HYPOTHESIS AND THEORY ARTICLE published: 15 November 2013 COMPUTATIONAL doi: 10.3389/fncom.2013.00162 Generalized activity for dynamics

Michael A. Buice 1 and Carson C. Chow 2*

1 Modeling, Analysis and Theory Team, Allen Institute for Brain Science, Seattle, WA, USA 2 Laboratory of Biological Modeling, NIDDK, National Institutes of Health, Bethesda, MD, USA

Edited by: Much progress has been made in uncovering the computational capabilities of spiking Julie Wall, Queen Mary, University neural networks. However, spiking will always be more expensive to simulate of London, UK compared to rate neurons because of the inherent disparity in time scales—the spike Reviewed by: duration time is much shorter than the inter-spike time, which is much shorter than Brent Doiron, University of Pittsburgh, USA any learning time scale. In numerical analysis, this is a classic stiff problem. Spiking G. Bard Ermentrout, University of neurons are also much more difficult to study analytically. One possible approach to Pittsburgh, USA making spiking networks more tractable is to augment mean field activity models with *Correspondence: some information about spiking correlations. For example, such a generalized activity Carson C. Chow, Laboratory of model could carry information about spiking rates and correlations between spikes Biological Modeling, NIDDK, National Institutes of Health, self-consistently. Here, we will show how this can be accomplished by constructing a Building 12A Room 4007, 12 South complete formal probabilistic description of the network and then expanding around a Drive, Bethesda, MD 20814, USA small parameter such as the inverse of the number of neurons in the network. The e-mail: [email protected] mean field theory of the system gives a rate-like description. The first order terms in the perturbation expansion keep track of covariances.

Keywords: mean field theory, theta model, fokker-planck, correlations, finite size networks, wilson-cowan model, population rate, fluctuations

INTRODUCTION methods we have developed to solve two different types of prob- Even with the rapid increase in computing power due to Moore’s lems. The first problem was how to compute finite system size law and proposals to simulate the entire human brain notwith- effects in a network of coupled oscillators. We adapted the meth- standing Ailamaki et al. (2012), a realistic simulation of a func- ods of the kinetic theory of gases and plasmas Ichimaru (1973); tioning human brain performing non-trivial tasks is remote. Nicholson (1993) to solve this problem. The method exploits the While it is plausible that a network the size of the human brain exchange symmetry of the oscillators and characterizes the phases could be simulated in real time Izhikevich and Edelman (2008); of all the oscillators in terms of a phase density η(θ, t), Eliasmith et al. (2012) there are no systematic ways to explore the where each oscillator is represented as a point mass in this den- parameter space. Technology to experimentally determine all the sity. We then write down a formal flux conservation parameters in a single brain simultaneously does not exist and of this density, called the Klimontovich equation, which com- any attempt to infer parameters by fitting to data would require pletely characterizes the system. However, because the density is exponentially more computing power than a single simulation. not differentiable, the Klimontovich equation only exists in the We also have no idea how much detail is required. Is it suffi- weak or distributional sense. Previously, e.g., Desai and Zwanzig cient to simulate a large number of single compartment neurons (1978); Strogatz and Mirollo (1991); Abbott and van Vreeswijk or do we need multiple-compartments? How much molecular (1993); Treves (1993) the equations were made usable by taking detail is required? Do we even know all the important biochemical the “mean field limit” of N →∞and assuming that the den- and biophysical mechanisms? There are an exponential number sity is differentiable in that limit, resulting in what is called the of ways a simulation would not work and figuring out which Vlasov equation. Instead of immediately taking the mean field remains computationally intractable. Hence, an alternative means limit, we regularize the density by averaging over initial condi- to provide appropriate prior distributions for parameter values tions and parameters and then expand in the inverse system size and model detail is desirable. Current theoretical explorations of N−1 around the mean field limit. This results in a system of cou- the brain utilize either abstract mean field models or small num- pled moment equations known as the BBGKY moment hierarchy. bers of more biophysical spiking models. The regime of large but In Hildebrand et al. (2007), we solved the moment equations for finite numbers of spiking neurons remains largely unexplored. It the Kuramoto model perturbatively to compute the pair corre- is not fully known what role spike time correlations play in the lation function between oscillators. However, the procedure was brain. It would thus be very useful if mean field models could be somewhat ad-hoc and complicated. We then subsequently showed augmented with some spike correlation information. in Buice and Chow (2007) that the BBGKY moment hierarchy This paper outlines a scheme to derive generalized activ- could be recast in terms of a density functional of the phase den- ity equations for the mean and correlation dynamics of a fully sity. This density functional could be written down explicitly as deterministic system of coupled spiking neurons. It synthesizes an integral over all possible phase histories, i.e., a Feynman-Kac

