PHYSICAL REVIEW E 77, 041915 共2008兲 Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train Aaditya V. Rangan,1 Gregor Kovačič,2 and David Cai1 1 Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012-1185, USA 2 Mathematical Sciences Department, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180, USA 共Received 23 August 2007; revised manuscript received 29 December 2007; published 18 April 2008兲 We present a kinetic theory for all-to-all coupled networks of identical, linear, integrate-and-fire, excitatory point neurons in which a fast and a slow excitatory conductance are driven by the same spike train in the presence of synaptic failure. The maximal-entropy principle guides us in deriving a set of three 共1 + 1兲-dimensional kinetic moment equations from a Boltzmann-like equation describing the evolution of the one-neuron probability density function. We explain the emergence of correlation terms in the kinetic moment and Boltzmann-like equations as a consequence of simultaneous activation of both the fast and slow excitatory conductances and furnish numerical evidence for their importance in correctly describing the coarse-grained dynamics of the underlying neuronal network. DOI: 10.1103/PhysRevE.77.041915 PACS number共s兲: 87.19.L⫺, 05.20.Dd, 84.35.⫹i I. INTRODUCTION Attempts to understand the enormous complexity of neuronal processing that takes place in the mammalian brain, supported by the ever-increasing computational power used in the modeling of the brain, have given rise to greatly increased sophistication in mathematical and computational modeling of realistic neuronal networks 关1–7兴. Striking manifestations of spatiotemporal neuronal dynamics, such as patterns of spontaneous activity in the primary visual cortex 共V1兲 关8,9兴 and motion illusions 关10兴, which take place on length scales of several millimeters and involve millions of neurons, can now be computed using large-scale, coupled, point-neuron models 关11,12兴. The ability to describe still more complicated neuronal interactions in yet larger portions of the brain, such as among multiple areas or layers of the visual cortex, may be significantly enhanced by appropriate coarse graining. Coarse-grained neuronal network models can describe network firing rates using the average membrane potential alone 关13–25兴, or they can also take into account its fluctuations 关26–29兴. The latter models are applicable to networks in which neuronal firing occurs solely due to membrane potential fluctuations while the average membrane potential stays below the firing threshold 关30–36兴, as well as to those that operate in the mean-driven regime in which the slaving potential that drives the neurons to fire is consistently above the firing threshold. A particularly fruitful use of coarsegrained models is in combination with point-neuron models, forming so-called embedded networks 关37兴. Such models are especially appropriate for describing neuronal dynamics in the brain because of the many regular feature maps in the laminar structure of the cerebral cortex. The primary visual cortex alone has a number of neuronal preference maps, such as for orientation or spatial frequency, laid out in regular or irregular patterns 关38–44兴. Certain subpopulations which contain sufficiently many neurons, but are small enough that their response properties can be treated as constant across the subpopulation, may be replaced by coarse-grained patches, while other subpopulations may be represented by point neu1539-3755/2008/77共4兲/041915共13兲 rons embedded in this coarse-grained background. In the simplest case of all-to-all coupled neuronal networks, the initial step is to use a nonequilibrium statistical physics framework and develop a Boltzmann-type kinetic equation for the network probability density function of the voltage and various conductances 关26–29,45–57兴. Since this is a partial differential equation in three or more dimensions, it is advantageous to further project the dynamics to one dimension and obtain a reduced kinetic theory in terms of the voltage alone. For purely excitatory networks, as well as networks with both excitatory and inhibitory neurons and both simple and complex cells, such a theory was developed in 关28,29兴. It achieves dimension reduction by means of a novel moment closure, which was shown to follow from the maximum-entropy principle 关58兴. Efficient numerical methods for solving the resulting set of kinetic equations were developed in 关59兴. The predominant excitatory neurotransmitter in the central nervous systems of vertebrates is glutamate, which binds to a number of different types of post-synaptic receptors 关60兴. Two main classes of glutamate-binding receptors are the AMPA and NMDA receptors. Activation of the AMPA receptors gives rise to fast post-synaptic conductance dynamics with decay times of about 3–8 ms, while the activation of the NMDA receptors gives rise to slow conductance dynamics with decay times of about 60–200 ms, respectively 关61–64兴. AMPA and NMDA receptors are frequently colocalized, such as in the rat visual cortex 关65兴 and hippocampus 关63,66兴 or in the superficial layers of cat V1 关67兴, and may thus respond to the same spikes. This colocalization of receptors with different time constants motivates the present study. We will examine its theoretical description in terms of a kinetic theory and investigate its dynamical consequences. From the viewpoint of kinetic theory, an important aspect of neuronal networks with conductances activated by fastand slow-acting receptors, colocalized on the same synapses, is the nature of the statistical correlations arising due to the two types of conductances being driven by 共parts of兲 the same network spike train. Therefore, in this paper, we develop a kinetic theory for purely excitatory neuronal net- 041915-1 ©2008 The American Physical Society PHYSICAL REVIEW E 77, 041915 共2008兲 RANGAN, KOVAČIČ, AND