An augmented moment method for stochastic ensembles with delayed couplings: II. FitzHugh-Nagumo model

TL;DR

This paper develops an augmented moment method to analyze stochastic FitzHugh-Nagumo neuron ensembles with delayed couplings, revealing how parameters influence oscillation and synchronization, with results aligning well with direct simulations.

Contribution

The paper introduces a semi-analytical augmented moment method for delayed stochastic neuron ensembles, extending previous work to handle delay effects and large ensemble dynamics.

Findings

Oscillations can be induced by parameter tuning.

Synchronization is enhanced near transition points.

AMM results agree with direct simulation data.

Abstract

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 semi-analytical augmented moment method (AMM) which has been proposed in a recent paper [H. Hasegawa, E-print: cond-mat/0311021]. For $N$-unit FN neuron ensembles, AMM transforms original $2N$-dimensional {\it stochastic} delay differential equations (SDDEs) to infinite-dimensional {\it 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 (AMM$m$), 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 {\it fluctuation-induced} enhancement at the transition between non-oscillating and oscillating states. Results calculated by AMM5 are in fairly good agreement with those obtained by direct simulations.