Phase Response Curves (PRCs)

Abstract

Phase response functions are the central tool in the mathematical analysis of pulse-coupled oscillators. When an oscillator receives a brief input pulse, the phase response function specifies how its phase is altered, as a function of the phase at which the input arrives. When the pulse is weak, it is customary to linearize the dependence of the response on pulse strength. The result is called the infinitesimal phase response function. These ideas have been used extensively in theoretical biology and also in some areas of engineering. I give an example in which the infinitesimal phase response function predicts that two oscillators, as they exchange pulses back and fourth, will converge to synchrony. Using the true phase response, I prove this prediction to be false for all positive interaction strengths. For short, the analogue of the Hartman-Grobman theorem that one might, at first sight, expect to hold is invalid.