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Author: Nekorkin, Vladimir ×

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    Embedding the dynamics of a single delay system into a feed-forward ring
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Klinshov, Vladimir; Shchapin, Dmitry; Yanchuk, Serhiy; Wolfrum, Matthias; D'Huys, Otti; Nekorkin, Vladimir
    We investigate the relation between the dynamics of a single oscillator with delayed selffeedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where stability of periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example we demonstrate how the complex bifurcation scenario of simultaneously emerging multi-jittering solutions can be transferred from a single oscillator with delayed pulse feedback to multi-jittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHughNagumo type.
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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.