Synthetic analysis of periodically stimulated excitable and oscillatory membrane models

Many excitable neuronal membranes become oscillatory when stimulated by large enough dc currents. In this paper we investigate how the transition from excitable to oscillatory regimes affects the response of the membrane to periodic pulse trains. To this end, we examine how the dynamics of periodically stimulated FitzHugh-Nagumo neuron model changes as the system switches from excitability to oscillation. We show that, despite the important change in the asymptotic dynamics of the unperturbed model, $p:q$ phase-locking (i.e., the model membrane discharges q times in p interstimulus intervals and q input-output intervals repeat periodically) regions in the stimulus period-stimulus amplitude parameter plane (Arnold tongues) change continuously when the model changes from excitable to oscillatory. We provide further evidence for the continuous change of the Arnold tongues by using an analytically tractable one-dimensional map that approximates the Poincaré map of the forced system. We argue that the smooth change in the Arnold tongues results from the fact that, despite the qualitative difference between the asymptotic dynamics of unforced excitable and oscillatory regimes, other aspects of the dynamics such as the wave form of individual action potentials, are similar in the two regimes.