Abstract

We study excitation waves on a Newman–Watts small-world network model of coupled excitable elements. Depending on the global coupling strength, we find differing resilience to the added long-range links and different mechanisms of propagation failure. For high coupling strengths, we show agreement between the network and a reaction-diffusion model with additional mean-field term. Employing this approximation, we are able to estimate the critical density of long-range links for propagation failure.

Highlights

  • Excitable media are well studied model systems in a variety of applications ranging from chemical [1] to electronic systems [2] and lasers [3] and from heart-muscle tissue [4] to neural systems [5, 6]

  • As a generic model of excitable dynamics we use the FitzHugh-Nagumo model [36, 37] on an undirected, unweighted network, where neighboring nodes are coupled by the difference in the activator concentrations

  • We find the same behavior as expected by the dispersion relation of Eq (6)

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Summary

Introduction

Excitable media are well studied model systems in a variety of applications ranging from chemical [1] to electronic systems [2] and lasers [3] and from heart-muscle tissue [4] to neural systems [5, 6]. In recent years dynamical systems coupled in complex network architectures have attracted a lot of attention [16, 17, 18, 19, 20] Those systems can occur in a wide variety of applications ranging from power grids [21] to biological networks [22]. Scenarios with excitable elements coupled in a chain-like one-dimensional topology have been suggested as a mechanism for the occurrence of traveling waves of activity in the visual cortex [11]. We attempt to do so by using a generic model of excitability, the well-known FitzHugh-Nagumo model [36, 37], combined with a Newman-Watts small-world architecture [17] as well as techniques from spatially continuous systems to study the behavior of excitation waves.

Dynamics
Traveling wave solutions
Wave-like solutions on small-world networks
Numerical observations
High D — continuum limit
Approximate boundary of wave propagation
Conclusion
A Excitability in the FitzHugh-Nagumo system
B Dispersion relation of traveling waves in the continuous system
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