On Phase Models for Oscillators

Oscillators have been a research focus for decades in many disciplines such as electronics and biology. The time keeping capability of oscillators is best described by the scalar quantity phase. Phase computations and equations describing phase dynamics have been useful in understanding oscillator behavior and designing oscillators least affected by disturbances such as noise. In this paper, we present a unified theory of phase equations assimilating the work that has been done in electronics and biology for the last seven decades. We first provide a review of isochrons, which forms the basis of a generalized phase notion for oscillators. We present a general framework for phase equations and derive an exact phase equation that is practically unusable but facilitates the derivation of usable ones based on linear (already known) and quadratic (new and more accurate) approximations for isochrons. We discuss the utility of these phase equations in performing (semi) analytical phase computations and also describe simpler and more accurate phase computation schemes. Numerical experiments on several examples are presented comparing the accuracy of the various phase equations and computation schemes described in this paper.