Firing Patterns In Coupled Neural System With Time-Delay

Time-delay is ubiquitous and it makes the finite dimensional dynamical system into the infinite dimensional system. Time-delay systems exhibit more complex nonlinear dynamic characteristics. So, the effects of time-delay on neural synchronization have received considerable attention of researchers. Currently, the research mainly focuses on the time-delay on the synchronous behavior of neural network, but the effects of time-delay on the firing patterns in two coupled neurons are rarely investigated. Therefore, it is important significance to investigate the transition mechanism of the firing patterns in the time-delay coupled neuronal system with the change of the interior parameters or external stimuli.In this paper, using the theory knowledge of nonlinear dynamics and the mathematical tools (such as MATLAB, xppall), we investigate transition mechanism of firing activities under the effects of time-delay. When the coupled neural systems reach complete synchronization, we study the effects of time-delay on the firing patterns by the numerical simulation. Firstly, based on the coupled Hindmarsh-Rose neurons achieved almost complete synchronization under a certain coupling strength without time-delay, we study the effects of time-delay on the firing patterns in nonidentical coupled Hindmarsh-Rose system. Secondly, based on the Morris-Lecar neuron exhibiting different firing patterns with the change of reversal potential of Ca2+, we explore the dynamics behaviors of the nonidentical coupled Morris-Lecar neurons with time-delay, and find the effects of time-delay on the firing patterns when the coupled system reach the almost complete synchronization.The results show that the nonidentical electrically coupled neurons can achieve almost complete synchronization under a certain coupling strength. As the time-delay changing within a certain region, the coupled neural systems exhibit different firing patterns. With the time-delay increased gradually, time-delay not only can induce the spiking in the bursting of the coupled HR neural system, but also can promote the coupled ML neural system exhibit the transition from the bursting synchronization to the spiking synchronization, and the transition is periodic. These results wili contribute to understand the important influence of time-delay on the firing pattern in the coupled neural system.