{"title": "Exact differential equation population dynamics for integrate-and-fire neurons", "book": "Advances in Neural Information Processing Systems", "page_first": 205, "page_last": 212, "abstract": null, "full_text": "Exact differential equation population \n\ndynamics for Integrate-and-Fire neurons \n\nJulian Eggert * \n\nHONDA R&D Europe (Deutschland) GmbH \n\nFuture Technology Research \n\nCarl-Legien-StraBe 30 \n\n63073 Offenbach/Main, Germany \n\njulian. eggert@hre-ftr.f.rd.honda.co.jp \n\nBerthold Bauml \n\nInstitut fur Robotik und Mechatronik \n\nDeutsches Zentrum fur Luft und Raumfahrt (DLR) \n\no berpfaffenhofen \n\nBerthold.Baeuml@dlr.de \n\nAbstract \n\nIn our previous work, integral equation formulations for \n\nMesoscopical, mathematical descriptions of dynamics of popula(cid:173)\ntions of spiking neurons are getting increasingly important for the \nunderstanding of large-scale processes in the brain using simula(cid:173)\ntions. \npopulation dynamics have been derived for a special type of spik(cid:173)\ning neurons. For Integrate- and- Fire type neurons, these formula(cid:173)\ntions were only approximately correct. Here, we derive a math(cid:173)\nematically compact, exact population dynamics formulation for \nIntegrate- and- Fire type neurons. It can be shown quantitatively \nin simulations that the numerical correspondence with microscop(cid:173)\nically modeled neuronal populations is excellent. \n\n1 \n\nIntroduction and motivation \n\nThe goal of the population dynamics approach is to model the time course of the col(cid:173)\nlective activity of entire populations of functionally and dynamically similar neurons \nin a compact way, using a higher descriptionallevel than that of single neurons and \nspikes. The usual observable at the level of neuronal populations is the population(cid:173)\naveraged instantaneous firing rate A(t), with A(t)6.t being the number of neurons \nin the population that release a spike in an interval [t, t+6.t). Population dynamics \nare formulated in such a way, that they match quantitatively the time course of a \ngiven A(t), either gained experimentally or by microscopical, detailed simulation. \n\nAt least three main reasons can be formulated which underline the importance \nof the population dynamics approach for computational neuroscience. First, it \nenables the simulation of extensive networks involving a massive number of neurons \n\n\fand connections, which is typically the case when dealing with biologically realistic \nfunctional models that go beyond the single neuron level. Second, it increases the \nanalytical understanding of large-scale neuronal dynamics, opening the way towards \nbetter control and predictive capabilities when dealing with large networks. Third, \nit enables a systematic embedding of the numerous neuronal models operating at \ndifferent descriptional scales into a generalized theoretic framework, explaining the \nrelationships, dependencies and derivations of the respective models. \n\nEarly efforts on population dynamics approaches date back as early as 1972, to the \nwork of Wilson and Cowan [8] and Knight [4], which laid the basis for all current \npopulation-averaged graded-response models (see e.g. [6] for modeling work using \nthese models). More recently, population-based approaches for spiking neurons were \ndeveloped, mainly by Gerstner [3, 2] and Knight [5]. In our own previous work [1], \nwe have developed a theoretical framework which enables to systematize and sim(cid:173)\nulate a wide range of models for population-based dynamics. It was shown that \nthe equations of the framework produce results that agree quantitatively well with \ndetailed simulations using spiking neurons, so that they can be used for realistic \nsimulations involving networks with large numbers of spiking neurons. Neverthe(cid:173)\nless, for