Complicated, but rational, phase-locking responses of an ideal time-base oscillator

Abstract

A model is advanced to account for the complicated phase-locking responses of a prototypical time-base oscillator driven by a train of excitory pulses. The model predicts that the intervals of stability for different ratios of phase locking should occur in specific sequences as functions of the parameters. With respect to variation of one parameter (a strength of coupling), these sequences comprise the odd convergents and associated semiconvergents of the simple continued fraction for the fixed parameter (a natural frequency ratio). With respect to variation of this second parameter, the sequences comprise a previously undescribed arithmetical series, which is closely related to the Farey series. An experimental test of the model is described. The results are by and large confirmatory. However, there are narrow intervals on the parameter space where pseudo-randomness versus the effects of true noise are an unresolved issue.