Emergent dynamics of the generalized Winfree model]{Emerging asymptotic patterns in a Winfree ensemble with higher-order couplings

TL;DR

This paper explores how higher-order couplings in the Winfree model influence emergent synchronization patterns, providing new theoretical conditions and numerical validation to understand complex collective behaviors.

Contribution

It introduces sufficient frameworks for asymptotic patterns in the Winfree model with higher-order couplings, extending analysis beyond first-order interactions.

Findings

Conditions for asymptotic patterns are independent of oscillator number

Higher-order couplings produce behaviors closer to the Peskin model

Numerical simulations validate analytical predictions

Abstract

The Winfree model is a phase-coupled synchronization model which simplifies pulse-coupled models such as the Peskin model on pacemaker cells. It is well-known that the Winfree ensemble with the first-order coupling exhibits discrete asymptotic patterns such as incoherence, locking and death depending on the coupling strength and variance of natural frequencies. In this paper, we further study higher-order couplings which makes the dynamics more close to the behaviors of the Peskin model. For this, we propose several sufficient frameworks for asymptotic patterns compared to the first-order coupling model. Our proposed conditions on the coupling strength, natural frequencies and initial data are independent of the number of oscillators so that they can be applied to the corresponding mean-field model. We also provide several numerical simulations and compare them with analytical results.