Stochastic Neurodynamics and the System Size Expansion

this paper, we introduce another master equation based approach to go beyond the mean field approximation. We concentrate on the macroscopic behavior of a network of two--state neurons, and introduce a master equation for the number of active neurons in the network at time t. We use a more systematic expansion of the master equation than hitherto, the "system size expansion"[11]. The expansion parameter is the inverse of the total number of the neurons in the network. We truncate the expansion at second order and obtain an equation for fluctuations about the mean number of active neurons, which is itself coupled to the equation for the average number of active neurons at time t. These equations show non--monotonic approaches to equilibrium values near critical points, a feature which is not seen in the mean field approximation. Monte Carlo simulations of the master equation itself show qualitatively similar non--monotonic behavior. 2 Master Equation and the System Size Expansio