CAI works of this type. For simplicity, we derive this kinetic theory for a linear, conductance-driven, all-to-all-coupled, integrate-and-fire 共IF兲 network of identical excitatory point neurons that incorporates synaptic failure 关68–73兴. Once we find an appropriate set of conductance variables, which are driven by disjoint spike trains, we can derive the kinetic equation as in 关29兴 and use the maximum-entropy principle of 关58兴 to suggest the appropriate closure that will achieve a reduction of independent variables to time and voltage alone. However, this closure is more general than that of 关58兴 insofar as it involves using the dynamics of the conductance 共co兲variances computed from a set of ordinary differential equations. Adding inhibitory neurons and space dependence of the neuronal connections to the kinetic description developed in this paper is a straightforward application of the results in 关28,29,58兴. The paper is organized as follows. In Sec. II, we describe the equations for a linear IF excitatory neuronal network with a fast and slow conductance. In Sec. III, we introduce a set of conductance-variable changes such that the resulting conductances are driven by disjoint spike trains. In Sec. IV, we present the Boltzmann-like kinetic equation that describes statistically the dynamics of the membrane potential and these new conductance variables, develop a diffusion approximation for it, and recast it in terms of the voltage and the fast and slow conductances alone. In Sec. V, we impose the boundary conditions for the kinetic equation obtained in Sec. IV in terms of the voltage and conductance variables and describe how the resulting problem becomes nonlinear due to the simultaneous presence of the firing rate as a selfconsistency parameter in the equation and the boundary conditions. In Sec. VI, we find an equivalent statistical description of the network dynamics in terms of an infinite hierarchy of equations for the conditional conductance moments, in which the independent variables are the time and membrane potential alone. In Sec. VII we describe the maximum-entropy principle which we use to guide us in discovering an appropriate closure for the infinite hierarchy of equations from Sec. VI. In Sec. VIII, we discuss the dynamics of the conductance moments and 共co兲variances used in the closure and also their role in determining the validity of the conductance boundary conditions imposed in Sec. V. In Sec. IX we postulate the closure and derive three closed kinetic equations for the voltage probability density and the first conditional conductance moments as functions of time and membrane potential. We also derive the boundary conditions for these equations. In Sec. X we consider the limit in which the decay rate of the fast conductance becomes infinitely fast and the additional limit in which the decay rate of the slow conductance becomes infinitely slow. In Sec. XI, we present conclusions and discuss the agreement of our kinetic theory with direct, full numerical simulations of the corresponding IF model. II. INTEGRATE-AND-FIRE NEURONAL NETWORK WITH FAST AND SLOW EXCITATORY CONDUCTANCES The kinetic theory we develop in this paper describes statistically the dynamics exhibited by linear, IF networks com- posed of excitatory point neurons with two types of postsynaptic conductances: one mediated by a fast and another by a slow receptor channel. In a network of N identical, excitatory point neurons, the membrane potential of the ith neuron is governed by the equation 冉 冊 冉 冊 Vi − r Vi − E d − Gi共t兲 , Vi = − dt 共1兲 where is the leakage time constant, r is the reset potential, E is the excitatory reversal potential, and Gi共t兲 is the neuron’s total conductance. The voltage Vi evolves according to Eq. 共1兲 as long as it stays below the firing threshold VT. When Vi reaches VT, the neuron fires a spike and Vi is reset to the value r. In the absence of a refractory period, the dynamics of Vi then immediately becomes governed by Eq. 共1兲 again. The total conductance Gi共t兲 of the ith neuron in the network 共1兲 is expressed as the sum Gi共t兲 = GAi 共t兲 + 共1 − 兲GNi 共t兲 共2兲 = Gif 共t兲 + G共1兲 i 共t兲 + 共1 − 兲Gi 共t兲, 共2a兲 in which GAi 共t兲 = Gif 共t兲 + G共1兲 i 共t兲 and GNi 共t兲 = G共2兲 i 共t兲 共2b兲 are the fast and slow conductances, respectively. The con共2兲 ductances G共1兲 i 共t兲 and Gi 共t兲 arise from the spikes mediated by the synaptic coupling among pairs of neurons within the network, while Gif 共t兲 is the conductance driven by the external stimuli. The parameter denotes the percentage of fast receptor contribution to the total conductance Gi共t兲 关74–76兴. If the coupling for the network 共1兲 is assumed to be allto-all, the conductances in 共2b兲 can be modeled by the equations d 1 Gif = − Gif + f 兺 ␦共t − ti兲, dt 共3a兲 d S̄ 1 G共1兲 = − G共1兲 兺 兺 p共1兲␦共t − tkl兲, i + dt i Np k⫽i l kl 共3b兲 d S̄ 共2兲 2 G共2兲 兺 兺 p共2兲␦共t − tkl兲, i = − Gi + dt Np k⫽i l kl 共3c兲 where we have assumed that the rising time scales of the conductances are infinitely fast. In Eqs. 共3a兲–共3c兲, ␦共¯兲 is the Dirac delta function, 1 and 2 are the time constants of the fast and slow synapses, f is the synaptic strength of the external inputs, and S̄ is the network synaptic strength. The ␦ functions on the right-hand sides of 共3a兲–共3c兲 describe the spike trains arriving at the neuron in question. In particular, the time ti is that of the th external input spike delivered to the ith neuron and tkl is the lth spike time of the kth neuron in network. Note that the network neuron spikes 兵tkl其 are 共2兲 common to G共1兲 i and Gi due to the all-to-all nature of the network couplings. 共2兲 The coefficients p共1兲 kl and pkl in Eqs. 