WIAS Preprint No. 2126, (2015)

Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback


  • Klinshov, Vladimir
  • Lücken, Leonhard
  • Shchapin, Dmitry
  • Nekorkin, Vladimir
  • Yanchuk, Serhiy

2010 Mathematics Subject Classification

  • 37G15 37N20 92B25

2008 Physics and Astronomy Classification Scheme

  • 87.19.ll, 05.45.Xt, 87.19.lr, 89.75.Kd


  • Phase oscillator, delayed feedback, pulsatile feedback, jitter, degenerate bifurcation, PRC


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 oscillator's 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.

Appeared in

  • Phys. Rev. E, 92 pp. 042914/1--042914/15.

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