Stability of Remote Synchronization in Star Networks of Kuramoto Oscillators

TL;DR

This paper investigates the stability of remote synchronization in star networks of Kuramoto oscillators, revealing the importance of coupling symmetry and phase shifts in maintaining stable synchronization, with theoretical and numerical validation.

Contribution

It provides new analytical conditions for the stability of remote synchronization in directed star networks of Kuramoto oscillators, considering coupling symmetry and phase delays.

Findings

Symmetry of coupling strengths is crucial for stable remote synchronization.

Phase shifts due to delays affect the stability conditions.

Numerical simulations confirm the theoretical stability criteria.

Abstract

Synchrony of neuronal ensembles is believed to facilitate information exchange among cortical regions in the human brain. Recently, it has been observed that distant brain areas which are not directly connected by neural links also experience synchronization. Such synchronization between remote regions is sometimes due to the presence of a mediating region connecting them, e.g., \textit{the thalamus}. The underlying network structure of this phenomenon is star-like and motivates us to study the \textit{remote synchronization} of Kuramoto oscillators, {modeling neural dynamics}, coupled by a directed star network, for which peripheral oscillators get phase synchronized, remaining the accommodating central mediator at a different phase. We show that the symmetry of the coupling strengths of the outgoing links from the central oscillator plays a crucial role in enabling stable remote synchronization. We also consider the case when there is a phase shift in the model which results from synaptic and conduction delays. Sufficient conditions on the coupling strengths are obtained to ensure the stability of remotely synchronized states. To validate our obtained results, numerical simulations are also performed.