Abstract
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 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 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.
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