Correlation Functions in a Large Stochastic Neural Network

Abstract

Most theoretical investigations of large recurrent networks focus on the properties of the macroscopic order parameters such as popu(cid:173) lation averaged activities or average overlaps with memories. How(cid:173) ever, the statistics of the fluctuations in the local activities may be an important testing ground for comparison between models and observed cortical dynamics. We evaluated the neuronal cor(cid:173) relation functions in a stochastic network comprising of excitatory and inhibitory populations. We show that when the network is in a stationary state, the cross-correlations are relatively weak, i.e., their amplitude relative to that of the auto-correlations are of or(cid:173) der of 1/ N, N being the size of the interacting population. This holds except in the neighborhoods of bifurcations to nonstationary states. As a bifurcation point is approached the amplitude of the cross-correlations grows and becomes of order 1 and the decay time(cid:173) constant diverges. This behavior is analogous to the phenomenon of critical slowing down in systems at thermal equilibrium near a critical point. Near a Hopf bifurcation the cross-correlations ex(cid:173) hibit damped oscillations.