Global bifurcations of a periodically forced biological oscillator

Abstract

A single brief current pulse delivered to a spontaneously beating aggregate of cardiac cells from embryonic chick heart will reset the rhythm of the aggregate. The resetting depends on both the magnitude of the stimulus and the phase in the cycle at which the stimulus is delivered. Experimental data on resetting are fitted to an analytic function. This function, in turn, is used to construct a first return or Poincaré map which can be iterated to predict the effects of periodic pulsatile stimulation for any particular combination of frequency and amplitude of the stimulation. As the stimulation strength is increased the Poincaré map changes from a monotonic circle map of degree 1 to a nonmonotonic circle map of degree 1. The bifurcations in the frequency-amplitude parameter space are determined numerically by iterating the Poincaré map, and are compared with bifurcations in a simple model map and with those experimentally observed. These systems display period-doubling, intermittent, and quasiperiodic dynamics. Universal features of the bifurcations in the frequency-amplitude parameter space are described.