states have been analysed nume rically, and the time evolution of the nondecay\nprobability of a metastable state is found to be really close to the exp onent even for the case when the potential\nbarrier height is comparable or smaller than the noise intensity if the e xact mean decay time (14) is substituted into\nthe factor of exponent.\nFor all investigated examples, the exponential approximation gives an adequate behavior of the probability. This\napproximation may be used in a wide range of parameters, enough fo r solution of many practical tasks, but it is\nnecessary to remark, that the exponential approximation may lea d to a significant error in the case of extremely large\nnoise intensity, and in the case when the noise intensity is small, the po tential is tilted, and the barrier is absent\n(purely dynamical motion slightly modulated by noise perturbations) .\nVI. ACKNOWLEDGMENTS\nAuthors wish to thank Prof. V.Belykh, Prof. P.Talkner, Prof. P.Jun g and Dr. W.Nadler for discussions and\nconstructive comments. This work has been supported by the Rus sian Foundation for Basic Research (Project N 96-\n02-16772-a, Project N 96-15-96718 and Project N 97-02-1692 8), by Ministry of High Education of Russian Federation\n(Project N 3877) and in part by Grant N 98-2-13 from the Interna tional Center for Advanced Studies in Nizhny\nNovgorod.\n[1] C.W.Gardiner, Handbook of Stochastic methods (Springe r-Verlag, 1985).\n[2] A.Szabo, K.Schulten and Z.Schulten, Journ. Chem. Phys. , 72, 4350 (1980).\n[3] W.Nadler and K.Schulten, Journ. Chem. Phys., 82, 151 (1985).\n[4] A.N.Malakhov and A.L.Pankratov, Physica C 269, 46 (1996).\n[5] A.N.Malakhov, CHAOS 7, 488 (1997).\n[6] L.A.Pontryagin, A.A.Andronov and A.A.Vitt, Zh. Eksp. T eor. Fiz. 3, 165 (1933) [translated by J.B.Barbour and repro-\nduced in ”Noise in Nonlinear Dynamics”, 1989, edited by F.Mo ss and P.V.E. McClintock (Cambridge University Press,\nCambridge) Vol. 1, p.329].\n[7] P.Hanggi, P.Talkner and M.Borkovec, Rev.Mod.Phys., 62, 251 (1990).\n[8] H.Risken, The Fokker-Planck equation (Springer Verlag , Berlin, 1985).\n[9] A.L.Pankratov, Physics Letters A 234, 329 (1997).\n[10] S.R.Shenoy and G.S.Agarwal, Phys. Rev. A, 29, 1315 (1984).\n[11] A.N.Malakhov, Cumulant analysis of random Non-Gaussi an processes and its transformations (Sovetskoe Radio, Mos cow,\n1978, in Russian).\n[12] R.Roy, R.Short, J.Durnin and L.Mandel, Phys. Rev. Lett ., 45, 1486 (1980).\n[13] K.Lindenberg and B.West, J. Stat. Phys., 42, 201 (1986).\n\n0.60\n0.80\n1.00\n/G33/G0B/G57/G0C\nkT = 3\nkT = 1\nkT = 0.5\nFIG. 1. Evolution of the nondecay probability for the potent ial Φ (x) = ax2 − bx3 for different values of noise intensity; the\n\n0.60\n0.80\n1.00\n/G33/G0B/G57/G0C\nFIG. 2. Evolution of the nondecay probability for the potent ial Φ (x) = ax4 − bx5 for different values of noise intensity.\n\n0.0 20 .0 40 .0 60 .0 80 .0 100 .0\n0.60\n0.80\n1.00\n/G33/G0B/G57/G0C\nFIG. 3. Evolution of the nondecay probability for the potent ial Φ (x) = 1 − cos(x) − ax for different values of noise intensity.\n\n0.0 3.0 6.0 9.0 12 .0 15 .0\n0.60\n0.80\n1.00\n/G33/G0B/G57/G0C\nkT = 0 .1\nkT = 1\nkT = 5\nFIG. 4. Evolution of the nondecay probability for the potent ial Φ (x) = − bx3 for different values of noise intensity; the"}
{"id": "adap-org/9807003", "meta": {"categories": ["adap-org", "cs.NE", "nlin.AO", "q-bio.PE"], "created": "1998-07-17", "extraction": {"body_chars": 25754, "cleaning": {"detected_repeated_margin_lines": ["3"], "page_count": 8, "removed_boilerplate_lines": 5}, "method": "pypdf_no_ocr", "source_pdf_bytes": 196239, "text_chars": 26703}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9807003", "primary_category": "adap-org", "source": "arxiv", "title": "Development and Evolution of Neural Networks in an Artificial Chemistry", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9807003"}, "text": "Development and Evolution of Neural Networks in an Artificial Chemistry\n\nAbstract\nWe present a model of decentralized growth for Artificial Neural Networks (ANNs) inspired by the development and the physiology of real nervous systems. In this model, each individual artificial neuron is an autonomous unit whose behavior is determined only by the genetic information it harbors and local concentrations of substrates modeled by a simple artificial chemistry. Gene expression is manifested as axon and dendrite growth, cell division and differentiation, substrate production and cell stimulation. We demonstrate the model's power with a hand-written genome that leads to the growth of a simple network which performs classical conditioning. To evolve more complex structures, we implemented a platform-independent, asynchronous, distributed Genetic Algorithm (GA) that allows users to participate in evolutionary experiments via the World Wide Web.\n\narXiv:adap-org/9807003v1 17 Jul 1998\nDevelopment and Evolution of Neural Networks in an Artificia l\nChemistry\nJens C. Astor ∗ and Christoph Adami\nComputation and Neural Systems and Kellogg Radiation Laboratory\nCalifornia Institute of Technology\nPasadena, CA, 91125 (USA)\nAbstract\nWe present a model of decentralized growth for Artifi-\ncial Neural Networks (ANNs) inspired by the develop-\nment and the physiology of real nervous systems. In\nthis model, each individual artificial neuron is an au-\ntonomous unit whose behavior is determined only by\nthe genetic information it harbors and local concentra-\ntions of substrates modeled by a simple artificial chem-\nistry. Gene expression is manifested as axon and den-\ndrite growth, cell division and differentiation, substrate\nproduction and cell stimulation. We demonstrate the\nmodel’s power with a hand-written genome that leads\nto the growth of a simple network which performs clas-\nsical conditioning. To evolve more complex structures,\nwe implemented a platform-independent, asynchronous,\ndistributed Genetic Algorithm (GA) that allows users to\nparticipate in evolutionary experiments via the World\nWide Web .\n1 Introduction\nEver since computational neuroscience was born with the\nintroduction of the abstract neuron by McCulloch and\nPitts in 1943 [1], we have witnessed a gap between the\nmathematical modeling of neurons—inspired by Turing’s\nnotions of universal computation—and the physiology of\nbiological neurons and the networks they form. The cur-\nrent state of affairs reflects this dichotomy: neurophysio-\nlogical simulation test beds [2] cannot solve engineering\nproblems, while sophisticated ANN models [3] do not\nexplain the miracle of biological information processing.\nCompared to real nervous systems, classical ANN\nmodels have a serious shortcoming owing to the fact\nthat they are engineered to solve particular classifica-\ntion problems, and analyzed according to standard the-\nory based mainly on statistics and global error reduc-\ntion. As such, they can hardly be considered universal.\nHence, such models define the network architecture a\npriori which is in most cases a fixed structure of homo-\ngeneous computation units.\nSome models support problem-dependent network\n∗⋆On leave from Dept. of Medical Informatics, University\nof Heidelberg/School of Technology, Heilbronn, Germany,\njastor@stud.fh-heilbronn.de\nchanges during simulation [4, 3]. In these models, global\ndecisions lead to network structures adapted to the prob-\nlem at hand. Other approaches try to shape networks\nfor a particular problem by evolving ANNs either di-\nrectly [5], or indirectly via a growth process [6]. More\nrecently, approaches like Ref. [7] include a kind of ar-\ntificial chemistry which allows a more natural develop-\nment. Still, in these models neurons are unevolvable\nhomogeneous structures in a more or less fixed architec-\nture which, we believe, limits their relevance to natural\nnervous systems.\nIn this paper we investigate the idea that interesting\ninformation-processing structures can be grown from a\nmodel which follows four basic principles of molecular\nand evolutionary biology, listed below. While models for\nANNs currently exist that implement a selection of them,\nthe inclusion of all four opens the possibility that, given\nenough evolutionary time, novel structures can emerge\nthat are comparable to natural nervous systems.\n• Coding. The model should encode ANNs in such way\nthat evolutionary principles can be applied.\n• Development. The model should be capable of\ngrowing an ANN by a completely decentralized de-\nvelopmental process, based exclusively on the cell and\nits interactions.\n• Locality. Each neuron must act autonomously and\nbe determined only by its genetic code and the state\nof its local environment.\n• Heterogeneity. The model must have the capabil-\nity to describe different, heterogeneous neurons in the\nsame ANN.\nOne of the key features of a model implementing those\nprinciples will be the absence of explicit activity func-\ntions, learning rules, or connection structures. Rather,\nsuch characteristics should emerge in the adaptive pro-\ncess and lead to ANNs with open architectures and more\nuniversal artificial neurogenesis.\nWhile keeping in mind that the model is not designed\nto reproduce real neural systems, we posit that an ad-1\n\nherence to the fundamental tenets of molecular and evo-\nlutionary principles—albeit in an artificial medium—\nrepresents the most promising unexplored avenue in the\nsearch for intelligent information-processing structures.\n2 Model\nIn this section we introduce our model of neurogenesis\nstarting with the artificial physics and biochemistry, and\ngo on to explain how local gene expression ultimately\nresults in information-processing structures. This gene\nexpression takes place exclusively in artificial neurons\nwhich are embedded in a tissue-like structure. As this\nmodel is inspired by the concepts of molecular cell biol-\nogy, we use the nomenclature of this science unabashedly\nwhile issuing the caveat that they are analogical in na-\nture only.\n2.1 Artificial Physics\nThe physical world is a two-dimensional grid of\nhexagons. Each such site harbors certain concentrations\nof substrates, measured as percentage values of satura-\ntion between 0 and 1. As cells are equidistant in a hexag-\nonal lattice, the diffusion of substrate k in cell i can be\nmodeled discretely as\nCik(t + 1) = D\n6∑\nj=1\n(\nCik(t) − CNi,jk(t)\n)\n(1)\nwhere Cik(t) is the concentration of substrate k in site\ni, D is a diffusion coefficient ( D < 0. 5 to avoid substrate\noscillation), and Ni,j represents the jth neighbor of grid\nelement i.\nFigure 1: Hexagonal grid with boundary elements. Diffusion\noccurs from a local concentration peak at grid element N .\nAccordingly, a local concentration of substrate will dif-\nfuse through the tissue under conservation of mass. The\ntissue itself is surrounded by special boundary elements\nwhich absorb substrates (Figure 1), thus modeling diffu-\nsion in infinite space. We caution at this point that the\nhexagons are sites that may harbor cells, but otherwise\nonly represent a convenient equidistant discretization of\nspace to facilitate the distribution of chemicals via dif-\nfusion.\n2.2 Artificial Biochemistry\nWe distinguish four different classes of substrates:\n• External proteins : Diffusive substrates which can\nbe produced by neurons if expressed.\n• Internal proteins : Produced by neurons, but non-\ndiffusive as they cannot cross cell membranes.\n• Cell-type proteins : Each neural cell harbors an\nexternal protein that defines its type. Like any exter-\nnal protein it is diffusive.\n• Neurotransmitter: Special type of internal pro-\ntein used for directed information exchange between\nneurons.\n2.3 Artificial Cell\nCell types We distinguish three kinds of neurons on\nthe cellular level: actuator cells, sensor cells, and com-\nmon neurons. The first two types are special versions of\nthe third and are used as interface to a (simulated) en-\nvironment to which the network adapts and on which it\ncomputes. These neurons can be excited to a real-valued\nlevel between 0 and 1, and take part in the information\ntransfer via dendritic or axonal connections, respectively\n(see below). Each type of cell is also characterized by its\nown cell-type protein, which it produces continuously at\na certain rate. These cell-type proteins diffuse over the\ntissue (Figure 1) and therefore signal cell existence to\nother cells. They can be compared to growth factors\nknown from the development of real nervous systems.\nActuator and sensor cells do not carry genetic infor-\nmation; they are used solely as interfaces to the environ-\nment (input-output units, see Section 2.4.) They rep-\nresent sources and sinks of signal. Consequently, their\nbehavior is hardwired and does not depend on transcrip-\ntion as for common neurons. The latter can receive a\nflux of neurotransmitter from dendrites with a particu-\nlar weight. However, this does not imply an automatic\nstimulation of activity unless such behavior is explicitly\nencoded in the neuron’s genome. Table 1 summarizes\nthe cell types and how they interact with other compu-\ntational elements used in our model.\nGenetic code and gene expression in artificial\nneurons Each neural cell carries a genome which com-\npletely encodes its behavior. Genomes consist of genes\nwhich can be viewed as a genetic program that can ei-\nther be executed ( expressed) or not, depending on a gene\ncondition (akin to the regulator/operator genes in the2\n\nType [1] [2] [3] [4] [5] [6]\nNeuron x x x x x x\nSensor x x x x\nActuator x x x x\nGrid element x\nBoundary element\nTable 1: Features of different cell types building the ar-\ntificial organic tissue: [1] participates in diffusion, [2]\ncan be stimulated, [3] depends on gene expression, [4]\ncan have axons, [5] can have dendrites, [6] produces a\ndiffusible cell-type protein.