Chemical synaptic coupling-induced delay-dependent synchronization transitions in scale-free neuronal networks

Chemical synaptic couplings are more common than electric(gap junction) connections in neurons.In this paper,the firing synchronizations induced by chemical synaptic coupling in chemically delayed scale-free networks of modified Hodgkin-Huxley neurons have been studied.It was found that the chemical coupling-induced synchronization transitions are delay-dependent and much different for various delay lengths.In the absence of delay,the neurons exhibit a transition from chaotic bursting(CB) to bursting synchronization(BS) with desynchronized spikes in each burst;for smaller delay lengths,the firing evolves from CB to spiking synchronization(SS),but for larger delay lengths,there are transitions from CB to intermittently multiple SS behaviors.These findings show that the chemical coupling-induced firing synchronization transitions strongly depend on the chemical delay lengths,and intermittently multiple SS can only occur for larger delay lengths.This result would be helpful for better understanding the joint roles of the chemical coupling and chemical delay in the firing activity of the neurons.

Standing Wave Solutions in Nonhomogeneous Delayed Synaptically Coupled Neuronal Networks

The authors establish the existence and stability of standing wave solutions of a nonlinear singularly perturbed system of integral differential equations and a non- linear scalar integral differential equation. It will be shown that there exist six standing wave solutions （（u（x,t）,w（x,t）） = （U（x）,W（x）） to the nonlinear singularly perturbed system of integral differential equations. Similarly, there exist six standing wave so- lutions u（x,t） = U（x） to the nonlinear scalar integral differential equation. The main idea to establish the stability is to construct Evans functions corresponding to several associated eigenvalue problems.