共3b兲 and 共3c兲 model the stochastic nature of the synaptic release 关68–73兴. Each 041915-2 KINETIC THEORY FOR NEURONAL NETWORKS WITH … p共i兲 kl , i = 1 , 2, is taken to be an independent Bernoullidistributed stochastic variable upon receiving a spike at the time tkl; i.e., p共i兲 kl = 1 with probability p and 0 with probability 1 − p, where p is the synaptic release probability. The stochastic nature of the synaptic release will play an important role in the next two sections in helping us determine the appropriate conductance variables, which are driven by disjoint spike trains and can be used in deriving the correct Boltzmann-like kinetic equation. Moreover, synaptic noise often appears to be the dominant noise source in neuronal network dynamics 关77–79兴. Other types of neuronal noise, such as thermal 关78兴 and channel noise 关79,80兴, can also be incorporated in the kinetic theory developed here or the kinetic theory with only fast excitatory conductances, developed in 关28,29,58兴. In the remainder of the paper, we assume that the train of the external input spiking times ti is a realization of a Poisson process with rate 0共t兲. While it need not be true that the spike times of an individual neuron, tki with a fixed i, are Poisson distributed, all the network spiking times tkl can be considered as Poisson distributed with the rate Nm共t兲 when the number of neurons, N, in the network is sufficiently large, the firing rate per neuron is small, and each neuronal firing event is independent of all the others 关81兴. This Poisson approximation for the spike trains arriving at each neuron in the network is used in deriving the kinetic equation. The division of the spiking terms in Eqs. 共3b兲 and 共3c兲 by the number of neurons, N, provides for a well-defined network coupling in the large-network limit N → ⬁. PHYSICAL REVIEW E 77, 041915 共2008兲 d S̄ 共2兲 2 Y 共2兲 兺 兺 p共2兲共1 − p共1兲 i = − Yi + kl 兲␦共t − tkl兲. dt Np k⫽i l kl 共4d兲 We have chosen these variables so that the dynamics of X共1兲 i and Y 共1兲 i is driven by the synaptic release on both the fast and is driven by the release slow receptors simultaneously, X共2兲 i on the fast receptors without release on the slow receptors, and Y 共2兲 i is driven by the release on the slow receptors without release on the fast receptors. By adding the pairs of Eqs. 共4a兲, 共4b兲 and 共4c兲, 共4d兲, re共1兲 共2兲 共1兲 共2兲 and G共2兲 spectively, we find that G共1兲 i = Xi + Xi i = Yi + Yi satisfy Eqs. 共3b兲 and 共3c兲. From Eqs. 共4a兲 and 共4c兲, we like共1兲 is wise compute that the conductance Ai = 1X共1兲 i + 2Y i driven by the synaptic release on both the fast and slow 共1兲 receives receptors simultaneously, while Bi = 1X共1兲 i − 2Y i no synaptic driving at all. From Eqs. 共1兲, 共3a兲, 共4b兲, and 共4d兲, we thus finally collect a set of equations in which all the conductance variables have been chosen so that they are driven by disjoint spike trains. This set is 冉 冉 冊 共5a兲 d 1 Gif = − Gif + f 兺 ␦共t − ti兲, dt 共5b兲 1 1 2S̄ d Ai = − Ai − Bi + 兺 兺 p共1兲p共2兲␦共t − tkl兲, dt + − Np k⫽i l kl kl III. CONDUCTANCES DRIVEN BY DISJOINT SPIKE TRAINS 共5c兲 The fact that two different sets of conductances with distinct time constants are driven by spikes belonging to the same spike train prevents us from being able to derive a kinetic representation of the neuronal network in the present case by using a straightforward generalization of the results in 关28,29,58兴. Therefore we must take advantage of the stochastic nature of the synaptic release and first introduce a set of conductance variables which are all driven by disjoint spike trains. We find this set in two steps. As a first step, we introduce four auxiliary conductance 共2兲 共1兲 共2兲 variables X共1兲 i , Xi , Y i , and Y i , which obey the dynamics d S̄ 共1兲 1 X共1兲 兺 兺 p共1兲p共2兲␦共t − tkl兲, i = − Xi + dt Np k⫽i l kl kl 冊 Vi − r Vi − E d − Gi共t兲 , Vi = − dt 1 1 d Bi = − Bi − Ai , dt + − 共5d兲 d S̄ 共2兲 1 X共2兲 兺 兺 p共1兲共1 − p共2兲 i = − Xi + kl 兲␦共t − tkl兲, dt Np k⫽i l kl 共5e兲 d S̄ 共2兲 2 Y 共2兲 兺 兺 p共2兲共1 − p共1兲 i = − Yi + kl 兲␦共t − tkl兲, dt Np k⫽i l kl 共4a兲 共5f兲 with the total conductance Gi given by d S̄ 1 X共2兲 = − X共2兲 兺 兺 p共1兲共1 − p共2兲 i + kl 兲␦共t − tkl兲, dt i Np k⫽i l kl 共2兲 Gi共t兲 = Gif 共t兲 + +Ai + −Bi + X共2兲 i + 共1 − 兲Y i . 共5g兲 共4b兲 In Eqs. 共5a兲–共5g兲, the constants ⫾ and ⫾ are + = d S̄ 2 Y 共1兲 = − Y 共1兲 兺 兺 p共1兲p共2兲␦共t − tkl兲, i + dt i Np k⫽i l kl kl 共4c兲 and 041915-3 2 1 2 , 2 + 1 − = 2 1 2 2 − 1 共6兲 PHYSICAL REVIEW E 77, 041915 共2008兲 RANGAN, KOVAČIČ, AND CAI + = 冉 冊 1 1− + , 2 1 2 − = 冉 冊 1 1− − , 2 1 2 ⬅ 共v,g f , , ,x2,y 2,t兲 共7兲 =E respectively. In other words, the conductances Gif , Ai, X共2兲 i , in Eqs. 共5a兲–共5g兲 jump in a mutually exclusive fashand Y 共2兲 i ion in that no two of them can jump as the result of the same spike. In what is to follow, we still treat the spike trains that 共2兲 drive the conductance variables Gif , Ai, X共2兲 i , and Y i in Eqs. 共5a兲–共5g兲 as Poisson for the same reason as explained at the end of Sec. II. v − r + 关g f + + + − + x2 + 共1 − 兲y 2兴 册 冋冉 冊 册 冊册 冋 册 − 共v,g f , , ,x2,y 2兲 + + 冋冉 i ␦„v − Vi共t兲…␦„g f − Gif 共t兲…␦„ − Ai共t兲… 册 共8兲 for the voltage and the conductance variables in Eqs. 共5a兲–共5g兲. Here E represents the expectation over all possible realizations of the Poisson spike trains and initial conditions. An argument nearly identical to that presented in Appendix B of 关29兴 shows that the evolution of the probability density 共v , g f , , , x2 , y 2兲 for the voltages and conductances evolving according to Eqs. 