CONTINUATION OF ARNOLD TONGUES IN MATHEMATICAL MODELS OF PERIODICALLY FORCED BIOLOGICAL OSCILLATORS

Periodic stimulation of spontaneously oscillating physiological rhythms has powerful effects on the intrinsic rhythm. As the frequency and amplitude of the periodic stimulus are varied, a large number of different coupling patterns are set up between the stimulus and the spontaneous oscillator. In one class of rhythms, there are a fixed number, N, of cycles of the stimulus for each M cycles of the spontaneous rhythm, and the spontaneous oscillation occurs at fixed phase (or phases) of the periodic stimulus. We call such rhythms N:M phase locking. In addition to phase-locked rhythms, it is also possible to observe irregular or aperiodic rhythms in which fixed phase relationships and regular repeating cyclic patterns are not observed. The following generalizations are applicable to a large number of experiments on periodic forcing of biological oscillators (Guevara et al., 1981; Glass et al., 1984; Petrillo and Glass, 1984).