Eur. Phys. J. B (2012) 85: 407
https://doi.org/10.1140/epjb/e2012-30810-x

Regular Article

Synchronisation in networks of delay-coupled type-I excitable systems

¹ Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
² Department of Physics, Indian Institute of Technology Bombay, Mumbai, India
³ Bernstein Center for Computational Neuroscience, Humboldt-Universität zu Berlin, Philippstraße 13, 10115 Berlin, Germany
⁴ Center for Complex Network Research, Northeastern University, 110 Forsyth Street, Boston, MA 02115, USA

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

Received: 5 September 2012
Received in final form: 4 October 2012
Published online: 13 December 2012

Abstract

We use a generic model for type-I excitability (known as the SNIPER or SNIC model) to describe the local dynamics of nodes within a network in the presence of non-zero coupling delays. Utilising the method of the Master Stability Function, we investigate the stability of the zero-lag synchronised dynamics of the network nodes and its dependence on the two coupling parameters, namely the coupling strength and delay time. Unlike in the FitzHugh-Nagumo model (a model for type-II excitability), there are parameter ranges where the stability of synchronisation depends on the coupling strength and delay time. One important implication of these results is that there exist complex networks for which the adding of inhibitory links in a small-world fashion may not only lead to a loss of stable synchronisation, but may also restabilise synchronisation or introduce multiple transitions between synchronisation and desynchronisation. To underline the scope of our results, we show using the Stuart-Landau model that such multiple transitions do not only occur in excitable systems, but also in oscillatory ones.