Preprints are manuscripts made publicly available before they have been submitted for formal peer review and publication. They might contain new research findings or data. Preprints can be a draft or final version of an author's research but must not have been accepted for publication at the time of submission.
posted on 03.02.2006by S. Coombes, Andrew H. Osbaldestin
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, quasi-periodic
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.
This is a pre-print. The definitive version: COOMBES, S. and OSBALDESTIN, A.H., 2000. Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator. Physical Review E, 62(3), pp.4057-4066 Part B.