Eur. Phys. J. B (2015) 88: 104
https://doi.org/10.1140/epjb/e2015-50869-y

Regular Article

Excitation waves on a minimal small-world model

¹ Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
² Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität zu Berlin, Philippstraße 13, 10115 Berlin, Germany

^a e-mail: phoevel@physik.tu-berlin.de

Received: 18 December 2014
Received in final form: 13 March 2015
Published online: 20 April 2015

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.