Periodic solutions for a pair of delay-coupled excitable theta neurons

We consider a pair of identical theta neurons in the excitable regime, each coupled to the other via a delayed Dirac delta function with the same delay. This simple network can support different periodic solutions, and we concentrate on two important types: those for which the neurons are perfectly synchronous, and those where the neurons are exactly half a period out of phase and fire alternatingly. Owing to the specific type of pulsatile feedback, we are able to determine these solutions and their stability analytically. More specifically, (infinitely many) branches of periodic solutions of either type are created at saddle-node bifurcations, and they gain stability at symmetry-breaking bifurcations when their period as a function of the delay is at its minimum. We also determine the respective branches of symmetry-broken periodic solutions and show that they are all unstable. We demonstrate by considering smoothed pulse-like coupling that the special case of the Dirac delta function can be seen as a sort of normal form: the basic structure of the different periodic solutions of the two theta neurons is preserved, but there may be additional changes of stability along the different branches.