Influence of Time Delay in Signal Transmission on Synchronization between Two Coupled FitzHugh-Nagumo Neurons

In this paper, the energy method is employed to analytically investigate the influence of time delay in signal transmission on synchronization between two coupled FitzHugh-Nagumo (FHN) neurons. Unlike pre-existing methods that deal with synchronization problems, our major idea is to consider the change rate of the energy of the synchronization error system, since the original system's synchronization is equivalent to the disappearance of the energy of the error system. In rewriting the original coupled system in the corresponding energy coordinates based on the energy method, we find that the change rate of energy of the error system can be divided into two parts (periodic and non-periodic). The synchronization criterion for the original system can then be obtained by letting the non-periodic part of the change rate of the energy be less than zero. The correctness of the analysis is illustrated with numerical simulations. Our analytical results show that time delay in signal transmission has very significant effects on the synchronization between two FHN neurons. If the time delay in signal transmission is not taken into account in the two coupled FHN neurons, synchronous spikes cannot be achieved in the system for any given coupling strength. By adjusting the value of the time delay in signal transmission, the neural system can freely switch between neural rest and synchronous spikes. This means that time delay in signal transmission is crucial for the occurrence of synchronous spikes in the FHN neural system, which contributes to our understanding of the interaction between neurons. We analytically show the influence of the time delay on the synchronization between two FHN neurons, which was seldom considered by other researchers.