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 parame- ters is the ratio ε of two time scales, which takes values between 0.001 and 0.1 in typical simulations of nerve axons. Based on the existence of a (singular) limit at ε = 0, it has been shown that the FitzHugh–Nagumo equation admits a stable traveling pulse solution for sufficiently small ε > 0. Here we prove the existence of such a solution for ε = 0.01, both for circular axons and axons of infinite length. This is in many ways a completely dif- ferent 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.