Noise-controlled oscillations and their bifurcations in coupled phase oscillators

Zaks, Michael A.; Neiman, Alexander; Feistel, Susanne; Schimansky‐Geier, Lutz

doi:10.1103/physreve.68.066206

Cited by 72 publications

(71 citation statements)

References 44 publications

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“…Based on previous findings and numerical observations it is reasonable to approximate the phase distribution by a Gaussian with time-dependent mean and variance [30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Cooperative behavior between oscillatory and excitable units: the peculiar role of positive coupling-frequency correlations

et al. 2014

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Abstract. We study the collective dynamics of noise-driven excitable elements, so-called active rotators. Crucially here, the natural frequencies and the individual coupling strengths are drawn from some joint probability distribution. Combining a mean-field treatment with a Gaussian approximation allows us to find examples where the infinite-dimensional system is reduced to a few ordinary differential equations. Our focus lies in the cooperative behavior in a population consisting of two parts, where one is composed of excitable elements, while the other one contains only self-oscillatory units. Surprisingly, excitable behavior in the whole system sets in only if the excitable elements have a smaller coupling strength than the selfoscillating units. In this way positive local correlations between natural frequencies and couplings shape the global behavior of mixed populations of excitable and oscillatory elements.

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“…Based on previous findings and numerical observations it is reasonable to approximate the phase distribution by a Gaussian with time-dependent mean and variance [30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Cooperative behavior between oscillatory and excitable units: the peculiar role of positive coupling-frequency correlations

et al. 2014

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“…The active rotator is a well-known, convenient phenomenological model that can describe both excitable and oscillatory dynamics. Systems of interacting active rotators have been extensively studied and their collective dynamics have been analyzed [9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Collective phase description of globally coupled excitable elements

2011

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We develop a theory of collective phase description for globally coupled noisy excitable elements exhibiting macroscopic oscillations. Collective phase equations describing macroscopic rhythms of the system are derived from Langevin-type equations of globally coupled active rotators via a nonlinear Fokker-Planck equation. The theory is an extension of the conventional phase reduction method for ordinary limit cycles to limit-cycle solutions in infinite-dimensional dynamical systems, such as the time-periodic solutions to nonlinear Fokker-Planck equations representing macroscopic rhythms. We demonstrate that the type of the collective phase sensitivity function near the onset of collective oscillations crucially depends on the type of the bifurcation, namely, it is type-I for the saddle-node bifurcation and type-II for the Hopf bifurcation.

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“…That is why excitable dynamical systems per se are in nonequilibrium and the acting noise is unbalanced with respect to dissipative forces. Hence, a variation of noise often may induce qualitative changes in the performance and functioning of excitable systems which can be characterized in terms of the dynamics of moments and their bifurcations [29,30,31,32,33,34].…”

Section: Introductionmentioning

confidence: 99%

Noise induced complexity: From subthreshold oscillations to spiking in coupled excitable systems

Zaks

Sailer

Schimansky‐Geier

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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103

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We study stochastic dynamics of an ensemble of N globally coupled excitable elements. Each element is modeled by a FitzHugh-Nagumo oscillator and is disturbed by independent Gaussian noise. In simulations of the Langevin dynamics we characterize the collective behavior of the ensemble in terms of its mean field and show that with the increase of noise the mean field displays a transition from a steady equilibrium to global oscillations and then, for sufficiently large noise, back to another equilibrium. Diverse regimes of collective dynamics ranging from periodic subthreshold oscillations to large-amplitude oscillations and chaos are observed in the course of this transition. In order to understand details and mechanisms of noise-induced dynamics we consider a thermodynamic limit N → ∞ of the ensemble, and derive the cumulant expansion describing temporal evolution of the mean field fluctuations. In the Gaussian approximation this allows us to perform the bifurcation analysis; its results are in good agreement with dynamical scenarios observed in the stochastic simulations of large ensembles.

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Noise-controlled oscillations and their bifurcations in coupled phase oscillators

Cited by 72 publications

References 44 publications

Cooperative behavior between oscillatory and excitable units: the peculiar role of positive coupling-frequency correlations

Cooperative behavior between oscillatory and excitable units: the peculiar role of positive coupling-frequency correlations

Collective phase description of globally coupled excitable elements

Noise induced complexity: From subthreshold oscillations to spiking in coupled excitable systems

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