Synchronization in heterogeneous FitzHugh-Nagumo networks with hierarchical architecture

TL;DR

This paper develops a control method to achieve synchronization in heterogeneous FitzHugh-Nagumo networks, especially in hierarchical topologies, overcoming the limitations caused by node heterogeneity.

Contribution

It introduces a controller that ensures synchronization in heterogeneous networks with hierarchical architecture, extending understanding of neural network synchronization.

Findings

Controller achieves synchronization where natural dynamics fail

Hierarchical topologies are effectively synchronized with the proposed control

Stability analysis supports the controller's effectiveness

Abstract

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.