Emergence of self-sustained patterns in small-world excitable media

Motivated by recent observations that long-range connections (LRC) play a role in various brain phenomena, we have observed two distinct dynamical transitions in the activity of excitable media where waves propagate both between neighboring regions and through LRC. When LRC density $p$ is low, single or multiple spiral waves are seen to emerge and cover the entire system. This state is self-sustaining and robust against perturbations. At $p=p_c^l$ the spirals are suppressed and there is a transition to a spatially homogeneous, temporally periodic state. Finally, above $p=p_c^u$, activity ceases after a brief transient.