共5a兲–共5g兲 is described by the kinetic equation To arrive at a statistical description of the neuronal network dynamics, we begin by considering the probability density 再冋冉 冊 1 N 共2兲 ⫻␦„ − Bi共t兲…␦„x2 − X共2兲 i 共t兲…␦„y 2 − Y i 共t兲… IV. KINETIC EQUATION = t v 冋兺 冉 冊册 冎 冋 册 冋 冉 冊 冋冉 冊 冋冉 v − E + gf f + 0共t兲 v,g f − , , ,x2,y 2 g f 1 1 1 2S̄ 1 , ,x2,y 2 − 共v,g f , , ,x2,y 2兲 + + m共t兲p2N v,g f , − + − Np 冊 册 册 冋 册 1 1 x2 y2 S̄ + + + m共t兲p共1 − p兲N v,g f , , ,x2 − ,y 2 − 共v,g f , , ,x2,y 2兲 + + − x2 1 Np1 y 2 2 冋冉 + m共t兲p共1 − p兲N v,g f , , ,x2,y 2 − 冊 册 S̄ − 共v,g f , , ,x2,y 2兲 . Np2 Here, 0共t兲 is the Poisson spiking rate of the external input and m共t兲 is the network firing rate normalized by the number of neurons, N. This normalization is used to prevent m共t兲 from increasing without bounds in the large-N limit. Equivalently, the firing rate m共t兲 is the population-averaged firing rate per neuron in the network. We here give a brief, intuitive description of the derivation process leading from the dynamical equations 共5a兲–共5g兲 to the kinetic equation 共9兲. To this end, we consider the dynamics of a neuron in the time interval 共t , t + ⌬t兲 and analyze the conditional expectations for the values of the voltage Vi共t + ⌬t兲 and conductances Gif 共t + ⌬t兲, Ai共t + ⌬t兲, Bi共t + ⌬t兲, 共2兲 X共2兲 i 共t + ⌬t兲, and Y i 共t + ⌬t兲 at time t + ⌬t, given their values 共2兲 f Vi共t兲, Gi 共t兲, Ai共t兲, Bi共t兲, X共2兲 i 共t兲, and Y i 共t兲 at time t. If the time increment ⌬t is chosen to be sufficiently small, at most one spike can arrive at this neuron with nonzero probability during the interval 共t , t + ⌬t兲. Five different, mutually exclusive events can take place during this time-interval. The first event is that either no spike arrives at the neuron or else a spike arrives, but activates none of the conductances because of synaptic failure. This event occurs with probability 关1 − 0共t兲⌬t兴关1 − p2m共t兲N⌬t兴关1 − p共1 − p兲m共t兲N⌬t兴2 + O共⌬t2兲 The neuron’s = 1 − 关0共t兲 + 共2p − p2兲m共t兲N兴⌬t + O共⌬t2兲. 共9兲 dynamics is then governed by the smooth 共streaming兲 terms in Eqs. 共5a兲–共5g兲. The other four events consist of an external input or network spike arriving and activating one of the 共2兲 conductances Gif , Ai, X共2兲 i , or Y i , with the respective 2 2 probabilities 0共t兲⌬t + O共⌬t 兲, p m共t兲N ⌬t + O共⌬t2兲, and p共1 − p兲m共t兲N ⌬t + O共⌬t2兲 for the last two possible events. Due to our variable choice, no two of these conductances can be activated simultaneously. The corresponding conductance jumps are f / 1, 2S̄ / Np, S̄ / 1Np, and S̄ / 2Np, respectively. This argument lets us compute the single-neuron conditional probability density function at time t + ⌬t. We then average over all possible values of Vi共t兲, Gif 共t兲, Ai共t兲, Bi共t兲, X共2兲 i 共t兲, and Y 共2兲 i 共t兲, as well as all neurons, initial conditions, and possible spike trains. This leads to the right-hand side of the kinetic equation 共9兲 being the coefficient multiplying ⌬t in the ⌬t expansion of the density 共v , g f , , , x2 , y 2 , t + ⌬t兲, which implies 共9兲. The details of the calculation are similar to those given in Appendix B of 关29兴. Assuming the conductance jump values f / 1, S̄ / Np, S̄ / Np1, and S̄ / Np2 to be small, we Taylor-expand the difference terms in 共9兲 and thus obtain the diffusion approximation to the kinetic equation 共9兲, given by 041915-4 KINETIC THEORY FOR NEURONAL NETWORKS WITH … = t v 再冋冉 冊 v − r 再冋 再冋 + + y2 PHYSICAL REVIEW E 77, 041915 共2008兲 冋冉 冊册 册冎 再冋 再冉 冊 冎 再冋 + 关g f + + + − + x2 + 共1 − 兲y 2兴 册冎 1 1 2 共 − ¯兲 + + 4p2¯2 2 + + − 册冎 v − E gf 册冎 2 2 1 共g f − ḡ兲 + 2f 2 1 1 g f 册冎 ¯2 2 1 共x2 − x̄2兲 + p共1 − p兲 2 2 1 1 x2 ¯2 2 1 共y 2 − ȳ 2兲 + p共1 − p兲 2 2 . 2 2 y 2 ḡ共t兲 = f 0共t兲, ¯共t兲 = 2+ pS̄m共t兲, 共10兲 2f 共t兲 JV共v,gA,gN,t兲 = − 1 = f 20共t兲, 2 ¯2共t兲 = S̄2 m共t兲, 2Np2 共11兲 with ⫾ and ⫾ as defined in 共6兲 and 共7兲, respectively. We now reverse the conductance variable changes performed in Sec. III, used to derive Eqs. 共5a兲–共5g兲, in order to derive a kinetic equation 共in the diffusion approximation兲 for the time evolution of the probability density function involving only the voltage v and the fast and slow conductances gA and gN, respectively: ⬅ 共v , gA , gN兲. This kinetic equation is 再冋冉 冊 v − r + 关gA + 共1 − 兲gN兴 冉 冊册 冎 v − E + 1 1 兵关gA − ḡA共t兲兴其 + 兵关gN − ḡN共t兲兴其 1 gA 2 gN + A2 2 2p20 2 20 2 + + , 21 gA2 12 gA gN 22 gN2 共12兲 with ḡA共t兲 = f 0共t兲 + S̄m共t兲, 1 A2 共t兲 = f 20共t兲 + 20共t兲, 2 ḡN共t兲 = S̄m共t兲, 20共t兲 = 冋冉 冊 冉 冊册 ⫻ x̄2共t兲 = ȳ 2共t兲 = 共1 − p兲S̄m共t兲, + 1 1 + + + − x2 In this equation the 共time-dependent兲 coefficients are = t v S̄2 m共t兲. 2Np 共13兲 Note that the cross-derivative term in the last line of the kinetic equation 共12兲 is due to the correlation between the jumps in fast and slow conductances GA and GN when both synaptic releases occur with probability p2. Had we treated the spike trains in Eqs. 共3b兲 and 共3c兲 as if they were distinct, this cross term would be missing. V. BOUNDARY CONDITIONS IN TERMS OF VOLTAGE AND CONDUCTANCES The first term in Eq. 共12兲 is the derivative of the probability flux in the voltage direction: v − r v − E + 关gA + 共1 − 兲gN兴 共v,gA,gN,t兲. 共14兲 Since we have assumed that the neurons in the network 共1兲 and 共3a兲–共3c兲 have no refractory period, a neuron’s voltage is reset to r as soon as it crosses the threshold value VT, before any change to the neuron’s conductance can occur. This implies the voltage-flux boundary condition JV共VT,gA,gN,t兲 = JV共r,gA,gN,t兲. 共15兲 With the conductance boundary conditions, we must proceed a bit more cautiously. Specifically, in the IF system 共1兲 and 共3a兲–共3c兲, the spike-induced jumps in the conductances are positive. In conjunction with the form of Eqs. 共3a兲–共3c兲, this fact implies that a neuron’s conductances GAi and GNi must always be positive and that there cannot be any conductance flux across the boundary of the first quadrant in the 共GAi , GNi 兲 conductance plane in either direction. One would therefore expect the probability density 共v , gA , gn , t兲 to be nonzero only when gA ⬎ 0 and gN ⬎ 0 as a result of the natural conductance dynamics, and not as a result of any boundary conditions. The approximation leading from the difference equation 共9兲 to the diffusion equation 共10兲, however, replaces the conductance jumps by a transport and a diffusion term, and one should expect that this combination of terms may imply some, at least local, downward conductance flux even across the half-rays gA = 0, gN ⬎ 0 and gN = 0, gA ⬎ 0. Therefore, a no-flux boundary condition across these two half-rays would have to be enforced, essentially artificially, as part of the diffusion approximation. In addition, due to the crossderivative term in Eq. 