Periodic solutions in an array of coupled FitzHugh-Nagumo cells

TL;DR

This paper investigates the dynamics of a symmetric array of coupled FitzHugh-Nagumo cells arranged in a torus, describing oscillation patterns and their dependence on coupling signs via bifurcation analysis.

Contribution

It characterizes the oscillation patterns in a symmetric coupled cell system and links pattern types to coupling signs through bifurcation analysis.

Findings

Patterns depend on coupling signs

Hopf bifurcation leads to specific oscillation modes

Symmetry constrains possible oscillation patterns

Abstract

We analyse the dynamics of an array of $N^2$ identical cells coupled in the shape of a torus. Each cell is a 2-dimensional ordinary differential equation of FitzHugh-Nagumo type and the total system is $\mathbb{Z}_N\times\mathbb{Z}_N-$symmetric. The possible patterns of oscillation, compatible with the symmetry, are described. The types of patterns that effectively arise through Hopf bifurcation are shown to depend on the signs of the coupling constants, under conditions ensuring that the equations have only one equilibrium state.