\nCondition Description\nADD [EP] Φ new=Φ before + [EP]\nMUL [eNT] Φ new=Φ before × [eNT]\nSUP [CTP0] suppresses gene if cell not of type CTP0\nAND [IP] fuzzy AND: Φ new = max([IP], Φ before)\nTable 2: Condition atoms and their interpretation. Con-\ndition atoms build a gene condition, which is obtained by\nevaluating its condition atoms in the given order. Here,\n[XY] means ‘the current local concentration of substrate\nXY inside of the cell to which the gene condition belongs’.\nΦ is the evaluation result of this gene condition.\nJacob-Monod-model). A gene condition is a combina-\ntion of several condition atoms, usually related to local\nconcentrations of substrates. The expression of a gene\ncan result in different behaviors such as the production of\na protein, cell division, axon/dendrite growth, cell stim-\nulation, etc. Figure 2 clarifies the structure of the ge-\nnetic code. Thus, gene conditions model the influence\nof external concentrations on the expression level of the\ngene, i.e., they model activation and suppression sites.\nTo evaluate a gene condition, each element of its chain of\nCondition 3\nExpression 1 Expression 2\nCondition 1 Condition 2\nGene 2\nGene 1\nGene 4\nGene 3\nGene 2\nN\nGenome\nFigure 2: Genetic structure of neural cells. Local con-\ncentrations of substrates (condition atoms) trigger gene\nexpression.\ncondition atoms contributes to obtain a real-valued ex-\npression level between 0 and 1, describing the strength\nwith which the respective gene expression will take place.\nt=0 cycles:\nt=10 cycles:\nt=6 cycles:\nt=8 cycles:\nFigure 3: Development of the network for classical condi-\ntioning.\n\nExpression Command Description Influence of\nCondition Value\nPRD[XY] produce substrate XY production quantity\nGDR[XY] grow dendrite following gradient of XY growing probability\nGRA[XY] grow axon following gradient of XY growing probability\nSPL[CTPx] divide to CTPx-type cell probability\nEXT excitatory stimulus increase rate\nINH inhibitory stimulus decrease rate\nMOD+ increase connection weights strengthening factor\nMOD- decrease connection weights weakening factor\nTable 3: Overview of expression commands. Growing axons/dendrit es follow the substrate gradient until the local\nmaximum is reached, then connect to the cell (if it exists). Strengt hening/weakening is a percentage increase/decrease\nof connection weights, determined by the product of the last neur otransmitter influx at each connection and the value\nof the gene condition. The cell-type protein assigned to a cell division command determines the future type of the\noffspring cell. In this example, the daughter will be of type CTPx and therefore produce cell-type protein CTPx\ncontinuously.\nConsider for example substrates with local concen-\ntrations [ep0, ip, ep1]=(0.3, 0.5, 0.5 ). Then, the\nevaluation of gene condition ADD[ep0] ADD[ip] MUL[ep1]\nwould lead to an overall expression level 0.4 (this value is\nmodded back into range between 0 and 1 if it falls outside\nit). Table 2 illustrates a few examples of such conditions,\nwhile Table 3 gives an overview about the different gene\nexpression commands. The evaluation result of the gene\ncondition has different meanings depending on the gene\nexpression command.\n2.4 Simulation of the Artificial Organism\nThe tissue of cells produced by gene expression and cell\ngrowth is termed the artificial organism . It receives in-\nput from the environment (the outside world) and can\nact on it by signaling to the environment via its actu-\nators. In the simplest case, thus, the organism receives\nand generates patterns.\nA simulation always starts by creating sensor and ac-\ntuator cells. Their number is determined only by the\ncomplexity of the outside world and is not coded for in\nthe genome. In other words, these cells really represent\npossible signals and actuations in the world, not actual\nsignals and actuations performed by the organism. An\norganism chooses to receive input or perform an actua-\ntion by connecting to these cells. If needed, an additional\nreinforcement cell can be created. This is a special sen-\nsor cell (with its own cell-type protein) used to provide a\nreinforcement signal from the world about the behavior\nof the organism. Whether or not this signal is used is\ndetermined by the organism’s genome. Furthermore, at\nthe start of each simulation, one initial neuron is placed\nin the center of the grid. After initialization, the simula-\ntion can begin. Input from the world is provided to the\nsensor cells, diffusion of produced cell-type proteins and\nexternal proteins takes place, and neurons execute their\ngenetic code synchronously.\nDepending on its gene expression, a neuron starts\ngrowing axons and dendrites, produces offspring cells\nand might initiate cell differentiation. Gene expressions\nmay lead to protein production cascades, stimulation,\nand ultimately information exchange between neurons.\nAfter every simulation cycle the network’s ‘fitness’ is de-\ntermined by comparing any inputs and outputs to what\nis expected in this particular world, producing a real-\nvalued reinforcement signal between 0 (punishment) and\n1 (reward). This signal can be used by the organism if\na reinforcement sensor is present and if the organism\nchooses to connect to it.\n3 An Example Genome\nFigure 3 documents the development of a simple ANN\nfrom a hand-written genome. Starting from a single ini-\ntial neuron, cell division takes place and connections (ax-\nons and dendrites) start to grow along the gradient of dif-\nfusing cell-type proteins. After a while, sensors, neurons\nand actuator cells are connected in a particular manner.\nIn fact, this network displays conditioned reflex behav-\nior as in Pavlov’s classical experiment [10]. Suppose the\nsensor on the lower left side in Figure 3 is stimulated\n(active) at the sound of a bell. Further, suppose the\nupper left sensor is an optical stimulus representing the\npresence (or absence) of food. Finally, let us imagine\nthat the actuator on the right side triggers a salivary\ngland if food is present. This behavior is the hardwired\nunconditioned reflex. The above network can learn to\nassociate the reflex with a condition: the sound of the\nbell. If presence of food and the ringing of the bell are\nassociated repeatedly, the network will learn to trigger\nthe gland even if only the bell rings. If the bell rings\nafter the conditioning without the presence of food, the\nassociation will gradually, but steadily, weaken. Such a4\n\n1. NNY(ip) SUP(cpt) ANY(spt0) -> SPL(acpt0) PRD(ip) SPL(ac pt2) GDR(spt0) DFN(NT1)\n2. NNY(ip) SUP(acpt0) ANY(spt1) ANY(cpt) -> PRD(ip) GDR(sp t1) GRA(cpt) DFN(NT1)\n3. ANY(spt1) SUP(acpt2) NNY(ip) ANY(apt0) -> SPL(acpt1) GD R(spt1) PRD(ip) GRA(apt0)\n4. ANY(acpt2) SUP(acpt1) ANY(spt0) NNY(ip) -> GRA(acpt2) G DR(spt0) GDR(cpt) PRD(ip)\n5. ANY(ip) -> PRD(ip)\n6. NSUP(cpt) NSUP(acpt1) ADD(eNT) -> EXT\n7. SUP(acpt1) ADD(NT1) MUL(eNT) -> EXT\n8. ADD(eNT) -> PRD(ip1)\n9. ADD(ip1) -> PRD(ip2)\n10. SUP(cpt) ADD(NT1) MUL(ip2) -> PRD(ep)\n11. SUP(cpt) ADD(ep) -> EXT\nFigure 4: Genome for development and behavior of network exhibitin g classical conditioning\nbehavior can be modelled using different kinds of cell-\ntypes. The “C” cell is activated if the network is in\nthe conditioned state, which means that acoustical and\noptical stimuli have been present together before. Cell\n“E” is activated if the acoustical stimulus is currently\npresent and the network is in the conditioned state at\nthe same time. If so, cell “G” representing the trigger of\nthe salivary gland is activated. Of course, cell “G” is also\nactivated if only food is present. This is the “hardwired”\nreflex. A schematical drawing of the network is shown in\nFigure 5. The genome which encodes the development\nand behavior of this network is shown in Fig. 4.\nBell\nFood\nE\nG\nC eNT\nSensor cells Actuator\neNT\neNT\nspt0\nacpt1\nacpt2 apt0\neNT\nacpt0eNT\nspt1\nNT-switch\nNT1\neNT\ncpt\nNT1\nFigure 5: A schematical representation of the network for\nclassical conditioning. The types of neurotransmitter use d\nare shown next to the axons. The cell-type protein used by\neach cell is indicated near the cell body.\nIt is beyond the scope of this paper to go into the\ndetails of this genome and its function (see [8, 9] for a\nmore thorough description). However, the explanation\nthat follows still gives an idea of the type of information\nnecessary to grow networks with particular characteris-\ntics.\nThe genome consists of 11 genes, each of which has\nits condition (left-hand side) and its expression (right-\nhand side). Genes 1 to 4 control cell division into the\ndifferent types that are needed, as well as the growth of\naxons and dendrites. The first gene is only expressed by\nthe initial cell, and only if no internal protein ip is present\n(the sequence NNY(ip) SUP(cpt) ). In addition, gene 1 is\nonly expressed if it senses nonzero concentrations of cell-\ntype protein spt0 (ANY(spt1) ANY(spt0) ) which is emitted\nby one of the sensor cells and has diffused. Under these\ncircumstances the initial cell will divide and produce off-\nspring of type acpt0 and acpt2 (SPL(acpt2) SPL(acpt0) ),\ngrow a dendrite that follows the gradient of the sensor\nprotein spt0 (GDR(spt0)), and produce the internal pro-\ntein ip. Once ip is produced, this will continue to happen\n(gene 5) which prevents that gene 1 can ever be turned on\nagain. Genes 2 to 4 work just as gene 1, but for other cell\ntypes. While gene 5 takes care of the hardwired-reflex\nstimulation, genes 10 and 11 control the conditioning\n(expressed only by cell-type cpt). If food is present and\nthe bell rings, gene 10 is expressed. It produces certain\namounts of external protein ep. The concentration of ep\ninfluences cell stimulation (gene 11 exhibits stimulation\nvia EXT). Due to diffusion, ep diminishes over time, so\nthe conditioning decreases accordingly. As pathways to\nthe “C” cell are not equally long, a production cascade of\ninternal proteins in genes 8, 9 and 10 is necessary that\ndelays the input of the acoustical sensor. The behav-\nior of the resulting phenotype network is documented in\nFigures 6, 7 and 8.\n4 WWW-based Genetic Algorithm:\nCommunity of Artificial Organisms\nWhile it is not difficult to write genomes which lead\nto simple networks with desired characteristics, one of\nthe main features of the system is its evolvability. Cer-\ntainly, the search space for such genomes is immense,\nand it is unreasonable to hope that interesting genomes\ncan be found without massive parallelism. Rather than\nchoosing to implement this system on supercomputers,\nwe opted to allow users on the Internet to donate their\nCPU time by participating in a global evolutionary ex-\nperiment.\nUsing Sun’s JavaTM technology, we developed an asyn-\nchronous, distributed GA system which allows a massive\nparallel search for new genomes based on evolutionary\nprinciples [8, 9]. It consists mainly of a central server5\n\nt=0..19\nt\n1.0\n0.0\nt\n1.0\n0.0\nt\n1.0\n0.0\nt\n1.0\n0.0\nt\n0.0\n1.0\nFood\nBell\nE-Cell\nGland\nC-CellFood\nBell\nSensor cells\nActuator\nC\nE\nG\nFigure 6: First, only food is present. This triggers the gland because of the hardwired reflex, while the “C”-cell and “E”-cell\nremain inactivated. Later, only the bell signal is present. Due to the fact that the network is not yet conditioned, none o f the\ncells becomes active as a result.\nt=16..38\nt\n1.0\n0.0\nt\n0.0\n1.0\nt\n1.0\n0.0\nt\n1.0\n0.0\nt\n1.0\n0.0\nBell\nFood\nC-Cell\nE-Cell\nGland\nFood\nBell\nC\nE\nG\nSensor cells\nActuator\nFigure 7: Both sensors, food and sound, are stimulated. The ANN become s conditioned (“C”-cell) and the gland is triggered\ndue to the presence of food.\nt=50..64\nt\n1.0\n0.0\nt\n1.0\n0.0\nt\n0.0\n1.0\nt\n1.0\n0.0\nt\n1.0\n0.0\nFood\nBell\nE-Cell\nC-Cell\nGland\nFood\nBell\nC\nE\nG\nSensor cells\nActuator\nFigure 8: Being in the conditioned state, the food sensor suddenly bec omes deactivated while the bell keeps on ringing.\nThus, the activation of the “C”-cell becomes weaker. This im plies a decrease of activation of the “E”-cell which finally r esults\nin a decline of gland activity.\n\nSun SPARC 141.7.12.105\nOrganism, e.g. on\nsomewhere\nan i86-PC\nOrganism as Applet\nin a browser, on\narchitecture 'foo',\non Sun SPARC\n131.215.48.50\nCentral GA-server, e.g.\nOrganism, e.g. on\nevaluation request for\nregister request\nnew genotype or\n'go on' reply\ngenotype reply:ζ\nnew client central GA-server\nΦ(ζ)\nFigure 9: Left: genotype evaluation in clients of different architect ure, using TCP/IP for communication with the\nasynchronous genetic algorithm. Right: communication between th e GA-server and a client hosting an organism.\napplication and clients, each of which hosting one indi-\nvidual of the current GA population.\nAs Java is supposed to be platform independent,\nclients can be started from every computer for which\nan accurate Java 1.1 virtual machine or browser exists\n(Figure 9). The clients are hybrids, which means that\nthey can be started as Java Applet by choosing the html\npage from our WWW-server, or with the help of a boot-\nloader program which dynamically downloads the client\nand starts it as a Java application (no browser neces-\nsary).\nA client automatically sends a request-to-register to\nthe central GA-server after it was started, and receives\nfrom the server a genotype ζ. The client then starts\nup a simulation as described in Section 2.4. After a\ncertain number of simulation cycles, it sends the geno-\ntype’s fitness Φ( ζ) (average reinforcement signal during\nsimulation) to the server. By comparing Φ( ζ) to the\nfitness of other genotypes in the database, the server de-\ncides if it is worth to keep this genotype or if the client\nshould be assigned a new one. If the server has to send a\nnew genotype, it either takes a suspended one out of its\ndatabase, or constructs one through the processes of re-\ncombination and/or mutation from genotypes of known\nfitness already present in the population (Figure 9). Fit-\nter genotypes are more likely to be selected for recombi-\nnation and/or asexual copying then genotypes of lower\nfitness. This leads to an increase of the average fitness\nover time.