共12兲, the individual conductance fluxes cannot be defined uniquely in the 共gA , gN兲 conductance variables, but only in the transformed conductance variables in which the second-order-derivative terms in 共12兲 become diagonalized. Employing such a transformation, taking the dot product of the flux vector with the appropriate normal, and then transforming that expression back in the 共gA , gN兲 conductance variables would make the already somewhat artificial no-flux boundary conditions also extremely unwieldy to compute. Alternatively, in Eq. 共12兲, we allow for the probability density 共v , gA , gN , t兲 to be a non-negative function everywhere and take instead the smallness of 共v , gA , gN , t兲 for 041915-5 PHYSICAL REVIEW E 77, 041915 共2008兲 RANGAN, KOVAČIČ, AND CAI gA ⱕ 0 or gN ⱕ 0 as part of the diffusion approximation; its loss may limit the temporal validity range of this approximation. The boundary conditions that we assume for the conductances in the solution of Eq. 共12兲 are thus simply that no neuron have infinitely large or small conductances—that is, 共v,gA → ⫾ ⬁,gN,t兲 → 0, 共v,gA,gN → ⫾ ⬁,t兲 → 0, 冕冕 ⬁ JV共VT,gA,gN,t兲dgAdgN . A共v兲 ⬅ 1,0共v兲, A共2兲共v兲 ⬅ 2,0共v兲, 共17兲 t ⬁ 共v,gA,gN,t兲dgAdgN r,s共v兲 = 冕冕 −⬁ v 冕冕 冕冕 f dgAdgN + −⬁ t ⬁ 共2兲 AN 共v兲 ⬅ 1,1共v兲. −⬁ ⬁ − −⬁ ⬁ fJV dgAdgN −⬁ ⬁ f JV dgAdgN −⬁ v 共g − ḡA兲 f dgAdgN 1 gA − 共g − ḡN兲 f dgAdgN 2 gN ⬁ A2 2 f 2 2 dgAdgN −⬁ 1 gA + ⬁ + −⬁ 2p20 2 f dgAdgN 1 2 g A g N ⬁ 20 2 f 2 2 dgAdgN . −⬁ 2 gN + 共22兲 Letting f = gArgNs in 共22兲, with r , s = 0 , 1 , 2 , . . ., and using 共14兲, we now derive a set of equations for the moments 冕冕 共18兲 ⬁ gArgNs共v,gA,gN兲dgAdgN = 共v兲r,s共v兲. 共23兲 −⬁ and the conditional moments gArgNs共兩gA,gN兩v兲dgAdgN , f dgAdgN + ⬁ −⬁ ⬁ 冕冕 冕冕 冕冕 冕冕 冕冕 冕冕 冕冕 ⬁ = VI. HIERARCHY OF MOMENTS 冕冕 N共2兲共v兲 ⬅ 0,2共v兲, 共21a兲 In order to obtain the equations that describe the dynamics of the conditional moments 共19兲, we begin by, for any given function f共v , gA , gN , t兲, using Eq. 共12兲 to derive an equation for the evolution of this function’s projection onto the v space alone. We multiply 共12兲 by f共v , gA , gN , t兲 and integrate over the conductance variables. Integrating by parts and taking into account the boundary conditions 共16兲 stating that must vanish together with all its derivatives at gA → ⫾ ⬁ and gN → ⫾ ⬁, faster than any power of gA or gN, we thus arrive at the equation Note that this firing rate feeds into the definitions 共13兲 for the parameters in the kinetic equation 共12兲. Equation 共12兲, boundary conditions 共15兲 and 共16兲, and the expression for the firing rate, Eq. 共17兲, which reenters Eq. 共12兲 as a self-consistency parameter via Eqs. 共13兲, provide a complete statistical description 共in the diffusion approximation兲 of the dynamics taking place in the neuronal network 共1兲 and 共3a兲–共3c兲. The simultaneous occurrence of the firing rate m共t兲 in both Eq. 共12兲 and the integral 共17兲 of the voltage boundary condition makes this description highly nonlinear. Solving a nonlinear problem involving a partial differential equation in four dimensions is a formidable task, and it is therefore important to reduce the dimensionality of the problem to 2 by eliminating the explicit dependence on the conductances from the description. We describe this reduction process in the subsequent sections. 共v,t兲 = N共v兲 ⬅ 0,1共v兲, 共21b兲 −⬁ In this section, we derive a description of the dynamics in terms of functions of voltage alone. In other words, we seek to describe the evolution of the voltage probability density 共20兲 We derive a hierarchy of equations for the conditional moments 共19兲, for which we will subsequently devise an approximate closure via the maximal entropy principle. For clarity of exposition, we rename 共16兲 sufficiently rapidly, together with all its derivatives. As a parenthetical remark, we should point out that, technically, the adoption of the approximate boundary conditions 共16兲 is also advantageous in eliminating unwanted boundary terms from the conductance-moment equations to be derived in Secs. VI and VIII, and thus significantly simplifying the resulting reduced kinetic theory. In Sec. VIII we will also present a conductance-moment-based validity criterion for the approximation made in adopting the conditions 共16兲. The population-averaged firing rate per neuron, m共t兲, is the total voltage flux 共14兲 across the firing threshold VT that includes all values of the conductances and is thus given by the integral m共t兲 = 共v,gA,gN,t兲 = 共v,t兲共兩gA,gN,t兩v兲. These equations read 共19兲 −⬁ where the conditional density 共兩gA , gN兩v兲 is defined by the equation − t v 041915-6 再冋冉 冊 冉 冊 v − r + v − E 册冎 关A + 共1 − 兲N兴 = 0, 共24a兲 KINETIC THEORY FOR NEURONAL NETWORKS WITH … 共A兲 − t v 再冋冉 冊 冉 冊 册冎 再冋冉 冊 冉 冊 册冎 v − r A + v − r N + + 共1 − 兲N共2兲兴 = − and, for r + s ⱖ 2, 共r,s兲 − t v + v − E 共24b兲 r,s + v − E 共24c兲 关r+1,s r 共r,s − ḡAr−1,s兲 1 2 s 共r,s − ḡNr,s−1兲 + A2 r共r − 1兲r−2,s 2 1 2p20 1 2 rsr−1,s−1 + 20 s共s 22 册 − 1兲r,s−2 , 冉 冊 册 + 共1 − 兲r,s+1共VT兲兴 共VT兲 − 冉 冊 0共gA,gN兲 = where M= ␣= e 冉 冊 冉 冊 冉 冊 ␣  ,  ␥ g= 1共 1 + 2兲 , 2D 2 gA , gN = g= ḡA共t兲 ḡN共t兲 , p 1 2共 1 + 2兲 , D 共v,gA,gN,t兲 共v,gA,gN,t兲dgAdgN , 0共gA,gN兲 subject to three constraints: 共18兲, and also 共23兲 with r = 1 and s = 0, and 共23兲 with r = 0 and s = 1, which are 冕冕 ⬁ gA共v,gA,gN兲dgAdgN = 共v兲A共v兲, −⬁ 冕冕 ⬁ gN共v,gA,gN兲dgAdgN = 共v兲N共v兲. 共30兲 −⬁ ˆ 共v,gA,gN兲 = 0共gA,gN兲exp关a0共v兲 + aA共v兲gA + aN共v兲gN兴, 共31兲 where the multipliers a0共v兲, aA共v兲, and aN共v兲 are to be determined. Solving the constraints 共18兲 and 共30兲, we find 共26兲 , ln 共29兲 When the Poisson rate 0 of the external input is independent of time, Eq. 共12兲 possesses an invariant Gaussian solution 0共gA , gN兲 given by the formula −共g−ḡ兲·M共g−ḡ兲 ⬁ −⬁ 共25兲 VII. MAXIMUM-ENTROPY PRINCIPLE 冑det M 冕冕 Applying Lagrange multipliers to maximizing the entropy 共29兲 under the constraints 共18兲 and 共30兲, we find for the maximizing density function ˆ 共v , gA , gN兲 to have the form r − E 关r+1,s共r兲 + 共1 − 兲r,s+1共r兲兴共r兲 = 0. 