Phase reduction of a limit cycle oscillator perturbed by a strong amplitude-modulated high-frequency force

TL;DR

This paper develops a phase reduction method for limit cycle oscillators under strong amplitude-modulated high-frequency forces, deriving conditions for entrainment and optimal perturbation waveforms, demonstrated on model systems.

Contribution

It introduces a new effective phase response curve and derives an optimal waveform for minimal power entrainment of oscillators.

Findings

Entrainment depends on the sign of the effective phase response curve.

Higher (lower) envelope frequency entrains the oscillator if the curve is positive (negative).

Optimal perturbation waveforms can be derived using Pontryagin's maximum principle.

Abstract

The phase reduction method for a limit cycle oscillator subjected to a strong amplitude-modulated high-frequency force is developed. An equation for the phase dynamics is derived by introducing a new, effective phase response curve. We show that if the effective phase response curve is everywhere positive (negative), then an entrainment of the oscillator to an envelope frequency is possible only when this frequency is higher (lower) than the natural frequency of the oscillator. Also, by using the Pontryagin maximum principle, we have derived an optimal waveform of the perturbation that ensures an entrainment of the oscillator with minimal power. The theoretical results are demonstrated with the Stuart-Landau oscillator and model neurons.