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

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.

preprint version - for final version please see journal reference

References in corpus (3)

• Computing Slow Manifolds of Saddle Type
• Homoclinic Orbits of the FitzHugh-Nagumo Equation: The Singular-Limit
• A novel canard-based mechanism for mixed-mode oscillations in a neuronal model

Cited by in corpus (11)

• Six decades of the FitzHugh-Nagumo model: A guide through its spatio-temporal dynamics and influence across disciplines
• Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems
• From First Lyapunov Coefficients to Maximal Canards
• Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations
• Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts
• Zero-Hopf bifurcation in the FitzHugh-Nagumo system
• A study on nonnegativity preservation in finite element approximation of Nagumo-type nonlinear differential equations
• Instability of small-amplitude periodic waves from fold-Hopf bifurcation
• Analysis and numerics of traveling waves for asymmetric fractional reaction-diffusion equations
• Computation of saddle type slow manifolds using iterative methods
• An iterative method for the approximation of fibers in slow-fast systems