Cluster Synchronization of Kuramoto Oscillators and Brain Functional Connectivity

Abstract

The recent progress of functional magnetic resonance imaging techniques has unveiled that human brains exhibit clustered correlation patterns of their spontaneous activities. It is important to understand the mechanism of cluster synchronization phenomena since it may reflect the underlying brain functions and brain diseases. In this paper, we investigate cluster synchronization conditions for networks of Kuramoto oscillators. The key analytical tool that we use is the method of averaging, and we provide a unified framework of stability analysis for cluster synchronization. The main results show that cluster synchronization is achieved if (i) the inter-cluster coupling strengths are sufficiently weak and/or (ii) the natural frequencies are largely different among clusters. Moreover, we apply our theoretical findings to empirical brain networks. Discussions on how to understand brain functional connectivity and further directions to investigate neuroscientific questions are provided.