Correlation Functions in a Large Stochastic Neural Network

Most theoretical investigations of large recurrent networks focus on the properties of the macroscopic order parameters such as population averaged activities or average overlaps with memories. However, 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 correlation 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 order 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-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 exhibit damped oscillations.