{"title": "SpikeAnts, a spiking neuron network modelling the emergence of organization in a complex system", "book": "Advances in Neural Information Processing Systems", "page_first": 379, "page_last": 387, "abstract": "Many complex systems, ranging from neural cell assemblies to insect societies, involve and rely on some division of labor. How to enforce such a division in a decentralized and distributed way, is tackled in this paper, using a spiking neuron network architecture. Specifically, a spatio-temporal model called SpikeAnts is shown to enforce the emergence of synchronized activities in an ant colony. Each ant is modelled from two spiking neurons; the ant colony is a sparsely connected spiking neuron network. Each ant makes its decision (among foraging, sleeping and self-grooming) from the competition between its two neurons, after the signals received from its neighbor ants. Interestingly, three types of temporal patterns emerge in the ant colony: asynchronous, synchronous, and synchronous periodic foraging activities - similar to the actual behavior of some living ant colonies. A phase diagram of the emergent activity patterns with respect to two control parameters, respectively accounting for ant sociability and receptivity, is presented and discussed.", "full_text": "SpikeAnts, a spiking neuron network modelling the\n\nemergence of organization in a complex system\n\nSylvain Chevallier\nTAO, INRIA-Saclay\n\nUniv. Paris-Sud\n\nF-91405 Orsay, France\nsylchev@lri.fr\n\nH\u00b4el`ene Paugam-Moisy\n\nLIRIS, CNRS\nUniv. Lyon 2\n\nF-69676 Bron, France\n\nhpaugam@liris.cnrs.fr\n\nMich`ele Sebag\nTAO, LRI \u2212 CNRS\nUniv. Paris-Sud\n\nF-91405 Orsay, France\n\nsebag@lri.fr\n\nAbstract\n\nMany complex systems, ranging from neural cell assemblies to insect societies,\ninvolve and rely on some division of labor. How to enforce such a division in a\ndecentralized and distributed way, is tackled in this paper, using a spiking neuron\nnetwork architecture. Speci\ufb01cally, a spatio-temporal model called SpikeAnts is\nshown to enforce the emergence of synchronized activities in an ant colony. Each\nant is modelled from two spiking neurons; the ant colony is a sparsely connected\nspiking neuron network. Each ant makes its decision (among foraging, sleeping\nand self-grooming) from the competition between its two neurons, after the signals\nreceived from its neighbor ants. Interestingly, three types of temporal patterns\nemerge in the ant colony: asynchronous, synchronous, and synchronous periodic\nforaging activities \u2212 similar to the actual behavior of some living ant colonies.\nA phase diagram of the emergent activity patterns with respect to two control\nparameters, respectively accounting for ant sociability and receptivity, is presented\nand discussed.\n\n1\n\nIntroduction\n\nThe emergence of organization is at the core of many complex systems, from neural cell assemblies\nto living insect societies. For instance, the emergence of synchronized rhythmical activity has been\nobserved in many social insect colonies [2, 4, 5, 7], where synchronized patterns of activity may\nindeed contribute to the collective ef\ufb01ciency in various ways. But how do ants proceed to tem-\nporally synchronize their activity? As suggested by Cole [4], the synchronization of activity is a\nconsequence of temporal coupling between individuals. It thus comes naturally to investigate how\nspiking neuron networks (SNNs), also based on temporal dynamics, enable to model the emergence\nof collective phenomena, speci\ufb01cally synchronized activities, in complex systems. The reader\u2019s\nfamiliarity with SNNs, inspired from the mechanisms of information processing in the brain, is\nassumed in the following, referring to [18] for a comprehensive presentation.