On the equivalence of phase-oscillator and integrate-and-fire models

TL;DR

This paper demonstrates a near-perfect quantitative equivalence between different oscillator models, including integrate-and-fire and phase models, in the weak-coupling limit, and shows the generality of self-consistent partial synchronization across models.

Contribution

It provides a detailed comparison showing the models' agreement and introduces a method to transform integrate-and-fire models with arbitrary pulse shapes into simpler delta-pulse models.

Findings

Models agree almost perfectly in the weak-coupling limit.

Self-consistent partial synchronization occurs across all model classes.

Integrate-and-fire models can be transformed into delta-pulse models with a suitable phase response curve.

Abstract

A quantitative comparison of various classes of oscillators (integrate-and-fire, Winfree, and Kuramoto-Daido type) is performed in the weak-coupling limit for a fully connected network of identical units. An almost perfect agreement is found, with only tiny differences among the models. We also show that the regime of self-consistent partial synchronization [SCPS] is rather general and can be observed for arbitrarily small coupling strength in any model class. As a by-product of our study, we are able to show that an integrate-and-fire model with a generic pulse shape can be always transformed into a similar model with $\delta$-pulses and a suitable phase response curve.