共28c兲 From 共13兲, we see that A2 ⬎ 20, and since 0 ⬍ p ⱕ 1, it is clear that D ⬎ 0. We also calculate that det M = ␣␥ − 2 = 12共1 + 2兲2 / 4D20 ⬎ 0, where the inequality holds because all the factors are positive. The inequalities det M ⬎ 0 and ␣ ⬎ 0, which follow from 共28a兲–共28c兲, imply that the matrix M is positive definite. The solution 共26兲 is normalized to unity over the 共gA , gN兲 space; its normalization over the 共v , gA , gN兲 space would be VT − r. Using the equilibrium solution 0共gA , gN兲 in 共26兲, we now formulate the maximal entropy principle, which will guide us in determining the correct closure conditions for the hierarchy 共24a兲–共24d兲. According to this principle, we seek the density function ˆ 共v , gA , gN , t兲 which maximizes the entropy 共24d兲 VT − r VT − E r,s共VT兲 + 关r+1,s共VT兲 共28b兲 and S关, 0兴共v,t兲 = − where two zero subscripts in any of the moments should be interpreted as making this moment unity and a negative subscript as making it zero. Equations 共24a兲–共24d兲 indeed form an infinite hierarchy, which needs to be closed at a finite order if we are to simplify the problem by eliminating the dependence on the conductances. Noting the form of the voltage-derivative terms in 共22兲 and using the voltage boundary condition 共15兲 and the flux definition 共14兲, we find the boundary conditions 冋冉 冊 A2 2共1 + 2兲2 2D 20 D = A2 共1 + 2兲2 − 4p22012 . 共2兲 关AN 再冋冉 冊 冉 冊 册冎 冋 v − r ␥= 1 共N − ḡN兲, 2 + 共1 − 兲r,s+1兴 = − − 关A共2兲 1 共A − ḡA兲, 1 共2兲 + 共1 − 兲AN 兴 =− 共N兲 − t v v − E PHYSICAL REVIEW E 77, 041915 共2008兲 共27兲 ˆ 共v,gA,gN,t兲 = 冑det M 共32兲 where = 共A共v兲 , N共v兲兲T is the vector of the first conditional moments, defined by 共21a兲, 共21b兲, and 共19兲. To find the relations between the first and second conditional moments of the probability density function ˆ 共v , gA , gN , t兲 in 共32兲, which maximizes the entropy 共29兲 under the constraints 共18兲 and 共30兲, we first put x = gA − A共v兲, 共28a兲 共v兲e−共g−兲·M共g−兲 , y = gN − N共v兲. 共33兲 We then use the definition of the matrix M in 共27兲 to rewrite the normalization of the function 共26兲 in the form 041915-7 PHYSICAL REVIEW E 77, 041915 共2008兲 RANGAN, KOVAČIČ, AND CAI 冕冕 ⬁ e−共␣x 2+2xy+␥ y 2兲 dx dy = −⬁ 冑␣␥ −  2 冉 which we differentiate upon ␣, , and ␥, respectively, and from 共32兲 obtain the expressions for the second moments of the conductances x and y in 共33兲 with respect to the maximizing density ˆ 共v , gA , gN , t兲. Using 共23兲 and 共28a兲–共28c兲, for the 共co兲variances ⌺A2 共v兲 ⬅ A共2兲共v兲 − 关A共v兲兴2 , 共2兲 共v兲 − A共v兲N共v兲, CAN共v兲 ⬅ AN ⌺N2 共v兲 ⬅ N共2兲共v兲 − 关N共v兲兴2 , 共34兲 + CAN共v兲 = − 2g共t兲 = 具gA2 典 − 具gA典2 , N2 共t兲 = 具gN2 典 − 具gN典2 , 2p20  = , 共␣␥ − 2兲 1 + 2 22 2 d 2 g = − 2g + 2A , dt 1 1 共35兲 冉 r VT = r gArgNs共v,gA,gN兲dv dgAdgN −⬁ 共v兲r,s共v兲dv , 共36兲 with 0 ⱕ r ⱕ 2, 0 ⱕ s ⱕ 2, and 1 ⱕ r + s ⱕ 2. These moments are the averages of the conditional moments A共v兲 through N共2兲共v兲 in 共21a兲 and 共21b兲 over the voltage distribution 共v兲. Integrating 共24a兲–共24d兲 over the voltage interval r ⬍ v ⬍ VT and using the boundary conditions 共25兲, we find for these moments the ordinary differential equations 1 d 具gA典 = − 关具gA典 − ḡA共t兲兴, dt 1 共37a兲 1 d 具gN典 = − 关具gN典 − ḡN共t兲兴, dt 2 共37b兲 22 2 d 2 具gA典 = − 关具gA2 典 − 具gA典ḡA共t兲兴 + 2A , dt 1 1 共37c兲 共39a兲 冊 2p20 1 1 d cAN = − + cAN + , dt 1 2 1 2 共39b兲 22 2 d 2 N = − N2 + 20 . dt 2 2 共39c兲 Note that when the parameters A2 and 20 do not depend on time, Eqs. 共39a兲–共39c兲 have the unique attracting equilibrium points 2g = More generally we seek the closure in terms of the moments 冕 冕冕 冕 共38兲 Eqs. 共37a兲–共37e兲 yield the equations VIII. DYNAMICS OF CONDUCTANCE MOMENTS ⬁ 共37e兲 cAN共t兲 = 具gAgN典 − 具gA典具gN典, Equations 共34兲 and 共35兲 furnish the expressions for the second-order conditional moments in term of the first-order conditional moments and amount to closure conditions based on the maximum-entropy principle. With the aid of these closure conditions we can close the hierarchy 共24a兲–共24d兲 at first order when the external-input and network firing rates are constant in time. VT 共37d兲 The initial conditions for these equations can be computed from the initial probability density 共v , gA , gN , t = 0兲 or its marginal counterpart over the conductances alone. For the 共co兲variances A2 ␥ = , 2共␣␥ − 2兲 1 20 ␣ = . ⌺N2 共v兲 = 2共␣␥ − 2兲 2 具gArgNs典 = 2p20 1 具gA典ḡN共t兲 + , 2 1 2 22 2 d 2 具gN典 = − 关具gN2 典 − 具gN典ḡN共t兲兴 + 20 . dt 2 2 we thus obtain the relations ⌺A2 共v兲 = 冊 1 1 1 d 具gAgN典 = − + 具gAgN典 + 具gN典ḡA共t兲 dt 1 2 1 , A2 , 1 cAN = 2p20 , 1 + 2 N2 = 20 , 2 共40兲 whose values are identical to the respective right-hand sides of the maximal-entropy relations 共35兲. This observation casts the 共co兲variances 共38兲 even in the time-dependent case as suitable candidates for replacing the right-hand sides of 共35兲 in a general second-order closure scheme, which we will postulate in the next section. We need to stress, however, that even if we were to solve the moment equations 共37a兲–共37e兲 共and their higher-order counterparts兲 explicitly, we would not obtain an explicit solution to the moment problem for the density 共v , gA , gN兲. This is because Eqs. 共37a兲–共37e兲, and so also their solutions, depend on the as-yet-unknown firing rate m共t兲. How to find this rate will be explained in the next section. Here, we discuss another important aspect of the conductance moment dynamics—namely, their significance in the validity of the diffusion approximation leading to Eq. 共12兲 and the boundary conditions 共16兲. As pointed out in Sec. V, the probability density function 共v , gA , gN兲 must be negligibly small in the region outside the first quadrant gA ⬎ 0, gN ⬎ 0, in order for this approximation to hold. 关This smallness can, for example, be expressed in terms of the integral of 共v , gA , gN兲 over that region and the voltage being small.兴 It 041915-8 KINETIC THEORY FOR NEURONAL NETWORKS WITH … is clear that the necessary condition for this smallness is that the ratio of the variances, 2g共t兲 and N2 共t兲, versus the first moments, 具gA典 and 具gN典, as computed from Eqs. 