Inter-layer coupling effects on synchronization in multiplex neuronal networks with different topologies

Synchronization in neuronal networks is fundamental to both normal and pathological brain activity. This study investigates the synchronization of two-layer multiplex networks of Hindmarsh-Rose neurons, focusing on the effects of unidirectional and bidirectional inter-layer coupling. A particularly interesting aspect explored here is how adding a control layer—identical to the target network—and connecting each node in the first layer to its corresponding node in the second layer can enhance synchronization. This approach is non-trivial because simply duplicating the network as an additional layer actually reduces the overall network density compared to the original single-layer network, leading to a smaller second eigenvalue of the Laplacian matrix (λ₂), which typically worsens synchronization rather than improving it. We find that unidirectional coupling between a synchronous first layer and an asynchronous second layer induces synchronization in the second layer, even without the coupling strength needed for synchronization of the isolated second layer. This highlights the strong influence of different layers on each other in driving synchronization. In contrast, bidirectional coupling allows both layers to influence each other, resulting in strikingly different network dynamics. For instance, we observed that when the inter-layer coupling strength exceeds a certain threshold, the system shifts toward oscillation death, where neuronal activity is suppressed throughout the network. Furthermore, we found that one layer does not need to be synchronous to help the other layer with intra-layer synchronization. Even when both layers are initially asynchronous, their bidirectional interactions can lead to intra-layer synchronization in both layers. Additionally, phase synchronization across the entire multiplex network emerged under both unidirectional and bidirectional inter-layer coupling cases, but complete synchronization was not achieved due to non-identical layers and the nature of inter-layer coupling. These results emphasize the significant role of inter-layer coupling directionality and the asynchronous-to-synchronous influence between layers in controlling network behavior.