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Time delay, bifurcation and synchronization in brain networks
We consider a neural field model for a brain network which is a network of Wilson-Cowan nodes with homeostatic adjustment of the inhibitory coupling strength and time delayed, excitatory coupling. Without time delay, the system has been show to exhibit rich dynamics including oscillations, mixed-mode oscillations, and chaos. We explore how synchronization of the nodes depends on both the connectivity structure of the network and the presence of time delays in the connections between nodes. We show that Hopf bifurcations induced by the excitatory coupling, the connectivity structure and the delay lead to different patterns of phase-locked oscillations, either synchronized or desynchronized. We show how interaction between different Hopf bifurcations can lead to complex solutions, such as intermittent synchronization.