Hamiltonian energy analysis of a multilayer Hindmarsh–Rose neuronal network

Complex networks require energy balance to maintain cohesion between interconnected nodes. The analysis of energy exchange inside neuronal networks is an essential aspect. This research investigates the Hamiltonian energy of a multilayer network of Hindmarsh–Rose neurons, consisting of two ring layers and a central hub that serves as a mediator. The study focuses on three types of synapses: chemical, electrical, and electrochemical. The findings indicate that synchronization cannot occur through small inter- and intra-layer couplings for any type of synapse. However, increasing these couplings causes an intra-layer energy balance. In electrical coupling scenarios, when all nodes have identical dynamics, the hub’s mean energy is approximately equal to that of the layers, and with large enough couplings, the network reaches an energy balance. In contrast, distinct hub dynamics result in remote synchronization for large enough couplings. Conversely, large enough couplings lead to distinct hub dynamics. In the case of a periodic hub, the hub’s mean energy is much larger than the layers’, and the remotely synchronized manifold of the layers is chaotic. A chaotic hub, on the other hand, has significantly lower mean energy than that of the layers, and the remote synchrony manifold of the layers has period-2 dynamics. In two other cases, chemical and electrochemical synapses, large enough couplings cause layers and hub to achieve distinct quiescent states, regardless of the nodes’ dynamics. In both cases, the hub’s energy is much greater than that of the layers.