\n5 Conclusion\nWe introduced a developmental and behavioral model\nbased on artificial gene expression which shares key prop-\nerties with natural neural development. Within this\nmodel we succeeded to construct simple systems with\nproperties which are believed to be essential [11, 12]\nfor higher self-organizing information processing sys-\ntems, such as deterministic structure development, self-\nlimiting growth, growth following diffusion gradients,\ncomputation of logical functions, pacemaker behavior\nand simple adaptation (sensitization, habituation, as-\nsociative classical conditioning) [8, 9]. Furthermore, we\nshowed how an evolutionary search for genomes coding\nfor information-processing network structures can be dis-\ntributed in a platform-independent manner such that\nthe unused CPU power of the Internet can be tapped\nto search for ANNs that reduce the gap between the ab-\nstract models and neurophysiology.\nReferences\n[1] W. Pitts and W. S. McCulloch. A logical calculus\nof the ideas immanent in nervous activity. Bulletin\nof Mathematical Biophysics , 5:115–133, 1943.\n[2] J. M. Bower. The Book of GENESIS : Exploring Re-\nalistic Neural Models with the GEneral NEural SIm-\nulation System. Santa Clara, CA : TELOS Springer-\nVerlag, 1995.\n\n[3] S. E. Fahlman and C. Lebi` ere. The cascade-\ncorrelation learning architecture. In Advances in\nNeural Information Processing Systems 2 , edited by\nD. S. Touretzky, pp. 524–532. Morgan Kaufmann,\n1990.\n[4] B. Fritzke. Growing cell structures: A self-\norganizing network in k dimensions. In Artificial\nNeural Networks II , edited by I. Aleksander and\nJ. Taylor, pp. 1051–1056. Elsevier, 1992.\n[5] D. H. Ackley and M. L. Littman. Interactions be-\ntween learning and evolution. In Artificial Life II ,\nedited by C. G. Langton, C. Taylor, J. D. Farmer,\nand S. Rasmussen, pp. 487–509. Addison Wesley,\n1992.\n[6] F. Gruau. Genetic synthesis of boolean neural net-\nworks with a cell rewriting developmental process.\nIn Combination of Genetic Algorithms and Neural\nNetworks, edited by D. Whitley and J. D. Schaffer,\nIEEE Computer Society Press, 1992.\n[7] O. Michel. An artificial life approach for the synthe-\nsis of autonomous agents. In Artificial Evolution ,\nedited by E. Lutton, J. M. Alliot and E. Ronald.\nLecture Notes in Computer Science 1063: 220–231,\n1996.\n[8] J. C. Astor. A Developmental Model for the Evolu-\ntion of Artificial Neural Networks: Design, Imple-\nmentation, Evaluation . Diploma thesis in Medical\nInformatics, University of Heidelberg, 1998.\n[9] J. C. Astor and C. Adami. In preparation.\n[10] I. P. Pavlov. Conditioned Reflexes: An Investigation\nof the Physiological Activity of the Cerebral Cortex .\nLondon: Oxford University Press, 1927.\n[11] E. R. Kandel and J. H. Schwartz. Principles of\nNeural Science. Elsevier, 1991.\n[12] C. Koch and T. Poggio. Biophysics of computa-\ntion: Neurons, synapses and membranes. In Synap-\ntic Function , edited by G. Edelman, W. Gall and\nW. Cowan, pp. 637–697. J. Wiley, 1982."}
{"id": "adap-org/9807005", "meta": {"categories": ["adap-org", "nlin.AO", "q-bio"], "created": "1998-07-27", "extraction": {"body_chars": 89565, "cleaning": {"detected_repeated_margin_lines": ["October 24, 2018 Page 1"], "page_count": 20, "removed_boilerplate_lines": 59}, "method": "pypdf_no_ocr", "source_pdf_bytes": 316047, "text_chars": 91308}, "license": null, "license_family": "unspecified_or_default_arxiv", "pdf_url": "https://arxiv.org/pdf/adap-org/9807005", "primary_category": "adap-org", "source": "arxiv", "title": "A dynamical theory of speciation on holey adaptive landscapes", "updated": "2009-11-30", "url": "https://arxiv.org/abs/adap-org/9807005"}, "text": "A dynamical theory of speciation on holey adaptive landscapes\n\nAbstract\nThe metaphor of holey adaptive landscapes provides a pictorial representation of the process of speciation as a consequence of genetic divergence. In this metaphor, biological populations diverge along connected clusters of well-fit genotypes in a multidimensional adaptive landscape and become reproductively isolated species when they come to be on opposite sides of a ``hole'' in the adaptive landscape. No crossing of any adaptive valleys is required. I formulate and study a series of simple models describing the dynamics of speciation on holey adaptive landscapes driven by mutation and random genetic drift. Unlike most previous models that concentrate only on some stages of speciation, the models studied here describe the complete process of speciation from initiation until completion. The evolutionary factors included are selection (reproductive isolation), random genetic drift, mutation, recombination, and migration. In these models, pre- and post-mating reproductive isolation is a consequence of cumulative genetic change. I study possibilities for speciation according to allopatric, parapatric, peripatric and vicariance scenarios. The analytic theory satisfactorily matches results of individual-based simulations reported by Gavrilets et al. (1998). It is demonstrated that rapid speciation including simultaneous emergence of several new species is a plausible outcome of the evolutionary dynamics of subdivided populations. I consider effects of population size, population subdivision, and local adaptation on the dynamics of speciation. I briefly discuss some implications of the dynamics on holey adaptive landscapes for molecular evolution.\n\narXiv:adap-org/9807005v1 27 Jul 1998\nA dynamical theory of speciation\non holey adaptive landscapes\nSergey Gavrilets\nDepartments of Ecology and Evolutionary Biology\nand Mathematics,\nUniversity of Tennessee,\nKnoxville, TN 37996-1610, USA\nphone: (423) 974-8136\nfax: (423) 974-3067\ne-mail: gavrila@tiem.utk.edu\nkey words: evolution, speciation, holey adaptive land-\nscapes, mathematical models\nrunning head: Dynamical Theory of Speciation\nABSTRACT: The metaphor of holey adaptive land-\nscapes provides a pictorial representation of the process of\nspeciation as a consequence of genetic divergence. In this\nmetaphor, biological populations diverge along connected\nclusters of well-fit genotypes in a multidimensional adap-\ntive landscape and become reproductively isolated species\nwhen they come to be on opposite sides of a “hole” in the\nadaptive landscape. No crossing of any adaptive valleys is\nrequired. I formulate and study a series of simple models\ndescribing the dynamics of speciation on holey adaptive\nlandscapes driven by mutation and random genetic drift.\nUnlike most previous models that concentrate only on\nsome stages of speciation, the models studied here de-\nscribe the complete process of speciation from initiation\nuntil completion. The evolutionary factors included are\nselection (reproductive isolation), random genetic drift,\nmutation, recombination, and migration. In these mod-\nels, pre- and post-mating reproductive isolation is a conse-\nquence of cumulative genetic change. I study possibilities\nfor speciation according to allopatric, parapatric, peri-\npatric and vicariance scenarios. The analytic theory sat-\nisfactorily matches results of individual-based simulations\nreported by Gavrilets et al. (1998). It is demonstrated\nthat rapid speciation including simultaneous emergence\nof several new species is a plausible outcome of the evo-\nlutionary dynamics of subdivided populations. I consider\neffects of population size, population subdivision, and lo-\ncal adaptation on the dynamics of speciation. I briefly\ndiscuss some implications of the dynamics on holey adap-\ntive landscapes for molecular evolution.\nSpeciation has traditionally been considered to be one\nof the most important and intriguing processes of evolu-\ntion. In spite of this consensus and significant advances\nin both experimental and theoretical studies of evolution,\nunderstanding speciation still remains a major challenge\n(Mayr 1982a; Coyne 1992). The main reason for such\na discouraging situation is that direct experimental ap-\nproaches, which are widely used for solving other prob-\nlems of evolutionary biology, are not effective for studying\nspeciation because of the time scale involved. Experi-\nmental work necessarily concentrates on distinct parts of\nthe process of speciation intensifying and simplifying the\nfactors under study (Rice and Hostert 1993; Templeton\n1996). In situations where direct experimental studies are\ndifficult or impossible, mathematical modeling has proved\nto be indispensable for providing a unifying framework.\nAlthough numerous attempts to model parts of the pro-\ncess of speciation have been made, a quantitative theory\nof the dynamics of speciation is still missing. Currently,\nverbal theories of speciation are far more advanced than\nmathematical foundations. As often is the case with ver-\nbal theories (both scientific and otherwise), different de-\nduced (or induced) aspects of speciation are emphasized\nby different workers resulting in confusion and contro-\nversy. The situation is not helped by the absence of gen-\neral agreement on a species definition (e.g., Claridge et\nal. 1997).\nHere, I attempt to develop some foundations for a gen-\neral dynamical theory of speciation. One possible ap-\nproach to this goal would be to begin with a species defini-\ntion, then to define speciation accordingly and to develop\nan appropriate dynamical model. I do not think such ap-\nproach would be very useful, due to a lack of generality.\nMy models are not based on a specific “species concept“.\nI reason that species are “different” with respect to some\ncharacteristics, and that whatever these differences, they\nhave a genetic basis. Thus, modeling the dynamics of spe-\nciation is equivalent to modeling the dynamics of genetic\ndivergence. I use a bottom-up approach: begin with a\nmodel incorporating a range of factors thought to lead to\nspeciation (e.g. selection, mutation, population subdivi-\nsion etc.) and then try to interpret its dynamic behavior\nin terms of different “species concepts”. As expected,\nmany aspects of speciation that are emphasized by dif-\nferent species concepts (such as reproductive isolation,\nseparate genotypic clusters or common evolutionary tra-\njectories) emerge from the same processes. This clearly\nindicates that different species concepts are not mutually\nexclusive.\nThe choice of a modeling approach depends upon the\npurpose of the model. A common view in (evolution-\nary) biology is that mathematical models are mainly use-\nful for making predictions that can be used in experi-\nmental work. Although such a pragmatic approach is\nprobably what should be expected in contemporary so-\nciety, a model’s testable predictions are not necessarily\nits main contribution to science. Insights provided by\nmodels, their ability to train our intuition about complex\nphenomena, to provide a framework for studying such\nphenomena and to identify key components in complex\n\nsystems are at least as important as specific predictions.\nFor these purposes the most useful tools are simple mod-\nels and metaphors.\nSewall Wright’s (1932) metaphor of “rugged adaptive\nlandscapes” is a well-known and widely used metaphor\nin evolutionary biology. In the standard interpretation,\na rugged adaptive landscape is a surface in a multidi-\nmensional space that represents the mean fitness of the\npopulation as a function of gamete (or allele) frequencies\nwhich characterize the population state (see Fig.1a). It is\nenvisioned that this surface has many peaks and valleys\ncorresponding to different adaptive and maladaptive pop-\nulation states, respectively. The population is imagined\nas a point on the surface which is driven by selection up\nhill but can get stuck on a local peak. Two general points\nabout scientific metaphors should be kept in mind. The\nfirst is that specific metaphors (as well as mathematical\nmodels) are good for specific purposes only. The second is\nthat accepting a specific metaphor necessarily influences\nand defines the questions that are considered to be im-\nportant. The metaphor of “rugged adaptive landscapes”\nis very useful for thinking about adaptation. However,\nits utility for understanding speciation is questionable.\nFrom a pragmatic point of view, the process of splitting a\npopulation into two different species is impossible to de-\nscribe in a framework where a population is the smallest\nunit. Finer resolution is necessary for describing the split-\nting of populations. Accepting the metaphor of rugged\nadaptive landscapes immediately leads to a problem to\nbe solved: how can a population evolve from one adap-\ntive peak to another across an adaptive valley when selec-\ntion opposes any changes away from the current adaptive\npeak? Wright’s solution to this problem, his shifting-\nbalance theory (Wright, 1931, 1982), does not seem to\nbe satisfactory (Gavrilets 1996; Coyne et al. 1997).\nProvine (1986), Barton and Rouhani (1987), Whitlock\net al. (1995), Gavrilets (1997a) and Coyne et al. (1997)\ndiscuss other weaknesses of Wright’s metaphor. I have\nargued elsewhere (Gavrilets 1997a) that his metaphor of\nrugged adaptive landscapes with its emphasis on adap-\ntive peaks and valleys is, to a large degree, a reflection of\nthe three-dimensional world we live in. Both genotypes\nand phenotypes of biological organisms differ in numerous\ncharacteristics, and, thus, the dimension of “real” adap-\ntive landscapes is much larger than three. Properties of\nmultidimensional adaptive landscapes are very different\nfrom those of low dimension. Consequently, it may be\nmisleading to use three-dimensional analogies implicit in\nthe metaphor of rugged adaptive landscapes in a multi-\ndimensional context. I believe that understanding speci-\nation requires a different metaphor.\nThe metaphor of “holey” adaptive landscapes . An in-\ndividual organism can be considered as a combination of\ngenes. All possible combinations of genes form a geno-\nfitness\ngenotype space\nfitness\na\nb\npopulation state\nFigure 1: Adaptive landscapes. a. A rugged\nadaptive landscape. The main emphasis is on\n“peaks” and “valleys” corresponding to different\nwell-adapted and maladaptive genotypes. b. A\nholey adaptive landscape. The main emphasis\nis on clusters of well-fit genotypes that extend\nthroughout the genotype space. All other geno-\ntypes are treated as “holes”.