Ordered Dynamics Of Neuronal Networks With Partially Coupled Time Delays

Neurons are the most basic structural and functional units of the biological neuronal network.The time lag effect is inevitable in the neuronal system due to the limited speed of information transmission and the existence of synaptic gaps.And the propagation speed of neuronal information is different,which depends on the type of neuron,the size of the diameter and the length of the axon.It may thus be inferred that the information transmission between some neurons is instantaneous or with some negligible delay,which makes there is only partial neuronal connections are delayed.And the case can be referred to as partial coupling delay.Under the influence of the concentration fluctuation of the calcium,sodium and potassium ions,the distribution of electromagnetic fields inside and outside of nerve cells changes correspondingly.That is to say,there is electromagnetic induction in the neuronal system.At the same time,neurons are generally subjected to various noises.The neuronal firing activity is closely related to the complex environmental inside and outside of the neuronal system.Therefore,this paper,based on the neuronal dynamics theory and stochastic simulation method,studies the ordered firing dynamics of a neuronal network with coupling delay,especially partial coupling delay,under the random noise and electromagnetic induction.The main contents and conclusions of this paper are as follows:1.Considering the characteristics of modules in cortical neuronal systems,we construct a modular neuronal network model that is closer to the real neuronal system.The modular neuronal network is composed of two subnetworks,which exhibit small-world and scale-free features,respectively.This chapter explores the effect of partial coupling delay on the spatiotemporal dynamics of a modular neuronal network consisting of Rulkov neurons.Numerical simulation results show that when there is no partial coupling delay in the neuronal network,Gaussian white noise with appropriate intensity can induce coherence resonance in a modular neuronal network,which occurs largely independent of the inter-coupling strength.Then we introduce partial coupling delay in the coupling,and find that multiple spatiotemporal orders can be induced by partial coupling delay.Moreover,the study also reveals that the phenomenon of multiple spatiotemporal orders could be the result of a locking between the length of coupling delay and the inherent oscillation period of the modular neuronal network.Finally,the study also show that the multiple spatiotemporal orders are independent of the changes of the probability of delayed connection as well as the key parameters of network architecture such as the rewiring probability,the average degree of small-world subnetwork,the initial nodes of scale-free subnetwork and the size of the modular neuronal network.2.Due to the existence of electromagnetic induction in the neuronal system,on the basis of the first part,this part first constructs a NW small-world neuronal network composed of FitzHugh-Nagumo neurons under electromagnetic induction.And the irregular electromagnetic radiation is simulated by Gaussian white noise.Then by virtue of analyzing coherence resonance and spatial synchronization,focuses on exploring the effect of electromagnetic radiation and coupling delay on the ordered firing patterns of neurons.Numerical simulation results show that when there is no coupling delay in the neuronal network,for the positive feedback gain of the induced current there is a moderate noise intensity that can make the ordered firing patterns appear.However,for the negative feedback gain of the induced current,the ordered firing patterns can occur at a wide range of suitable noise intensity.Then,the coupling delay,especially partial coupling delay is introduced in the coupling,the results show that the noise-induced ordered firing patterns can appear intermittently.Furthermore,it is further confirmed that the ordered firing patterns is not affected by the average degree of neuronal network and the change of the probability of delayed connection.