Bifurcation analysis and multistability detection of two delay-coupled FHN neurons

TL;DR

This study analyzes the complex dynamics of two delay-coupled FitzHugh-Nagumo neurons, revealing bifurcation phenomena, multistability, and various oscillatory behaviors influenced by coupling strength and delay.

Contribution

It provides a comprehensive bifurcation analysis of coupled non-identical neurons with delay, identifying new stability regions and multistability patterns not previously characterized.

Findings

Identification of bifurcation points including saddle-node, transcritical, Hopf, and Bautin bifurcations.

Discovery of multistability with synchronous and anti-phase solutions.

Delay-dependent stability regions and complex bifurcation structures.

Abstract

This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with quadratic term and delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics. Bifurcation diagrams are obtained numerically or analytically from the mathematical model, and the parameter regions of different behaviors are clarified. The neural system exhibits a unique rest point or three ones by employing the saddle-node bifurcation, when strong coupling is applied in the system. Also the trivial rest point shows transcritical bifurcation with one of the new rest points. The asymptotic stability and possible Hopf and Bautin bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Fold cycle, torus, fold-torus, and big homoclinic bifurcations of limit cycles, together with $1:1$ and $1:4$ resonances are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf, Hopf-transcritical, and double-zero bifurcation points can be obtained from the intersection of different bifurcation branches. Various patterns of multistability have been observed, both for small and large values of delay. The system may exhibit synchronous and anti-phase periodic solutions, which occur due to Hopf, fold cycle and torus bifurcations. By increasing the range of parameter $\tau$ other branches of fold and torus bifurcations appear, which lead to creation of more stable periodic solutions.