Large time behaviour and synchronization of complex networks of reaction–diffusion systems of FitzHugh–Nagumo type

Abstract

We focus on the long time behavior of complex networks of reactiondiffusion (RD) systems. We prove the existence of the global attractor and a L ∞ -bound for a network of n RD systems with d variables each. This allows us to prove the identical synchronization for general class of networks and establish the existence of a coupling strength threshold value that ensures such a 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 relatively to the number of nodes and the network topology, and discuss the link between spatial dimension and synchronization.

Propagation of Bursting Oscillations in Coupled Non-homogeneous Hodgkin–Huxley Reaction–Diffusion Systems

“…The functions H i represent excitatory nonlinear coupling (i.e. chemical synaptic coupling) between neurons, see [5,6,8,9] and references therein cited. Without loss of generality, we have set the constant C equal to 1.…”

Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type

“…As we have already pointed out, this question arises naturally from neuroscience since each neuron or group of neurons can be represented by a RD system, whereas coupling terms take account of synaptic interactions between these neurons. Note that we use the result proved here, namely the existence of the attractor and the L ∞ -bound, to study, in [4], theoretically and numerically, the synchronization phenomenon, for complex networks of RD systems. We present here results for a network of n partially diffusive systems with d equations.…”

“…graph connectivity which here, represents the synaptic connectivity) of the network is. The existence of the attractor and the L ∞ -bound appear crucially in the related article [4] in which we study the synchronization phenomena for these networks.…”

Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model

“…A huge number of studies have been devoted to complex networks of dynamical systems given by ordinary differential equations (ODE), but only a few works are studying complex networks of dynamical systems given by partial differential equations (PDE). Those studies are motivated by numerous applications of great interest, including neural networks, epidemiological networks or geographical networks (see for instance [2,5,7,8,40]). Complex networks can also be applied to ecological models, since they can reproduce the heterogeneity of biological environments fragmented by urban and industrial expansion, which threats the natural equilibrium of biodiversity [19].…”

“…Recently, the asymptotic behavior of solutions of Keller-Segel equations in network shaped domains has been studied in [23], where the convergence towards stationary solutions is investigated. In another recent paper, conditions of synchronization have been obtained in [2] for a neural network built with the FitzHugh-Nagumo reaction-diffusion system. Thus it appears essential to develop a novel approach in order to analyze the asymptotic behavior of complex networks in the case of infinite dimension and to generalize what has been proved in particular cases.…”