Partial Phase Cohesiveness in Networks of Networks of Kuramoto Oscillators

Partial, instead of complete, synchronization has been widely observed in various networks, including, in particular, brain networks. Motivated by data from human brain functional networks, in this article, we analytically show that partial synchronization can be induced by strong regional connections in coupled subnetworks of Kuramoto oscillators. To quantify the required strength of regional connections, we first obtain a critical value for the algebraic connectivity of the corresponding subnetwork using the incremental two-norm. We then introduce the concept of the generalized complement graph, and obtain another condition on the node strength by using the incremental $\infty$-norm. Under these two conditions, regions of attraction for partial phase cohesiveness are estimated in the forms of the incremental two- and $\infty$-norms, respectively. Our result based on the incremental $\infty$-norm is the first known criterion that applies to noncomplete graphs. Numerical simulations are performed on a two-level network to illustrate our theoretical results; more importantly, we use real anatomical brain network data to show how our results may contribute to a better understanding of the interplay between anatomical structure and empirical patterns of synchrony.

I. Introduction

Neuronal synchronization across cortical regions of human brain, which has been widely detected through recording and analyzing brain waves, is believed to facilitate communication among neuronal ensembles [1], and only closely correlated oscillating neuronal ensembles can exchange information effectively [2]. In healthy human brain, it is frequently observed that only a part of its cortical regions are synchronized [3], and such a phenomenon is commonly referred to as partial phase cohesiveness or partial synchronization. In contrast, in pathological brain of a patient, such as an epileptic, excessive synchronization of neural activities takes place across the brain [4]. These observations suggest that healthy brain has powerful regulation mechanisms that are not only able to render synchronization, but also capable of preventing unnecessary synchronization among neuronal ensembles. Partly motivated by these experimental studies, researchers are interested in theoretically studying cluster or partial synchronization [5]–[8] and chimera states [9], even though analytical results are much more difficult to obtain, whereas analytical results for complete synchronization are ample, e.g., [10]–[12].