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path integral. The advantage of using this density functional for- The equations are remarkably stiff with time scales spanning malism is that the moments to arbitrary order in 1/N could be orders of magnitude from milliseconds for ion channels, to sec- computed as a steepest-descent expansion of the path integral, onds for adaptation, and from hours to years for changes in which can be expressed in terms of Feynman diagrams. This made synaptic weights and connections. Parameter values must be the calculation more systematic and mechanical. We later applied assigned for 1011 neurons with 104 connections each. Here, we the same formalism to synaptically coupled spiking models Buice present a formalism to derive a set of reduced activity equations and Chow (2013b). directly from a network of deterministic spiking neurons that cap- Concurrently with this line of research, we also explored the ture the spike rate and spike correlation dynamics. The formalism question of how to generalize population activity equations, such first constructs a density functional for the firing dynamics of all as the Wilson-Cowan equations, to include the effects of correla- the neurons in a network. It then systematically marginalizes the tions. The motivation for this question is that the Wilson-Cowan unwanted degrees of freedom to isolate a set of self-consistent equations are mean field equations and do not capture the effects equations for the desired quantities. For heuristic reasons, we of spike-time correlations. For example, the gain in the Wilson- derive an example set of generalized activity equations for the first Cowan equations is fixed, (which is a valid approximation when and second cumulants of the firing dynamics of a simple spiking the neurons fire asynchronously), but correlations in the firing model but the method can be applied to any spiking model. times can change the gain Salinas and Sejnowski (2000).Thus, A convenient form to express spiking dynamics is with a phase it would be useful to develop a systematic procedure to aug- oscillator. Consider the quadratic integrate-and-fire ment population activity equations to include spike correlation effects. The approach we took was to posit plausible microscopic dVi = I + V2 + α u(t) (1) stochastic dynamics, dubbed the spike model, that reduced to the dt i i i Wilson-Cowan equations in the mean field limit and compute the ( ) self-consistent moment equations from that microscopic theory. where I is an external current and u t are the synaptic cur- α Buice and Cowan (2009) showed that the solution of the mas- rents with some weight i. The spike is said to occur when ter equation of the spike model could be expressed formally in V goes to infinity whereupon it is reset to minus infinity. The terms of a path integral over all possible spiking histories. The quadratic non-linearity ensures that this transit will occur in a = (θ/ ) random variable in the path integral is a spike count whereas in finite amount of time. The substitution V tan 2 yields the the path integral for the deterministic phase model we described theta model Ermentrout and Kopell (1986): above, the random variable is a phase density. To generate a sys- dθi tem of moment equations for the microscopic stochastic system, = 1 − cos θi + (1 + cos θi)(Ii + αiu) (2) we transformed the random spike count variable in the path dt integral into moment variables Buice et al. (2010).Thisisaccom- which is the normal form of a Type I neuron near the bifur- plished using the effective action approach of field theory, where cation to firing Ermentrout (1996).Thephaseneuronisan the exponent of the cumulant generating functional, called the adequate approximation to spiking dynamics provided the inputs action, which is a function of the random variable is Legendre are not overly strong as to disturb the . The phase neu- transformed into an effective action of the cumulants. The desired ron also includes realistic dynamics such as not firing when the generalized Wilson-Cowan activity equations are then the equa- input is below threshold. Coupled phase models arise naturally in tions of motion of the effective action. This is analogous to the weakly coupled neural networks Ermentrout and Kopell (1991); transformation from Lagrangian variables of position and veloc- Hoppensteadt and Izhikevich (1997); Golomb and Hansel (2000). ity to Hamiltonian variables of position and momentum. Here, They include the Kuramoto model Kuramoto (1984),whichwe we show how to apply the effective action approach to a deter- have previously analyzed Buice and Chow (2007); Hildebrand ministic system of synaptically coupled spiking neurons to derive et al. (2007). a set of moment equations. Here, we consider the phase dynamics of a set of N coupled phase neurons obeying APPROACH ˙ Consider a network of single compartment conductance-based θi = F(θ, γi, u(t)) (3) neurons u˙(t) =−βu(t) + βν(t) (4) n   N N    dVi 1 C =− g xr (V − v ) + g s (t) ν( ) = δ − l r i i r ij j t t tj (5) dt = N r 1 j = 1 j = 1 l dxr τr i = ( , ) θ i f Vi xi whereeachneuronhasaphase i that is indexed by i, u is a dt global synaptic drive, F(θ, γ, u) is the phase and synaptic drive   dsj dependent frequency, γ represents all the parameters for neuron τ = h V , s i j dt j j i drawn from a distribution with density g(γ), ν is the popula-   l dgij tion firing rate of the network,tj is the lth firing time of neuron j τ = φ g , V π g dt ij and a neuron fires when its phase crosses . In the present paper,

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we consider all-to-all or global coupling through a synaptic drive that are all prepared with different initial conditions and param- variable u(t). However, our basic approach is not restricted to eter values. Here, we show how a perturbation theory in N−1 global coupling. can be developed to expand around the mean field solution. This We can encapsulate the phase information of all the neurons requires the construction of the probability density functional into a neuron density function Buice and Chow (2007, 2011, over the ensemble of spiking neural networks. We adapt the tools 2013a,b); Hildebrand et al. (2007). of statistical field theory to perform such a construction.

