Effective phase dynamics of noise-induced oscillations in excitable systems

Schwabedal, Justus T. C.; Pikovsky, Arkady

doi:10.1103/physreve.81.046218

Cited by 29 publications

(40 citation statements)

References 30 publications

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“…Recently, Schwabedal and Pikovsky [28,29] studied somewhat similar issues on effective phase description of noiseinduced oscillations in excitable systems. While their study treated a one-body problem for an ensemble of statistically independent excitable elements at the microscopic level, the present study focuses on globally coupled noisy excitable elements exhibiting collective oscillations, which can be considered a one-body problem at the macroscopic level.…”

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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Section: Introductionmentioning

confidence: 99%

Collective phase description of globally coupled excitable elements

2011

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“…Finally, we emphasize that our corresponding considerations are closely related to the problem of finding an optimal phase definition for a chaotic system. [80][81][82][83][84] Achieving a better understanding of the linkage between both geometric and optimal phase approaches will be a subject of future research. …”

Section: -11mentioning

confidence: 99%

Phase coherence and attractor geometry of chaotic electrochemical oscillators

Zou

Donner

Wickramasinghe

et al. 2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Chaotic attractors are known to often exhibit not only complex dynamics but also a complex geometry in phase space. In this work, we provide a detailed characterization of chaotic electrochemical oscillations obtained experimentally as well as numerically from a corresponding mathematical model. Power spectral density and recurrence time distributions reveal a considerable increase of dynamic complexity with increasing temperature of the system, resulting in a larger relative spread of the attractor in phase space. By allowing for feasible coordinate transformations, we demonstrate that the system, however, remains phase-coherent over the whole considered parameter range. This finding motivates a critical review of existing definitions of phase coherence that are exclusively based on dynamical characteristics and are thus potentially sensitive to projection effects in phase space. In contrast, referring to the attractor geometry, the gradual changes in some fundamental properties of the system commonly related to its phase coherence can be alternatively studied from a purely structural point of view. As a prospective example for a corresponding framework, recurrence network analysis widely avoids undesired projection effects that otherwise can lead to ambiguous results of some existing approaches to studying phase coherence. Our corresponding results demonstrate that since temperature increase induces more complex chaotic chemical reactions, the recurrence network properties describing attractor geometry also change gradually: the bimodality of the distribution of local clustering coefficients due to the attractor’s band structure disappears, and the corresponding asymmetry of the distribution as well as the average path length increase.

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“…While the derivation of optimum phase variables has been recently attracted considerable interest [63][64][65][66], we restrict our attention in this work to the standard analytical signal approach. Here, a scalar signal x(t) is extended to the complex plane using the Hilbert transform…”

Section: Quantifying Phase Coherence Of Chaotic Oscillators a Pmentioning

confidence: 99%

Geometric and dynamic perspectives on phase-coherent and noncoherent chaos

Zou

Donner

Kurths

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Statistically distinguishing between phase-coherent and noncoherent chaotic dynamics from time series is a contemporary problem in nonlinear sciences. In this work, we propose different measures based on recurrence properties of recorded trajectories, which characterize the underlying systems from both geometric and dynamic viewpoints. The potentials of the individual measures for discriminating phase-coherent and noncoherent chaotic oscillations are discussed. A detailed numerical analysis is performed for the chaotic Rössler system, which displays both types of chaos as one control parameter is varied, and the Mackey-Glass system as an example of a time-delay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral-and screw-type chaos, a common route from phase-coherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given. PACS numbers:Oscillatory processes can be frequently observed in natural and technological systems. Often, the corresponding dynamics is not strictly periodic, but shows more complex temporal variability patterns characterized by a fast divergence of trajectories with arbitrarily close initial conditions [1-3]. There are numerous examples of such chaotic oscillators for which long-term predictions of amplitudes and phases are not possible. Therefore, studying their phase dynamics has recently attracted particular interest, e.g., regarding the process of phase synchronization between different coupled systems [4,5]. However, most existing methods suitable for this purpose require the explicit definition of an appropriate phase variable, which can become a non-trivial problem in the case of noncoherent chaotic oscillations. Therefore, studying the phase coherence properties of chaotic systems has become an important problem in both theoretical and experimental studies [6]. In this work, we propose some methods based on the concept of recurrences in phase space, which allow studying complementary aspects of chaotic oscillators relating to the geometric structure of, and the dynamics on the attractor. Specifically, we derive a detailed characterization of changes of the geometric structure of complex systems in phase space with varying control parameter, which accompany transitions from phase-coherent to noncoherent dynamics.

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Effective phase dynamics of noise-induced oscillations in excitable systems

Cited by 29 publications

References 30 publications

Collective phase description of globally coupled excitable elements

Collective phase description of globally coupled excitable elements

Phase coherence and attractor geometry of chaotic electrochemical oscillators

Geometric and dynamic perspectives on phase-coherent and noncoherent chaos

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