Existence of periodic solutions of the Fitzhugh-Nagumo equations for an explicit range of the small parameter (Q2819093)

The authors study the classical slow-fast FitzHugh-Nagumo reaction-diffusion model for the propagation of nerve impulses in axons [\textit{J. Müller} and \textit{C. Kuttler}, Methods and models in mathematical biology. Deterministic and stochastic approaches. Berlin: Springer-Verlag (2015; Zbl 1331.92002)], with the singular perturbation parameter \(\varepsilon\) representing the ratio of time-scales in the model. While the existence of periodic travelling wave solutions for sufficiently small and positive \(\varepsilon\in(0,\varepsilon_0]\) is well-known, corresponding proofs are usually perturbative in nature [\textit{C. C. Conley}, Lect. Notes Phys. 38, 498--510 (1975; Zbl 0316.34063)] and do not yield explicit values for \(\varepsilon_0\).NEWLINENEWLINEHere, the authors prove the existence of a periodic orbit in the FitzHugh-Nagumo model explicitly for \(\varepsilon\in(0,\varepsilon_0]\), where \(\varepsilon_0=0.0015\). Moreover, they show that the resulting range of \(\varepsilon\)-values is sufficiently wide for its upper bound to be attainable by standard validated continuation procedures; in particular, they perform a rigorous continuation in the sub-range of \(\varepsilon\in[0.00015,0.0015]\) -- without specific recourse to the slow-fast structure of the model -- and they confirm that, for \(\varepsilon=0.0015\), the classical interval Newton-Moore method [\textit{A. Neumaier}, Interval methods for systems of equations. Cambridge etc.: Cambridge University Press (1990; Zbl 0715.65030)] can be applied successfully to a sequence of Poincaré maps. Their approach is based on computer-assisted rigorous computation and, specifically, on a novel combination of the topological techniques of covering relations [\textit{P. Zgliczyński} and \textit{M. Gidea}, J. Differ. Equations 202, No. 1, 32--58 (2004; Zbl 1061.37013)] and isolating segments [\textit{R. Srzednicki}, Zesz. Nauk. Uniw. Jagielloń. 750, Acta Math. 26, 183--190 (1987; Zbl 0662.34046)]; a self-contained exposition of the underlying theory is provided in the article. Finally, they argue that the techniques developed therein can be adapted to other systems which exhibit a similar slow-fast structure.