Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model.

Dynamics of FitzHugh-Nagumo (FN) neuron ensembles with time-delayed couplings subject to white noises, has been studied by using both direct simulations and a semianalytical augmented moment method (AMM) which has been proposed in a preceding paper [H. Hasegawa, Phys. Rev. E 70, 021911 (2004)]. For N-unit FN neuron ensembles, AMM transforms original 2N-dimensional stochastic delay differential equations (SDDEs) to infinite-dimensional deterministic DEs for means and correlation functions of local and global variables. Infinite-order recursive DEs are terminated at the finite level m in the level-m AMM (AMMm), yielding 8(m+1)-dimensional deterministic DEs. When a single spike is applied, the oscillation may be induced if parameters of coupling strength, delay, noise intensity and/or ensemble size are appropriate. Effects of these parameters on the emergence of the oscillation and on the synchronization in FN neuron ensembles have been studied. The synchronization shows the fluctuation-induced enhancement at the transition between nonoscillating and oscillating states. Results calculated by AMM5 are in fairly good agreement with those obtained by direct simulations.