\n\n1.1 Related work\n\nIn computational neuroscience, SNNs are well known for generating a rich variety of dynamical\npatterns of activity, e.g. synchrony of cell assemblies [9], complete synchrony [17], transient syn-\nchrony [10], order-chaos phase transition [20] or polychronization [11]. For instance, a mesoscopic\nmodel [3] explains the emergence of a rhythmic oscillation at the network level, resulting from the\ncompetition of excitatory and inhibitory connections between neurons. In computer science, the\n\ufb01eld of reservoir computing (RC) [13] focuses on analyzing and exploiting the echos generated by\nexternal inputs in the dynamics of sparse random networks. The proposed SpikeAnts model features\none distinctive characteristics compared to the state of the art in RC and SNNs: its only aim is to\n\n1\n\n\fmodel an emergent property in a complex closed system; it does neither receive any external inputs\nnor involve any learning rule. To our best knowledge, current models of emergence are mostly based\non statistical physics, involving differential equations and mean \ufb01eld approaches [19], or mathemat-\nics and computer science, using random Markov \ufb01elds, cellular automata or multi-agent systems.\n\n1.2 Target of the SpikeAnts model\n\nThe SpikeAnts model implements a distributed decision making process in a population of agents,\nsay an ant colony. The phenomenon to analyze is the division of labor. The model relies on the\nspatio-temporal interactions of spiking neurons, where each ant agent is accounted for by two neu-\nrons.\nA simpli\ufb01ed scheme is proposed, inspired from [2] and [16]: Each agent may be in one out of\nfour states, Observing, Foraging, Sleeping or self-Grooming (Fig. 1). The interactions take place\nduring the observation round. Each agent a observes its environment and if it perceives none or too\nfew working agents, a goes foraging for a given time and eventually goes to sleep. Otherwise, if\na perceives \u201csuf\ufb01ciently many\u201d agents engaged in foraging, it goes back to the nest for less vital\ntasks (the grooming state) before returning to observation after a while. Each state lasts for a \ufb01xed\nduration (resp. tO, tF , tS and tG), with an exception for the observation state. The observation\nperiod is only subject to an upper bound tO. If the agent sees suf\ufb01ciently many other foraging ants\nbefore the end of the observation period, it can switch at once to the self-grooming state.\n\nG\n\nO\n\nF\n\nS\n\nF\nS\nO\n\noraging\n\nleeping (long) or\n\nbservation\n\nG\n\nrooming (short)\n\ntime\n\nFigure 1: (Left) Transitions between the four agent states: Grooming, Observing, Foraging and\nSleeping states. Black arrows denote transitions and the dotted arrow indicates an inhibitory mes-\nsage. (Right) An example of agent schedule.\n\nThe agent decisions only depend on the information exchanged between them, through agent neu-\nrons sending spikes to (respectively, receiving spikes from) other agents in the population. It must\nbe emphasized that the proposed decision process does not assume the agent ability to \u201ccount\u201d (here\nthe number of its foraging neighbors). In the meanwhile, this process is deterministic, contrasting\nwith the threshold-based probabilistic models used in [1, 2, 7].\n\n2 The SpikeAnts spiking neuron network\n\nThis section describes the structure of the SpikeAnts model. Each ant agent is modelled by two\nspiking neurons. Any two agents (i, j) are connected with an average density \u03c1 (0 (cid:54) \u03c1 (cid:54) 1). The\nant colony thus de\ufb01nes a sparsely connected network of spiking neurons, referred to as SNN.\n\n2.1 Spiking neuron models\n\nAn agent is modelled by two coupled spiking neurons, respectively a Leaky Integrate-and-Fire (LIF)\nneuron [6, 14] and a Quadratic Integrate-and-Fire (QIF) neuron [8, 15]. These models of neuron are\nbiologically plausible and they have been thoroughly studied. We shall show that their coupling\nachieves a frugal control of the agent behavior.\nA LIF neuron \ufb01res a spike if its potential Vp exceeds a threshold \u03d1. Upon \ufb01ring a spike, Vp is reset\n\nto Vreset. Formally: (cid:26) dVp\n\ndt = \u2212\u03bb(Vp(t) \u2212 Vrest) + Iexc(t),\nelse \ufb01res a spike and Vp is set to V p\n\nreset\n\nif Vp < \u03d1\n\n,\n\n(1)\n\nwhere \u03bb is the relaxation constant. Iexc(t) models instantaneous synaptic interactions. Let Pre\ndenote the set of presynaptic neurons (such that there exists an edge from every neuron in Pre and\n\n2\n\n\fthe current neuron), and let Traini denote the spike trains of the ith neuron in Pre; then,\n\n(cid:88)\n\n(cid:88)\n\n\u03b4(t \u2212 ti\nj),\n\n(2)\n\nIexc(t) = w\n\ni\u2208Pre\n\nj\u2208Traini\n\nwhere w is a synaptic weight controlling the dynamics of the SNN (more in section 3.1), \u03b4(.) is\nDirac distribution and ti\nThe QIF neuron is described by the evolution of the potential Va, compared to the resting potential\nVrest and an internal threshold Vthres. Additionally, it receives an internal signal Iclock modelling a\ngap junction connection:\n\nj is the \ufb01ring time of the jth spike from the ith presynaptic neuron.