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

TL;DR

This paper investigates how long-range connections influence wave patterns in small-world excitable media, revealing two critical transitions from spiral waves to homogeneous activity and then to activity cessation as LRC density increases.

Contribution

It identifies and characterizes two dynamical transitions in excitable media driven by varying long-range connection density, highlighting their impact on wave propagation and system stability.

Findings

Low LRC density supports self-sustained spiral waves.

Transition at critical LRC density suppresses spirals, leading to homogeneous oscillations.

High LRC density causes activity to cease after transients.

Abstract

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.