On the Existence and Stability of Fast Traveling Waves in a Doubly-Diffusive FitzHugh-Nagumo System

Abstract

The FitzHugh-Nagumo equation, which was derived as a simplification of the Hodgkin-Huxley model for nerve impulse propagation, has been extensively studied as a paradigmatic activator-inhibitor system. We consider the version of this system in which two agents diffuse at an equal rate. Using geometric singular perturbation theory, we prove the existence and stability of fast traveling pulses. The stability proof makes use of the Maslov index--an invariant of symplectic geometry--to count unstable eigenvalues for the linearization about the wave. The calculation of the Maslov index is carried out by tracking the evolution of the unstable manifold of the rest state using the timescale separation. This entails a careful consideration of how the transition from fast to slow dynamics occurs in the tangent bundle over the wave. Finally, we observe in the calculation that the Maslov index lacks monotonicity in the spatial parameter, which distinguishes this application of the Maslov index from similar analyses of Hamiltonian systems.

arXiv comment3 figures

References in corpus (3)

• Opening the Maslov Box for Traveling Waves in Skew-Gradient Systems
• The Maslov and Morse indices for Schrödinger operators on $\mathbb{R}$
• A Stability Index for Traveling Waves in Activator-Inhibitor Systems

Cited by in corpus (8)

• Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions
• Travelling Waves for Reaction-Diffusion Equations Forced by Translation Invariant Noise
• Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations
• Stability of Travelling Waves for Reaction-Diffusion Equations with Multiplicative Noise
• Near-Pulse Solutions of the FitzHugh-Nagumo Equations on Cylindrical Surfaces
• Instability of small-amplitude periodic waves from fold-Hopf bifurcation
• Index theory for traveling waves in reaction diffusion systems with skew gradient structure
• The FitzHugh-Nagumo system on undulated cylinders: spontaneous symmetrization and effective system