1 N FORMALISM η(θ, γ, t) = δ(θ − θ (t)) δ(γ − γ ) (6) N i i The complete description of the system given by equations i = 1 (4, 7, 8) can be written as  where δ(·) is the Dirac delta functional, and θ (t) is a solution to i ˙( ) + β ( ) − β γ (π, γ, ( ))η(π, γ, ) = system (3–5). The neuron density gives a count of the number u t u t d F u t t 0(9) of neurons with phase θ and synaptic strength γ at time t. Using ∂ ∂ the fact that the Dirac delta functional in (5) can be expressed η(θ, γ, t) + [F(θ, γ, u(t))η(θ, γ, t)]≡Lη = 0 (10) δ( − l) = θ˙ δ(π − θ ) ∂t ∂θ as l t tj j j , the population firing rate can be rewritten as The probability density functional governing the system specified  by the synaptic drive and Klimontovich equations (9) and (10) ν( ) = γ (π, γ, ( ))η(π, γ, ) t d F u t t (7) given initial conditions (η0, u0) can be written as  The neuron density formally obeys the conservation equation P[η, u]= Du0(t) Dη0(θ, γ) P[η, u|η0, u0] P0[η0, u0, γ] (11) ∂ ∂ η(θ, γ, t) + [Fη(θ, γ, t)]=0(8) η, |η , ∂t ∂θ where P [ u 0 u0] is the conditional probability density func- tional of the functions (η, u),andP0 [η0, u0] is the density with initial condition η(θ, γ, t0) = η0(θ, γ) and u(t0) = u0. functional over initial conditions of the system. The integral is Equation (8) is known as the Klimontovich equation Ichimaru a Feynman-Kac path integral over all allowed initial condition (1973); Liboff (2003). The Klimontovich equation, the equation functions. Formally we can write P [η, u|η0, u0] as a point mass for the synaptic drive (4), and the firing rate expressed in terms (Dirac delta) located at the solutions of (9) and (10) given the of the neuron density (7), fully define the system. The system is initial conditions: still fully deterministic but is now in a form where various sets of reduced descriptions can be derived. Here, we will produce δ [Lη − η0δ(t − t0)] anexampleofasetofreducedequationsorgeneralizedactiv-  ity equations that capture some aspects of the spiking dynamics. δ u˙ + βu − β dγ F(π, γ, u(t))η(π, γ, t) − u0δ(t − t0) The path we take toward the end will require the introduction of some formal machinery that may obscure the intuition around the approximations. However, we feel that it is useful because it The probability density functional (11) is then provides a systematic and controlled way of generating averaged  quantities that can be easily generalized. P[η, u]= Du0(t) Dη0(θ, γ) δ [Lη − η0δ(t − t0)] For finite N, (8) is only valid in the weak or distributional  sense since η is not differentiable. In the N →∞limit, it has × δ ˙ + β − β γ (π, γ, ( ))η(π, γ, ) been argued that η will approach a smooth density ρ that evolves u u d F u t t according to the Vlasov equation that has the same form as (8) but with η replaced by ρ Ichimaru (1973); Desai and Zwanzig (1978); − u0δ(t − t0) P0 [η0, u0, γ] (12) Strogatz and Mirollo (1991); Nicholson (1993); Hildebrand et al. (2007). This has been proved rigorously in the case where noise is added using the theory of coupled diffusions McKean Jr (1966); Equation (12) can be made useful by noting that the Fourier rep- Faugeras et al. (2009); Baladron et al. (2012); Touboul (2012). resentation of a Dirac delta is given by δ(x) ∝ dk eikx.Usingthe This N →∞ limit is called mean field theory. In mean field infinite dimensional Fourier functional transform then gives theory, the original microscopic many body neuronal network  is represented by a smooth macroscopic density function. In − [η,η˜, ,˜] P[η, u]= Dη˜Due˜ NS u u . other words, the ensemble of networks prepared with different microscopic initial conditions is sharply peaked at the mean field solution. For large but finite N,therewillbedeviationsaway The exponent S[η, u] in the probability density functional is from mean field Buice and Chow (2007); Hildebrand et al. (2007); called the action and has the form Buice and Chow (2013a,b). These deviations can be characterized in terms of a distribution over an ensemble of coupled networks S = Su + Sϕ + S0 (13)

