Synchrony and Asynchrony for Neuronal Dynamics Defined on Complex Networks

We describe and analyze a model for a stochastic pulse-coupled neuronal network with many sources of randomness: random external input, potential synaptic failure, and random connectivity topologies. We show that different classes of network topologies give rise to qualitatively different types of synchrony: uniform (ErdAs-R,nyi) and small-world networks give rise to synchronization phenomena similar to that in all-to-all networks (in which there is a sharp onset of synchrony as coupling is increased); in contrast, in scale-free networks the dependence of synchrony on coupling strength is smoother. Moreover, we show that in the uniform and small-world cases, the fine details of the network are not important in determining the synchronization properties; this depends only on the mean connectivity. In contrast, for scale-free networks, the dynamics are significantly affected by the fine details of the network; in particular, they are significantly affected by the local neighborhoods of the hubs in the network.