Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator

The response of an excitable neuron to trains of electrical spikes is relevant to the understanding of the neural code. In this paper, we study a neurobiologically motivated relaxation oscillator, with appropriately identified fast and slow coordinates, that admits an explicit mathematical analysis. An application of geometric singular perturbation theory shows the existence of an attracting invariant manifold, which is used to construct the Fenichel normal form for the system. This facilitates the calculation of the response of the system to pulsatile stimulation and allows the construction of a so-called extended isochronal map. The isochronal map is shown to have a single discontinuity and be of a type that can admit three types of response: mode-locked, quasiperiodic, and chaotic. The bifurcation structure of the system is seen to be extremely rich and supports period-adding bifurcations separated by windows of both chaos and periodicity. A bifurcation analysis of the isochronal map is presented in conjunction with a description of the various routes to chaos in this system.