\n\n(cid:26) dVa\ndt = \u03bb(Va(t) \u2212 Vrest)(Va(t) \u2212 Vthres) + Iinh(t) + Iclock(t),\nelse \ufb01res a spike and Va is set to V a\n\nif Va < \u03d1\n\n(3)\n(cid:62) Vthres), the\nDepending on whether the reset threshold is greater than the internal threshold (V a\nreset\nQIF neuron is bistable [12], which motivated the choice of this neuron model. If V a\nreset < Vthres,\nthe membrane potential Va stabilizes on Vrest when there is no external perturbation, and the neuron\n(cid:62) Vthres, the neuron displays a bursting behavior\nthus exhibits an integrator behavior. When V a\nand \ufb01res periodically.\n\nreset\n\nreset\n\n.\n\n2.2 The ant agent model\n\nEach SpikeAnts agent mimics an ant. Its behavior is controlled after the competition between two\ncoupled spiking neurons, an active one (QIF, Eq. (3)) and a passive one (LIF, Eq. (1)). The agent\nadditionally involves an internal unit providing the Iclock signal.\nDuring the observation round, the ant makes its decision (whether it goes foraging) based on the\ncompetition between its active and passive neurons (Fig. 2). Both neurons are aware of the forag-\ning neighbor ants. The signal emitted by these neighbors is an excitatory signal (respectively an\ninhibitory signal) for the passive (resp. active) neuron: Iinh(t) = \u2212Iexc(t). The active neuron\nadditionally receives the excitatory signal Iclock(t) of the internal clock unit.\nIn the case where the ant agent does not see too many foraging ants, the internal excitatory signal\nIclock(t) dominates the inhibitory signal Iinh(t), the active neuron \ufb01res \ufb01rst and drives the ant to\n\nActive neuron\nPassive neuron\n\n\u03d1\n\n1.5\n\n1\n\n0.5\n\n0\n\n0\n\n20\n\n40\n\n60\n\nTime (ms)\n\n80\n\n100\n\n120\n\nS\n\nO\n\nF\n\nS\n\nO\n\nG\n\nO\n\nF\n\nS\n\n)\n\nV\nm\n\n(\n\nl\na\ni\nt\nn\ne\nt\no\np\n\ne\nn\na\nr\nb\nm\ne\nM\n\ne\nt\na\nt\nS\n\nFigure 2: Membrane potentials of active (in dark/red) and passive (in grey/green) neurons. The\ndashed line indicates the threshold \u03d1. The \ufb01rst observation state starts at 20ms: the active neuron\n\ufb01res before the passive one, the agent thus goes foraging and the active neuron continues sending\nspikes during the whole foraging period (signalling its foraging behavior to other agents). After a\nsleep period (from circa 50 to 70ms), starts a second observation round. This time the passive neuron\n\ufb01res before the active one. The agent thus goes self-grooming, and switches to the observation state\nthereafter. During the last observation round, the active neuron wins again against the passive one,\nand the agent goes foraging.\n\n3\n\n\fforaging (\ufb01rst and last episode in Fig. 2). When foraging, the active neuron enters in a bursting\nphase and periodically sends a spike to the ant neighbors. Note that these spikes are only meaningful\nfor the ants in observation state. After a foraging period (duration tF ), the ant goes to sleep (duration\ntS). The sleeping state is triggered by a delayed connection between the internal unit and the active\nneuron.\nQuite the contrary, if the ant sees many other foraging ants, the excitatory signal Iexc(t) drives the\npassive neuron to \ufb01re before the active one (second episode in Fig. 2), and the ant accordingly\nsets in a self-grooming state (duration tG). The decision making of the ant agent thus relies on the\ncompetition between its active and passive neurons. In particular, the number of spikes needed for\nan ant to go foraging or self-grooming depends on the temporal dynamics of the system; it varies\nfrom one observation episode to another. After some rest (self-grooming or sleeping states, with\nrespective durations tG and tS, tG< tS), the ant returns to the observation state.\nAs above-mentioned, incoming spikes are only relevant to the active and passive neurons of an\nobserving ant. During the foraging and resting states, presynaptic spikes have no in\ufb02uence, which\ncan be thought of as an intrinsic plasticity mechanism [21] driven by the internal unit. The internal\nunit can indeed be seen as the ant biological clock. In a further model, it will be replaced by a neural\ngroup interacting with active and passive neurons through intrinsic plasticity, e.g. using a transient\nincrease of \u03bb for LIF and QIF neurons.