Homoclinic Orbits of the FitzHugh-Nagumo Equation: Bifurcations in the Full System

TL;DR

This paper explores the bifurcation structure of homoclinic orbits in the FitzHugh-Nagumo equation using numerical and geometric methods, revealing complex bifurcation phenomena including sharp turns, multi-pulse orbits, and canard explosions.

Contribution

It provides a detailed numerical and geometric analysis of homoclinic bifurcations in the FitzHugh-Nagumo model, highlighting new bifurcation features and predicting complex orbit behaviors.

Findings

Identification of sharp turns in homoclinic bifurcation curves.

Visualization of invariant manifolds in phase space.

Prediction of multi-pulse and homoclinic orbits, as well as canard explosions.

Abstract

This paper investigates travelling wave solutions of the FitzHugh-Nagumo equation from the viewpoint of fast-slow dynamical systems. These solutions are homoclinic orbits of a three dimensional vector field depending upon system parameters of the FitzHugh-Nagumo model and the wave speed. Champneys et al. [A.R. Champneys, V. Kirk, E. Knobloch, B.E. Oldeman, and J. Sneyd, When Shilnikov meets Hopf in excitable systems, SIAM Journal of Applied Dynamical Systems, 6(4), 2007] observed sharp turns in the curves of homoclinic bifurcations in a two dimensional parameter space. This paper demonstrates numerically that these turns are located close to the intersection of two curves in the parameter space that locate non-transversal intersections of invariant manifolds of the three dimensional vector field. The relevant invariant manifolds in phase space are visualized. A geometrical model inspired by the numerical studies displays the sharp turns of the homoclinic bifurcations curves and yields quantitative predictions about multi-pulse and homoclinic orbits and periodic orbits that have not been resolved in the FitzHugh-Nagumo model. Further observations address the existence of canard explosions and mixed-mode oscillations.