| The study of networks of coupled elements is vital to many branches of science, e.g. neural networks, social networks or metabolic networks, to name a few. Examining collective phenomena in such networks on an abstract level can give understanding and inspiration to these fields. Of those collective phenomena, synchronization has been thoroughly studied in our group, whereas the aspect of 'traveling' patterns on complex networks is just starting to receive attention [1,2]. We investigate the FitzHugh-Nagumo model, a paradigmatic model of excitable dynamics, on a complex network. Unlike in the reaction-diffusion case, it is difficult to define the notion of a traveling wave or spot on a generic complex network. We approach the problem by considering a small-world construction starting from a ring, where in analogy to a 1D-reaction-diffusion system a traveling wave is well-defined. We also investigate the effect of additional nonlocal coupling given by a kernel function. Traveling patterns on this ring show sensitivity to the 'small-world' perturbations, i.e., adding long-range links, with a well-defined threshold for their death. [1] Roxin, A., Riecke, H. and Solla, S. A., Self-sustained activity in a small-world network of excitable neurons, Phys. Rev. Lett. 92, 198101 (2004). [2] Sinha, S., Saramäki, J. and Kaski, K., Emergence of self-sustained patterns in small-world excitable media, Phys. Rev. E 76, 015101(R), (2007). |
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