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

We investigate the mechanism of abrupt transition between small and large amplitude oscillationsin fast-slow piecewise-linear (PWL) models of FitzHugh-Nagumo (FHN) type. In the context ofneuroscience, these oscillatory regimes correspond to subthreshold oscillations and action potentials(spikes) respectively. The minimal model that shows such phenomenon has a cubic-like nullcline (forthe fast equation) with two or more linear pieces in the middle branch and one piece in the left andright branches. Simpler models with only one linear piece in the middle branch or a discontinuitybetween the left and right branches (McKean model) show a single oscillatory mode. As the numberof linear pieces increases, PWL models of FHN type approach smooth FHN-type models. For theminimal model, we investigate the bifurcation structure, we describe the mechanism that leads tothe abrupt, canard-like transition between subthreshold oscillations and spikes, and we provide an∗Also, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology. E-mail: [email protected]