Beyond . a pacemaker ’ s entrainment limit : phase walk-through

maker's entrainment limit: phase walk-through. occurs when a pacemaker is externally stimulated at too high a rate, beyond the entrainment limit. One may then observe phase walk-through: the pacemaker and stimulus phase dif'ference, rather than lock to a fixed pattern, may cycle repeatedly through all values. By means of a simple but nonlinear one-variable model we describe qualitatively the properties of entrainment, loss of entrainment, and phase walk-through for rhythmically stimulated pacemakers. We obtain an expression for the cycle length (beat period) of the repetitive phase walk-through. This shows, in contrast to the simplest treatment, that in general beat frequency is not a simple linear function of the difference between the pacemaker and stimulus frequencies. Our results are illustrated in the context of a model experimental pacemaker, the rhythmically flashing firefly. We discuss how these modeling results apply in a much more general setting. desynchronization; phase-locking; biological oscillators; beat frequency; firefly NUMEROUS EXAMPLES (2, 5, 13, 15, 16) illustrate the ability of an external periodic stimulus to entrain a. biological pacemaker. One-for-one entrainment is typically observed over a range of stimulus frequencies that includes the pacemaker's intrinsic frequency. When the stimulus frequency is tuned beyond the pacemaker's entrainment limit, desynchronization may occur. In many cases this loss of entrainment is only transient, and the pacemaker may again lock into a fixed-phase relationship with the stimulus but not in a one-for-one fashion. In other cases, e.g., for a weaker stimulus, de-synchronization may persist. The pacemaker and stimulus phase difference, rather than lock to a fixed simple pattern, may cycle repeatedly through all values. This has been called phase walk-through (5). We will focus on this latter phendmenon and present, a simple but nonlin-ear, one-variable model that qualitatively describes the properties of entrainment, loss of entrainment, and phase walk-through for rhythmically stimulated pacemakers. Moreover we obtain a quantitative expression R102 (Eq. 4), for the cycle length (beat period) of the repetitive phase walk-through. We illustrate these results in the context of a model experimental pacemaker, the rhythmically flashing firefly (2, 3, 5). The modeling results we present are robust. They apply in a much more general setting for the case of a weak periodic stimulus acting on a strongly attracting oscillator. Since this treatment does not include the special case of a weakly nonlinear oscil-lator that is weakly driven, we do not find the desyn-chronization phenomenon of " phase-trapping " (see 7). …