Rotating waves in the Theta model for a ring of synaptically connected neurons

We study rotating waves in the Theta model for a ring of synaptically-interacting neurons. We prove that when the neurons are oscillatory, at least one rotating wave always exists. In the case of excitable neurons, we prove that no travelling waves exist when the synaptic coupling is weak, and at least two rotating waves, 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 wave velocities. We also study the special case of uniform coupling, in which complete analytical results on the rotating waves can be achieved.