Bifurcation structure of two coupled FHN neurons with delay

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 bifurcations (Neimark-Sacker bifurcation in the Poincare map of a limit cycle), of which detection was impossible without exact and systematic dynamical study. 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.

A Frequency Domain Analysis of the Excitability and Bifurcations of the FitzHugh–Nagumo Neuron Model

“…Nagumo et al formulated the model in an electrical analogy, including inductor and negative resistance elements. The resulting FitzHugh–Nagumo (FHN) model displays realistic neural dynamics like the cessation of repetitive spiking as the amplitude of the stimulus current increases. , It has also been broadly studied by its rich phase portraits, as described in books and review articles, ,− and it is computationally efficient for analyzing the dynamics of neural networks. , The dynamics of systems of coupled FHN neurons and their bifurcation properties have been amply studied in recent years. − …”

The Impedance of Spiking Neurons Coupled by Time‐Delayed Interaction

“…A frequently used model is the sigmoid coupling [ 20–22,24 ] …”

“…Equation () is a transcendental equation that determines the bifurcation properties of the system. [ 20–25 ]…”

“…The delay is yet an additional bifurcation parameter that has been amply studied in the literature. [ 20–25 ]…”

“…[ 12,15–19 ] With the introduction of the time delay interaction the coupled FHN neurons display a transition from the original disordered motions to periodic ones, which is accompanied by complex bifurcation properties. [ 20–25 ]…”