Renewal Approach to the Analysis of the Asynchronous State for Coupled Noisy Oscillators

TL;DR

This paper introduces a renewal theory framework to analyze the asynchronous state in coupled noisy oscillators, enabling stability analysis and extension to more complex systems, and demonstrates a Hopf bifurcation leading to synchronization.

Contribution

It presents a novel renewal approach for analyzing asynchronous states in pulse-coupled oscillators, including stability and bifurcation analysis, with extensions to systems with additional variables.

Findings

The framework effectively models the asynchronous state.

The system exhibits a super-critical Hopf bifurcation.

Synchronization emerges through bifurcation analysis.

Abstract

We develop a framework in which the activity of nonlinear pulse-coupled oscillators is posed within the renewal theory. In this approach, the evolution of inter-event density allows for a self-consistent calculation that determines the asynchronous state and its stability. This framework, can readily be extended to the analysis of systems with more state variables. To exhibit this, we study a nonlinear pulse-coupled system, where couplings are dynamic and activity dependent. We investigate stability of this system and we show it undergoes a super-critical Hopf bifurcation to collective synchronization.