\n\n2.3 Model parameters\n\nOverall, the SpikeAnts model is controlled by three types of parameters, respectively related to\nspiking neuron models, to ant agents (state durations) and to the whole population (size and connec-\ntivity of the SNN). The default parameter values used in the simulations are displayed in Table 1.\nThe values of state durations are such that their ratio are not integers, in order to avoid spurious\nsynchronizations. Note that state duration timescale is not signi\ufb01cant at the ant colony level.\n\nParameter type\n\nSymbol\n\nNeural\n\nAgent\n\nPopulation\n\n\u03bb\n\nVrest\n\n\u03d1\nV p\nreset\nVthres\nV a\nreset\nIclock\n\nw\ntF\ntO\ntS\ntG\n\u03c1\n\nM\n\nDescription\nMembrane relaxation constant\nResting potential\nSpike \ufb01ring threshold\nPassive neuron reset potential\nActive neuron bifurcation threshold\nActive neuron reset potential\nActive neuron constant input current\nSynaptic weight\nForaging duration\nMaximum observation duration\nSleeping duration\nSelf-grooming duration\nConnection probability\npopulation size\n\nValue\n\nmV\nmV\nmV\nmV\nmV\nmV\n\n(units)\n0.1 mV\u22121\n0.0\n1.0\n-0.1\n0.5\n0.55\n0.1\n0.01 mV\u22121\n47.1\n10.5\n45.7\n16.7\n0.3\n150\n\nms\nms\nms\nms\n\nagents\n\nTable 1: Neural, model and population parameters used in simulations.\n\n3 Experiments\nThis section reports on the experimental study of the SpikeAnts model, \ufb01rst describing the exper-\nimental setting and the goals of experiments. The population behavior is measured after a global\nindicator, and the sensitivity thereof w.r.t. the SpikeAnts parameters is studied. Two compound con-\ntrol parameters, summarizing the model parameters and governing the emergent synchronization of\nthe system are proposed. A consistent phase diagram depicting the global synchronization in the\nplane de\ufb01ned from both control parameters is displayed and discussed.\n\nGoals of experiments A \ufb01rst goal of experiments is to measure the global activity of the popula-\ntion, denoted F and de\ufb01ned as the overall time spent foraging:\n\nnF (t)\n\n(4)\n\nF =\n\n(cid:88)\n\nt\n\n4\n\n\fwhere nF (t) is the number of foraging agents at time t. The study focuses on the sensitivity of F\nw.r.t. the model parameters.\nThe second and most important goal of experiments is to study the temporal structure of the pop-\nulation activity. A synchronization indicator will be proposed and its sensitivity w.r.t.\nthe model\nparameters will be examined.\n\nExperimental settings Each run starts with all ants initially sleeping. Each ant wakes up after\nsome time uniformly drawn in ]0, 2tS ]. Spiking neurons are simulated using a discrete time scheme:\nnumerical simulations of the spiking neuron network are based on a clock-driven simulator, us-\ning Runge-Kutta method for the approximation of differential equations, with a small time step of\n0.1ms to enforce numerical stability. Each run lasts for 100,000 time steps. All reported results are\naveraged over 10 independent runs.\n\n3.1 Sensitivity analysis of the foraging effort\nThis section \ufb01rst examines how the overall foraging effort F depends on the size M of the popu-\nlation, the connection rate \u03c1 and two neural parameters, the active neuron reset potential V a\nreset and\nthe synaptic weight w. The average \u00afF is reported with its standard deviation in Fig. 3.\n\n\u00afF\n\n1000\n800\n600\n400\n200\n\n280\n\n\u00afF\n\n240\n\n200\n\n0\n\n200\n\n400\n\n600\n\n800\n\n1000\n\nM\n\n\u00afF\n\n\u00afF\n\n700\n600\n500\n400\n300\n200\n\n1500\n\n1000\n\n500\n\n0\n\n0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1\n\n\u03c1\n\n0.1\nw\n\n0.15\n\n0.2\n\n0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95\n\n0\n\n0.05\n\nV a\n\nreset\n\nreset: the closer V a\n\nFigure 3: Sensitivity analysis of the average foraging effort \u00afF, versus population size M (top left),\nreset (bottom left) and synaptic\nconnection probability \u03c1 (top right), active neuron reset potential V a\nweight w (bottom right).