Large time behaviour and synchronization of complex networks of reactiondiffusion systems of FitzHughNagumo type

We focus on the long-time behaviour of complex networks of reaction-diffusion (RD) systems. In a previous work, we have proved the existence of the global attractor and the L-infinity-bound for these networks. Here, we discuss the synchronization phenomenon and establish the existence of a coupling strength threshold value that ensures this synchronization. Then, we apply these results to some particular networks with different structures (i.e. different topologies) and perform numerical simulations. We found out theoretical and numerical heuristic laws for the minimal coupling strength needed for synchronization with respect to the number of nodes and the network topology. We also discuss the link between spatial heterogeneity and synchronization. Our main conclusion is that some of widespread heuristic laws known for synchronization of ordinary differential equations remain valid for networks of RD systems, i.e. networks in which each node has its own spatial heterogeneity.