\ntype space (which, mathematically, can be represented\nby a hypercube). In discussing the evolution of popula-\ntions, it is useful to visualize each individual as a point in\nthis genotype space. Accordingly, a population will be a\ncloud of points, and different populations (or species) will\nbe represented by different clouds. Selection, mutation,\nrecombination, random drift and other factors change the\nsize, location and structure of these clouds. To construct\nan adaptive landscape one assigns “fitness” to each geno-\ntype (or each pair of genotypes) in genotype space. Dif-\nferent forms of selection and reproductive isolation can\nbe treated within this conceptual framework. For exam-\nple, fitness can be a genotype’s viability (in the case of\nviability selection); it can be fertility or the probability\nof successful mating between a pair of genotypes (in the\ncase of fertility selection or premating isolation, respec-\ntively). A finite population subject to mutation is likely\nto be represented by genotypes with fitnesses within a\nfitness band determined by the balance of mutation, se-\nlection and random drift. A general property of adap-\ntive landscapes with a very large number of dimensions is\nthat genotypes with fitnesses within a specified band form\nconnected “clusters” that extend throughout the geno-\n\ntype space (Gavrilets 1997; Gavrilets and Gravner 1997).\nThus, populations can evolve and diverge along bands of\nhighly-fit genotypes without going across the states with\na large number of low-fit genotypes (that is without cross-\ning any adaptive valleys).\nThe metaphor of “holey” adaptive landscapes puts spe-\ncial emphasis on these clusters of highly-fit genotypes, dis-\nregarding fitness differences between them and treating\nall other genotypes as “holes” (Gavrilets 1997; Gavrilets\nand Gravner 1997). The justification for the latter is a\nbelief that selection will be effective in moving the popu-\nlation away from these areas of genotype space on a time\nscale that is much faster than the time scale for speciation.\nAccordingly, microevolution and local adaptation can be\nviewed as the climbing of the population from a “hole” to-\nwards the holey adaptive landscape, whereas macroevolu-\ntion can be viewed as a movement of the population along\nthe holey landscape with speciation taking place when the\ndiverging populations come to be on opposite sides of a\n“hole” in the adaptive landscape. In this scenario, there is\nno need to cross any “adaptive valleys”; reproductive iso-\nlation between populations evolves as an inevitable side\neffect of accumulating different mutations. The metaphor\nof holey adaptive landscapes can be traced to a verbal\ntwo-locus two-allele model of reproductive isolation pro-\nposed by Dobzhansky (1937) and similar ideas discussed\nby Bateson (cited in Orr, 1997), Muller (1942), Maynard\nSmith (1970, 1983), Nei (1976), Barton and Charlesworth\n(1984), Kondrashov and Mina (1986). For more discus-\nsion of this metaphor see Gavrilets (1997ab) and Gavrilets\nand Gravner (1997). Orr (1995) and Gavrilets (1997a)\ngives a summary of relevant experimental evidence. The\nmetaphor of “holey” adaptive landscapes is illustrated\ngraphically in Figure 1b.\nMathematical models for holey adaptive landscapes.\nHere I briefly review previously published work on the\nevolutionary dynamics on “holey” adaptive landscapes.\nNei (1976) and Wills (1977) were the first to present for-\nmal analyses of the Dobzhansky model. Nei et al. (1983)\nstudied one- and two-locus multi-allele models with step-\nwise mutations and considered both postmating and pre-\nmating reproductive isolation. In their models geno-\ntypes were reproductively isolated if they were different\nby more than 1 or 2 mutational steps. In these situations,\nspeciation was very slow. They conjectured, however,\nthat increasing the number of loci may significantly in-\ncrease the rate of speciation. Bengtsson and Christiansen\n(1983) presented a deterministic analysis of mutation-\nselection balance in the Dobzhansky model. Bengts-\nson (1985), Barton and Bengtsson (1986) and Gavrilets\n(1997b) analyzed the properties of hybrid zones arising\nunder Dobzhansky-type epistatic selection. Wagner et al.\n(1994) considered a two-locus, two-allele model of stabi-\nlizing selection acting on an epistatic character. For a\nspecific set of parameters, the interaction of epistasis in\nthe trait and stabilizing selection on the trait resulted\nin a fitness “ridge”. The existence of this ridge sim-\nplified stochastic transitions between alternative equilib-\nria. Gavrilets and Hastings (1996) formulated a series\nof two- and three-locus Dobzhansky-type viability selec-\ntion models as well as models for selection on polygenic\ncharacters. They studied these models in the context of\nfounder effect speciation and noticed that the existence\nof ridges in the adaptive landscape made stochastic di-\nvergence much more plausible. Similar conclusions were\nreached by Gavrilets and Boake (1998) who studied the\neffects of premating reproductive isolation on the plau-\nsibility of founder effect speciation. Higgs and Derrida\n(1991, 1992) proposed a model where the probability of\nmating between two haploid individuals is a decreasing\nfunction of the proportion of loci at which they are differ-\nent. Here, any two sufficiently different genotypes can be\nconsidered as sitting on opposite sides of a hole in a ho-\nley adaptive landscape. These authors as well as Manzo\nand Peliti (1994) studied this model numerically assum-\ning that the number of the loci is infinite, the loci are\nunlinked and highly mutable, and mating is preferential.\nOrr (1995) and Orr and Orr (1996) studied speciation\nin a series of models in which viability of a diploid or-\nganism depends on the number of heterozygous loci. All\nthese papers postulated the existence of ridges of highly-\nfit genotypes. Gavrilets and Gravner (1997) studied a\ngeneral class of multilocus selection models and showed\nthe existence of ridges to be inevitable under fairly gen-\neral conditions. Independently, a similar conclusion was\nreached in Reidys et al. (1997). Most previous studies of\nthe dynamics of speciation on holey adaptive landscapes\nwere numerical. To develop a dynamical theory of speci-\nation it is desirable to have a simple model that can be\ntreated analytically.\nThe Model\nI consider finite populations of haploid individuals with\ndiscrete, non-overlapping generations. I assume that re-\nproduction involves gene exchange (amphimixis) between\nindividuals. The restriction to haploids is for algebraic\nsimplicity. Models for diploids will be discussed later. In-\ndividuals are different with respect to L possibly linked\ndiallelic loci. Without any loss of generality each individ-\nual’s genotype can be represented as a sequence of 0’s and\n1’s. Letlα = (lα\n1,...,l α\nL) wherelα\ni is equal to 0 or 1, be such\na sequence for an individual α . In standard population\ngenetics models, the population state is usually described\nin terms of gamete frequencies. In systems with many\nloci such an approach is not practical. For instance, with\n10 diallelic loci there are 2 10 different gametes. Thus, one\nwould need to analyze more than 1000 coupled equations.\n\nAnother complication follows from the fact that even in\nvery large populations with hundreds of thousands indi-\nviduals each specific genotype is represented only by a\nsmall number of copies or is not represented at all. Thus,\nthe notion of a gamete frequency in multilocus evolution\nmight be very difficult to justify. Here, I will be inter-\nested in the levels of genetic variation within subpopu-\nlations and genetic divergence between subpopulations.\nBoth can be characterized in terms of genetic distance\nd defined as the number of loci at which two individu-\nals are different. More formally, the genetic distance dαβ\nbetween individuals α and β is\ndαβ =\nL∑\ni=1\n(lα\ni − lβ\ni )2. (1)\nGenetic distance d is the standard Hamming distance.\nIt is analogous to the number of segregating sites in a\nsample of two gametes, which is widely used in molecu-\nlar evolutionary genetics (Li, 1997), and to the number\nof heterozygous loci in a diploid organism. Genetic dis-\ntanced is also closely related to the notion of the overlap\nq between two sequences, d = L\n2 (1 − q), which is com-\nmonly used in statistical physics (e.g., Derrida and Peliti\n1991). I model the expected dynamics of average genetic\ndistances within and between populations, using Dw for\nthe former and Db for the latter.\nI assume that reproductive isolation is caused by cumu-\nlative genetic change. I will use a very simple symmetric\nmodel which is closely related to the models discussed\nabove and which allows one to treat both pre- and post-\nmating isolation within the same framework. I posit that\nan encounter of two individuals can result in viable and\nfecund offspring only if the individuals are different at\nno more than K loci. Otherwise the individuals do not\nmate (premating reproductive isolation) or these offspring\nare inviable or sterile (postmating reproductive isolation).\nMore formally, I assign “fitness” w to each pair of individ-\nuals depending on the genetic distance d between them\nw(d) =\n{\n1 for d ≤ K,\n0 for d>K. (2)\n(see Appendix for an outline of more complicated ap-\nproaches). In this formulation, any two genotypes differ-\nent at more than K loci can be conceptualized as sitting\non opposite sides of a hole in a holey adaptive landscape\n(cf. Higgs and Derrida 1991, 1992). At the same time, a\npopulation can evolve to any reproductively isolated state\nby a chain of single locus substitutions. The adaptive\nlandscape corresponding to this model is both “holey”\nand “correlated”. The latter means that the probability\nthat two genotypes are reproductively isolated correlates\nwith the genetic distance between them. In Nei et al.\n(1983) and Gavrilets and Boake (1998) models individ-\nuals separated by more than one mutational step were\nreproductively isolated which corresponds to K = 1. The\nneutral case (no reproductive isolation) corresponds to K\nequal to the number of loci.\nThe mathematical model presented above was inter-\npreted as describing sexual haploid populations with fit-\nnesses assigned to pairs of individuals depending on the\ngenetic distance between them. However, there is an al-\nternative interpretation in that the model describes ran-\ndomly mating diploid populations. In the diploid case,\nthe genetic distance (1) between the two gametes forming\nan individual is equivalent to individual’s heterozygosity,\nand fitness function (2) specifies fitness as a function of\nindividual heterozygosity. Therefore, most conclusions of\nthis paper will also be applicable to situations when post-\nmating reproductive isolation is in the form of reduced (or\nzero) viability of hybrids due to incompatibility of the\ngenes they receive from their parents (Wu an Palopoli\n1994).\nDynamics in the neutral case\nBefore developing a theory for the dynamics of specia-\ntion in the above model, it is illuminating to start with\nthe neutral case. Here I summarize some relevant results\nthat are presented in (or follow directly from) classical\npapers (Watterson 1975; Li 1976; Slatkin 1987; Strobeck\n1987). Let µ be the probability of mutation per locus per\ngeneration. The approximations below assume that mat-\ning is random, the number of loci L is large, but µ is very\nsmall so that the probability of mutation per individual\nper generation v ≡ Lµ<< 1. The migration rate m and\nthe inverse of the population size 1 /N are small as well.\nGenetic variation within an isolated population . Let\nus consider an isolated population of size NT . The ex-\npected change in the average genetic distance within the\npopulation per generation is\n∆ Dw = 2v − Dw\nNT\n, (3)\nwhere the first term in the right-hand side is the contribu-\ntion of mutation whereas the second term is the random\ndrift reduction of Dw. Asymptotically, a mutation-drift\nequilibrium is reached with\nD∗\nw =θ ≡ 2vNT. (4)\nGenetic divergence between isolated populations . Let\nus consider several isolated populations of arbitrary size.\nThe probability that a specific mutation gets fixed in a\npopulation is one over the population size. Different mu-\ntations will get fixed in different populations resulting in\ntheir genetic divergence. The average genetic distance\n\nbetween any two of them increases with the rate equal\ntwice the mutation rate per gamete\n∆ Db = 2v. (5)\nThe rate of neutral divergence does not depend on pop-\nulation sizes. In particular, it is the same independent of\nwhether there are many small populations or a few large\npopulations. Because the number of loci L is finite, an\nindefinite increase of Dw, which is implied by equation\n(5) is impossible. This equation as well as equation (3)\nabove and equations (6) below approximate the dynamics\nwhen genetic distances Dw and Db are small relative to\nthe number of loci L. To treat the general case, one has to\nsubstitute v for v(1 − 2Dw/L ) in equations (3) and (6a)\nand for v(1 − 2Db/L ) in equations (5) and (6b). With\na finite number of loci, the genetic distance Db between\nisolated populations approaches L/ 2 asymptotically.\nSubdivided populations. The effect of migration on the\naverage genetic distances depends on the spatial structure\nof populations. Assume that a population of size NT is\nsubdivided into n subpopulations of size N =NT/n and\nthat a proportion m> 0 of individuals migrate to any of\nthe other n − 1 subpopulations. The expected changes\nin the average genetic distances within and between sub-\npopulations are\n∆ Dw = 2v + 2m(Db − Dw) − Dw\nN , (6a)\n∆ Db = 2v + 2m\nn − 1 (Dw − Db). (6b)\nEquations (6) assume that v,m and 1 /N are small.