Bifurcation structure of two coupled FHN neurons with delay

TL;DR

This paper analyzes the complex bifurcation structures of two coupled FitzHugh-Nagumo neurons with delay, revealing rich dynamics including multiple bifurcations, stability regions, and periodic solutions through analytical and numerical methods.

Contribution

It provides a comprehensive bifurcation analysis of coupled non-identical neurons with delay, identifying various bifurcations and stability regions that were not previously characterized.

Findings

Identification of bifurcation points including Hopf, double-Hopf, and pitchfork bifurcations.

Mapping of delay-dependent stability regions in parameter space.

Discovery of multiple periodic solutions due to bifurcations.

Abstract

This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Homoclinic, fold, and pitchfork bifurcations of limit cycles are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf bifurcation points can be obtained from the intersection points of two branches of Hopf bifurcation. The dynamical behavior of the system may exhibit one, two, or three different periodic solutions due to pitchfork cycle and torus (Neimark-Sacker) bifurcations. In addition, Hopf, double-Hopf, and torus bifurcations of the non trivial rest points are found. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behaviors are clarified.