neuronal populations composed of Integrate-and-Fire (I&F) neurons, this \nframework was only correct in an approximation. In this paper, we derive the exact \npopulation dynamics formulation for I&F neurons. This is achieved by reducing \nthe I&F population dynamics to a point process and by taking advantage of the \nparticular properties of I&F neurons. \n\n2 Background: Integrate-and-Fire dynamics \n\n2.1 Differential form \n\nWe start with the standard Integrate- and- Fire (I&F) model in form of the well(cid:173)\nknown differential equation [7] \n\n(1) \n\nwhich describes the dynamics of the membrane potential Vi of a neuron i that is \nmodeled as a single compartment with RC circuit characteristics. The membrane \nrelaxation time is in this case T = RC with R being the membrane resistance and C \nthe membrane capacitance. The resting potential v R est is the stationary potential \nthat is approached in the no-input case. The input arriving from other neurons is \ndescribed in form of a current ji. \n\nIn addition to eq. (1), which describes the integrate part of the I&F model, the \nneuronal dynamics are completed by a nonlinear step. Every time the membrane \npotential Vi reaches a fixed threshold () from below, Vi is lowered by a fixed amount \nLl > 0, and from the new value of the membrane potential integration according to \neq. (1) starts again. \n\nif Vi(t) = () (from below) . \n\n(2) \n\nAt the same time, it is said that the release of a spike occurred (i.e., the neuron \nfired), and the time ti = t of this singular event is stored. Here ti indicates the \ntime of the most recent spike. Storing all the last firing times, we gain the sequence \nof spikes {t{} (spike ordering index j, neuronal index i). \n\n\f2.2 \n\nIntegral form \n\nNow we look at the single neuron in a neuronal compound. We assume that the \ninput current contribution ji from presynaptic spiking neurons can be described \nusing the presynaptic spike times tf, a response-function ~ and a connection weight \nW\u00b7 . \n',J \n\nji(t) = l: Wi ,j l: ~(t - tf) \n\n(3) \n\nIntegrating the I&F equation (1) beginning at the last spiking time tT, which de(cid:173)\ntermines the initial condition by Vi(ti) = vi(ti - 0) - 6., where vi(ti - 0) is the \nmembrane potential just before the neuron spikes, we get 1 \n\nj \n\nf \n\nVi(t) = vRest + fj(t - t:) + l: Wi ,j l: a(t - t:; t - tf) , \n\nj \n\nf \n\nwith the refractory function \n\nfj(s) = -\n\n(v Rest - Vi(t:)) e- S / T \n\nand the alpha-function \n\na(s; s') = r ds\" e-[sf -S\"J/T ~(s\") . \n\nSf \n\nJSI_S \n\n(4) \n\n(5) \n\n(6) \n\nIf we start the integration at the time ti* of the spike before the last spike, then for \nti* :::; t < ti the membrane potential is given by an expression like eq. (4), where ti \nis replaced by t:i* . Especially we can now express v( ti - 0) and therefore the initial \ncondition for an integration starting at tT in terms of ti* and v(ti* - 0). In this \nway, we can proceed repetitively and move back into the past. After some simple \nalgebra this results in \n\nVi(t) = vRest + l:ry(t-t{)+ l:Wi,j l:a(oo ;t - tf) , \n\n(7) \n\nf \n\nj \n\nf \n\n~ ~-------y~------~ \n\nvfef(t) \n\nv~yn(t) \n\nwith a refractory function wich differs in the scale factor from that in eq. (5) \n\n(8) \nThe components vref(t) and v?n(t) to the membrane potential indicate refractory \nand synaptic components to the neuron i, respectively, as normally used in the \nSpike- Response- Model (SRM) notation [2]. \n\nry(s) = -6. e- S / T \n\n\u2022 \n\nBoth equations (4) and (7) formulate the neuronal dynamics using a refractory \ncomponent, which depends on the own spike releases of a neuron, and a synaptic \ncomponent, which comprises the integrated input contribution to the membrane \npotential by arrival of spikes from other neurons 2. The synaptic component is based \non the alpha-function