Detecting couplings between interacting oscillators with time-varying basic frequencies: Instantaneous wavelet bispectrum and information theoretic approach

Jamsek, Janez; Paluš, Milan; Stefanovska, Aneta

doi:10.1103/physreve.81.036207

Cited by 53 publications

(44 citation statements)

References 59 publications

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“…When treated in an inverse approach such systems are usually considered as stochastic. In an attempt to cope with the problem, several methods for the inverse approach were introduced, including wavelet-based decomposition [14], bispectral analysis [15], harmonic detection [16] and phase coherence [17], and Bayesian-based inference [18] methods.…”

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confidence: 99%

Chronotaxic Systems: A New Class of Self-Sustained Nonautonomous Oscillators

2013

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Nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies have often been treated as stochastic, inappropriately. We therefore formulate them as a new class and discuss how they generate complex behavior. We show how to extract the underlying dynamics, and we demonstrate that it is simple and deterministic, thus paving the way for a diversity of new systems to be recognized as deterministic. They include complex and nonautonomous oscillatory systems in nature, both individually and in ensembles and networks. DOI: 10.1103/PhysRevLett.111.024101 PACS numbers: 05.45.Xt, 05.65.+b, 05.90.+m, 89.75.Fb Dynamical systems are generally seen as being either deterministic or stochastic. The advent of dynamical chaos several decades ago attracted much attention and illustrated that even complex dynamics can be deterministic. Dynamical systems can also be classified as autonomous (self-contained) or nonautonomous (subject to external influences). In reality, nonautonomous systems are the commoner, but they are far harder to treat. Until now they were mostly treated as stochastic or, alternatively, attempts were made to reformulate them as autonomous. Neither approach captures the characteristic properties of these systems.In this Letter we propose a new class of nonautonomous systems and name them chronotaxic to characterize oscillatory systems with time-varying, but stable, amplitudes and frequencies. Nonautonomous oscillatory systems appear in various fields of research including neuroscience [1,2], cardiovascular dynamics [3], climate [4,5] and evolutionary science [6], as well as in complex systems and networks [7][8][9]. Although we are witnessing a rapid development of the theory of nonautonomous [10,11] and random dynamical systems [12,13], nonautonomous oscillatory systems with stable but time-varying characteristic frequencies have to date not been addressed. When treated in an inverse approach such systems are usually considered as stochastic. In an attempt to cope with the problem, several methods for the inverse approach were introduced, including wavelet-based decomposition [14] Common to all these systems is that they are oscillatory, have stable amplitudes, and frequencies that are resistant to external perturbations. The variety of systems with these characteristics suggests that their dynamics are generated from a universal basis. To date, the description of stable oscillatory dynamics has been based on the model of autonomous self-sustained limit cycle oscillators [23]. While this model provides stable amplitude dynamics, frequencies of oscillations within this model can be easily changed even by weakest external perturbations.The new class of nonautonomous oscillatory dynamical systems that we now propose account for such dynamics. The novelty of these systems is that not only are the amplitude dynamics stable but also the frequencies of the oscillations are time dependent and stable-i.e., their timedependent values cannot be easily altered by external perturbations. Their c...

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Chronotaxic Systems: A New Class of Self-Sustained Nonautonomous Oscillators

2013

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“…In a theoretical treatment the coupling strength is clearly the scaling parameter of the coupling functions. There is great interest in being able to evaluate the coupling strength for which many effective methods have been designed (Mormann et al, 2000;Rosenblum and Pikovsky, 2001;Paluš and Stefanovska, 2003;Marwan et al, 2007;Bahraminasab et al, 2008;Staniek and Lehnertz, 2008;Chicharro and Andrzejak, 2009;Smirnov and Bezruchko, 2009;Jamšek, Paluš, and Stefanovska, 2010;Faes, Nollo, and Porta, 2011;Sun, Taylor, and Bollt, 2015). The dominant direction of influence, i.e., the direction of the stronger coupling, corresponds to the directionality of the interactions.…”

Section: Coupling Strength and Directionalitymentioning

confidence: 99%

Coupling functions: Universal insights into dynamical interaction mechanisms

et al. 2017

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The dynamical systems found in nature are rarely isolated. Instead they interact and influence each other. The coupling functions that connect them contain detailed information about the functional mechanisms underlying the interactions and prescribe the physical rule specifying how an interaction occurs. A coherent and comprehensive review is presented encompassing the rapid progress made recently in the analysis, understanding, and applications of coupling functions. The basic concepts and characteristics of coupling functions are presented through demonstrative examples of different domains, revealing the mechanisms and emphasizing their multivariate nature. The theory of coupling functions is discussed through gradually increasing complexity from strong and weak interactions to globally coupled systems and networks. A variety of methods that have been developed for the detection and reconstruction of coupling functions from measured data is described. These methods are based on different statistical techniques for dynamical inference. Stemming from physics, such methods are being applied in diverse areas of science and technology, including chemistry, biology, physiology, neuroscience, social sciences, mechanics, and secure communications. This breadth of application illustrates the universality of coupling functions for studying the interaction mechanisms of coupled dynamical systems.

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“…In the time-frequency domain, wavelet phase coherence can be used to monitor phase relationships over time and frequency by utilising the phase information obtained from the continuous wavelet transform [15,17]. In a similar way, couplings between oscillators can be detected and quantified using wavelet bispectrum [22]. The ability of these methods to directly take into account the time-varying characteristics of data makes them ideal for application to nonautonomous systems.…”

Section: Inverse Approaches To Nonautonomous Dynamical Systemsmentioning

confidence: 99%

Detecting Chronotaxic Systems from Single-Variable Time Series with Separable Amplitude and Phase

Lancaster

Clemson

Suprunenko

et al. 2015

Entropy

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The recent introduction of chronotaxic systems provides the means to describe nonautonomous systems with stable yet time-varying frequencies which are resistant to continuous external perturbations. This approach facilitates realistic characterization of the oscillations observed in living systems, including the observation of transitions in dynamics which were not considered previously. The novelty of this approach necessitated the development of a new set of methods for the inference of the dynamics and interactions present in chronotaxic systems. These methods, based on Bayesian inference and detrended fluctuation analysis, can identify chronotaxicity in phase dynamics extracted from a single time series. Here, they are applied to numerical examples and real experimental electroencephalogram (EEG) data. We also review the current methods, including their assumptions and limitations, elaborate on their implementation, and discuss future perspectives.

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Detecting couplings between interacting oscillators with time-varying basic frequencies: Instantaneous wavelet bispectrum and information theoretic approach

Cited by 53 publications

References 59 publications

Chronotaxic Systems: A New Class of Self-Sustained Nonautonomous Oscillators

Chronotaxic Systems: A New Class of Self-Sustained Nonautonomous Oscillators

Coupling functions: Universal insights into dynamical interaction mechanisms

Detecting Chronotaxic Systems from Single-Variable Time Series with Separable Amplitude and Phase

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