We study synchronization in heterogeneous FitzHugh-Nagumo networks. It is well known that heterogeneities in the nodes hinder synchronization when becoming too large. Here, we develop a controller to counteract the impact of these heterogeneities. We first analyze the stability of the equilibrium point in a ring network of heterogeneous nodes. We then derive a sufficient condition for synchronization in the absence of control. Based on these results we derive the controller providing synchronization for parameter values where synchronization without control is absent. We demonstrate our results in networks with different topologies. Particular attention is given to hierarchical (fractal) topologies, which are relevant for the architecture of the brain.

We study synchronization in delay-coupled neural networks of heterogeneous nodes. It is well known that heterogeneities in the nodes hinder synchronization when becoming too large. We show that an adaptive tuning of the overall coupling strength can be used to counteract the effect of the heterogeneity. Our adaptive controller is demonstrated on ring networks of FitzHughNagumo systems which are paradigmatic for excitable dynamics but can also -depending on the system parameters -exhibit self-sustained periodic firing. We show that the adaptively tuned time-delayed coupling enables synchronization even if parameter heterogeneities are so large that excitable nodes coexist with oscillatory ones.

The problem of synchronization in networks of linear systems with nonlinear diffusive coupling and a connected undirected graph is studied. By means of a coordinate transformation, the system is reduced to the form of mean-field dynamics and a synchronization-error system. The network synchronization conditions are established based on the stability conditions of the synchronization-error system obtained using the circle criterion, and the results are used to derive the condition for synchronization in a network of neural-mass-model populations with a connected undirected graph. Simulation examples are presented to illustrate the obtained results.