Synchronization among neurons is critical for many processes in the nervous system, ranging from the processing of sensory information to the onset of pathological conditions such as epilepsy. Here, we study synchronization in an array of neurons, each modeled by a set of nonlinear ordinary differential equations. We find that an array of 20x20 coupled neurons undergoes a series of alternating low and high synchronization states, as measured by phase-locking and frequency entrainment, as the coupling constant is tuned. The role of long-range connections in inducing "small-world networks" has recently been of great interest in many physical and biological problems. Since long-range connections do exist in the brain, we investigated the role of such connections in our neural array. Introducing a biologically realistic percentage of long-range connections has no significant effect on synchronization. We find that it is rather the type of coupling and the total number of connections that determine the synchronization state of the array. We also show that some coupling conditions can lead to frustration in the system, resulting from an inability to simultaneously satisfy conflicting phase requirements. This frustration leads to a drift in the overall behavior of the network, which may offer an explanation for transitions between different types of neural oscillations observed experimentally.
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