On the resonance structure in a forced excitable system

James C. Alexander, Eusebius J. Doedel, Hans G. Othmer

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The dynamics of forced excitable systems are studied analytically and numerically with a view toward understanding the resonance or phase-locking structure. In a singular limit the system studied reduces to a discontinuous flow on a two-torus, which in turn gives rise to a set-valued circle map. It is shown how to define rotation numbers for such systems and derive properties analogous to those known for smooth flows. The structure of the phase-locking regions for a Fitzhugh-Nagumo system in the singular limit is also analyzed. A singular perturbation argument shows that some of the general results persist for the nonsingularly-perturbed system, and some numerical results on phase-locking in the forced Fitzhugh-Nagumo equations illustrate this fact. The results explain much of the phase-looking behavior seen experimentally and numerically in forced excitable systems, including the existence of threshold stimuli for phase-locking. The results are compared with known results for forced oscillatory systems.

Original languageEnglish (US)
Pages (from-to)1373-1418
Number of pages46
JournalSIAM Journal on Applied Mathematics
Issue number5
StatePublished - Jan 1 1990


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