Relay synchronization in a multiplex memristive neuronal network with electrical and field couplings

In complex network analysis, relay synchronization is a critical phenomenon that enables distant coordination. Relay synchronization, or remote synchronization, occurs in multilayer networks where specific layers, called relay layers, mediate synchronization between remote layers without direct connections. This paper presents a detailed investigation of a triplex memristive Hindmarsh-Rose neuronal network, where layers are connected through field couplings, and within each layer, electrical or generalized electrical synapses define the quality of neuronal connections. The results suggest that neurons can achieve complete synchronization within each layer as intralayer connections strengthen. Additionally, at lower values of intralayer and interlayer connections, neurons from different layers can coordinate their temporal behavior. Under certain conditions, relay synchronization can be observed, where neurons from remote layers synchronize while maintaining an asynchronous pattern within their respective layers. Furthermore, considering the effect of electromagnetic induction within the intralayer couplings as a generalized version of the electrical synapses, more conditions were identified under which remote layer neurons can achieve synchrony.