Patterns of Neural Activity in Networks with Complex Connectivity

An understanding of emergent dynamics on complex networks requires investigating the interplay between the intrinsic dynamics of the node elements and the connectivity of the network in which they are embedded. In order to address some of these questions in a specific scenario of relevance to the dynamical states of neural ensembles, we have studied the collective behavior of excitable model neurons in a network with small-world topology. The small-world network has local lattice order, but includes a number of randomly placed connections that may provide connectivity shortcuts. This topology bears a schematic resemblance to the connectivity of the cerebral cortex, in which neurons are most strongly coupled to nearby cells within fifty to a hundred micrometers, but also make projections to cells millimeters away. We find that the dynamics of this small-world network of excitable neurons depend mostly on both the density of shortcuts and the delay associated with neuronal projections. In the regime of low shortcut density, the system exhibits persistent activity in the form of propagating waves, which annihilate upon collision and are spawned anew via the re-injection of activity through shortcut connections. As the density of shortcuts reaches a critical value, the system undergoes a transition to failure. The critical shortcut density results from matching the time associated with a recurrent path through the network to an intrinsic recovery time of the individual neurons. Furthermore, if the delay associated with neuronal interactions is sufficiently long, activity reemerges above the critical density of shortcuts. The activity in this regime exhibits long, chaotic transients composed of noisy, large-amplitude population bursts.