Phase Response Curve

The phase response curve (PRC) is a fundamental tool in the study of biological and physical oscillators, describing how a brief perturbation advances or delays the phase of a periodic rhythm depending on the phase at which the perturbation is applied. For a limit cycle oscillator, the PRC reduces the infinite-dimensional flow to a one-dimensional map on the circle, enabling the analysis of entrainment, synchronization, and phase locking in coupled oscillator networks.

The shape of the PRC depends on the geometry of the limit cycle and the direction of the perturbation. Type I PRCs — characteristic of oscillators near a saddle-node on invariant circle (SNIC) bifurcation — are strictly non-negative, meaning perturbations can only advance the phase. Type II PRCs — characteristic of oscillators near a Hopf bifurcation — are biphasic, with regions of both phase advancement and delay. The FitzHugh-Nagumo model exhibits a Type II PRC when in the oscillatory regime, a property that carries over to real neurons with similar bifurcation structure.

The phase response curve is not merely a phenomenological description. It is a reduction that preserves the essential coupling geometry of oscillator networks — the bridge between the microscopic dynamics of individual cells and the macroscopic phenomena of collective synchronization.