\nThe overall foraging effort F was expected to linearly increase with the population size M. While\nit indeed increases with M, it displays a breaking down around M=600 (Fig. 3, top left); this\nunexpected change will be explained in section 3.2, and related to the increased variability of the\npopulation synchronization. F was expected to exponentially decrease with the connectivity \u03c1, and\nit does so (Fig. 3, top right): the more neighbors, the more likely an ant will see other foraging ants,\nand will thus avoid go foraging itself. Along the same line, F was expected to decrease with the\nreset potential V a\nreset to \u03d1, the more spikes a foraging ant will sent, exciting other\nants\u2019 passive neuron and thereby sending these ants to rest (Fig. 3, bottom left; the value of \u03d1 is 1,\nand F indeed goes to 0 as V a\nThe most surprising result regards the in\ufb02uence of the synaptic weight w (Fig. 3, bottom right). It\nwas expected that high w values would favor the triggering of passive neurons, and thus adversely\naffect the foraging effort. High w values however mostly result in a high variance of F. The in-\nterpretation proposed for this fact goes as follows. For low w values, an ant behaves as a \u201cgood\nstatistician\u201d, meaning that its decision is based on observing many other foraging agents. Accord-\ningly, the foraging/resting ratio is very stable along time and across runs. As w increases however,\nit makes it possible for an ant to take decisions based on few cues and the behavioral variability\nincreases. More precisely, the F variance is low for small w values (an ant makes its decision based\non about 80 spikes for w = 0.01). The variance dramatically increases in a narrow region around\nw = 0.15; an ant makes its decision based on circa 6 spikes and small variations in the received\nspike trains might thus lead to different decisions, explaining the high variance of F. For higher w\n\nreset goes to 1).\n\n5\n\n\fvalues however, the F variance decreases again. A close look at the experimental results reveals the\nexistence of different temporal regimes with abrupt transitions among these, explaining the breaking\ndown around M = 600 ants and the abrupt increase and decrease of F variance.\n\n3.2 Emergent synchronization: Control parameters and phase transitions\n\nA\n\nAsynchronous\n\nB\n\nSynchronous aperiodic\n\nC\n\nSynchronous periodic\n\n80\n70\n60\n50\n40\n30\n20\n10\n0\n\n80\n70\n60\n50\n40\n30\n20\n10\n0\n\n)\nt\n(\nF\nn\n\n)\n1\n+\n\nt\n(\nF\nn\n\n0\n\n1000\n\n500\n1500\nSimulated time t\n\n2000\n\n)\nt\n(\nF\nn\n\n250\n\n200\n\n150\n\n100\n\n50\n\n0\n\n0\n\n250\n\n200\n\n150\n\n100\n\n50\n\n)\n1\n+\n\nt\n(\nF\nn\n\n1000\n\n500\n1500\nSimulated time t\n\n2000\n\n1000\n900\n800\n700\n600\n500\n400\n300\n200\n100\n0\n\n1000\n\n800\n\n600\n\n400\n\n200\n\n)\nt\n(\nF\nn\n\n)\n1\n+\n\nt\n(\nF\nn\n\n0\n\n1000 1500 2000\n\n500\nSimulated time t\n\n0 10 20 30 40 50 60 70 80\n\nnF (t)\n\n50\n\n100 150 200 250\nnF (t)\n\n200 400 600 800 1000\n\nnF (t)\n\nFigure 4: (Top row) Asynchronous, synchronous aperiodic and synchronous periodic patterns of ac-\ntivity (number of foraging ants versus time for t = 1 . . . 2, 000). (Bottom row) Temporal correlation\nof the activity for the above three patterns, for t = 1 . . . 100, 000.\n\nThe emergence of three synchronization patterns appears in the experimental results. The \ufb01rst one,\nreferred to as asynchronous (Fig. 4, left), depicts a situation where each ant (almost) independently\nmakes its own decisions. The second one, referred to as synchronous (Fig. 4, middle) displays\nsome coordination among the ants; speci\ufb01cally, the number of foraging ants is piecewise constant,\nthough varying from a time interval to another. The third pattern, referred to as periodic synchronous\n(Fig. 4, right) involves two stable subpopulations which forage alternatively; the population enters a\nbi-phase mode, as actually observed in some ants colonies [4, 5].\nThe difference between the three patterns of activity is most visible from the phase diagram plotting\nnF (t + 1) vs nF (t) (Fig. 4, bottom row; transient states are removed in the synchronized periodic\nand aperiodic regime for the sake of clarity). The orbit of the synchronous aperiodic activity indi-\ncates the presence of at least one attractor whereas the synchronous periodic activity displays a \ufb02ip\nbifurcation.