Stochastic Neural Field Theory and the System-Size Expansion

Abstract

Abstract. We analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons. The state of the network is specified by the fraction of active or spiking neurons in each population, and transition rates are chosen so that in the thermodynamic or deterministic limit (N → ∞) we recover standard activity-based or voltage-based rate models. We derive the lowest order corrections to these rate equations for large but finite N using two different approximation schemes, one based on the Van Kampen system-size expansion and the other based on path integral methods. Both methods yield the same series expansion of the moment equations, which at O(1/N ) can be truncated to form a closed system of equations for the first-and second-order moments. Taking a continuum limit of the moment equations while keeping the system size N fixed generates a system of integrodifferential equations for the mean and covariance of the corresponding stochastic neural field model. We also show how the path integral approach can be used to study large deviation or rare event statistics underlying escape from the basin of attraction of a stable fixed point of the mean-field dynamics; such an analysis is not possible using the system-size expansion since the latter cannot accurately determine exponentially small transitions.Key words. neural field theory, master equations, stochastic processes, system-size expansion, path integrals AMS subject classifications. Primary, 92; Secondary, 60 DOI. 10.1137/090756971 1. Introduction. Continuum models of neural tissue have been very popular in analyzing the spatiotemporal dynamics of large-scale cortical activity (reviewed in [24,35,7,16]). Such models take the form of integrodifferential equations in which the integration kernel represents the spatial distribution of synaptic connections between different neural populations, and the macroscopic state variables represent populationaveraged firing rates. Following seminal work in the 1970s by Wilson, Cowan, and Amari [72,73,2], many analytical and numerical results have been obtained regarding the existence and stability of spatially structured solutions to continuum neural field equations. These include electroencephalogram (EEG) rhythms [55,46,67], geometric visual hallucinations [25,26,70,8], stationary pulses or bumps [2,62,50,51,30], traveling fronts and pulses [27,61,6,74,17,32], spatially localized oscillations or breathers [31,33,59], and spiral waves [44,49,71,48].One potential limitation of neural field models is that they only take into account mean firing rates and thus do not include any information about higher-order statistics such as correlations between firing activity at different cortical locations. It is well known that the spike trains of individual cortical neurons in vivo tend to be very noisy, having interspike interval distributions that are close to Poisson [68]. However, it is far less clear how noise at the single cell...

Balanced Networks of Spiking Neurons with Spatially Dependent Recurrent Connections

“…Spatially extended stochastic neural field models are typically constructed by appending additive noise to a deterministic model [30,32], similar to the practice of augmenting reaction diffusion systems with additive or multiplicative noise [1]. Analysis of neural-field models driven by external stochastic forcing shows that the spatiotemporal structure of noise is a critical determinant of the ensuing stochastic dynamics [26][27][28].…”

“…There has been substantial theoretical work on the spatiotemporal dynamics of phenomenological "neural-field-type" macroscale models of cortex [25]. However, treatments of neural variability in these frameworks have either assumed an external source of fluctuations [26][27][28] or that neurons are intrinsically Markovian [29,30]. In both cases, the stochastic aspects of the microscale system are imposed, in contrast to the internally generated variability in balanced networks.…”

Large-scale neural dynamics: Simple and complex

“…In other situations equations going beyond the mean-field approach have been proposed that govern second-order correlations [100,101,102,103]. Indeed there has been a recent upsurge of interest in this area adapting methods from non-equilibrium statistical physics to determine corrections to mean-field theory involving equations for two-point and higher-order cumulants [104,105]. One immediate, yet potentially tractable, challenge would be to develop a framework for understanding networks of synaptically interacting nonlinear integrate-and-fire networks.…”

The role of fluctuations in coarse-grained descriptions of neuronal networks

“…Particularly interesting are scale-free networks for which 2 < γ ≤ 3. In such networks, the mean, µ, of the distribution P in (n) remains finite, while its second moment, 19) as computed from Equation (3.17), diverges as the size of the network, N , increases. A guess that the firing rate m k of the k-neurons is a linear function of the neuronal incoming degree k when Equation (3.18) holds leads to the solution of Equation (3.16) given by…”