An Operational Definition of Phase Characterizes the Transient Response of Perturbed Limit Cycle Oscillators

Phase reduction is an essential tool used in the study of perturbed limit cycle oscillators. Usually, phase is defined with respect to some asymptotic limit, often making phase reduction's ability to predict transient behavior during the approach to a limit cycle inadequate. In this work, we present an operational notion of phase, defined with respect to a distinct feature of the oscillator, for example, the timing of a periodically firing neuron's action potential. We derive relationships between standard and operational phase reduction in order to make comparisons between these two approaches. In theoretical and numerical examples, we show how operational phase reduction accurately predicts both the asymptotic and transient behavior due to perturbations without relying on the notion of higher order phase response curves which are valid only for specific perturbations. Furthermore, we develop a strategy for direct measurement of the necessary terms of the operational phase reduction when the full dynamical equations are unknown.