A geometric approach to Phase Response Curves and its numerical computation through the parameterization method

TL;DR

This paper introduces a geometric and numerical method based on the parameterization approach to compute Phase Response Curves (PRCs) and amplitude response functions (ARCs) in oscillatory systems, especially for large perturbations.

Contribution

The paper develops a novel geometric framework and numerical technique for calculating PRCs and ARCs using the parameterization method, extending applicability to large perturbations.

Findings

The method accurately computes PRCs for classical neuroscience models.

It links invariant curve properties to PRC waveform changes.

The approach extends to large amplitude perturbations beyond traditional limits.

Abstract

The Phase Response Curve (PRC) is a tool used in neuroscience that measures the phase shift experienced by an oscillator due to a perturbation applied at different phases of the limit cycle. In this paper we present a new approach to PRCs based on the parameterization method. The underlying idea relies on the construction of a periodic system whose corresponding stroboscopic map has an invariant curve. We demonstrate the relationship between the internal dynamics of this invariant curve and the PRC, which yields a method to numerically compute the PRCs. Moreover, we link the existence properties of this invariant curve as the amplitude of the perturbation is increased with changes in the PRC waveform and with the geometry of isochrons. The invariant curve and its dynamics will be computed by means of the parameterization method consisting of solving an invariance equation. We show that the method to compute the PRC can be extended beyond the breakdown of the curve by means of introducing a modified invariance equation. The method also computes the amplitude response functions (ARCs) which provide information on the displacement away from the oscillator due to the effects of the perturbation. Finally, we apply the method to several classical models in neuroscience to illustrate how the results herein extend the framework of computation and interpretation of the PRC and ARC for perturbations of large amplitude and not necessarily pulsatile.