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Subject: PRC × WGL Institute: WIAS ×

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    Multistable jittering in oscillators with pulsatile delayed feedback
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Klinshov, Vladimir; Lücken, Leonhard; Shchapin, Dmitry; Nekorkin, Vladimir; Yanchuk, Serhiy
    Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in recent years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. At the bifurcation point numerous regimes with non-equal interspike intervals emerge. We show that the number of the emerging, so-called jittering regimes grows exponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the multi-jitter bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit.
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    Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Klinshov, Vladimir; Lücken, Leonhard; Shchapin, Dmitry; Nekorkin, Vladimir; Yanchuk, Serhiy
    Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillators phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous jittering regimes with non-equal interspike intervals (ISIs). The number of the emergent solutions increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how each periodic solution consisting of different ISIs implies the appearance of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback.