Heterogeneity of time delays determines synchronization of coupled oscillators

TL;DR

This paper develops a theoretical framework to analyze how the heterogeneity of transmission delays in network-coupled oscillators affects their synchronization patterns, with applications to brain oscillations.

Contribution

It introduces a method to decompose complex delay distributions into patterns, simplifying stability analysis of synchronization in delay-structured networks.

Findings

Delay heterogeneity leads to various phase-locked states.

Analytical stability boundaries for different synchronization patterns.

Applicable to understanding brain oscillation dynamics.

Abstract

Network couplings of oscillatory large-scale systems, such as the brain, have a space-time structure composed of connection strengths and signal transmission delays. We provide a theoretical framework, which allows treating the spatial distribution of time delays with regard to synchronization, by decomposing it into patterns and therefore reducing the stability analysis into the tractable problem of a finite set of delay-coupled differential equations. We analyse delay-structured networks of phase oscillators and we find that, depending on the heterogeneity of the delays, the oscillators group in phase-shifted, anti-phase, steady, and non-stationary clusters, and analytically compute their stability boundaries. These results find direct application in the study of brain oscillations.