On the Resonance Structure in a Forced Excitable System

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-locking 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.