Periodic travelling waves in the theta model for synaptically connected neurons

We study periodic travelling waves in the Theta model for a linear continuum of synaptically-interacting neurons. We prove that when the neurons are oscillatory, at least one periodic travelling of every wave number always exists. In the case of excitable neurons, we prove that no periodic travelling waves exist when the synaptic coupling is weak, and at least two periodic travelling waves of each wave-number, a `fast' one and a `slow' one, exist when the synaptic coupling is sufficiently strong. We derive explicit upper and lower bounds for the `critical' coupling strength as well as for the wave velocities. We also study the limits of large wave-number and of small wave-number, in which results which are independent of the form of the synaptic-coupling kernel can be obtained. Results of numerical computations of the periodic travelling waves are also presented.