Collective Behavior in Neuronal Networks with Electrical and Hybrid Synapses: Frequency Responses of Oscillations

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We study synchronization through the combination of electrical synapse and hybrid synapse, and analyse frequency responses in a network constituted of two subnetworks (layers). We prove the onset of synchronization for two coupled nondelayed neurons when we set properly the coupling strength of electrical and chemical synapses, and neurotransmitter parameters of chemical synapse with constant initial conditions. The chemical synapse takes into consideration inhibitory and excitatory behaviors of the neuron. We further prove that for random initial conditions of few sets of identical nodes which constitute a network with two layers, global synchronization is possible for specific values of coupling strength and inner matrix configuration. To ensure great accuracy of the global synchronization, we add a controller and for all settings of electrical and chemical parameters, we always observe global synchronization in both layers and then, the associated network. Five different network topologies were used for the analysis. We finally verify the robustness of the global synchronization by using an external sinusoidal stimuli and observe that the control action is very powerful to maintain the global synchronization dynamic of the network. We observe equally that the instantaneous frequency can toggle between delta oscillations, awake state of the brain, up to gamma oscillations which are the frequencies of intensive activities of the brain.