Phase Response Curves and the Role of Coordinates

TL;DR

This paper reveals that phase response curves depend on the full coordinate system and introduces a coordinate-free definition, enabling better reconstruction and analysis of PRCs in biological systems like neurons.

Contribution

It provides a coordinate-free framework for defining PRCs and demonstrates methods to reconstruct PRCs considering full coordinate dependence, enhancing analysis accuracy.

Findings

PRC depends on full coordinate choice

Coordinate-free PRC definition simplifies analysis

Reconstruction of PRCs via delay coordinates is possible

Abstract

The "Phase Response Curve" (PRC) is a common tool used to analyze phase resetting in the natural sciences. We make the observation that the PRC with respect to a coordinate $y\in\mathbb{R}$ actually depends on the full choice of coordinates $(x,y)$, $x\in\mathbb{R}^d$. We give a coordinate-free definition of the PRC making this observation obvious. We show how by controlling $y$, using delay coordinates of $y$, and postulating the dynamics of $x$ as a function of $x$ and $y$, we can sometimes reconstruct the PRC with respect to the $(x,y)$ coordinates. This suggests a means for obtaining the PRC of, e.g., a neuron via a voltage clamp.