characteristic of isolated arriving spikes, with an increase of \nthe membrane potential after spike arrival and a subsequent exponential decrease. \n\n1 Strictly speaking, the constants vRest, T, () and ,6, and the function 1]( s) may vary for \neach neuron, so that they should be written with a subindex i [similarly for n(s; s') , which \nmay vary for each connection j -+ i so that we should write it with subindices i, j]. For \nthe sake of clarity, we omit these indices here. \n\n2S0 the I&F model can be formulated as a special case of the Spike- Response- Model, \n\nwhich defines the neuronal dynamics in the integral formulation. \n\n\fThe comparison of the equivalent expressions eq. (4) and eq. (7) reveals an interest(cid:173)\ning property of the I&F model. They look very similar, but in eq. (4), the refractory \ncomponent depends only on the time elapsed since the last spike (thus reflecting a \nrenewal property, sometimes also called a short term memory for refractory proper(cid:173)\nties), whereas in eq. (7), it depends on a sum of the contributions of all past spikes. \nThe simpler form of the refractory contribution in eq. (4) is achieved at the cost \nof an alpha-function that now depends on the time t - ti elapsed since the last \nown spike in addition to the times t - tf elapsed since the release of spikes at the \npresynaptic neurons j that provide input to i. In eq. (7) , we have a more complex \nrefractory contribution, but an alpha-function that does not depend on the last own \nspike time any more. \n\n2.3 Probabilistic spike release \n\nProbabilistic firing is introduced into the I&F model eq. (4) resp. (7) by using \nthreshold noise. The spike release of each neuron is controlled by a hazard function \n>.(v), so that \n>.(v)dt = Prob. that a neuron with membrane potential v spikes in [t , t + dt) \n\n(9) \nWhen a neuron spikes, we proceed as usual: The membrane potential is reset by a \nfixed amount 6. and the I&F dynamics continues. \n\n3 Population dynamics \n\n3.1 Density description \n\nDescriptions of neuronal populations usually assume a neuronal density function \np(t; X) which depends on the state variables X of the neurons. The density quan(cid:173)\ntifies the likelihood that a neuron picked out of the population will be found in the \nvicinity of the point X in state space, \n\np(t; X) dX = Portion of neurons at time t with state in [X, X + dX) \n\n(10) \n\nIf we know p(t; X) , the population activity A(t) can be easily calculated. Using the \nhazard function >'(t; X), the instantaneous population activity (spikes per time) can \nbe calculated by computing the spike release averaged over the population, \n\nA(t) = J dX >.(t; X) p(t; X) \n\n(11) \n\nThe population dynamics is then given by the time course of the neuronal den(cid:173)\nsity function p(t; X), which changes because each neuron evolves according to its \nown internal dynamics, e.g. after a spike release and the subsequent reset of the \nmembrane potential. \n\nThe main challenge for the formulation of a population dynamics resides in selecting \na low-dimensional state space [for an easy calculation of A(t)] and a suitable form \nfor gtp(t; X). \nAs an example, for the population dynamics for I&F neurons it would be straight(cid:173)\nforward to use the membrane potential v from eq. (1) as the state variable X. But \nthis leads to a complicated density dynamics, because the dynamics for v(t) consist \nof a continuous (differential equation (1)) and a discrete part (spike generation). \nTherefore, here we concentrate on an alternative description that allows a compact \nformulation of the desired I&F density dynamics. \n\n\f3.2 Exact population dynamics for I&F neurons \n\nWhich is the best state space for a population dynamics of I&F neurons? For the \nformulation of a population dynamics, it is usually assumed that the synaptic