Frequency-dependent organization of the brain's functional network through delayed-interactions.

The structure of the brain network exhibits modularity at multiple spatial scales. The effect of the modular structure on the brain dynamics has been the focus of several studies in recent years but many aspects remain to be explored. For example, it is not well-known how the delays in the transmission of signals between the neurons and the brain regions interact with the modular structure to determine the brain dynamics. In this paper, we show an important impact of the delays on the collective dynamics of brain networks with modular structure; that is, the degree of the synchrony between different brain regions depends on the oscillating frequency. In particular, we show that when increasing the frequency of the nodes the network transits from a global synchrony state to an asynchronous state, through a transition region over which the local synchrony inside the modules is stronger than the global synchrony. When the delays depend on the distance between the nodes, the modular structure of different spatial scales appears in the correlation matrix over different specific frequency bands, so that, finer spatial modular structures reveal in higher frequency bands. The results are corroborated by a simple theoretical argument and elaborated by simulations on several simplified modular networks and the connectome with different spatial resolutions.