Networks of diffusively time-delay coupled systems: Conditions for synchronization and its relation to the network topology

•Conditions for boundedness of solutions of networks of time-delay coupled systems.•Conditions for local and global synchronization of time-delay coupled systems.•All synchronization conditions are related to the network topology.

We consider networks of time-delayed diffusively coupled systems and relate conditions for synchronization of the systems in the network to the topology of the network. First we present sufficient conditions for the solutions of the time-delayed coupled systems to be bounded. Next we give conditions for local synchronization and we show that the values of the coupling strength and time-delay for which there is local synchronization in any network can be determined from these conditions. In addition we present results on global synchronization in relation to the network topology for networks of a class of nonlinear systems. We illustrate our results with examples of synchronization in networks with FitzHugh–Nagumo model neurons and Hindmarsh–Rose neurons.