The Effect Of Time-delay On The Bursting Firing Pattern Of The System With Multi-time Scale

The biological nerve system is a highly nonlinear complex system, due to the major difference between its physiological cycle activity and the circumstances it faces, the dynamical activity of neuronal system is frequently characterized by the multi-time scale. In addition, the existence of synaptic gaps, through which the information communication among the neurons is completed, make the time-delay of information delivery unavoidable, hence the system is also a time-delay system.The coupled Hindmarsh-Rose (HR) neuron system is a nonlinear system with fast-slow time scale. It can generate bursting activity, emulate the action potential and frequency characteristic, and thereby can be treated as a reasonable neuron model to perform dynamical analysis. Taking the time-delay underling the neuronal system into account, we investigated in this paper the effect of time-delay on the bursting activity in the system with fast-slow time scale by the coupled HR model. Our study will, to some extent, have guiding significance in understanding the neurons’physiological response and mutual action in that bursting is the major firing pattern of the nerve system.This paper mainly contains these five parts as follows:The first section is the introduction. The second section presents basic knowledge relevant to the study of this paper and summarizes the research progress related with our subject. In the section three, we primarily consider the HR excitable system with chemical delayed coupling. For this system, we analyzed the nonlinear dynamical activity of its stationary point, and based on the two-parameter plane composed of time-delay and coupling strength, we simulated in the distinct delay regions the excitability and synchronous pattern transition of the coupled neuronal system. In the section four, we chiefly researched the general HR coupled system. In this study, we based on the same two-parameter plane explored the synchronization and synchronous pattern transition of the coupled system under the collective effect of the delay and coupling strength. The results of these two parts just mentioned above reveal that delay and coupling strength can respectively and collectively results in the regular variation of the synchronization and synchronous pattern transition of the coupled HR system whether the system is excitable or general, reflecting the nonlinearity and complexity of the nerve system. The last section is the conclusion and prospect of this paper.Synchronous pattern is a common firing pattern in the nerve system and also one of the characteristics of the nonlinear dynamical system. Even though our results can lead us to comprehend the effect of interior system parameter on the exterior physiological reaction, it is still necessary and urgent to be resolved for some significant problems such as the dynamical nature of the synchronization transition of which we still have less knowledge.