con(cid:173)\ntributions to the membrane potential are identical for all neurons. This is the case \nif we group all neurons of the same dynamical type and with identical connectivity \npatterns into one population. That is, we say that neurons i and i' belong to the \nsame population if Wi,j = Wi',j for all j (for simulations of realistic networks of \nspiking neurons, this will of course never be exactly the case, but it is reasonable \nto assume that a grouping of neurons into populations can be achieved to a good \napproximation) . \nIn our case, looking at eq. (4), we see that, since o:(s, s') depends on s = t - ti and \ntherefore on the own last spike time, the synaptic contribution to the membrane \npotential differs according to the state of the neuron. Thus we regard eq. (7). Here, \nwe see that for identical connectivity patterns Wi,j, the synaptic contributions are \nthe same for all neurons, because 0:(00, s') does not depend on the own spike time \nany more. Which are then the state variables of eq. (1) for the density description? \nWe see that, for a fixed synaptic contribution, the membrane potential Vi is fully \ndetermined by the set of the own past spiking times {tf}. But this would mean \nan infinite-dimensional density for the state description of a population, and, ac(cid:173)\ncordingly, a computationally overly expensive calculation of the population activity \nA(t) according to eq. (11). \n\nTo avoid this we take advantage of a particular property of the I&F model. Accord(cid:173)\ning to eq. (8), the single spike refractory contributions 'TJ(s) are exponential. Since \nany sum of exponential functions with common relaxation constant T can be again \nexpressed as as an exponential function with the same T , we can write instead of \nvrf(t) from eq. (7) \n\n(12) \nNow the membrane potential Vi(t) only depends on the time of the last own spike \nti and the refractory contribution amplitude modulation factor at the last spike 'TJi . \nThat is, we have transferred the effect of all spikes previous to the last one into 'TJi. \nIn addition, we have to care about updating of ti and 'TJi when a neuron spikes so \nthat we get 3 \n\n(13) \n\n'TJi \nti \n\n--+ \n--+ \n\n'TJi = 1 + 'TJie-(t-tn!T , \nti = t . \n\nThe effect of taking into account more than the most recent spike ti in the refractory \ncomponent vief(t) leads to a modulation factor 'TJi greater than 1, in particular if \nspikes come in a rapid succession so that refractory contributions can accumulate. \nInstead of using a modulation factor 'TJi the effect of previous spikes can also be \ntaken into account by introducing an effective last spiking time ii. \n\nvi\"f(t) = 'TJ(t - in = 'TJi'TJ(t - tn , \n\n(14) \n\nwhere ii and 'TJi are connected by \n\ni; = t; + TIn'TJi \nThe effect of i* is sort of funny. Because of 'TJi \n::::: 1 it holds for the effective last \nspiking time ii ::::: ti. This means, that, while at a given time t it is allways ti :::; t, \nit happens that ii ::::: t, meaning the neurons behave as if they would spike in the \nfuture. \n\n(15) \n\n3Here, the order of reemplacement matters; first we have to reemplace 1]:, then ti. \n\n\fFor the membrane potential we get now instead of eq. (7) \n\nVi(t) = vRest + ry(t - tn + 2..: Wi ,j 2..: 0:(00; t - t;) \nand for the update rule for the effective last spiking time t; follows \n\nf \n\nj \n\ntA* \ni -+ i= \n\ntA* \n\nf (t tA*) \n\n' i ' \n\nwith \n\n(16) \n\n(17) \n\n(18) \n\nTherefore we can regard the dimensionality of the state space of the I&F dynamics \nas 1-dimensional in the description of eq. (16). The dynamics of the single I&F \nneurons now turns out to be very simple: Calculate the membrane potential Vi(t) \nusing eq. (16) together with the state variable t;, and check if Vi(t) exceeds the \nthreshold. If not, move forward in time and calculate again. If the membrane \npotential exceeds threshold, update t; according to eq. (17) and then proceed with \nthe calculation of Vi(t) as normal. \nUsing this single neuron dynamics , we can now proceed to gain a population dy(cid:173)\nnamics using a density p(t; t*). The time t is here the explicit time dependence, \nwhereas t* denote the state variable of the population. By fixing t* and the synap(cid:173)\ntic contribution vsyn(t) to the membrane potential, the state of a neuron is fully \ndetermined and the hazard function can be written as ,X[vsyn(t); t*]. \n\nThe dynamics of the density p(t; t*) is then calculated as follows. Changes of p(t; t*) \noccur when neurons spike and t* is updated according to eq. (17). The hazard \nfunction controls the spike release, and, therefore, the change of p(t; t*). For chosen \nstate variables, p(t; t*) decreases due to spiking of the neurons with the fixed t*, \nand increases because neurons with other t'* spike and get updated in just that \nway that after updating their state variable falls around t*. This occurs according \nto the reemplacement rule eq. (17) when \n\nf(t, t'*) = t* . \n\nTaking all together the dynamics of the density p(t; t*) is given by \ndecrease due to same state t* spiking \n\nA \n\n'-,X[vsyn(t); t*]p(t; t*)' \n\n-ftp(t;t*) = \n\n1+ 00 \n\n+ -00 dt'* 8[J(t, t'*) - t*] ,X[vsyn(t); t'*] p(t; t'*) \n\nincrease due to spiking of neurons with other states t'* \n\nThe population activity can then be calculated using the density according to \neq. (11) as follows \n\n1+00 \n\nA(t) = \n\n- 00 dt* ,X[vsyn(t); t*] p(t; t*) \n\n(19) \n\n(20) \n\n(21) \n\n(22) \n\nRemark that the expression for the density dynamics (eq. 20) automatically con(cid:173)\n\nserves the norm of the density, so that 1+00 \n\n- 00 dt* p( t ; t*) = const , \n\nwhich is a necessary condition because the number of neurons participating in the \ndynamics must remain constant. \n\n\f4 Simulations \n\nThe dynamics of a population of I&F neurons , represented by the time course \nof their joint activity, can now be easily calculated in terms of the differential \nequation (20) , if the neuronal state density of the neuronal population p(t; i*) and \nthe synaptic input vsyn(t) are known. This means that all we have to store is the \ndensity p(t; i*) for past and future effective last spiking times i* 4 . Favorably for \nnumerical simulations, only a limited time window of i* around the actual time t \nis needed for the dynamics. The activity A(t) only appears as an auxiliary variable \nthat is calculated with the help of the neuronal density. \n\nIn figure 1 the simulation results for populations of of spiking neurons are shown. \nThe neurons are uncoupled and a hazard function \n\nA(V) = ~ e2,B(v-e) , \n\nTO \n\n(23) \n\nwith spike rate at threshold liTO = 1.0ms- 1 , a kind of inverse temperature (3 = 2.0, \nwhich controls the noise level, and the threshold e = 1.0. The other parameters of \nthe model in eq. (1) are: resting potential vRest = 0, jump in membrane potential \nafter spike release ~ = 1 and time constant T = 20ms. This parameters are chosen \nto be biologicaly plausible. \n\nA (spikes/ms) \n\n0.14 \n\n0.12 \n\n0.1 \n\n0.08 \n\n0.06\n\n0.04 \n\n0.02 \n\nr-------\n\nvsyr'i-' _ _ 1_00 __ 15_0 __ 20_0 _ _ 2_50 __ 30_0 ----,1 (ms) \n\no~ I c) \n\n100 \n\n150 \n\n200 \n\n250 \n\n300 \n\nI (ms) \n\n: \n= \n\nb) \n\nII \" \" \nn \n\" \" \n:: :: \n!l \n: ~ \n! \\_, .. ----: \n: \n_______ J \n\nI \n1\\ \n\n1l \n* : \\ \n! \\_ .. ----2 '-1 \n! ....... j ! \n~ .. ' \n\n~,' \n\nI:, \n\nr-------\n\nFigure 1: Activity A(t) of simulated populations of neurons. The neurons are \nuncoupled and to each neuron the same synaptic field vsyn(t), ploted in c) and d), \nis applied. a) shows the activity A(t) for a population of I&F neurons simulated \non the one hand as N = 10000 single neurons (solid line) using eq. (7) and