Canard-Like Explosion of Limit Cycles in Two-Dimensional Piecewise-Linear Models of FitzHugh–Nagumo Type

Abstract

We investigate the mechanism of abrupt transition between small- and large amplitude oscillations in fast-slow piecewise-linear (PWL) models of FitzHugh–Nagumo (FHN) type. In the context of neuroscience, these oscillatory regimes correspond to subthreshold oscillations and action potentials (spikes), respectively. The minimal model that shows such phenomena has a cubic-like nullcline (for the fast equation) with two or more linear pieces in the middle branch and one piece in the left and right branches. Simpler models with only one linear piece in the middle branch or a discontinuity between the left and right branches (McKean model) show a single oscillatory mode. As the number of linear pieces increases, PWL models of FHN type approach smooth FHN-type models. For the minimal model we investigate the bifurcation structure; we describe the mechanism that leads to the abrupt, canard-like transition between subthreshold oscillations and spikes; and we provide an analytical way of predicting the amplitude regime of a given limit cycle trajectory which includes the approximation of the canard critical control parameter. We extend our results to models with a larger number of linear pieces. Our results for PWL-FHN-type models are consistent with similar results for smooth FHN-type models. In addition, we develop tools that are amenable for the investigation of a variety of related, and more complex, problems including forced, stochastic, and coupled oscillators.