  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.