\nAsymptotically, a mutation-migration-drift equilibrium is\nreached with\nD∗\nw =θ, (7a)\nD∗\nb =θ + (n − 1) 2v\nm, (7b)\nwhere θ is given by equation (4). The average genetic\ndistance within a subpopulation (of size N ) does not de-\npend on the number of subpopulations n and migration\nratem and is the same as is expected in a single popula-\ntion with size NT . The average genetic distance between\nsub-populations increases with the population subdivi-\nsion and decreasing migration. Figure 2 illustrates the\ndynamics of neutral divergence in a system of two sub-\npopulations.\nPeripheral population . Assume a “peripheral” popu-\nlation of size N is receiving migrants from a very large\n“main” population. Genetic variation in the main popu-\nlation is assumed to be constant (and not influenced by\nmigration from the peripheral population). The expected\nchanges in the average genetic distances within the pe-\n0 500 1000 1500 2000\ngeneration\n40genetic distances Dw\nDw\nDb\nDb\nFigure 2: Dynamics of Dw and Db in the neutral case.\nPopulation size N = 100 (solid lines) and N = 200 (dashed\nlines). The rate of migration is m = 0 . 01, the mutation rate\nper individual is v = . 0384. Initially, Dw = Db = 0.\nripheral population, Dw, and between the peripheral and\nmain populations, Db are\n∆ Dw = 2v + 2m(Db − Dw) − Dw\nN , (8a)\n∆ Db =v +m(D0 − Db) (8b)\nwhereD0 is the average genetic distance within the main\npopulation and m is the proportion of individuals in the\nperipheral population replaced by migrants from the main\npopulation. Asymptotically, a mutation-migration-drift\nequilibrium is reached with\nD∗\nw = 2Nm\n1 + 2Nm D0, (9a)\nD∗\nb =D0 + v\nm, (9b)\nwhere the former equation assumes that genetic variation\nin the main population is sufficiently large ( D0 >> vN)\nand the number of migrants, Nm , is not too small. The\naverage genetic distance within the peripheral population\nis always larger than that for an isolated population of\nits size ( D∗\nw > 2vN ). If the number of migrants is large\n(Nm>> 1), the average genetic distance within the pe-\nripheral population is about the same as in the “main”\npopulation.\nDynamics with reproductive isolation\nThe main feature of the model for reproductive isola-\ntion introduced above and other models of holey adaptive\nlandscapes is the existence of chains of equally-fit combi-\nnations of genes separated by single substitutions that\nextend throughout the genotype space. These chains can\nbe though of as ”neutral paths” in the adaptive land-\nscape. It is important to realize, however, that the exis-\n\ntence of “holes” in a holey adaptive landscape makes the\nactual dynamics of genetic divergence not neutral. In this\nsection I summarize some analytical results on the evo-\nlutionary dynamics in the case of reproductive isolation\ndescribed by equation (2). Gavrilets et al. (1998) have\nstudied the possibilities for speciation in this model nu-\nmerically. Details of the analytical methods used are out-\nlined in the Appendix. To derive the dynamic equations\nbelow, I have used the same assumptions as described\nabove at the beginning of the section on the neural case\nsubstituting the assumption of random mating for the\nassumption of random encounters. In addition, I have\nassumed that the distributions of genetic distances both\nwithin and between populations are Poisson. There are\nseveral sets of approximations resulting in Poisson distri-\nbution of genetic distances. In the present contest, the\nweakest set seems to be the assumption that genetic vari-\nation at each locus is small most of the time (rare-alleles\napproximation) and that the population is approximately\nat linkage equilibrium. These assumptions are standard\nin analyzing the dynamics of multilocus systems under\nthe joint action of selection, mutation, and random drift\n(e.g., Barton 1986; Barton and Turelli 1987; B¨ urger et al.\n1989; Gavrilets and de Jong 1993). The fit of individual-\nbased simulations with analytic predictions is satisfactory\nboth at qualitative and quantitative levels (see below and\nGavrilets et al. 1998).\nGenetic variation within an isolated population . After\nthe population becomes polymorphic at K loci, new mu-\ntations are selected against when rare because individuals\ncarrying them have a reduced probability of producing vi-\nable and fecund offspring. Selection experienced by indi-\nvidual loci underlying reproductive isolation is frequency-\ndependent (and is similar to that arising in the case of\nunderdominant selection on a diploid locus). The change\nin Dw per generation in an isolated population of size N\nis approximately\n∆ Dw = −sDw + 2v − Dw\nN , (10)\nwhere\ns = e− DwDK\nw\nΓ(K + 1,D w) (11)\nand Γ( x,y ) is an incomplete gamma function (e.g.,\nGradshteyn and Ryzhik, 1994). The value of D∗\nw at\nthe mutation-drift-selection equilibrium can be found by\nequating the right-hand side of (10) with zero and solving\nfor Dw. Figure 3a illustrates the dependence of D∗\nw on\nthe parameters of the model. This figure indicates that\nthe equilibrium values of Dw are close to the correspond-\ning neutral predictions (4) if K is larger than 2 or 3 times\nθ (whereθ = 2Nv is the average genetic distance within a\nfinite population in the neutral case). Figure 3b gives the\nvalues of the effective selection coefficient s. With moder-\nately large K (that is with K ≥ 10),s is very small. The\neffective selection coefficient s can also be thought of as\nthe strength of induced selection on each locus underlying\nreproductive isolation. Figure 3b shows that very strong\nselection on the whole genotype (implied by the existence\nof complete reproductive isolation at finite values of K)\nresults in very weak selection at the level of individual\nloci.\n0 10 20 30 40 50\nK\n0.00\n0.01\n0.02\n0.03\n0.04\n0.05\nGenetic load\n-4.0\n-3.0\n-2.0\n-1.0\n0.0\nS\n40Dw\nv=0.0384\nv=0.00384\nv=0.00384\nv=0.0384\nN=100\nN=200\nN=400\nN=800a\nb\nc\nFigure 3: (a) Average genetic distance Dw maintained by\nmutation-selection-drift balance in an isolated populati on of\nsize N as a function of K for v = 0. 0384. The circles, squares,\ndiamonds and triangles give estimates from individual-bas ed\nsimulations for N = 100, 200, 400 and 800, respectively (thirty\nruns for each parameter configuration). (b) Effective select ion\ncoefficient s in the case of infinite population size for two values\nof v. (c) The genetic load 1 − w for parameters values as in\nFigure b.\nThe mean fitness of the population, ww, can be defined\nas the proportion of pairs of individuals that can mate\nand produce fertile and viable offspring (cf., Nei et al.\n1983). For a population with an average genetic distance\nDw,\nww = Γ(K + 1,D w)\nΓ(K + 1) , (12)\nwhere Γ( x + 1) is a gamma function (e.g., Gradshteyn\nand Ryzhik, 1994. For integer x, Γ(x + 1) = x!). Figure\n3c shows that in spite of relatively high levels of genetic\nvariation maintained in the population, the genetic load\n(that is the proportion of reproductively isolated pairs\nof individuals, 1 −\nww) is very low. This seems to be a\ngeneral property of holey adaptive landscapes (cf. Wills\n\n1977; Bengtsson and Christiansen 1983).\nGenetic divergence between isolated populations . Even\nafter the genetic distance within an isolated population\nhas reached an equilibrium level, the population keeps\nevolving as different mutations get fixed. As a conse-\nquence, isolated populations will continuously diverge ge-\nnetically. The asymptotic rate of divergence of two iso-\nlated populations of size N each is\n∆ Db = 2vR, (13a)\nwhere\nR = 2e− S√\nS√ π erf(\n√\nS)\n(13b)\nis the rate of divergence relative to the neutral case. Here,\nS = Ns/ 2, s is defined by equation (11) with Dw corre-\nsponding to the mutation-selection-drift equilibrium, and\nerf (x) is the error function (= 2 / √\nπ\n∫x\n0 exp(−y2)dy). In\nthe neutral case, s = 0, U = 1/N and equation (13a) re-\nduces to equation (5). Figure 4 illustrates the dependence\nof the relative rate of divergence R on model parame-\nters. In the neutral case, the rate of genetic divergence\n∆ Db does not depend on the population size (equation\n5). In contrast, with reproductive isolation the rate of\ndivergence decreases with increasing population size. Af-\nter the population becomes polymorphic at K loci, new\nmutations are selected against when rare. Genetic drift\noperating in finite populations overcomes the effect of se-\nlection and allows genetic divergence. For example with\nK = 20 and v =. 0384, a population of size N = 800 will\naccumulate about 5 substitutions per 1000 generations.\nA few thousand generations will be sufficient for Db to\nexceed K significantly. In contrast, very large randomly\nmating populations will diverge very slowly.\nFigure 4 indicates that the rate of substitutions is close\nto the corresponding neutral predictions if K is larger\nthan 2-3 times θ (θ = 2Nv ). Note that as in the neutral\ncase considered above, an implicit assumption in equation\n(13) is that genetic distance Db is small relative to the\nnumber of loci L. In the general case, ∆ Db = 2v(1 −\n2Db/L )R and Db approachesL/ 2 asymptotically.\nAt what moment can the two diverging population be\nconsidered as two different species? The answer obviously\ndepends on what one means by a species. Let us say that\nthe two populations are different species if the proportion,\nwb, of encounters between individuals from different pop-\nulations that can result in mating and viable and fertile\noffspring is less than a small number γ. (This definition\nuses the biological species concept.) During initial stages\nof divergence, this proportion can be approximated by\nthe right-hand side of equation (12) with Db taking the\nplace of Dw. Figure 5 shows the minimum genetic dis-\ntance between populations required for speciation as a\nfunction of K for several values of γ. One can see that a\n0 10 20 30 40 50\nK\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nR\nN=100\nN=200\nN=400\nN=800\nFigure 4: The rate of difergence R relative to the neu-\ntral case in an isolated population of size N as a function of\nK for v = . 0384. The circles, squares, diamonds and tri-\nangles give estimates from individual-based simulations f or\nN = 100 , 200, 400 and 800, respectively (thirty runs for each\nparameter configuration).\ngenetic distance between the populations on the order of\n2 or 3 times K will be sufficient for the status of separate\n“biological” species. Note that there is very little effect\nof the magnitude of γ.\nSpeciation in a subdivided population . In the determin-\nistic limit (that is with NT → ∞ ), the genetic variation\nof a subdivided population can be maintained by migra-\ntion. This can happen if initially alternative alleles are\nclose to fixation in different subpopulations and selection\nis sufficiently strong relative to migration (e.g., Karlin\nand McGregor 1972). Let k be the number of loci at\nwhich alternative alleles are close to fixation in different\nsubpopulations. Respectively, L − k will be the number\nof loci at which the same allele is close to fixation in dif-\nferent subpopulations. In the deterministic limit, k does\nnot change. In the n-island model the dynamics of Dw\nand Db are described by equations\n∆ Dw = −sDw + 2v + 2me(Db − Dw), (14a)\n∆ Db = −s(Db − k) + 2v + 2me\nn − 1 (Dw − Db), (14b)\nwhere s is defined by equation (11), and the “effective”\nmigration rate\nme =m wb\nww\n(15)\nif k ≤ K and me = 0 otherwise. Here, ww is given by\nequation (12) above whereas wb = Γ( K + 1 − k,D b −\n\n0 10 20 30 40 50\nK\n80Minimum distance for speciation\nFigure 5: Minimum genetic distance between populations\nfor speciation for γ = . 0001, . 001 and . 01 (solid lines from top\nto bottom). Also shown is the diagonal D = K (dashed line)\nk)/ Γ(K + 1− k) is the probability that two randomly cho-\nsen individuals from different populations are not repro-\nductively isolated. The effective migration rate me can\nbe thought of as half the probability of mating between\nindividuals from different subpopulations. With no re-\nproductive isolation (with very large K) or no genetic di-\nvergence between subpopulations (with Db ≈ Dw,k = 0),\nthe effective migration rate is equal to the actual migra-\ntion rate (me =m). Comparing equations (14) with their\nneutral analogs (8) shows that reproductive isolation re-\nsults in two effects. First, it directly reduces genetic vari-\nation within subpopulations and genetic divergence be-\ntween subpopulations. These effects are described by the\nfirst terms in the right-hand side of equations (14). Also,\nreproductive isolation reduces the gene flow between pop-\nulations. Given that Db ≥ Dw, then me ≤ m reflecting\nthe fact that genes brought by migrants have a smaller\nprobability of being incorporated in the resident popula-\ntion. In the deterministic limit, both Dw and Db always\nevolve to finite equilibrium values.\nRandom genetic drift results in two effects. First, it re-\nduces the genetic variation within sub-populations by the\namountDw/N . The dynamic equation for Dw becomes\n∆ Dw = −sDw + 2v + 2me(Db − Dw) − Dw\nN . (16)\nSecond, genetic drift might change k. The expected\nchange in k can be approximated as\n∆ k = 2vR2− Nm e − 2kmeR(e/ 2)Nm e, (17)\nwhere R is given by equation (13b). The first term in\nthe right-hand side of equation (17) can be thought of as\nthe rate at which an allele that is initially rare in both\nsub-populations becomes close to fixation in one of the\nsubpopulations. This rate was found by Lande (1979)\nusing a diffusion approximation and assuming that mi-\ngration is weak. The second term is the rate at which\nthe loci with different alleles initially close to fixation in\ndifferent subpopulations become fixed for the same allele\nin both of them. To find this term I used Barton and\nRouhani’s (1987) method.\nDepending on parameter values and initial conditions\nthere are two different dynamic regimes. In the first\nregime, both Dw and Db evolve to finite values, which\nare smaller than those in the neutral case (and which are\nmuch smaller than the number of loci L). Here, selec-\ntion (reproductive isolation) reduces genetic divergence\nboth within and between subpopulations. In the sec-\nond regime, Dw stays small (relative to L) whereas Db\nincreases effectively indefinitely (to values order L/ 2).\nHere, selection (reproductive isolation) reduces the ef-\nfective migration rate to zero resulting in speciation.\nThese dynamics can be understood in the following way.\nChanges in Dw andDb induced by selection are expected\nto happen on a faster time scale than changes in k in-\nduced by random genetic drift. Thus, Dw and Db values\nshould be close to the equilibrium values predicted by\nequations (14b,16) when k is treated as a constant. The\ndynamic behavior depends on whether k reaches a finite\nequilibrium value or keeps increasing. In the latter case,\nthe effective migration rate me reduces to zero, the rate\nof change of k approaches 2vR withDb increasing at the\nsame rate (cf. equation 13b).\nFigure 6 illustrates the dynamics observed by numer-\nically iterating the model equations. The iterations\nstarted with all N individuals identical. During the first\n1000 generations there were no restrictions on migra-\ntion between subpopulations and the whole population\nevolved as a single randomly mating unit (cf. Gavrilets\net al. 1998). The average genetic distance within the\npopulationDw evolves according to equation (10). Start-\ning with generation 1000, restrictions on migration were\nintroduced and the dynamics are described by equations\n(14b-17) afterwards. After generation 1000, each of these\nfigures has two sets of three curves corresponding to two\ndifferent values of the parameter(s) under consideration.\nThe curves within each set represent Db,D w andk. With\nmigration rate m = 0. 01, all these variables evolve to-\nwards finite equilibrium values (see Fig.6a) whereas with\na smaller migration rate ( m = 0. 001), Db and k increase\neffectively indefinitely signifying that speciation has taken\nplace. Thus, reducing migration makes speciation more\nplausible. Figure 6b shows that increasing mutation rate\n(from v = 0. 0384 to 5 times this value) has a similar ef-\n\nfect. These two figures describe the dynamics expected\nin a system of two subpopulations. Figure 6c compares\nthe dynamics observed in a population subdivided into 2\nand 4 subpopulations. This figure shows that increasing\npopulation subdivision makes speciation more plausible.\nNote that the process of genetic divergence described in\nFig.6c results in a simultaneous emergence of 4 species.\nIn the cases where speciation takes place (as signified by\ncontinuous increase in the genetic distance between sub-\npopulations), the curves representing Db and k are par-\nallel meaning that asymptotically the genetic divergence\nis due to fixation of different mutations in different sub-\npopulations. In the cases where speciation does not take\nplace, k is close to zero.\nAltogether, at the qualitative level the results presented\nin Fig.6 correspond to both biological intuition and the\nresults of individual-based simulations in Gavrilets et al.\n(1998). At the quantitative level, there is a very good\nfit between simulations and analytical predictions for lev-\nels of genetic variation maintained in subpopulations and\nthe asymptotic rate of divergence between subpopula-\ntions. However, the conditions for speciation as predicted\nby iterating equations (14b-17) appear to be more strict\nthan those observed in the individual-based simulations\nperformed by Gavrilets et al. (1998). For example, for\nparameter values used in Figure 6c, no speciation in a\nsystem of four subpopulations occurs if m >0. 0035. In\ncontrast, in individual-based simulations speciation was\nobserved form = 0. 01 (Figure 3b in Gavrilets et al. 1998).\nThe main reason for this disperancy seems to be an inad-\nequacy of equation (17) at moderate levels of migration\n(e.g. Lande 1979; Barton and Rouhani 1987).\nSpeciation in a peripheral population . Here I consider\nthe case of a peripheral population of size N receiving\nmigrants from a very large main population. The dynam-\nics of the average genetic distance within the peripheral\npopulation,Dw, the average genetic distance between the\nperipheral and main populations, Db, and the number of\ndiverged loci k are approximated by equations\n∆ Dw = − sDw + 2v + 2me(Db − Dw) − Dw\nN , (18a)\n∆ Db = − s\n2 (Db − k) +v +me(D0 − Db), (18b)\n∆ k =vR2− Nm e − kmeR(e/ 2)Nm e. (18c)\nHere D0 is the average genetic distance within the main\npopulation. Figure 7 illustrates the dynamics observed\nby numerically iterating equations (18). For D0 I used\nmutation-selection balance values for a very large isolated\npopulation predicted by equations (10, 11). The initial\nvalues of Dw and Db were equal to D0. The parameter\nvalues in Fig.7a and Fig.7b are the same as those that\nresulted in speciation in Fig.6a and 6b, respectively. The\noutcome of the dynamics is the same - speciation - but the\nrate of divergence is smaller than when all subpopulations\nare uniformly small. This is apparent from the level of\ngenetic distance between subpopulations achieved after\n1000 generations of divergence which are about twice as\nsmall in Figures 7a and 7b relative to those in Figures 6a\nand 6b.\nDiscussion\nThe theory developed above together with earlier numer-\nical simulations (see references above) show that rapid\nspeciation is a plausible outcome of the evolutionary dy-\nnamics in subdivided populations. Here speciation is a\nconsequence of two fundamental factors. The first fac-\ntor is the existence of various and possibly significantly\ndifferent highly-fit combinations of genes underlying di-\nverse solutions (genetical, ecological, behavioral, devel-\nopmental etc.) to the problem of survival and reproduc-\ntion. In multidimensional genotype space these combina-\ntions of genes tend to form connected clusters that extend\nthroughout genotype space. At the same time these geno-\ntypes are not mutually compatible - they are separated\nby “holes”. The second factor is mutation pressure. Be-\ncause the population size is finite and the number of loci\nis very large whereas the probability of a specific mu-\ntation is very small, different mutations tend to appear\n(and increase in frequency) in different subpopulations\n(cf., Barton 1989; Mani and Clarke 1990). Metaphori-\ncally speaking, mutation tends to tear apart the cloud\nof points representing the population in genotype space.\nCombining genes from two different organisms in one off-\nspring can counteract the disruptive effect of mutation,\nkeeping a single randomly mating population together in\ngenotype space. But restricting gene exchange as a conse-\nquence of limited migration between subpopulations gives\nmutation a significant advantage. Eventually the popula-\ntion cloud will be broken and smaller clouds representing\nthe subpopulations will drift apart in genotype space -\nan event representing speciation. Given sufficient genetic\ndivergence, restoring migration to high levels will not re-\nturn the system back to the state of free gene exchange\nbetween subpopulations which now can be considered as\ndifferent species. It is not necessary to invoke strong selec-\ntion for local adaptation to explain speciation in a subdi-\nvided population, as studied here, or after a founder event\n(Gavrilets and Hastings 1996; Gavrilets and Boake 1998).\nMutation is ubiquitous. Population size is never infinite\nand, thus, genetic drift is always present. Speciation as\ncaused by mutation and random drift should represent\na null model against which speciation as caused by local\nadaptation can be tested (cf., Nei 1976; Lande 1976).\n\nUnlike most previous models that concentrate only on\nsome stages of speciation, the model studied here de-\nscribes the complete process of speciation from initiation\nuntil completion. I assumed that reproductive isolation\nis caused by cumulative genetic change. The model is\ndescribed in terms of dynamic equations for the vari-\nables analogous to those used in molecular evolution-\nary biology - the average genetic distances between and\nwithin subpopulations. Average genetic distances within\n(sub)populations always evolve towards finite equilibrium\nvalues. Depending on parameter values and initial con-\nditions average genetic distances between subpopulations\neither converge to a finite equilibrium or increase effec-\ntively indefinitely. The former regime is interpreted as no\nspeciation. In the latter regime, three effects take place\nsimultaneously: (1) genetic distances between subpopula-\ntions significantly exceed genetic distances within them,\n(2) encounters between individuals from different sub-\npopulations do not result in viable and fertile offspring,\n(3) evolutionary changes in a subpopulation do not affect\nother subpopulations. Thus, subpopulations form sepa-\nrate genotypic clusters in genotype space, become repro-\nductively isolated and undertake changes as evolutionary\nindependent units. This regime is interpreted as specia-\ntion according to any of the species concept common in\nthe literature (e.g., Mallet 1995; Claridge et al. 1997).\nThe dynamic equations derived above describe the ex-\npected changes in the average genetic distances neglect-\ning stochastic fluctuations around the expected values.\nThe predicted dynamics have two clearly distinct regimes:\nconvergence towards a finite equilibrium (no speciation)\nor effectivly indefinite divergence (speciation). Stochas-\ntic fluctuations around the expected values, which are\npresent in natural populations (and individual-based sim-\nulations), make the boundary between these two regimes\nless strict and may result in the population escaping the\nfirst regime and entering the second regime after some\ntime (see Gavrilets et al. 1998). My analysis has been\nbased on approximations which are standard in study-\ning multilocus systems. I assumed that alleles are rare,\nthat linkage disequilibrium, mutation and migration rates\nare small and used a theory developed by Lande (1979),\nWalsh (1982) and Barton and Rouhani (1987) for de-\nscribing stochastic transitions driven by random genetic\ndrift. The analytic theory presented here fits satisfacto-\nrily with the results of individual-based simulations. The\nmodel can be used to evaluate qualitative effects of dif-\nferent factors on the dynamics of speciation, the order of\nmagnitude of parameters resulting in or preventing spe-\nciation, and the time scale involved. According to both\nbiological intuition and previous numerical simulations,\nincreasing mutation rate and decreasing migration pro-\nmote speciation. Increasing the number of loci has sig-\nnificantly increased the plausibility of speciation relative\nto that in earlier models (Nei et al. 1983; Wagner et al.\n1995; Gavrilets and Hastings 1996). Note that the actual\nnumber of loci influences the dynamics only through the\nmutation rate per gamete, v, and parameter K. For re-\nalistic parameter values the time scale for speciation can\nbe as short as a few thousands or even hundreds of gener-\nations. This is compatible with rates observed in several\ncases of rapid speciation in natural populations described\nrecently (Schluter and McPhail 1992; Yampolsky et al.\n1994; Johnson et al. 1996; McCune 1996, 1997) includ-\ning the most spectacular case - the origin of hundreds of\nspecies of Lake Victoria cichlids in 12,000 years (Johnson\net al. 1996). The model has demonstrated the plausibility\nof speciation with relatively low levels of both initial ge-\nnetic variation and new genetic variation introduced into\nthe population each generation (both supplied by muta-\ntion). With higher levels of the former (as in laboratory\nexperiments on speciation, reviewed by Rice and Hostert\n1993, and Templeton, 1996) or of the latter (for instance\nas a result of natural hybridization, reviewed by Bullini\n1994, Rieseberg 1995, Arnold 1997), the rate of speciation\nis expected to be even higher.\nLocal adaptation and speciation\nThe model analyzed above shows that rapid speciation in\na subdivided population can occur even without any dif-\nferences between selection regimes operating in different\nsubpopulations (that is without selection for local adap-\ntation). An important question is how genetic changes\nbrought about by selection for local adaptation would af-\nfect the dynamics of speciation (e.g., del Solar 1966; Ayala\net al. 1974; Kilias et al. 1980; Dodd 1989; Schluter 1996;\nGivnish and Sytsma 1997). These effects will depend on\nwhether the genes responsible for local adaptation are\ndifferent from or are the same as the genes underlying\nreproductive isolation.\nAssume first that the two sets of genes are completely\ndifferent. Let the strength of selection per locus induced\nby reproductive isolation be very small so that these loci\ncan be considered as effectively neutral. (For the model\nstudied here, this seems to be the case if K is larger that\n2-3 times θ, where θ is the average genetic distance main-\ntained by mutation in a finite population in the neutral\ncase.) Then, Birky and Walsh’s (1988) results tell us\nthat the rate of substitution in these loci will not be af-\nfected by selection on other loci independently of linkage.\nHowever, given that reproductive isolation is a result of\ngenetic incompatibilities, the loci underlying reproduc-\ntive isolation will be under frequency-dependent selection\nagainst rare alleles which is analogous to underdominant\nselection in diploid populations. Birky and Walsh (1988)\nhave shown that linkage to advantageous alleles slightly\nincreases the rate of fixation of detrimental mutations.