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

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 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 saddlenode 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.

Pulse-adding of temporal dissipative solitons: resonant homoclinic points and the orbit flip of case B with delay

“…Figure 2(a) reveals that these two one-pulse TDS are part of the same family of periodic orbits connected through a saddle-node bifurcation for small τ , where the (vivid) blue curve folds back on itself. Notice also that the slope along this family is approximately one, as it is characteristic for one pulse-TDS with scaling T/τ ≈ 1 for large τ as seen in experiments [18,19,38,49,63,67] and theory [33,50,72].…”

“…We have identified the link between codimension-two homoclinic bifurcations to a real saddle and the existence of bound two-pulse TDSs in DDEs. This link presents a novel pathway into the analysis of TDS in time-delayed feedback systems with large delays [18,19,33,38,41,45,49,50,56,63,[65][66][67]. Such analysis is challenging as systems with large delay can exhibit a high degree of multi-existence and multi-stability of various types of TDS (also bound multipulses), as exemplified by the Yamada model for a semiconductor laser with delayed feedback [63], which is potentially governed by a codimension-two homoclinic bifurcation point.…”

“…where f is a sufficiently smooth function in both arguments, are an important class of models for complex dynamical phenomena and the formation of spatiotemporal patterns [70] in various fields of science [16,60], including optics [66] and neuroscience [33]. Although not explicitly depending on spatial variables, time-delay systems often arise from feedback loops considering the finite speed of propagation or transport of a physical quantity along a distance.…”

“…In analogy to (and to differentiate from) periodic travelling pulses in spatially-extended systems, which are commonly referred to as (dissipative) solitons in physics [46,48], temporally localised solutions in DDEs are referred to as temporal dissipative solitons (TDSs) [72]. TDSs have been observed experimentally, for example, in semiconductor lasers and optical resonators [18,19,38,49,63], electrically coupled excitable biological cells [67], and various mathematical models of physical processes [33,41,45,50,56,65,66]. Of particular interest in spatially extended systems have been the underlying mechanisms that cause the existence of multi-peak travelling pulses [17,69].…”