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where exp(NW[J, K]) =    i N Dξ exp −NS[ξ]+N dx J (x)ξi(x) + Sϕ = dθdγdt ϕ˜ (x) [∂tϕ(x) + ∂θF(θ, γ, u(t))ϕ(x)] (14) 2   ij   dxdx ξi(x)K (x, x )ξj(x ) (17) represents the contribution of the transformed neuron density to the action, where J and K are moment generating fields, ξ1(x) = u(t),  ξ (x) = u˜(t), ξ (x) = ϕ(x), ξ (x) =˜ϕ(x),andx = (θ, γ, t). 1 2 3 4 Su = dt u˜(t) u˙(t) + βu(t) Einstein summation convention is observed beween upper and N lower indices. Unindexed variables represent vectors. The inte-  gration measure dx is assumed to be dt when involving indices − β γ (π, γ, ( ))[˜ϕ(π, γ, ) + ]ϕ(π, γ, ) d F u t t 1 t (15) 1 and 2. Covariances between an odd and even index corresponds to a covariance between a field and an auxiliary field. Based on the structure of the action S and (17) we see that this represents [˜ϕ ( ), ( )] represents the global synaptic drive, S0 0 x0 u0 t0 represents a linear propagator and by causality and the choice of the Ito = (θ, γ, ) the initial conditions, and x t . For the case where the convention is only non-zero if the time of the auxiliary field is neurons are considered to be independent in the initial state, we evaluated at an earlier time than the field. Covariances between have two even indices correspond to that between two auxiliary fields and are always zero because of the Ito convention. 1 ξ S0 [ϕ˜ 0(x0), u0 (t0)] =− u˜(t0)u0 (16) The mean and covariances of can be obtained by taking N [ , ]  derivatives of the action W J K in (17), with respect to J and K and setting J and K to zero: − ln 1 + dθdγϕ˜ 0 (θ, γ, t0) ρ0 (θ, γ, t0) δ W =ξ | i J, K = 0 where u is the initial value of the coupling variable and ρ (θ, γ, t) δJi 0 0  is the distribution from which the initial configuration is drawn δ  W = 1ξ ξ  for each neuron. The action includes two imaginary auxiliary i j  δKij 2 , = response fields (indicated with a tilde), which are the infinite J K 0 dimensional Fourier transform variables. The factor of 1/N appears to ensure correct scaling between the u and ϕ vari- Expressions for these moments can be computed by expanding ables since u applies to a single neuron while ϕ applies to the the path integral in (17) perturbatively around some mean field entire population. The full derivation is given in Buice and Chow solution. However, this can be unwieldy if closed form expres- (2013b) and a review of path integral methods applied to dif- sions for the mean field equations do not exist. Alternatively, the ferential equations is given in Buice and Chow (2010).Inthe moments at any order can be expressed as self-consistent dynami- course of the derivation we have made a Doi-Peliti-Jannsen trans- cal equations that can be analyzed or simulated numerically. Such formation Janssen and Täuber (2005); Buice and Chow (2013b), equations form a set of generalized activity equations for the =ξ  = [ξ ξ − ] given by means ai i ,andcovariancesCij N i j aiaj . We derive the generalized activity equations by Legendre trans- forming the action W, which is a function of J and K,to −˜η(x) ϕ(x) = η(x)e an effective action  that is a function of a and C.Justasa η˜( ) ϕ˜ (x) = e x − 1 Fourier transform expresses a function in terms of its frequen- cies, a Legendre transform expresses a convex function in terms of its derivatives. This is appropriate for our case because the In deriving the action, we have explicitly chosen the Ito conven- moments are derivatives of the action. The Legendre transform of tion so that the auxiliary variables only depend on variables in W[J, K] is the past. The action (13) contains all the information about the   statistics of the network. i 1  1 ij The moments for this distribution can be obtained by [a, C]=−W[J, K]+ dxJ ai + dxdx aiaj + Cij K 2 N taking functional derivatives of a moment generating func- (18) tional. Generally, the moment generating function for a ran- which must obey the constraints dom variable is given by the expectation value of the expo- nential of that variable with a single parameter. Because our δW goal is to transform to new variables for the first and second = ai δJi cumulants, we form a “two-field” moment generating func- tional, which includes a second parameter for pairs of random δW 1 1 = aiaj + Cij variables, δKij 2 N

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and which gives      δ , 1 ≡ i 00 = Ji + a Kij + Kji 0, ij j exp(−N [a, C]) = exp −NS[a]−Tr  Cij Dψ δai 2 1   δ , 1 ≡ 0 ij = Kij (19) N ij 0, ij δ exp − dxL [a]ψiψj + N dx  ψiψj Cij 2N 2 1

The generalized activity equations are given by the equations of where higher order terms in N−1 are not included. To lowest non- motion of the effective action, in direct analogy to the Euler- 0, ij = −10, ij  Lagrange equations of classical mechanics, and are obtained by zero order N 1 since 0 is only a function of a and setting Ji = 0andKij = 0 in (19). not C.Ifweset In essence, what the effective action does is to take a prob- 0, ij = ( / ) ij − ( / ) ij, abilistic (statistical mechanical) system in the variables ξ with 1 1 2 L 1 2 Q (23) action S and transform them to a deterministic (classical mechan- ical) system with an action . Our approach here follows that we obtain used in Buice et al. (2010) to construct generalized activity equa- tions for the Wilson Cowan model. However, there are major 1 1 exp(−N [a, C]) = exp −NS[a]− Tr LijC + Tr QijC differences between that system and this one. In Buice et al. 2 ij 2 ij (2010), the microscopic equations were for the spike counts of an   N ij inherently probabilistic model so the effective action and ensu- × Dψ exp − dx Q [a]ψiψj (24) ing generalized activity equations could be constructed directly 2 from the Markovian spike count dynamics. Here, we start from deterministically firing individual neurons and get to a proba- to order N−1. Qij is an unknown function of a and C, bilistic description through the Klimontovich equation. It would which we will deduce using self-consistency. The path inte- be straightforward to include stochastic effects into the spiking gral in (24), which is an infinite dimensional Gaussian that dynamics. can be explicitly integrated, is proportional to 1/ det Qij = Using (18) in (17) gives exp(−(1/2) ln det Qij) = exp(−(1/2)Tr ln Qij), using properties   of matrices. Hence, (24) becomes i exp(−N [a, C]) = Dψ exp −NS[ξ]+N dx J (ξi − ai) exp(−N [a, C])  N  1 ij 1 ij 1 ij 1 ij + dxdx ξiξj − aiaj − Cij K (20) = exp −NS[a]− Tr L C − Tr Q C + Tr ln Q 2 N 2 ij 2 ij 2 where J and K are constrainted by (19). We cannot com- and pute the effective action explicitly but we can compute it perturbatively in N−1. We first perform a shift ξ = a + i i [ , ]= [ ]+ 1 ij + 1 ij − 1 ij ψ [ + ψ]= [ ]+ ( i[ ]ψ + a C S a Tr L Cij Tr ln Q Tr Q Cij i, expand the action as S a S a dx L a i 2N 2N 2N  ij (1/2) dx L [a]ψiψj) +··· and substitute for J and K with the constraints (19) to obtain Taking the derivative of  with respect to Cij yields   (− [ , ]) = − [ ]− 0, ij ∂ ∂ exp N a C exp NS a NTr Cij 0, ij 1 ij −1 kl lk kl  = L + (Q ) Q − (Q Clk)    2N ∂Cij ∂Cij i 1  ij Dψ exp −N dx L [a]ψi + dx L [a]ψiψj 2 −   Self consistency with (23) then requires that Qij = (C 1)ij which i, 00 2  0, ij leadstotheeffectiveaction + N dx  ψi + N dxdx ψiψj (21)

1 −1 ij 1 ij [a, C]=S[a]+ Tr ln(C ) + Tr L Cij (25) where  2N 2N    Tr AijB = dxdx Aij(x, x )B (x, x ) (22) ij ij where  Our goal is to construct an expansion for  by collecting terms in  −1 ik   dx (C ) (x, x )Ckj(x , x0) = δijδ(x − x0) successive orders of N−1 in the path integral of (21). Expanding −1 −2  as [a, C]=0 + N 1 + N 2 and equating coefficients of N in (21) immediately leads to the conclusion that 0 = S[a], and we have dropped the irrelevant constant terms.

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 −1   δ [ ( )]   The equations of motion to order N are obtained from (19) 1[ ] ,  = S a x = θ γ γ ( )∂ ( )δ −  i ij L a x x  d d dt a4 x θa3 x t t with J and K set to zero: δa1 (t )      + 1 ( ) d δ −  + β  δ [ ] δ dt a2 t t t a2 t S a + 1 ij = N dt δ δ Tr L Cij 0 (26)  ai 2N ai     −  β γγ (π, γ, ) + (π, γ, ) 1 − a2 t d a4 t 1 a3 t [−(C 1)ij + Lij]=0 (27)  2N δ [ ( )] 2[ ]( , ) = S a x = 1 da1 + β ( ) − β L a x x   a1 t δa2(t ) N dt and (27) can be rewritten as        dγ I + γa1(t ) a4(π, γ, t ) + 1 a3(π, γ, t )   δ [ ( )]  ik   3  S a x    dx L (x, x )Ckj(x , x0) = δijδ(x − x0) (28) L [a](x, x ) = = dt a (θ , γ , t)∂ δ(t − t ) δa (x) 4 t 3      + dθa (θ, γ , t )∂θ(I + γ a (t ))δ(θ − θ ) Hence, given any network of spiking neurons, we can 4 1 write down a set of generalized activity equations for the    β      mean and covariance functions by (1) constructing a neu- − a2(t ) I + γ a1(t ) a4(π, γ , t ) + 1 ron density function, (2) writing down the conservation law N  (Klimontovich equation), (3) constructing the action and (4) × δ(π − θ )   using formulas (26) and (28). We could have constructed  δS[a(x)]     4[ ]( , ) = = ∂  ( ) + ∂  + γ ( ) ( ) these equations directly by multiplying the Klimontovich L a x x  t a3 x θ I a1 t a3 x δa4(x ) and synaptic drive equations by various factors of u and β   η and recombining. However, as we saw in Buice et al.       − a2(t ) I + γ a1(t ) a3(π, γ , t )δ(π − θ ) (2010) this is not a straightforward calculation. The effec- N tive action approach makes this much more systematic and (31) mechanical. The mean field equations are obtained by solving Li = 0using = = PHASE MODEL EXAMPLE (31). We immediately see that a2 a4 0 are solutions, which We now present a simple example to demonstrate the con- leaves us with cepts and approximations involved in our expansion. Our  goal is not to analyze the system per se but only to demon- a˙1 + βa1 − β dγ(I + γa1) a3(π, γ, t) = 0 (32) strate the application of our method in a heuristic setting. We begin with a simple non-leaky integrate-and-fire neu- ∂ta3 + (I + γa1) ∂θa3 = 0 (33) ron model, which responds to a global coupling variable. This is a special case of the dynamics given above, with F The mean field equations should be compared to those of the given by spike response model Gerstner (1995, 2000). We can also solve (33) directly to obtain [θ, γ, ]= ( ) + γ  F u I t u (29) t      a3(x, t) = ρ0 θ − dt I(t ) + γa1(t ) , γ, t0 The action from (14) and (15) is where ρ is the initial distribution. If the neurons are initially dis-  0 tributed uniformly in phase, then ρ0 = g(γ)/2π and the mean [ ]= θ γ ( ) ∂ ( ) + ∂ + γ ( ) ( ) S a d d dt a4 x [ ta3 x θ(I a1 t ) a3 x ] field equations reduce to   + 1 ( ) ˙ ( ) + β ( ) − β γ ( + γ ( )) β dt a2 t a1 t a1 t d I a1 t a˙ (t) + βa (t) − (I + γ¯a (t)) = 0 (34) N 1 1 2π 1

[a4(π, γ, t) + 1] a3(π, γ, t)) (30) which has the form of the Wilson-Cowan equation, with (β/2π) (I + γ¯a1) acting as a gain function. Hence, the Wilson- and we ignore initial conditions for now. Cowan equation is a full description of the infinitely large system In order to construct the generalized activity equations limit of a network of globally coupled simple phase oscillators we need to compute the first and second derivatives of in the asynchronous state. For all other initial conditions, the the action Li and Lij. Taking the first derivative of (30) one-neuron conservation equation (called the Vlasov equation in gives kinetic theory) must be included in mean field theory.

