Wave Propagation in Excitable Random Networks with Spatially Constrained Connections

Abstract

We study wave propagation in excitable Greenberg-Hastings cellular automaton on random (Erdos-Renyi) graph, where connections are spatially constrained within radius r_c. We show that governing equation resolved by parabolic Fisher-Kolmogorov equation fails to describe wave speed, but a hyperbolic reaction-diffusion equation provides adequate wave speed for arbitrary mean network degree k. In more general case, wave speed depends rather on ratio of network moments (second moment/first moment). The wave speed is strikingly similar in different network topologies, including constant-degree, exponential and power-law degree distributions. Inspired by problem of firing propagation in networks of electrically coupled pyramidal neurons, our results may suggest spatial mean-field solution for a wide class of similar problems, such as spread of epidemics or information through real-world networks of arbitrary topology.