Beyond global synchrony: Equivalence between Kuramoto oscillators and Wilson-Cowan model for large scale brain networks

Oscillations are ubiquitous in the nervous system, from single neurons to whole brain networks. The link between neural oscillations and cognition and behaviour is actively investigated by cognitive and computational neuroscience. Biophysically motivated computational models, such as Wilson-Cowan [W-C], have contributed to the understanding of the dynamics of oscillatory neuronal networks. W-C describes mean field interactions between excitatory/inhibitory neural populations. Using Malkin's Theorem we show the equivalence, under certain conditions, between the W-C model and the Kuramoto oscillators, with the advantage that the latter comprises fewer parameters. We construct a thirty-two nodes network of Kuramoto oscillators, coupled using two options: homogeneous (same strength in all connections) and heterogeneous (different values of coupling strengths). We characterized the Kuramoto network synchrony by measuring the Kuramoto order parameter, and the frequency spectrum of each oscillator using Welch's periodograms. We characterized those two features as a function of number of nodes, their intrinsic frequency, and the global coupling parameter. Using variable intrinsic frequency between oscillators, we found that as we increase the number of nodes of the system, the global synchrony becomes dependent on the global coupling strength. Also, as global coupling increases, the frequency spectrum of each oscillator converges to the mean intrinsic frequency, similar to the case when the intrinsic frequency is equal for all nodes. We conclude that the Kuramoto order parameter alone is not enough of characterizing network dynamics, and that a distribution of intrinsic node frequency is important to generate the sort of network dynamics observed in brain imaging data.