  The FitzHugh-Nagumo model describing propagation of nerve impulses in axon is
given by fast-slow reaction-diffusion equations, with dependence on a parameter
$\epsilon$ representing the ratio of time scales. It is well known that for all
sufficiently small $\epsilon>0$ the system possesses a periodic traveling wave.
With aid of computer-assisted rigorous computations, we prove 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 novel method of
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 $\epsilon \in
[1.5 \times 10^{-4}, 0.0015]$ we perform a rigorous continuation based on
covering relations and not specifically tailored to the fast-slow setting.
Moreover, we confirm that for $\epsilon=0.0015$ the classical interval
Newton-Moore method applied to a sequence of Poincar\'e maps already succeeds.
Techniques described in this paper can be adapted to other fast-slow systems of
similar structure.
