Browsing by Author "Krauskopf B"

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Periodic solutions for a pair of delay-coupled excitable theta neurons
(The Royal Society, 2025-05-28) Laing CR; Krauskopf B
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.
Theta neuron subject to delayed feedback: a prototypical model for self-sustained pulsing
(The Royal Society Publishing, 2022-10-26) Laing CR; Krauskopf B
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 the 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.