Statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks

Abstract

Dynamical stochastic models of single neurons and neural networks often take the form of a system of n\ensuremath{\ge}2 coupled stochastic differential equations. We consider such systems under the assumption that third and higher order central moments are relatively small. In the general case, a system of 1/2n(n+3) (generally) nonlinear coupled ordinary differential equations holds for the approximate means, variances, and covariances. For the general linear system the solutions of these equations give exact results---this is illustrated in a simple case. Generally, the moment equations can be solved numerically. Results are given for a spiking Fitzhugh-Nagumo model neuron driven by a current with additive white noise. Differential equations are obtained for the means, variances, and covariances of the dynamical variables in a network of n connected spiking neurons in the presence of noise. \textcopyright{} 1996 The American Physical Society.