Theta neuron subject to delayed feedback: a prototypical model for self-sustained pulsing

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Date
2022-10-26
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Publisher
The Royal Society Publishing
Rights
(c) 2022 The Author/s
CC BY 4.0
Abstract
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.
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Keywords
neuron dynamics, self-pulsations, delay differential equations, bifurcation analysis
Citation
Laing CR, Krauskopf B. (2022). Theta neuron subject to delayed feedback: a prototypical model for self-sustained pulsing. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 478. 2266.
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