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    To go beyond mean field theory we need to compute δ(γ − γ )δ(t − t ) ij   i   L (x, x , x ) = δL (x, x )/δaj(x ): L44[a]=0

11 L [a]=0 −1 The activity equations for the means to order N are given by 1 d (26). The only non-zero contributions are given by L13 and L31 L12[a]= − + β − β N dt resulting in     γγ (π, γ, ) + (π, γ, ) δ(  − ) 1 δ  d a4 t 1 a3 t t t L2 + dxdx (L13C + L31C ) = 0 2N δa 13 31  2  13[ ]= γ θ ( )δ(γ − γ)∂ δ(θ − θ) 1 δ  L a d a4 x θ L4 + dxdx (L13C + L31C ) = 0 2N δa 13 31 4   β        − γ a2(t ) a4(π, γ , t ) + 1 δ(π − θ ) δ(t − t ) since a = a = 0 and correlations involving response variables N 2 4 (even indices) will be zero for equal times. The full activity β 14        equations for the means are thus L [a]= γ ∂θ a3(x ) − γ a2(t )a3(π, γ , t )δ(π − θ ) N    ˙ δ(t − t ) a1 + βa1 − β dγ(I + γa1) a3(π, γ, t)     21[ ]= 1 d + β − β γγ (π, γ, ) + (π, γ, ) β L a  d a4 t 1 a3 t − dγγC(π, γ, t) = 0 (35) N dt N   δ(t − t ) 1 ∂ta3 + (I + γa1) ∂θa3 + γ∂θC(θ, γ, t) = 0 (36) L22[a]=0 N β   (θ, γ, ) = ( ; θ, γ, ) = (θ, γ, ; ) 23      where C t C13 t t C31 t t . L [a]=− (I + γ a (t )) a (π, γ , t )) + 1 δ(π − θ ) ij N 1 4 We can now use the L in (28) to obtain activity equations for C . There will be sixteen coupled equations in total but the δ(  − ) ij t t applicable non-zero ones are β 24         L [a]=− (I + γ a1(t ))a3(π, γ , t )]δ(π − θ )δ(t − t ) N d + β − β γγ (π, γ, ) ( ; )  d a3 t C11 t t0 β dt 31       L [a]= dθ a (θ, γ , t )γ ∂θδ(θ − θ ) − a (t )γ  4 N 2 − β dγ (I + γa ) C (π, γ, t; t ) 1 31 0         a4(π, γ , t ) + 1 δ(π − θ ) δ(t − t ) − β dγ (I + γa1(t)) a3(π, γ, t)C41(π, γ, t; t0) = 0 (37)    β       L32[a]=− I + γ a (t ) a (π, γ , t ) + 1 δ(π − θ ) N 1 4 d + β − β dγγa3(π, γ, t) C13(t; x0)   dt δ(t − t )  − β γ ( + γ ) (π, γ, ; ) L33[a]=0 d I a1 C33 t x0    34        L [a]= δ(θ − θ )∂  − ∂θ I + γ a (t ) δ(θ − θ ) t 1 − β γ ( + γ ( )) (π, γ, ) (π, γ, ; ) = d I a1 t a3 t C43 t x0 0 (38) β      − a2(t )(I + γ a1(t ))δ(π − θ )δ(π − θ ) N γ∂θa3(x)C11 (t; t0) + [∂t + (I + γa1) ∂θ] C31(x; t0) δ(γ − γ)δ(  − ) β t t − (I + γa (t)) a (π, γ, t)δ(π − θ)C (t, t ) = 0 (39) N 1 3 21 0 β 41        L [a]= ∂θ γ a3(x ) − a2(t )γ a3(π, γ , t )δ(π − θ ) γ∂θa (x)C (t; x ) + [∂ + (I + γa (t)) ∂θ] C (x, x ) N 3 13 0 t 1 33 0   β δ(t − t ) − (I + γa (t)) a (π, γ, t)δ(π − θ)C (t, x ) = 0 (40) N 1 3 23 0 β 42        L [a]=− (I + γ a1(t ))a3(π, γ , t )δ(π − θ )δ(t − t ) → N Adding (38) and (39) and taking the limit t0 t and setting   θ = θ, γ = γ gives 43       0 0 L [a]=∂  δ(x − x ) + ∂θ I + γ a (t ) δ(x − x ) t 1    β     ∂ (θ, γ, ) + β − β γ γ (π, γ, ) + ( + γ ) ∂ − a (t ) I + γa (t ) δ(π − θ )δ(π − θ ) tC t d a3 t I a1 θ N 2 1