\nThe ergodicity of the SpikeAnts system is \ufb01rst analyzed based on the Lyapunov exponents, after the\ncomputation algorithms proposed in [22]. On asynchronous patterns, the mean value of the 5,000\nLyapunov exponents found with an 8 dimension analysis is \u22120.01\u00b1 0.1. For synchronous aperiodic\npatterns, the mean value of the 3,500 Lyapunov exponents found with a 6 dimension analysis is\nalso \u22120.01\u00b1 0.1 (after discarding the transient states). Whereas the asynchronous and synchronous\naperiodic activities lie at the edge of chaos, the periodic synchronous regime only displays large\nnegative Lyapunov exponents, indicating a very stable behavior.\nAn entropy-based indicator is proposed to analyze the emergent synchronization of the SpikeAnts\nsystem. Let I denote the set of values nF (t) (after pruning all transient time steps such that nF (t) (cid:54)=\nnF (t + 1) and nF (t) (cid:54)= nF (t \u2212 1)); the foraging histogram is de\ufb01ned by associating to each value\nk in I, the number nk of time steps such that nF (t) = k. The synchronization of the population is\n\n6\n\n\f\ufb01nally measured from the histogram entropy H:\n\nH = \u2212(cid:88)\n\nnk(cid:80)\n\n(cid:19)\n\n(cid:18) nk(cid:80)\n\nm nm\n\nlog\n\n(5)\n\nk\u2208I\n\nm nm\n\n\u221a\n\nThe entropy of the asynchronous regime is zero, since all states are transient. The synchronous\nperiodic regime, where two subpopulations alternatively forage, gets a low entropy (< log 2). Fi-\nnally, the synchronous aperiodic regime which involves a few dozens of subpopulations, gets a high\nentropy value. The transition from one regime to another one is clearly related to the model pa-\nrameters. The goal thus becomes to identify the in\ufb02uential factors, best explaining the population\nbehavior.\nM and referred to as sociability, controls the amount of\nA \ufb01rst such in\ufb02uential factor, de\ufb01ned as \u03c1\ninteractions between the ants. A high sociability enables the ants to base their foraging decision on\nreliable estimates of the current foraging activity, thus entailing a low variance of the global foraging\neffort.\nA second in\ufb02uential factor, referred to as receptivity, is the ratio between the weight w of the input\nsignal and the subthreshold range (depending on the resting potential Vrest and the spike \ufb01ring\nthreshold \u03d1). This ratio\nw|\u03d1\u2212Vrest| indicates the amplitude of the depolarization induced by the input\nspike compared to the difference between rest and threshold. A high receptivity thus enables the ant\nto postpone its foraging decision based on few cues (i.e. visible foraging ants), thereby entailing a\nhigh variance of the global foraging effort.\nThe sociability and receptivity factors, referred to as control parameters, support a clear picture of\nthe asynchronous, synchronous aperiodic and periodic synchronous patterns. The entropy (Fig. 5,\nleft) and its variance (Fig. 5, right) are displayed in the 2D plane de\ufb01ned from the sociability and\nreceptivity of the SpikeAnts system, de\ufb01ning the phase diagram of the SpikeAnts system.\nFor a low sociability and a high receptivity (region A in Fig. 5), few interactions among ants take\nplace and each ant makes its decisions based on few cues. In this region, the population is a col-\nlection of quasi independent individuals, and few ants (60 on average on Fig. 4) are foraging at any\ngiven time step.\nFor a higher sociability and a low receptivity (region B in Fig. 5), ants see more of their peers and\nthey base their decisions on reliable estimates of the foraging activity. A synchronization of the ant\nactivities emerges, in the sense that many agents make their foraging decisions at the same time.\nStill, the synchronization remains aperiodic, i.e. the number of foraging ants varies from 50 to 240\n(Fig. 4).\nFor a high sociability and a high receptivity (region C in Fig. 5), ants see many of their peers and they\nmake their decisions based on few cues. In this case a periodic synchronized regime is observed,\nwhere two subpopulations alternatively go foraging (the \ufb01rst one involves \u223c 950 ants in Fig. 4).