Stochastic bifurcation of FitzHugh-Nagumo ensembles subjected to additive and/or multiplicative noises

TL;DR

This study analyzes how additive and multiplicative noises induce stochastic bifurcations in FitzHugh-Nagumo ensembles, revealing different bifurcation behaviors and synchronization properties, using an augmented moment method reformulated with Fokker-Planck equations.

Contribution

The paper introduces a reformulated augmented moment method with Fokker-Planck equations to analyze stochastic bifurcations in FitzHugh-Nagumo ensembles under noise.

Findings

Multiplicative noise causes wider oscillating regions than additive noise.

The bifurcation transition diagrams differ significantly between noise types.

Results from the AMM agree well with direct simulations.

Abstract

We have studied the dynamical properties of finite $N$-unit FitzHugh-Nagumo (FN) ensembles subjected to additive and/or multiplicative noises, reformulating the augmented moment method (AMM) with the Fokker-Planck equation (FPE) method [H. Hasegawa, J. Phys. Soc. Jpn. {\bf 75}, 033001 (2006)]. In the AMM, original $2N$-dimensional stochastic equations are transformed to eight-dimensional deterministic ones, and the dynamics is described in terms of averages and fluctuations of local and global variables. The stochastic bifurcation is discussed by a linear stability analysis of the {\it deterministic} AMM equations. The bifurcation transition diagram of multiplicative noise is rather different from that of additive noise: the former has the wider oscillating region than the latter. The synchronization in globally coupled FN ensembles is also investigated. Results of the AMM are in good agreement with those of direct simulations (DSs).