Cluster synchronization in an ensemble of neurons interacting through chemical synapses.

In networks of periodically firing spiking neurons that are interconnected with chemical synapses, we analyze a cluster state, where an ensemble of neurons are subdivided into a few clusters, in each of which neurons exhibit perfect synchronization. To clarify stability of the cluster state, we decompose linear stability of the solution into two types of stabilities, stability of a mean state and stabilities of clusters. Computing Floquet matrices for these stabilities, we clarify the total stability of the cluster state for any type of neuron and any strength of interaction even if the size of networks is infinitely large. First, we apply this stability analysis to investigating synchronization in the large ensemble of integrate-and-fire (IF) neurons. In one-cluster state we find the change of stability of a cluster, which elucidates that in-phase synchronization of IF neurons occurs with only inhibitory synapses. Then, we investigate entrainment of two clusters of IF neurons with different excitability. IF neurons with fast decaying synapses show low entrainment capability, which is explained by a pitchfork bifurcation appearing in the two-cluster state with change of synapse decay time constant. Second, we analyze a one-cluster state of Hodgkin-Huxley (HH) neurons and discuss the difference in synchronization properties between IF neurons and HH neurons.

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