共39a兲, 共39c兲, 共37a兲, and 共37b兲, respectively, must be small. If this is not the case, the diffusion approximation 共12兲, as well as the reduction to the voltage dependence alone discussed in the next section, may become invalid. PHYSICAL REVIEW E 77, 041915 共2008兲 ⫻ ⌺A2 共v兲 ⬅ 2g共t兲, JV共v,t兲 = − ⌺N2 共v兲 ⬅ N2 共t兲, CAN共v兲 ⬅ cAN共t兲, 共41兲 where the 共co兲variances ⌺A2 共v兲, CAN共v兲, and ⌺N2 共v兲 are defined as in 共34兲 and 2g共t兲, cAN共t兲, and N2 共t兲 are computed from Eqs. 共39a兲–共39c兲 with the appropriate initial conditions, as discussed after listing Eqs. 共37a兲–共37e兲. Since the 共co兲variances 2g共t兲, cAN共t兲, and N2 共t兲 no longer depend on the voltage v, Eqs. 共24a兲–共24c兲 for the voltage probability density function 共v兲 and the first conditional moments A共v兲 and N共v兲 under the closure assumption 共41兲 simplify to become 共v兲 = t v ⫻ 再冋冉 冊 v − r 冉 冊册 v − E 冎 再冉 冊 1 A共v兲 = − 共A共v兲 − ḡA兲 + t 1 共42a兲 v − r + 关A共v兲 + 共1 − 兲N共v兲兴 冉 冊冎 v − E 冋冉 冊 册 + 关cAN共t兲 + 共1 − v − E + 关A共v兲 + 共1 − 兲N共v兲兴 共v兲 = ⬁ JV共v,gA,gN,t兲dgAdgN , −⬁ 共43兲 where JV共v , gA , gN , t兲 is the probability flux 共14兲 in the voltage direction for the original kinetic equation 共12兲. The second equality in 共43兲 is obtained from the definitions 共14兲, 共21a兲, and 共21b兲 and Eq. 共23兲, just as in the derivation of Eq. 共25兲. From 共14兲 and the definition 共17兲 of the populationaveraged firing rate per neuron, m共t兲, we now immediately see that this firing rate can be expressed as the voltage flux through the firing threshold: m共t兲 = JV共VT,t兲. 共44兲 We now proceed to derive the boundary conditions for the reduced equations 共42a兲–共42c兲. First, from the definition of the voltage probability flux 共43兲 and the voltage boundary condition 共15兲, we immmediately derive the equation JV共VT,t兲 = JV共r,t兲, 共45兲 which is also the first equation of Eqs. 共25兲 and further gives the boundary condition 兵共VT − r兲 + 关A共VT兲 + 共1 − 兲N共VT兲兴共VT − E兲其共VT兲 共46a兲 Under the closure 共41兲, using the 共co兲variance definitions 共34兲, and the voltage flux probability definition 共43兲, the boundary condition 共45兲, and the firing rate expression 共44兲, we can transform Eqs. 共25兲 for r = 1, s = 0 and r = 0, s = 1, respectively, into the equations ⫻关共VT − E兲共VT兲 A共 v 兲 v − 共r − E兲共r兲兴, 共46b兲 ⫻关共VT − E兲共VT兲 共42b兲 − 共r − E兲共r兲兴. v − r + 关A共v兲 + 共1 − 兲N共v兲兴 v − r m共t兲关A共VT兲 − A共r兲兴 = 关2g共t兲 + 共1 − 兲cAN共t兲兴 再冉 冊 共42c兲 m共t兲关N共VT兲 − N共r兲兴 = 关cAN共t兲 + 共1 − 兲N2 共t兲兴 v − E 共v兲 , 1 N共v兲 = − 共N共v兲 − ḡN兲 + t 2 共v兲 . 冋冉 冊 冉 冊册 冕 冕 ⫻ + 关2g共t兲 + 共1 − 兲cAN共t兲兴 1 ⫻ 共v兲 v v − E = 关A共r兲 + 共1 − 兲N共r兲兴共r − E兲共r兲. + 关A共v兲 + 共1 − 兲N共v兲兴 共v兲 , 冋冉 冊 册 Equations 共42a兲–共42c兲 provide the desired reduced kinetic description of moments, and are the main result of this paper. Equation 共42a兲 is clearly in conservation form, with the corresponding voltage probability flux IX. CLOSURE AND REDUCED KINETIC MOMENT EQUATIONS In this section, we postulate a second-order closure for the hierarchy 共24a兲–共24d兲 of equations for the moments 共19兲 based on the discussion of the previous two sections. First, using Eq. 共24a兲, we rewrite Eqs. 共24b兲 and 共24c兲 in such a 共2兲 共v兲, and N共2兲共v兲 in way that the second moments A共2兲共v兲, AN them are expressed in terms of the first moments A共v兲 and N共v兲 and the 共co兲variances ⌺A2 共v兲, CAN共v兲, and ⌺A2 共v兲 via Eqs. 共34兲. In view of the discussion presented in the preceding two sections, for the closure in the general case, we consider it natural to postulate 1 共v兲 v 冉 冊冎 v − E N共 v 兲 v 共46c兲 Equations 共46a兲–共46c兲 provide the physiological boundary conditions for the kinetic moment equations 共42a兲–共42c兲. In passing, let us remark that, due to the definitions 共34兲, 共36兲, and 共38兲, the closure 共41兲 implies the moment relations 兲N2 共t兲兴 冕 VT r 041915-9 共v兲A2 共v兲dv = 冉冕 VT r 共v兲A共v兲dv 冊 2 , PHYSICAL REVIEW E 77, 041915 共2008兲 RANGAN, KOVAČIČ, AND CAI 冕 VT r 共v兲A共v兲N共v兲dv = 冕 r ⫻ 冕 VT r 共v兲N2 共v兲dv = 冉冕 VT VT r 冕 1 N共v兲 = − 关N共v兲 − ḡN共t兲兴 + t 2 共v兲A共v兲dv VT r 冊 + 关cAN共t兲 + 共1 − 2 ⫻ . Recapping the results of this section, the reduced statistical description for the voltage dynamics alone is furnished by Eqs. 共42a兲–共42c兲, the boundary conditions 共46a兲–共46c兲, the ordinary differential equations 共39a兲–共39c兲, the firing rate definition 共44兲, the parameter definitions 共13兲, and the initial conductance distribution or its linear and quadratic moments. This problem is highly nonlinear because of the nonlinear moment equations 共42b兲 and 共42c兲 and boundary condition 共46a兲 and because the firing rate m共t兲 enters the governing equations as well as the boundary conditions as a selfconsistency parameter. A2 共t兲 , 1 2p20共t兲 S̄2 = m共t兲. 2 N2 2 共t兲 A共v兲 = ḡA共t兲 + A 共v兲 v 冋冉 冊 册 v − E 共v兲 . 共48兲 In other words, the conditional moment A共v兲 relaxes on the O共1兲 time scale to being slaved to the dynamics of 共v兲 and N共v兲. In the limit of the instantaneous fast conductance time scale, as 1 → 0, we thus only have two equations governing the moments 共v兲 and N共v兲, 共v兲 = t v ⫻ 再冋冉 冊 v − r 冉 冊册 v − E + 关A共v兲 + 共1 − 兲N共v兲兴 冎 共v兲 , 共49a兲 N共 v 兲 v v − E 共v兲 , 共49b兲 ⌺N2 共v兲 = N2 共t兲, 2p20共t兲 . 2 共50兲 Due to relation 共48兲, Eq. 共49b兲 is now second-order in the voltage v. Recall that the terms in 共49a兲 and 共49b兲 multiplied by cAN共t兲 originate from the correlation effects due to the spikes that are common to both the fast and slow receptors. In a steady state, using 共40兲 and taking the limit 1 → 0, we have cAN → 2pN2 —i.e., a rather strong correlation effect if the synaptic failure is low. Because of 共47兲, as 1 → 0, the boundary condition 共46b兲 simplifies to become 共VT − E兲共VT兲 = 共r − E兲共r兲, 共47兲 The limits 共47兲 imply that 12g共t兲 → A2 共t兲 and 1cAN共t兲 → 0, and thus Eq. 共42b兲 becomes v − E 冋冉 冊 册 CAN共v兲 = In the limit of the instantaneous fast conductance time scale—i.e., when 1 → 0—regardless of whether the forcing terms A2 共t兲 and 20共t兲 depend on time or not, the solutions 2g共t兲 and cAN共t兲 of Eqs. 