\n\n0.2\n\n0.15\n\n0.1\n\n0.05\n\ny\nt\ni\nv\ni\nt\np\ne\nc\ne\nR\n\nA\n\nB\n\n5\n\n0\n\nC\n\n4.5\n4\n3.5\n3\n2.5\n2\n1.5\n1\n0.5\n0\n\n)\nn\na\ne\nm\n\n(\n\nH\n\n0.2\n\n0.15\n\n0.1\n\n0.05\n\ny\nt\ni\nv\ni\nt\np\ne\nc\ne\nR\n\nA\n\nB\n\nC\n\n1.6\n1.4\n1.2\n1\n0.8\n0.6\n0.4\n0.2\n0\n\n)\nn\no\ni\nt\na\ni\nv\ne\nd\n\nd\nr\na\nd\nn\na\nt\ns\n(\n\nH\n\n10\n\n15\n\n20\n\n25\n\nSociability\n\n0\n\n5\n\n10\n\n15\n\n20\n\n25\n\nSociability\n\nFigure 5: Emergence of synchronizations in the population activity: entropy H (left) and variance\nof H (right) versus the ant sociability and receptivity. The asynchronous pattern, with entropy\nH = 0 corresponds to a low sociability and high receptivity (region A). The synchronous aperiodic\npattern, with high entropy, corresponds to a medium sociability and low receptivity (region B).\nThe synchronous periodic pattern, H \u223c log 2, corresponds to both high sociability and receptivity\n(region C).\n\n7\n\n\f)\nt\n(\nF\nn\n\n250\n\n200\n\n150\n\n100\n\n50\n\n0\n\n0\n\n1000\n\n2000\n\n3000\n\n4000\n\n5000\n\nt\n\n6000\n\n7000\n\n8000\n\n9000\n\n10000\n\nFigure 6: A representative simulation: the global behavior switches from a synchronous aperiodic\nregime to an asynchronous one before stabilizing in a periodic synchronous regime.\n\nComplementary experiments show abrupt transitions between the different regimes in the border-\nline regions. Speci\ufb01cally, an asynchronous aperiodic regime (region B) is prone to evolve into an\nasynchronous (region A) or periodic synchronous (region C) regimes (Figure 6). Quite the contrary,\nthe periodic synchronous regime is stable, i.e. the population does not get back to any other regime\nafter the periodic synchronous regime is installed. The aperiodic synchronous regime, though less\nstable than the periodic one, is far more stable than the asynchronous one.\n\n4 Discussion\n\nThe main contribution of this paper is a local and parsimonious model, accounting for individual\ndecision making, which reproduces the emergence of synchronized activity in a complex system in\na realistic way: the three different regimes obtained in simulation are comparable to the different\npatterns of activity observed in social insect colonies [7, 5, 4]. The synchronization patterns that\nemerge at the macroscopic scale can be fully controlled by several model parameters ruling the so-\nciability of ants (whether an ant may observe many other ants) and their receptivity (whether an ant\nmakes its foraging decision based on a few cues). The synchronization patterns are endogenous,\nwith no external in\ufb02uence from the environment. Additionally, they do not rely on individual syn-\nchronizations, as each agent has a speci\ufb01c behavior, different from its neighbor and varying during\nsimulation time.\nTo our best knowledge, the SpikeAnts model is the \ufb01rst one accounting for a population behavior\nand based on spiking neurons. SpikeAnts captures both spatial and temporal features of the complex\nsystem in a deterministic way (as opposed to stochastic models). It does not require any external\nconstraints or data. Most importantly, it does not require the agent to feature sophisticated skills\n(e.g. \u201ccounting\u201d its foraging neighbors). It is worth noting that SpikeAnts does not involve the\nresolution of differential equations: While spiking neurons are modelled in continuous time, their\nbehavior is computed through \ufb01nite differences, parameterized from the user-speci\ufb01ed time step. In\nsummary, SpikeAnts demonstrates that SNNs can be used to model a simple self-organizing system.\nIt hopefully opens new perspectives for modelling emergent phenomena in complex systems.\nA \ufb01rst perspective for further research is to investigate the temporal dynamics of spike trains using\nstandard approaches from neuroscience. The underlying question is whether the population syn-\nchronization can be facilitated, e.g. in the transient regime, by making spiking neurons sensitive to\nthe synchrony of spike trains. The role of inhibition and the role of the excitation/inhibition balance\nin the emergence of synchronized patterns will be studied. In particular, the impact on the phase\ndiagram of individual parameter variations will be analyzed.\nA second perspective is to endow SpikeAnts with some learning skills, e.g. adapting the connections\nweights w with a local unsupervised learning rule (e.g. Spike-Timing-Dependent Plasticity), in order\nto optimize the collective ef\ufb01ciency of the population. Along the same line, the ability of SpikeAnts\nto cope with external perturbations (e.g. affecting the number of foraging ants) will be investigated.