Phase Response Curves of Coupled Oscillators

Many real oscillators are coupled to other oscillators and the coupling can affect the response of the oscillators to stimuli. We investigate phase response curves (PRCs) of coupled oscillators. The PRCs for two weakly coupled phase-locked oscillators are analytically obtained in terms of the PRC for uncoupled oscillators and the coupling function of the system. Through simulation and analytic methods, the PRCs for globally coupled oscillators are also discussed.

💡 Research Summary

The paper investigates how coupling between oscillators modifies their phase response curves (PRCs), which quantify the phase shift induced by a brief external stimulus. While PRCs are well‑studied for isolated or weakly interacting units, many natural and engineered systems consist of large ensembles of mutually coupled oscillators whose collective dynamics can differ dramatically from the sum of their parts. The authors address this gap by first deriving an analytical expression for the PRC of two weakly coupled, phase‑locked oscillators, and then extending the analysis to globally coupled networks using mean‑field theory.

Model formulation
Each oscillator’s phase (\theta_i) obeys a standard phase‑reduction equation:
\