  We consider a single theta neuron with delayed self-feedback in the form of a
Dirac delta function in time. Because the dynamics of a theta neuron on its own
can be solved explicitly -- it is either excitable or shows self-pulsations --
we are able to derive algebraic expressions for existence and stability of the
periodic solutions that arise in the presence of feedback. These periodic
solutions are characterized by one or more equally spaced pulses per delay
interval, and there is an increasing amount of multistability with increasing
delay time. We present a complete description of where these self-sustained
oscillations can be found in parameter space; in particular, we derive explicit
expressions for the loci of their saddle-node bifurcations. We conclude that
the theta neuron with delayed self-feedback emerges as a prototypical model: it
provides an analytical basis for understanding pulsating dynamics observed in
other excitable systems subject to delayed self-coupling.