\n\nAcknowledgments\n\nWe thank Mathias Quoy, Universit\u00b4e Cergy, for many fruitful discussions about complex systems, and helpful\nremarks about this paper. We thank Jean-Louis Deneubourg and Jos\u00b4e Halloy, Universit\u00b4e Libre de Bruxelles,\nfor many insights into the collective behavior of living systems. This work was supported by NSF grant No.\nPHY-9723972 and by the European Integrated Project SYMBRION.\n\n8\n\n\fReferences\n[1] E. Bonabeau, G. Theraulaz, and J.L. Deneubourg. Fixed response thresholds and the regulation of division\n\nof labor in insect societies. Bulletin of Mathematical Biology, 60(4):753\u2013807\u2013807, July 1998.\n\n[2] E. Bonabeau, G. Theraulaz, and J.L. Deneubourg. The synchronization of recruitement-based activities\n\nin ants. BioSystems, 45:195\u2013211, 1998.\n\n[3] N. Brunel and X.J. Wang. What determines the frequency of fast network oscillations with irregular\nneural discharges? I. Synaptic dynamics and excitation-inhibition balance. Journal of Neurophysiology,\n90(1):415\u2013430, 2003.\n\n[4] B.J. Cole. Short-term activity cycles in ants: Generation of periodicity by worker interaction. The Amer-\n\nican Naturalist, 137(2), 1991.\n\n[5] N.R. Franks and S. Bryant. Rhythmical patterns of activity within the nest of ants. Chemistry and Biology\n\nof Social Insects, pages 122\u2013123, 1987.\n\n[6] W. Gerstner and W. Kistler. Spiking Neuron Models: Single Neurons, Population, Plasticity. Cambridge\n\nUniversity Press, 2002.\n\n[7] S. Goss and J.L. Deneubourg. Autocatalysis as a source of synchronised rythmical activity in social\n\ninsects. Insectes Sociaux, 35(3):310\u2013315, 1988.\n\n[8] D. Hansel and G. Mato. Existence and stability of persistent states in large neuronal networks. Physical\n\nReview Letters, 86(18):4175\u20134178, April 2001.\n\n[9] D.O. Hebb. The Organization of Behaviour. Wiley, New York, 1949.\n[10] J.J. Hop\ufb01eld and C.D. Brody. What is a moment? Transient synchrony as a collective mechanism for\n\nspatiotemporal integration. Proc. Natl. Acad. Sci., 98(3):1282\u20131287, 2001.\n\n[11] E.M. Izhikevich. Polychronization: Computation with spikes. Neural Computation, 18(2):245\u2013282, 2006.\n[12] E.M. Izhikevich. Dynamical systems in neuroscience: the geometry of excitability and bursting, chapter\n\nOne-Dimensional Systems. MIT Press, 2007.\n\n[13] H. Jaeger, W. Maass, and J. Principe. Special issue on echo state networks and liquid state machines\n\n(editorial). Neural Networks, 20(3):287\u2013289, April 2007.\n\n[14] B.W. Knight. Dynamics of encoding in a population of neurons. The Journal of General Physiology,\n\n59(6):734\u2013766, June 1972.\n\n[15] P.E. Latham, B.J. Richmond, P.G. Nelson, and S. Nirenberg. Intrinsic dynamics in neuronal networks. i.\n\ntheory. Journal of Neurophysiology, 83(2):808\u2013827, February 2000.\n\n[16] W. Liu, A.F.T. Win\ufb01eld, J. Sa, J. Chen, and L. Dou. Towards energy optimisation: Emergent task alloca-\n\ntion in a swarm of foraging robots. Adaptive Behavior, 15(3):289\u2013305, 2007.\n\n[17] R.E. Mirollo and S.H. Strogatz. Synchronization of pulse-coupled biological oscillators. SIAM Journal\n\non Applied Mathematics, 50(6):1645\u20131662, 1990.\n\n[18] H. Paugam-Moisy and S.M. Bohte. Handbook of Natural Computing, chapter 10. Computing with Spiking\n\nNeuron Networks. Springer, 2010. (in press).\n\n[19] D. Phan, M.B. Gordon, and J.P. Nadal. Cognitive Economics, chapter Social interactions in economic\n\ntheory: An insight from statistical mechanics, pages 335\u2013358. Springer, 2004.\n\n[20] B. Schrauwen, L. B\u00a8using, and R. Legenstein. On computational power and the order-chaos phase transi-\ntion in reservoir computing. In D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, editors, Advances in\nNeural Information Processing Systems, pages 1425\u20131432. MIT Press, 2008.\n\n[21] J. Triesch. Synergies between intrinsic and synaptic plasticity mechanisms. Neural Computation,\n\n19(4):885\u2013909, 2007.\n\n[22] A. Wolf, J. Swift, H. Swinney, and J. Vastano. Determining lyapunov exponents from a time series.\n\nPhysica D: Nonlinear Phenomena, 16(3):285\u2013317, 1985.\n\n9\n\n\f", "award": [], "sourceid": 1134, "authors": [{"given_name": "Sylvain", "family_name": "Chevallier", "institution": null}, {"given_name": "H\u00e9l\\`ene", "family_name": "Paugam-moisy", "institution": null}, {"given_name": "Michele", "family_name": "Sebag", "institution": null}]}