\n\nThis suggests that selection on linked loci will increase\nthe rate of substitutions in the loci underlying reproduc-\ntive isolation and, thus, will promote speciation to some\ndegree. No results seem to be known on how linkage to\nadvantageous alleles increases the rate of fixation of un-\nderdominant mutations or alleles experiencing frequency-\ndependent selection. No quantitative predictions can be\nmade here, but most likely if the two sets of loci are not\nextremely tightly linked, effects of selection for local adap-\ntation on the rate of speciation will not be significant.\nAssume now that the loci under consideration\npleiotropically affect both survival in a given environment\nand reproductive isolation. For instance, this may be\nthe case if disruptive selection acts on habitat preferences\nwhich also define mating patterns (e.g., Rice 1984; Rice\nand Salt 1988) or if the probability of mating between\nindividuals depends on the difference in their morpho-\nlogical traits which are under direct selection. Let sLA\nbe the average strength of selection per locus induced by\nselection for local adaptation. Using Walsh’s (1982) re-\nsults, the relative rate of fixation of new mutations in an\nisolated population of size N can be approximated (see\nAppendix) as\nR = 4e− S(1− α )2√\nS\n√ π\n[\nerf (\n√\nS(1 +α )) +erf (\n√\nS(1 − α ))\n] , (19)\nwhere S = Ns/ 2, s is the strength of selection per lo-\ncus induced by reproductive isolation, and α = sLA/s .\nWith α = 0, equation (19) reduces to (13b). Figure 8\nillustrates the dependence of R on S and α . Increasing\nα always increases R. Thus, selection for local adapta-\ntion always increases the rate of substitutions and pro-\nmotes speciation. With sufficiently strong selection for\nlocal adaptation ( sLA > s), the net effect of new alleles\nwill be advantageous and their frequencies will tend to\nincrease even when rare. In the limit of large popula-\ntion size, the probability of fixation is 2( sLA − s). This\nis analogous to the classical results on the probability of\nsurvival of an advantageous mutant in a very large pop-\nulation (Haldane 1927; Walsh 1982). The rate of accu-\nmulation of genetic differences will be 2( sLA − s)Nv and\ncan be significant. Very strong artificial selection for local\nadaptation has been shown to result in rapid evolution of\nreproductive isolation (e.g., del Solar 1966; Kilias et al.\n1980; Dodd 1989). However, the changes brought about\nby moderately strong artificial selection may not exceed\nthose resulting from random genetic drift only (e.g. Ringo\net al. 1985).\nPopulation subdivision and speciation\nIn the models considered here, speciation is a by-product\nof fixation of different alleles in different subpopulations.\nIt is well known that the rate of fixation of neutral alleles\ndoes not depend on population size, that of advantageous\nalleles increases with population size, and that of delete-\nrious or underdominant alleles decreases with population\nsize (e.g., Gillespie 1991; Ohta 1992). At the level of\nindividual loci, selection induced by reproductive isola-\ntion in the form considered here is similar to underdom-\ninant selection (or frequency-dependent selection against\nrare alleles). Thus, in the absence of selection for local\nadaptation (or with independent loci controlling traits\nfor local adaptation) decreasing population size will in-\ncrease the rate of substitutions and promote speciation\n(see equation 13b). Effects of the population size on the\nplausibility of speciation will be similar even if the same\nloci control both reproductive isolation and locally ben-\neficial traits given that selection for local adaptation is\nnot too strong ( sLA < s, see equation 19 and Fig.9). In\nall these cases, speciation will be driven by mutation and\nrandom genetic drift and will be fastest if the popula-\ntion is subdivided into small subpopulations. This con-\nclusion about the effect of population subdivision on the\nprobability of speciation in Dobzhansky-type models dif-\nfers from that of Orr and Orr (1996). They argued that\nthe degree of population subdivision has no effect on the\nrate of speciation if speciation is caused by mutation and\nrandom drift. Orr and Orr did not consider the actual\nprocess of fixation of new mutations assuming that it\nwill be a simple neutral process. However, the existence\nof holes in the adaptive landscape makes the process of\nsubstitution non-neutral and new mutations are selected\nagainst when rare. Such mutations are fixed more easily\nin smaller subpopulations. For the discussion of the ex-\nisting experimental evidence regarding effects of random\ngenetic drift on the plausibility of speciation see Rice and\nHostert (1993) and Templeton (1996). The time scale\nfor speciation is short meaning that restrictions on mi-\ngration between subpopulations do not need to be long\nlasting; several hundreds of generations may be sufficient\nfor a significant divergence and evolution of reproductive\nisolation. It is quite possible that several new species\nwill emerge from a highly subdivided population within\na short period of time (see above; Gavrilets et al. 1998).\nThese theoretical conclusions are consistent with a verbal\n“micro-allopatric” model of speciation suggested for cich-\nlid fishes in the East African Great Lakes (e.g., Reinthal\nand Meyer, 1997). Hoelzer and Melnick (1994) have em-\nphasized that the possibility of simultaneous emergence\nof several new species should be incorporated more ex-\nplicitly in the contemporary methods for reconstructing\nphylogenies.\nIf the same loci control both reproductive isolation and\nlocally beneficial traits and selection for local adaptation\nis sufficiently strong ( sLA >s ), increasing the population\nsize will result in increasing the rate of substitutions (see\n\nFig.9). In this case, speciation will be driven by selection\nand will be fastest if the population is subdivided into\na small number (say, two) of large subpopulations (Orr\nand Orr 1996) as implied by the vicariance scenario (e.g.\nWiley 1988).\nMany species are thought to be represented by a few\nlarge populations and many smaller “peripheral” popula-\ntions. Mayr (1963, 1982b) proposed the theory of peri-\npatric speciation arguing that speciation is typically ini-\ntiated in small peripheral populations and he attributed\na special role to genetic drift in this process. Gavrilets\n(1996) has shown that an invasion of a new adaptive com-\nbination of genes is most successful if it is initiated in a\nperipheral population. The results presented here bear\nout Mayr’s argument(see Fig. 8). Small peripheral pop-\nulations will rapidly diverge genetically from the “main”\nlarge population and speciate. Although differences in se-\nlection regimes between peripheral and main populations\ncan accelerate divergence, random genetic drift will be the\nmost important factor. On the other hand, if a periph-\neral population is large enough and is under a selection\nregime that is sufficiently different from the one operating\nin “main” populations, then disruption of gene flow can\ncause evolutionary divergence, perhaps leading to rapid\nspeciation, in the absence of contributions from random\ngenetic drift (Garcia-Ramos and Kirkpatrick 1997).\nSummarizing, large randomly mating populations will\ndiverge genetically and speciate only if there is strong se-\nlection for local adaptation (for instance after a change\nin the environment). In contrast, small populations will\ndiverge and speciate even without differences in selec-\ntion regimes between them. Possibilities for speciation\nstrongly depend on the geographic structure of the pop-\nulation. Here, analysis was restricted to the island model\nand the continent-island model. Manzo and Peliti (1994)\nand Gavrilets et al. (1998) present numerical results for\nstepping-stone models.\nRelationship to other speciation models\nUsing genetic distance (1) implies the equivalence of loci.\nA general case of non-equivalent loci can be described by\nintroducing a ( L × L) matrix G = {Gij} of weights and\ndefining a generalized distance between individuals α and\nβ as\ndαβ = (lα − lβ )TG(lα − lβ ), (20)\nwhere lα and lβ are vectors defining the corresponding\ngenotypes and superscript T means transpose. Consid-\nering haploid populations and premating isolation only,\nthe model assumes that individuals can mate only if they\nare not too different genetically. Here, the degree of\nreproductive isolation was controlled by cumulative ge-\nnetic difference. However, using the generalized distance\n(20) allows one to treat models for reproductive isola-\ntion controlled by quantitative traits as well as models\nfor sexual selection within the same framework (see Ap-\npendix). The close relationship between the models of\nspeciation as a consequence of “quasi-neutral” divergence\nalong ridges in the adaptive landscapes and as a conse-\nquence of sexual selection was already recognized by Bar-\nton and Charlesworth (1984).\nA fundamental reason for speciation on a holey adap-\ntive landscape is mutation which tends to break the pop-\nulation into reproductively isolated pieces. Population\nsubdivision and the resulting reduction in gene exchange\nfacilitates this process. Here, migration rates compati-\nble with rapid speciation were small (that is speciation\nwas allopatric or parapatric). An interesting question is\nwhether speciation is possible with much higher migration\nrates. In other words, is sympatric speciation by muta-\ntion and random genetic drift on a holey adaptive land-\nscape possible? Numerical simulations of similar models\nof sympatric speciation where mutation rates were higher\n(Higgs and Derrida 1991, 1992) or the time span studied\nwas longer (Wu 1985) or the population size was smaller\n(Gavrilets and Boake 1998) than here, provide an affir-\nmative answer. Adding disruptive selection due to either\nabiotic factors (say, different resources) or biotic factors\n(competition) should create additional pressure on the\npopulation cloud which might result in rapid sympatric\nspeciation.\nBeyond holey landscapes\nGavrilets and Gravner’s (1997) results have suggested\nthat clusters of well-fit genotypes that extend through-\nout genotype space are plausible. If this is so, biological\npopulations are expected to evolve mainly within these\nclusters and consist most of the time of well-fit geno-\ntypes with fitnesses within some band. The metaphor\nof “holey” adaptive landscapes neglects the fitness differ-\nences between genotypes in the cluster but these differ-\nences are supposed to exist and should be apparent on a\nfiner scale. If one applies a finer resolution, the movement\nalong the cluster will be accompanied by slight increases\nor decreases in fitness. Evolution will proceed by fixa-\ntion of weakly selected alleles which can be advantageous,\ndeleterious, over- and underdominant, or apparently neu-\ntral depending on the specific area of genotype space the\npopulation passes through. Smaller populations will pass\nfaster through the areas of genotype space correspond-\ning to fixation of slightly deleterious mutations whereas\nlarger populations will pass faster through the areas cor-\nresponding to fixation of (compensatory) slightly advan-\ntageous mutations. This pattern of molecular evolution,\nas predicted from the general properties of multidimen-\nsional adaptive landscapes, is similar to the patterns re-\nvealed by the methods of experimental molecular biology,\n\nwhich form the empirical basis for the nearly neutral the-\nory of molecular evolution (Ohta 1992). From general\nconsiderations, one should not expect complete symme-\ntry of “real” adaptive landscapes which are supposed to\nhave areas varying with respect to the “width” and con-\ncentration of ridges of well-fit genotypes. Sexual popula-\ntions are expected to spend more time in areas of high\nconcentration of well-fit genotypes (Peliti and Bastolla\n1994). One of the biological manifestations of this effect\nwill be apparent reduction in the probability of harmful\nmutations, that is, evolution of genetic canalization (cf.,\nWagner 1996). The metaphor of holey adaptive land-\nscapes may be useful for thinking about these and other\nevolutionary problems.\nAcknowledgments.\nI am grateful to Chris Boake, Marc Camara, Mitch\nCruzan, Nick Barton and G¨ unter Wagner for very help-\nful comments and suggestions. Nick Barton and G¨ unter\nWagner forced me to extend the generality of the math-\nematical approximations developed here. Collaboration\nwith Hai Li and Michael Vose was crucial in developing\na computer program used to obtain results of individual-\nbased simulations presented in Figures 3 and 4. This\nwork was partially supported by grants from Universit´ e\nP. et M. Curie and ´Ecole Normale Sup´ erieur, Paris, and\nby National Institutes of Health grant GM56693.\nAppendix\nEffects of mutation, migration and drift on the dynam-\nics of the average genetic distances within and between\nsubpopulations have been previously studied thoroughly\n(e.g., Watterson 1975; Li 1976; Slatkin 1987; Strobeck\n1987). What is left is to add reproductive isolation (that\nis selection) to the model. I will use the deterministic\nframework assuming that the population size N → ∞ .\nThe distribution of Dw under rare-alleles and linkage\nequilibrium approximation. I will use the standard no-\ntations Ai and ai for alternative alleles at the i-locus\n(i = 1,...,L ). Let pi be the frequency of allele Ai at the\ni-th locus, qi = 1 − pi, and ψw,i = 2piqi. Variable ψw,i can\nbe thought of as the probability that two randomly cho-\nsen individuals (sequences) from the same subpopulation\nare different at the i-th locus. Let dw,i = (lα\ni − lβ\ni )2 be the\ngenetic distance at the i-th locus between two randomly\nchosen individuals α and β . Note that dw,i = 1 with\nprobability ψw,i and dw,i = 0 with probability 1 − ψw,i .