Existence of Periodic Solutions of the FitzHugh-Nagumo Equations for an Explicit Range of the Small Parameter

The FitzHugh–Nagumo model describing propagation of nerve impulses in axons is given by fast-slow reaction-diffusion equations, with dependence on a parameter representing the ratio of time scales. It is well known that for all sufficiently small > 0 the system possesses a periodic traveling wave. With the help of computer-assisted rigorous computations, we prove the existence of this periodic orbit in the traveling wave equation for an explicit range ∈ (0, 0.0015]. Our approach is based on a novel method of a combination of topological techniques of covering relations and isolating segments, for which we provide a self-contained theory. We show that the range of existence is wide enough, so the upper bound can be reached by standard validated continuation procedures. In particular, for the range ∈ [1.5× 10−4, 0.0015] we perform a rigorous continuation based on covering relations and not specifically tailored to the fast-slow setting. Moreover, we confirm that for = 0.0015 the classical interval Newton–Moore method applied to a sequence of Poincaré maps already succeeds. Techniques described in this paper can be adapted to other fast-slow systems of similar structure.