Frontiers in Computational Neuroscience www.frontiersin.org November 2013 | Volume 7 | Article 162 | 7 Buice and Chow Generalized activity equations

      they represent the dynamics of the system averaged over ini- C(θ, γ, t) − β dγ I + γ a C (π, γ , t; x) 1 33 tial conditions and unknown parameters. Hence, the solution of this PDE system replaces multiple simulations of the original − 2β (I + γa (t)) a (π, γ, t)δ(π − θ) + γ∂θa (x)C (t; t) = 0 1 3 3 11 system. In previous work, we required over a million simula-   tions of the original system to obtained adequate statistics Buice whereweusethefactthatC21(t, t ) = N and C43(x; x ) = δ(θ − θ)δ(γ − γ)  and Chow (2013b). There is also a possibility that simplify- in the limit of t approaching t from below and equal ing approximations can be applied to such systems. The system to zero when approaching from above. Adding (37) and (40) to has complete phase memory because the original system was themselves with t and t0 interchanged and taking the limit of t0 fully deterministic. However, the inclusion of stochastic effects approaching t gives will shorten the memory and possibly simplify the dynamics.  It will pose no problem to include such stochastic effects. In d fact, the formalism is actually more suited for stochastic systems + 2β − 2β dγγa3(π, γ, t)] C11(t; t) dt Buice et al. (2010).  − β γ ( + γ ) (π, γ, ) = 2 d I a1 C t 0 DISCUSSION The main goal of this paper was to show how to systematically [∂ + (I + γa (t)) ∂θ] C (x; x) + 2γ[∂θa (x)]C(x) = 0 t 1 33 3 derive generalized activity equations for the ensemble averaged moments of a deterministically coupled network of spiking neu- ( ; ) = ( ; ) = because C41 x t 0andC23 t x 0. Putting this all together, rons. Our method utilizes a path integral formalism that makes we get the generalized activity equations the process algorithmic. The resulting equations could be derived  using more conventional perturbative methods although possi- da1 bly with more calculational difficulty as we found before Buice + βa1(t) − β dγ (I + γa1(t)) a3(π, γ, t) dt et al. (2010). For example, for the case of the stochastic spike  β model, Buice et al. (2010) presumed that the Wilson-Cowan − dγγC(π, γ, t) = 0 (41) N activity variable was the rate of a Poisson process and derived a system of generalized activity equations that corresponded to ∂ (θ, γ, ) + ( + γ ) ∂ (θ, γ, ) ta3 t I a1 θa3 t deviations around Poisson firing. Bressloff (2010), on the other 1 hand, assumed that the Wilson-Cowan activity variable was a + γ∂θC(θ, γ, t) = 0 (42) N mean density and used a system-size expansion to derive an alter-  native set of generalized activity equations for the spike model.    ∂tC(θ, γ, t) + β − β dγ γ a3(π, γ , t) + (I + γa1) ∂θ The classical derivations of these two interpretations look quite  different and the differences and similarities between them are      not readily apparent. However, the connections between the two C(θ, γ, t) − β dγ I + γ a C (π, γ , t; θ, γ, t) 1 33 types of expansions are very transparent using the path integral formalism. − 2β (I + γa1(t)) a3(θ, γ, t)δ(π − θ) Here, we derived equations for the rate and covariances +γ∂θa3(θ, γ, t)C11(t; t) = 0 (43) (first and second cumulants) of a deterministic synaptically cou-  d pled spiking network as a system size expansion to first order. + 2β − 2β dγγa3(π, γ, t) C11(t; t) However, our method is not restricted to these choices. What dt  is particularly advantageous about the path integral formal- ism is that it is straightforward to generalize to include higher − 2β dγ (I + γa1) C(π, γ, t) = 0 (44) order cumulants, extend to higher orders in the inverse sys- tem size, or to expand in other small parameters such as the [∂t + (I + γa1(t)) ∂θ] C33(θ, γ, t; θ, γ, t) inverse of a slow time scale. The action fully specifies the sys- + γ∂ (θ, γ, ) (θ, γ, ) = 2 θa3 t C t 0 (45) tem and all questions regarding the system can be addressed with it. Initial conditions, which are specified in the action, are required To give a concrete illustration of the method, we derived the for each of these equations. The derivation of these equations self-consistent generalized activity equations for the rates and using classical means require careful consideration for each par- covariances to order N−1 for a simple phase model. The resulting ticular model. Our method provides a blanket mechanistic algo- equations consist of ordinary and partial differential equations. rithm. We propose that these equations represent a new scheme This is to be expected since the original system was fully deter- for studying neural networks. ministic and memory cannot be lost. Even mean field theory Equations 41–45 are the complete self-consistent generalized requires the solution of an advective partial differential equa- activity equations for the mean and correlations to order N−1.It tion. The properties of these and similar equations remain to is a system of partial differential equations in t and θ. These equa- be explored computationally and analytically. The system is pos- tions can be directly analyzed or numerically simulated. Although sibly simpler near the asynchronous state, which is marginally the equations seem complicated, one must bear in mind that stable in mean field theory like the Kuramoto model Strogatz and

Frontiers in Computational Neuroscience www.frontiersin.org November 2013 | Volume 7 | Article 162 | 8 Buice and Chow Generalized activity equations