共39a兲 and 共39b兲 relax on the O共1兲 time scale to their respective forcing terms, so that 冉 冊冎 兲N2 共t兲兴 1⌺A2 共v兲 = A2 共t兲, A. Instantaneous fast-conductance time scale cAN共t兲 → 1 共v兲 v while A共v兲 is expressed by the relation 共48兲. Here, the timedependent coefficients ḡA共t兲, ḡN共t兲, and A2 共t兲 can be found in 共13兲 and cAN共t兲 is computed in 共47兲. Note that all these quantities can now be computed from the external-input and network firing rates 0共t兲 and m共t兲 alone, without having to solve any additional differential equation. Only the coefficient N2 共t兲 is still computed from the ordinary differential equation 共39c兲, with 20共t兲 again given in terms of the firing rate m共t兲 in 共13兲. The closure 共41兲, in the limit 1 → 0, effectively becomes X. DISTINGUISHED LIMITS 2g共t兲 → v − r + 关A共v兲 + 共1 − 兲N共v兲兴 共v兲N共v兲dv , 共v兲N共v兲dv 再冉 冊 共51a兲 and therefore the condition 共46c兲 also simplifies to become N共VT兲 = N共r兲. 共51b兲 Using the boundary conditions 共51a兲 and 共51b兲 and relation 共48兲, we can finally simplify the boundary condition 共46a兲 to become 共VT − r兲共VT兲 + 2A2 共t兲共VT − E兲 = 2A2 共t兲共r − E兲 共r兲. v 共VT兲 v 共51c兲 Note that these boundary conditions are now linear in and N, but still contain the firing rate through the parameter A2 共t兲. To summarize, in the 1 → 0 limit, the simplified description of the problem is achieved by Eqs. 共49a兲 and 共49b兲, relation 共48兲, the boundary conditions 共51a兲–共51c兲, the single ordinary differential equation 共39c兲, the firing rate definition 041915-10 KINETIC THEORY FOR NEURONAL NETWORKS WITH … PHYSICAL REVIEW E 77, 041915 共2008兲 共44兲, the parameter definitions 共13兲, and the initial conductance distribution or its initial moments 具gN典共0兲 and 具gN2 典共0兲. If, in addition to 1 → 0, we now also consider the limit 2 → ⬁—i.e., if we consider the dynamics over the time scales with 1 t 2—we find cAN共t兲 = S̄2 m共t兲 → 0 N2 firing rate (spikes/s) 80 B. Instantaneous fast-conductance and infinite slowconductance time scales 共52兲 共53兲 where N2 共0兲 = 具gN2 典共0兲 − 具gN典2共0兲 and 具gN2 典共0兲 and 具gN典共0兲 are computed from their definitions in 共36兲 with the integrals taken against the initial probability density 共v , gA , gN , t = 0兲 or its conductance-dependent marginals. Using 共52兲 and 共53兲, we conclude that Eqs. 共49a兲 and 共49b兲 then reduce yet further to 共v兲 = t v ⫻ 再冋冉 冊 v − r + 关A共v兲 + 共1 − 兲N共v兲兴 冉 冊册 冎 再冉 冊 冉 冊冎 v − E 共v兲 , N共 v 兲 = t v − r ⫻ + N共 v 兲 v 共1 − 兲N2 共0兲 共v兲 v 冋冉 冊 册 v − E 共v兲 , 共54b兲 with A共v兲 again expressed by 共48兲. Because of 共52兲, the correlation term disappears from Eq. 共54b兲. The closure 共41兲, in the limit 1 → 0 and 2 → ⬁, effectively becomes 1⌺A2 共v兲 = A2 共t兲, 0 100 200 300 400 500 time (ms) FIG. 1. 共Color online兲 Accuracy of the kinetic theory for a network of neurons with fast and slow conductances driven by the same spike train: The time evolution of the population-averaged firing rate. Gray 共red online兲: simulation of the IF neuronal network described in the text, averaged over 1024 ensembles. Dark line 共blue online兲: numerical solution of the kinetic moment equations 共42a兲–共42c兲. Light line 共blue online兲: numerical solution of the kinetic moment equations 共42a兲–共42c兲, without the correlation terms—i.e., with cAN共t兲 = 0. A step stimulus is turned on at t = 0 with N = 20, = 20, r = 0, E = 4.67, VT = 1, = 0.5, 1 = 0.005, 2 = 0.01, f A = 0.05, A = 11, f N = 0.13, N = 0.13, p = 1, and S̄ = 0.857. 共54a兲 + 关A共v兲 + 共1 − 兲N共v兲兴 v − E 40 20 from 共47兲, as well as dN2 / dt = 0 from 共39c兲. It is therefore clear that N2 共t兲 = N2 共0兲, 60 ⌺N2 共v兲 = N2 共0兲, CAN共v兲 = 0. 共55兲 The complete description of the problem in this limit is achieved using the same ingredients as in the previous section, except that Eq. 共54b兲 replaces Eq. 共49b兲 and that the variance N2 共0兲 does not evolve, so there is no need to solve any of the ordinary differential equations 共39a兲–共39c兲. XI. DISCUSSION A comparison among a full simulation of IF neuronal network ensembles corresponding to different realizations of the Poisson inputs, and two corresponding numerical solutions of kinetic moment equations—one including and the other excluding the correlations cAN共t兲—is presented in Fig. 1. The results are depicted in gray 共red online兲, thick, and fine 共blue online兲 lines, respectively. The IF neuronal network simulated here is a straightforward generalization of Eqs. 共1兲 and 共3a兲–共3c兲, which includes external drive for the slow conductances GNi with the Poisson rate N and strength f N in addition to the drive for the fast conductances GAi with the Poisson rate A and strength f A. The kinetic moment equations are the corresponding generalizations of 共42a兲–共42c兲. It is apparent that the curve representing the results of the kinetic theory with the correlations included faithfully tracks the temporal decay of the firing rate oscillations computed using the IF network. On the other hand, the kinetic theory that does not take the correlations into account dramatically overestimates the amplitude of the firing rate and thus underestimates the decay rate of its oscillations. The results depicted in Fig. 1 convincingly illustrate how important it is to include explicitly in the kinetic theory the assumption that both the fast and slow conductances are driven by the same spike train. The correlation terms appearing as the consequence in the kinetic moment equations render an accurate statistical picture of the corresponding IF network dynamics. Finally, let us remark that developing a kinetic theory for more realistic neuronal networks, for instance, with neurons containing dendritic and somatic compartments and a voltage-dependent time course of the slow conductance mimicking the true NMDA conductance 关82–84兴, would proceed along the same lines as those employed in the current paper. 041915-11 PHYSICAL REVIEW E 77, 041915 共2008兲 RANGAN, KOVAČIČ, AND CAI ACKNOWLEDGMENTS We would like to thank S. Epstein, P. R. Kramer, and L. Tao for helpful discussions. 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