Regular Single-Spike Traveling Waves In A One-Dimesional Network Of Theta Neurons

In this paper, a one-dimensional network of theta neurons is studied from two aspects: the movement characteristics of neurons and modeling differential equations. At first, based on the characteristics of single-spike traveling waves, we approach two hypotheses for constructing the model and three conditions which single-spike traveling waves must satisfy. The first hypothesis is that the effect between neurons is transmitted through synapse in solitary direction, the second one is that the effect can just last for in finite time period. Then we select specific decaying weight function J(x) of the distance between two neurons in the network, and approximate the synaptic gates(t) by function α (t). Then by using shooting method we prove that there are two single-spike traveling solutions to the model equation. At the end, we discuss parameter dependence of solutions and extend the results of existence to general cases. This is useful for neuronal network research.