Excitation Waves on a Minimal Small-World Model

TL;DR

This paper studies how adding a long-range link to a ring network of FitzHugh-Nagumo nodes creates complex wave patterns, revealing mechanisms behind various spatio-temporal behaviors and their dependence on network parameters.

Contribution

It introduces a minimal small-world model to analyze excitation wave dynamics and provides a detailed phase diagram of pattern regimes with analytical insights.

Findings

Identification of multiple wave behaviors such as propagation failure and period multiplication.

Dependence of wave patterns on network parameters like distance d and coupling strength.

Analytical scaling laws for critical distances in wave propagation.

Abstract

We examine traveling-wave solutions on a regular ring network with one additional long-range link that spans a distance d. The nodes obey the FitzHugh-Nagumo kinetics in the excitable regime. The additional shortcut induces a plethora of spatio-temporal behavior that is not present without it. We describe the underlying mechanisms for different types of patterns: propagation failure, period decreasing, bistability, shortcut blocking and period multiplication. For this purpose, we investigate the dependence on d, the network size, the coupling range in the original ring and the global coupling strength and present a phase diagram summarizing the different scenarios. Furthermore, we discuss the scaling behavior of the critical distance by analytical means and address the connection to spatially continuous excitable media.