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Jittering regimes of two spiking oscillators with delayed coupling

Vladimir Klinshov

Vladimir Klinshov

Institute of Applied Physics of the Russian Academy of SciencesNizhny Novgorod, Russia
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Klinshov, Vladimir
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Oleg Maslennikov

Oleg Maslennikov

Institute of Applied Physics of the Russian Academy of SciencesNizhny Novgorod, Russia
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Maslennikov, Oleg
 and 
Vladimir Nekorkin

Vladimir Nekorkin

Institute of Applied Physics of the Russian Academy of SciencesNizhny Novgorod, Russia
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Nekorkin, Vladimir
  
Mar 14, 2016
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Published Online: Mar 14, 2016
Page range: 197 - 206
Received: Nov 12, 2015
Accepted: Mar 11, 2016
DOI: https://doi.org/10.21042/AMNS.2016.1.00015
Keywords
© 2016 Vladimir Klinshov, Oleg Maslennikov, Vladimir Nekorkin, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Dynamics of the oscillators with pulse delayed coupling and derivation of the discrete maps. Dots depict the moments of spikes emission. (a) Single oscillator with delayed feedback and (b) its dynamics. Each spike produced arrives to the oscillator after delay time τ. Tj are the inter-spike intervals. (c) Two oscillators with mutual delayed coupling and (d-e) their dynamics. A spike produced by each oscillator arrives to another one after delay time τ. (d) Near-in-phase dynamics, δj are the time lags between the oscillators. (c) Near-antiphase dynamics, Sj are cross-spike intervals.
Dynamics of the oscillators with pulse delayed coupling and derivation of the discrete maps. Dots depict the moments of spikes emission. (a) Single oscillator with delayed feedback and (b) its dynamics. Each spike produced arrives to the oscillator after delay time τ. Tj are the inter-spike intervals. (c) Two oscillators with mutual delayed coupling and (d-e) their dynamics. A spike produced by each oscillator arrives to another one after delay time τ. (d) Near-in-phase dynamics, δj are the time lags between the oscillators. (c) Near-antiphase dynamics, Sj are cross-spike intervals.

Fig. 2

The PRC for κ = 0.1, ε = 0.1, q = 40. The stars depict points with Z′(φ) = −1.
The PRC for κ = 0.1, ε = 0.1, q = 40. The stars depict points with Z′(φ) = −1.

Fig. 3

The bifurcation diagram of two oscillators with mutual delayed coupling. The observed ISIs of the established dynamical regimes are plotted versus the time delay. Blue lines stand for near-in-phase, red for near-antiphase regimes. Theoretically predicted regular regimes are plotted by thin lines, solid for stable and dashed for unstable regimes. Diamonds depict saddle-node or period-doubling bifurcations, while stars depict multi-jitter bifurcations. Thick dots depict the numerical results. Note that almost everywhere the stable branches predicted by the theory are covered by numerically observed dots.
The bifurcation diagram of two oscillators with mutual delayed coupling. The observed ISIs of the established dynamical regimes are plotted versus the time delay. Blue lines stand for near-in-phase, red for near-antiphase regimes. Theoretically predicted regular regimes are plotted by thin lines, solid for stable and dashed for unstable regimes. Diamonds depict saddle-node or period-doubling bifurcations, while stars depict multi-jitter bifurcations. Thick dots depict the numerical results. Note that almost everywhere the stable branches predicted by the theory are covered by numerically observed dots.

Fig. 4

Examples of jittering regimes of two oscillators: (a) antiphase jittering at τ = 1.95 and (b) in-phase jittering at τ = 2.4. The upper panels show the temporal dynamics of the ISIs for both oscillators, the bottom panels show the moments of spike emission. Note that in the case of in-phase jittering the ISIs for the both oscillators coincide, which means that they remain fully synchronized.
Examples of jittering regimes of two oscillators: (a) antiphase jittering at τ = 1.95 and (b) in-phase jittering at τ = 2.4. The upper panels show the temporal dynamics of the ISIs for both oscillators, the bottom panels show the moments of spike emission. Note that in the case of in-phase jittering the ISIs for the both oscillators coincide, which means that they remain fully synchronized.
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