Regular Traveling Waves in a One-Dimensional Network of Theta Neurons

This paper focuses on regular traveling waves, or waves that propagate smoothly, in a one-dimensional network of theta neurons. We show that when coupling strength is sufficiently large, there exist two traveling wave solutions for this network. Moreover, the cells in the network spike more than one time after joining the wave; that is, some form of synaptic depression or other adaptive mechanism must exist to attain single-spike traveling waves. We also present results about how the behavior of solutions to this network depends on the maximal synaptic coupling strength and the wave speed. This is relevant to consideration of uniqueness and to bounding the range of coupling strengths for which waves exist.