on the \nother hand using the density dynamics eq. (20) (dashed line). In b) the activity \nA(t) of a population ofI&F neurons (dashed line) and a population of SRM neurons \nwith renewal (solid line) are compared. For all simulations the same parameters as \nspecified in the text were used. \n\nThe simulations show that the density dynamics eq. (20) reproduces the activity \nA( t) of a population of single I&F neurons almost perfect, with the exception of the \nnoise in the single neuron simulations due to the finite size effects. This holds even \nfor the peaks occuring at the steps of the applied synaptic field vsyn (t), although the \ndensity dynamics is entirely based on differential equations and one would therefore \nnot expect such an excellent match for fast changes in activity. \n\n4 VSYll (t) only appears as a scalar in the dynamics, so that no integration over time takes \n\nplace here. \n\n\fThe simulations also show that there can be a big difference between I&F and SRM \nneurons with renewal. Because of the accumulation of the refractory effects of all \nformer spikes in the case of I&F neurons the activity A(t) is generaly lower than \nfor the SRM neurons with renewal and the higher the absolute actitvity level the \nbigger is the difference between both. \n\n5 Conclusions \n\nIn this paper we derived an exact differential equation density dynamics for a popu(cid:173)\nlation of I&F neurons starting from the microscopical equations for a single neuron. \nThis density dynamics allows a compuationaly efficient simulation of a whole pop(cid:173)\nulation of neurons. \n\nIn future work we want to simulate a network of connected neuronal populations. \nIn such a network of populations (indexed e.g. by x) , a self-consistent system of \ndifferential equations based on the population's p(x, t; i*) and A(x, t) emerges if \nwe constrain ourselves to neuronal populations connected synaptically according to \nthe constraints given by the pool definition [2]. In this case, two neurons i and j \nbelong to pools x and y, if Wi,j = W(x, y). This allows us to write for the synaptic \ncomponent of the membrane potential \n\n(24) \n\nvsyn(x,t) = 2: W (x , y) 100 ds'a(oo;s')A(y,t-s') \n\ny \n\n0 \n\nUsing the alpha-function a(oo ; s') as introduced in (6), and a \"nice\" response(cid:173)\nfunction ~ for the input current time course after a spike, we can write eq. (24) \nusing differential equations that use A(y, t) as input. This results in a system that \nis based entirely on differential equations and is very cheap to compute. \n\nReferences \n\n[1] J. Eggert and J.L. van Hemmen. Modeling neuronal assemblies: Theory and imple(cid:173)\n\nmentation. N eural Computation, 13(9):1923- 1974, 200l. \n\n[2] W. Gerstner. Population dynamics of spiking neurons: Fast transients, asynchronous \n\nstates and locking. Neural Computation, 12:43- 89, 2000. \n\n[3] W . Gerstner and J . L. van Hemmen. Associative memory in a network of 'spiking' \n\nneurons. Network, 3:139- 164, 1992. \n\n[4] B. W. Knight . Dynamics of encoding in a populations of neurons. J. Gen. Physiology, \n\n59:734- 766, 1972. \n\n[5] B. W. Knight. Dynamics of Encoding in Neuron Populations: Some General Mathe(cid:173)\n\nmatical Features. Neural Comput., 12:473- 518, 2000. \n\n[6] Z. Li. A neural model of contour integration in the primary visual cortex. Neural \n\nComput. , 10(4):903- 940, 1998. \n\n[7] H. C. Tuckwell. Introduction to Th eoretical N eurobiology. Cambridge University Press, \n\nCambridge, 1988. \n\n[8] H. R. Wilson and J. D. Cowan. Excitatory and inhibitory interactions in localized \n\npopulations of model neurons. Biophys. J., 12:1- 24, 1972. \n\n\f", "award": [], "sourceid": 2076, "authors": [{"given_name": "Julian", "family_name": "Eggert", "institution": null}, {"given_name": "Berthold", "family_name": "B\u00e4uml", "institution": null}]}