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 (Erdős-Rényi) 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.