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

The FitzHugh-Nagumo model describing nerve impulse propagation in an axon is given by a slow-fast reaction-diffusion equation depending on a timescale separation parameter $\epsilon$. It is well known that for $\epsilon>0$ small enough the system possesses a periodic traveling wave. With aid of computer-assisted rigorous computations we are able to show the existence of this periodic orbit in the traveling wave equation for an explicit range $\epsilon \in (0, 0.0015]$. Our approach is based on a combination of topological techniques of isolating segments and covering relations, and we focus on making the range of existence wide enough so the upper bound can be reached by the standard rigorous continuation procedures. In particular, for the range $\epsilon \in [1.5*10^{-4}, 0.0015]$ we are able to perform a rigorous continuation based on covering relations and not specifically tailored to the slow-fast nature of the system. Moreover, for the parameter upper bound $\epsilon=0.0015$ the interval Newton-Moore method for proving the existence of the orbit already succeeds.