Existence and stability of traveling pulse solutions of the FitzHugh–Nagumo equation

The FitzHugh–Nagumo model is a reaction–diffusion equation describing the propagation of electrical signals in nerve axons and other biological tissues. One of the model parameters is the ratio ϵϵ of two time scales, which takes values between 0.0010.001 and 0.10.1 in typical simulations of nerve axons. Based on the existence of a (singular) limit at ϵ=0ϵ=0, it has been shown that the FitzHugh–Nagumo equation admits a stable traveling pulse solution for sufficiently small ϵ>0ϵ>0. Here we prove the existence of such a solution for ϵ=0.01ϵ=0.01, both for circular axons and axons of infinite length. This is in many ways a completely different mathematical problem. In particular, it is non-perturbative and requires new types of estimates. Some of these estimates are verified with the aid of a computer. The methods developed in this paper should apply to many other problems involving homoclinic orbits, including the FitzHugh–Nagumo equation for a wide range of other parameter values.