Mirollo (1991) andliketheKuramotomodel,weconjecturethat Buice, M. A., and Chow, C. C. (2007). Correlations, fluctuations, and stability the finite size effects will stabilize the asynchronous state Buice of a finite-size network of coupled oscillators. Phys. Rev. E 76:031118. doi: and Chow (2007); Hildebrand et al. (2007). The addition of noise 10.1103/PhysRevE.76.031118 Buice, M. A., and Chow, C. C. (2010). Path integral methods for stochastic will also stabilize the asynchronous state. Near asynchrony could differential equations. arXiv 1009.5966. be exploited to generate simplified versions of the asynchronous Buice, M. A., and Chow, C. C. (2011). Effective stochastic behavior in dynami- state. cal systems with incomplete information. Phys.Rev.E84(5 Pt 1):051120. doi: We considered a globally connected network, which allowed 10.1103/PhysRevE.84.051120 us to assume that networks for different parameter values and Buice, M. A., and Chow, C. C. (2013a). Beyond mean field theory: statistical field theory for neural networks. J. Stat. Mech. Theor. Exp. 2013:P03003. doi: initial conditions converge toward a “typical” system in the large 10.1088/1742-5468/2013/03/P03003 N limit. However, this property may not hold for more realistic Buice, M. A., and Chow, C. C. (2013b). Dynamic finite size effects in spiking neural networks. While the formalism describing the ensemble average networks. PLoS Comput. Biol. 9:e1002872. will hold regardless of this assumption, the utility of the equa- Buice, M. A., Cowan, J. D., and Chow, C. C. (2010). Systematic fluctuation expan- tions as descriptions of a particular network behavior may suffer. sion for neural network activity equations. Neural Comput. 22, 377–426. doi: 10.1162/neco.2009.02-09-960 For example, heterogeneity in the connectivity (as opposed to the Desai, R., and Zwanzig, R. (1978). Statistical mechanics of a nonlinear stochastic global connectivity we consider here) may threaten this assump- model. J. Stat. Phys. 19, 1–24. doi: 10.1007/BF01020331 tion. This is the case with so called “chaotic random networks” Elgart, V., and Kamenev, A. (2004). Rare events statistics in reaction-diffusion Sompolinsky et al. (1988) in which there is a spin-glass transi- systems. Phys. Rev. E 70:041106. doi: 10.1103/PhysRevE.70.041106 Eliasmith, C., Stewart, T. C., Choo, X., Bekolay, T., DeWolf, T., Tang, Y., et al. tion owing to the variance of the connectivity crossing a critical (2012). A large-scale model of the functioning brain. Science 338, 1202–1205. threshold. This results in the loss of a “typical” system in the large doi: 10.1126/science.1225266 N limit requiring an effective stochastic equation which incorpo- Ermentrout, B. (1996). Type I membranes, phase resetting curves, and synchrony. rates the noise induced by the network heterogeneity. Whether Neural Comput. 8, 979–1001. doi: 10.1162/neco.1996.8.5.979 the expansion we present here is useful without further consider- Ermentrout, G. B. and Kopell, N. (1986). Parabolic in an excitable system coupled with a slow oscillation. SIAM J. Appl. Math. 46, 233–253. doi: 10.1137/ ation depends upon whether the network heterogeneity induces 0146017 this sort of effect. This is an area for future work. A simpler issue Ermentrout, G., and Kopell, N. (1991). Multiple pulse interactions and averag- arises when there are a small discrete number of “typical” sys- ing in systems of coupled neural oscillators. J. Math. Biol. 29, 195–217. doi: tems (such as with bistable solutions to the continuity equation). 10.1007/BF00160535 In this case, there are noise induced transitions between states. Faugeras, O., Touboul, J., and Cessac, B. (2009). A constructive mean-field analysis of multi-population neural networks with random synaptic weights While the formalism has a means of computing this transition and stochastic inputs. Front. Comput. Neurosci. 3:1. doi: 10.3389/neuro.10. Elgart and Kamenev (2004), we do not consider this case here. 001.2009 An alternative means to incorporate heterogeneous connec- Gerstner, W. (1995). Time structure of the activity in neural network models. Phys. tions is to consider a network of coupled systems. In such a Rev. E 51, 738–758. doi: 10.1103/PhysRevE.51.738 network, a set of generalized activity equations, such as those Gerstner, W. (2000). Population dynamics of spiking neurons: fast tran- sients, asynchronous states, and locking. Neural Comput. 12, 43–89. doi: derived here or simplified versions, would be derived for each 10.1162/089976600300015899 local system, together with equations governing the covariances Golomb, D., and Hansel, D. (2000). 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Strogatz, S., and Mirollo, R. (1991). Stability of incoherence in a population of Received: 29 April 2013; accepted: 23 October 2013; published online: 15 November coupled oscillators. J. Stat. Phys. 63, 613–635. doi: 10.1007/BF01029202 2013. Touboul, J. (2012). Mean-field equations for stochastic firing-rate neural fields with Citation: Buice MA and Chow CC (2013) Generalized activity equations for spik- delays: Derivation and noise-induced transitions. Phys. D Nonlin. Phenom. 241, ing neural network dynamics. Front. Comput. Neurosci. 7:162. doi: 10.3389/fncom. 1223–1244. doi: 10.1016/j.physd.2012.03.010 2013.00162 Treves, A. (1993). Mean-field analysis of neuronal spike dynamics. Netw. Comput. This article was submitted to the journal Frontiers in Computational Neuroscience. Neural Syst. 4, 259–284. doi: 10.1088/0954-898X/4/3/002 Copyright © 2013 Buice and Chow. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribu- tion or reproduction in other forums is permitted, provided the original author(s) Conflict of Interest Statement: The authors declare that the research was con- or licensor are credited and that the original publication in this journal is cited, in ducted in the absence of any commercial or financial relationships that could be accordance with accepted academic practice. No use, distribution or reproduction is construed as a potential conflict of interest. permitted which does not comply with these terms.

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