\nBecause dw,i is a Binomial random variable, its gener-\nating function is γdw,i(s) = ψw,is + 1 − ψw,i , which can\nbe approximated as eψ w,i(s− 1) if ψw,i << 1 (rare-alleles\napproximation). Under approximate linkage equilibrium,\nthe generating function of dw = ∑ dw,i is\nγdw (s) = Π ieψ w,i(s− 1) =e\n∑\niψ w,i(s− 1) =eDw (s− 1) (A1)\nwhere Dw = ∑\niψw,i . This shows that random variable\ndw has approximately Poisson distribution with parame-\nter Dw and, thus,\nP (dw =i) = exp(−Dw)Di\nw\ni! . (A2)\nSelection within an isolated population . Let w(d) be\nthe expected number of fertile and viable offspring that\ncan be produced by a pair of individuals different in d\nloci. The average fitness of the population is\nw =\n∑\nj\nw(j)P (d =j).\nThe dynamics of the general model of fertility selection\nand premating isolation in a haploid population consid-\nered here are identical to that of a symmetric viability\nselection model for a diploid population with viabilities\nw(d) depending on the number of heterozygous loci d.\nUnder approximate linkage equilibrium, changes in allele\nfrequencies are described by Wright’s equation\n∆ spi = piqi\n∂ lnw\n∂pi\n. (A3)\n(Wright 1969). Using the equalities ∂ lnw/∂p i = 2(qi −\npi)∂ lnw/∂ψ i and Dw = ∑\niψi, equation (A3) can be\nrewritten as\n∆ spi =spiqi(pi − qi), (A4a)\nwith\ns = d lnw\ndDw\n. (A4b)\nTo describe the dynamics of allele frequencies one needs\nto know the mean fitness of the population.\nTruncation selection. This is a selection scheme ana-\nlyzed in detail in the main body of the paper. Here\nw(d) =\n{\n1 for d ≤ K,\n0 for d>K. (A5)\nUsing the Poisson approximation (A1), the mean fitness\nis\nwthreshold =\nK∑\ni=0\nexp(−Dw)Di\nw\ni! = Γ(K + 1,D w)\nΓ(K + 1) ,\nwhere the last equality follows from equation (8.352) in\nGradshteyn and Ryzhik (1994), resulting in s given by\nequation (11). To find equation (10), one starts with\n(A4a) and proceeds using the fact that ∆ ψi ≈ 2(qi −\n\npi)∆ pi and that Dw = ∑ ψi.\nOther selection schemes can be considered in a simi-\nlar way, and some of them result in relatively compact\nexpressions for\nw and s.\nLinear selection. Here\nw(d) =\n{ 1 − ad for d ≤ K,\n0 for d>K. (A6)\nThe mean fitness is\nwlinear =wthreshold − aDwΓ(K,D w)\nΓ(K) .\nQuadratic selection. Here\nw(d) =\n{\n1 − ad − bd2 for d ≤ K\n0 for d>K (A7)\nThe mean fitness is\nw =wlinear − b [Dw(Dw − K)\n+ Dw(K + 1)Γ(K,D w)\nΓ(K)\n−exp(−Dw)DK+2\nw H(2,K + 2,D w)\nΓ(K + 2)\n]\n,\nwhereH is the hypergeometric function (Gradshteyn and\nRyzhik 1994).\nExponential selection. Here\nw(d) =\n{ exp(−ad) for d ≤ K,\n0 for d>K. (A8)\nThe mean fitness is\nw =exp(−Dw(1 − e− a)) Γ(K + 1,D we− a)\nΓ(K + 1) .\nI have not explored how assuming these selection schemes\nwould affect the outcome of the dynamics.\nStochastic transitions in an isolated population .\nAdding mutation results in equation\n∆ pi =spiqi(pi − qi) +µ (qi − pi), (A9)\nwhereµ is the rate of mutation (assumed to be equal for\nforward and backward mutations). Equation (A9) is sim-\nilar to the classical equation describing underdominant\nselection on a single locus in a diploid population. This\nallows one to use Lande’s results (1979; see also Hedrick\n1981; Walsh 1982; Barton and Rouhani 1987) to find the\nrate of stochastic divergence. This rate is twice the ex-\npected number, vN , of new mutations in a population\ntimes the probability that a given one will be fixed, U .\nUsing the diffusion approximation, U is defined by equa-\ntions (1a) and (2) in (Lande 1979). Lande used some\napproximations to evaluate U . However, the integrals in\nhis equation (1a) can be found exactly resulting in\nU = 1\n\n1 −\nerf\n[ √\nS(1 − 2\nN )\n]\nerf\n[ √\nS\n]\n\n, (A10)\nwhere S = Ns/ 2 (Walsh 1983). Expanding in a Taylor\nseries under the assumption that 1 /N << 1 results in\n(13b) which is equivalent to Lande’s (1979) formula. The\ndifference between Lande’s approximate formula and the\nexact equation (A10) is negligible.\nThe distribution of Db under rare-alleles and linkage\nequilibrium approximation. Let us consider two subpop-\nulations. Let pi and Pi be the frequencies of allele Ai\nin the first and second subpopulations, respectively. The\ngenetic distance db,i at the i-th locus between two ran-\ndomly chosen sequences from two different subpopula-\ntions is a Binomial variable taking values 1 and 0 with\nprobabilities ψb,i =piQi +qiPi and 1 − ψb,i , respectively\n(qi = 1 − pi,Q i = 1 − Pi). I will assume that genetic varia-\ntion within each subpopulation is low so that ψb,i is close\nto either 0 or 1. Let δi =db,i ifψb,i ≈ 0 and δi = 1 − db,i if\nψb,i ≈ 1. The genetic distance between individuals α and\nβ can be represented as db =k − ∑\n1δi + ∑\n2δi where the\nfirst sum is over k loci at which ψb,i ≈ 1 and the second\nsum is over L − k loci at which ψb,i ≈ 0. Using the as-\nsumption of linkage equilibrium, the generating function\nof db becomes\nγdb(s) =E{sk−\n∑\n1δi+\n∑\n2δi }\n=skΠk\ni=1e− φ i(s− 1)ΠL\ni=k+1eφ i(s− 1)\n= sk\nek(s− 1)eDb(s− 1),\nwhereφi is the expectation of δi andDb is the expectation\nof db. Using the properties of generating functions, the\ndistribution of db is\nP (db =i) =\n{\n0 if i<k,\n(Db− k)i− k\n(i− k)! e− (Db− k) if i ≥ k. (A11)\nWith fitness function (A5), the probability that two ran-\ndomly chosen individuals from different subpopulations\nare not reproductively isolated is\nwb =\nK∑\ni=0\nP (db =i) = Γ(K − k + 1,D b − k)\nΓ(K − k + 1) , (A12)\nif k ≤ G and wb = 0 if k>K .\nDeterministic dynamics in a subdivided population .\nWith no reproductive isolation and with equal forward\n\nand backward migration rates and equal population sizes,\nthe change in pi due to migration is ∆ mpi =m(Pi − pi).\nThe corresponding change in Dw is ∆ mDw = 2m(Db −\nDw). With reproductive isolation and given that Db ≥\nDw, individuals migrating from other subpopulations will\nhave reduced probability of mating. Let\nww and wb be\nthe expected numbers of fertile and viable offspring that\ncan be produced as a result of within and between sub-\npopulations encounters. For simplicity I will omit the in-\ndex specifying the locus under consideration. With equal\npopulation sizes and migration rates, the change in the\nallele frequency due to migration becomes\n∆ mp =me(P − p), (A13a)\nwhere the “effective” migration rate is\nme =m\nwb\nww\n. (A13b)\n(compare with the models with migration between popu-\nlations of unequal size where the effective migration rate\nism times the ratio of the population sizes, e.g. Gavrilets\n1996). The corresponding change in Dw is\n∆ mDw = 2me(Db − Dw). (A13c)\nChanges ∆ mpi can be thought of as changes in allele fre-\nquencies brought about by selection between groups of in-\ndividuals (migrants and residents) whereas the first term\nin the right-hand side of equation (A9) can be thought of\nas the change in p brought about by individual selection.\nThe dynamics of allele frequencies at a specific locus\nunder the joint action of selection, mutation and migra-\ntion are described by\n∆ p =spq(p − q) +me(P − p) +µ (q − p), (A14a)\n∆ P = ˜sPQ (P − Q) +me(p − P ) +µ (Q − P ).(A14b)\nWith s = ˜s = const and me = const and me < s/ 6,\ndynamic system (A14) has two types of stable equilibria:\nmutation-selection balance equilibria with p ≈ P, pq ≈\nµ/s, ψ w ≈ 2µ/s,ψ b ≈ 2µ/s and migration-selection equi-\nlibria with p ≈ Q, pq ≈ µ/s + m/s, ψ w ≈ 2µ/s +\n2m/s,ψ b ≈ 1 − 2µ/s − 2m/s . I assume that in the deter-\nministic limit, k out of L loci evolve towards migration-\nselection balance equilibria whereas the remaining L − k\nloci evolve towards mutation-selection balance equilibria.\nIn the latter L − k loci, the dynamics of ψw and ψb are\napproximated by equations\n∆ ψw = −sψw + 2µ + 2me(ψb − ψw), (A15a)\n∆ ψb = −sψb + 2µ + 2me(ψw − ψb). (A15b)\nIn the former k loci, the dynamics of ψw are described as\nbefore by (A15a) whereas the dynamics of ψb are approx-\nimated by equation\n∆ ψb =s(1 − ψb) − 2µ + 2me(ψw − ψb). (A16)\nSelection always reduces ψw whereas mutation always\nincreases it (see equation A15a). Selection and muta-\ntion have the same effects on ψb for the loci evolving\ntoward mutation-selection balance equilibria (see equa-\ntion A15b). However, for the loci evolving toward\nmigration-selection balance equilibria, selection increases\nψb whereas mutation decreases it (see equation A16).\nSumming up over all loci, one finds equations (14) of the\nmain text. Equations (18) are derived in a similar way as-\nsuming that the allele frequencies in the main population\ndo not change.\nStochastic transitions in a subdivided population . In\na subdivided population, migration tends to reduce ge-\nnetic differentiation. Given that migration is sufficiently\nstrong relative to selection, the same allele will be close\nto fixation in both subpopulations. If, by a chance, an\nalternative allele approaches fixation in one of the sub-\npopulations creating significant differentiation at a given\nlocus, such differentiation will be quickly lost. The num-\nber of loci k at which alternative alleles are close to fix-\nation in different subpopulations will be close to zero on\naverage. However, if migration is relatively weak, then\nthe differentiation created by random genetic drift will\nnot be lost quickly and actually can even accumulate.\nLet us consider a locus at which initially the same allele\nis close to fixation in both subpopulations (that is both\np ≈ 0 and P ≈ 0). Neglecting the changes in P , the de-\nterministic change in p due to selection and migration is\napproximately\n∆ p =spq(p − q) − mep. (A17)\nLande (1979; see also Barton and Rouhani 1987) has\nshown that the rate at which allele A becomes close to\nfixation in the first subpopulation while its frequency is\nabout zero in the second population is approximately\n2− Nm e times the rate of fixation in the absence of im-\nmigration. Assuming that alleles A are brought about\nby mutation at rate µ and summing up over L − k loci,\none finds the first term in the right-hand side of equation\n(17). Once alternative alleles are close to fixation in dif-\nferent subpopulations, random drift can remove genetic\ndifferentiation. Let us consider a locus at which initially\np ≈ 0 but P ≈ 1. Neglecting the changes in P , the deter-\nministic change in p due to selection and immigration is\napproximately\n∆ p =spq(p − q) − me(1 − p). (A18)\n\nUsing Barton and Rouhani’s (1987) method one finds that\nthe rate at which allele A becomes close to fixation in\nboth subpopulations is approximately (e/ 2)Nm e times the\nrate of fixation in the absence of immigration. Assuming\nthat alleles A are brought about by migration at rate me\nand summing up over k loci, one finds the second term in\nthe right-hand side of equation (17).\nStochastic divergence with local adaptation . Let us as-\nsume that the allele under consideration is favorable in\na given environment with selective advantage sLA. The\nchange in this allele frequency as defined by the joint ac-\ntion of selection induced by reproductive isolation and\nselection for local adaptation is\n∆ sp =spq(p − q) +sLApq. (A19)\nThis equation is identical to the one describing meiotic\ndrive in the Appendix of Walsh (1982). Following Walsh,\nthe fixation probability is\nU =\nerf\n[ √\nS(1 − α )\n]\n− erf\n[ √\nS(1 − α − 2\nN )\n]\nerf\n[ √\nS(1 − α )\n]\n+erf\n[ √\nS(1 +α )\n] , (A20)\nwhere S = Ns/ 2 and α = sLA/s . Expanding the nu-\nmerator in a Taylor series under the assumption that\n1/N << 1 and multiplying the results by the expected\nnumber of mutants, vN , results in the relative rate of\nfixation given by equation (19).\nGenetic distance (20) and some other models . Genetic\ndistance (1) is recovered by assuming that G is an iden-\ntity matrix. Assuming that G is a diagonal matrix with\nnon-equal diagonal elements is a simple way to introduce\nnon-equivalence of loci. The case when the probability of\nmating depends on the difference in a quantitative trait\ncan be treated within the same framework. Let ci be the\ncontribution of the i-th locus to a quantitative trait z.\nNeglecting microenvironmental effects, the trait values for\nindividualsα andβ arexα = ∑ cilα\ni andzβ = ∑ cilβ\ni , re-\nspectively. The square of the difference of zα andzβ is re-\ncovered from equation (20) by assuming that Gij =cicj.\nA common way to model sexual selection is to assume\nthat the probability of mating between a male and a fe-\nmale depends on the difference in a female phenotypic\ntrait, zf , and a male phenotypic trait, zm, which are\ncontrolled by two different sets of loci (e.g., Lande 1981;\nKirkpatrick 1982; Nei at al. 1983; Wu 1985; Turner and\nBurrows 1995). 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Variation of allozyme loci in endemic gam-\nmarids of Lake Baikal. Biol. J. Linn. Soc. 53: 309-323.\n0 500 1000 1500 2000\ngeneration\n50genetic distances\n50genetic distances\n50genetic distances\na\nb\nc\nDb\nk\nDw\nDb k\nDw\nDb\nk\nDw\nFigure 6: Dynamics of speciation in a subdivided popula-\ntion. Unless specified otherwise, K = 20 , v = 0 . 0384, n = 2.\na. Effects of migration rate. Stronger migration, m =\n0. 01 (dashed lines; no speciation), and weaker migration,\nm = 0 . 001 (solid lines; speciation). Total population size\nNT = 200. b. Effects of mutation rate. Weaker mutation,\nv = 0 . 0384 (dashed lines; no speciation), and stronger muta-\ntion, v = 5 × 0. 0768 (solid lines; speciation). Other param-\neters: NT = 400, m = 0 . 005. c. Effect of population sub-\ndivision. n = 2 subpopulations (dashed lines; no speciation)\nand n = 4 subpopulations (solid lines; speciation). Other pa-\nrameters: NT = 800, m = 0 . 0033. Dashed lines represent Db\n(top line), Dw (middle line) and k (bottom line), respectively.\nDuring the first 1000 generations there are no restrictions o n\nmigration.\n\n0.0 200.0 400.0 600.0 800.0 1000.0\ngeneration\n50genetic distances\n50genetic distances\na\nb\nDb\nk\nDw\nDb\nk\nDw\nFigure 7: Dynamics of speciation in a peripheral population.\na. Speciation with m = 0 . 001; N = 100; K = 20; v = 0 . 0384\n(cf. Fig.7a). b. Speciation with m = 0 . 005; N = 200; K =\n20; v = 5 × 0. 0768 (cf. Fig.7b).\n0.0 2.0 4.0 6.0 8.0 10.0\nS\n0.0\n2.0\n4.0\n6.0\n8.0\n10.0\nR\nα=2\nα=1\nα=0.5\nα=0\nFigure 8: Relative rate of fixation in the case with local\nadaptation."}