Living systems have time-evolving interactions that, until recently, could not be identified accurately from recorded time series in the presence of noise. Stankovski et al. Phys. Rev. Lett. 109, ...024101 (2012) introduced a method based on dynamical Bayesian inference that facilitates the simultaneous detection of time-varying synchronization, directionality of influence, and coupling functions. It can distinguish unsynchronized dynamics from noise-induced phase slips. The method is based on phase dynamics, with Bayesian inference of the time-evolving parameters being achieved by shaping the prior densities to incorporate knowledge of previous samples. We now present the method in detail using numerically generated data, data from an analog electronic circuit, and cardiorespiratory data. We also generalize the method to encompass networks of interacting oscillators and thus demonstrate its applicability to small-scale networks.
Structure and function go hand in hand. However, while a complex structure can be relatively safely broken down into the minutest parts, and technology is now delving into nanoscales, the function of ...complex systems requires a completely different approach. Here the complexity clearly arises from nonlinear interactions, which prevents us from obtaining a realistic description of a system by dissecting it into its structural component parts. At best, the result of such investigations does not substantially add to our understanding or at worst it can even be misleading. Not surprisingly, the dynamics of complex systems, facilitated by increasing computational efficiency, is now readily tackled in the case of measured time series. Moreover, time series can now be collected in practically every branch of science and in any structural scale—from protein dynamics in a living cell to data collected in astrophysics or even via social networks. In searching for deterministic patterns in such data we are limited by the fact that no complex system in the real world is autonomous. Hence, as an alternative to the stochastic approach that is predominantly applied to data from inherently non-autonomous complex systems, theory and methods specifically tailored to non-autonomous systems are needed. Indeed, in the last decade we have faced a huge advance in mathematical methods, including the introduction of pullback attractors, as well as time series methods that cope with the most important characteristic of non-autonomous systems—their time-dependent behaviour. Here we review current methods for the analysis of non-autonomous dynamics including those for extracting properties of interactions and the direction of couplings. We illustrate each method by applying it to three sets of systems typical for chaotic, stochastic and non-autonomous behaviour. For the chaotic class we select the Lorenz system, for the stochastic the noise-forced Duffing system and for the non-autonomous the Poincaré oscillator with quasi-periodic forcing. In this way we not only discuss and review each method, but also present properties which help to clearly distinguish the three classes of systems when analysed in an inverse approach—from measured, or numerically generated data. In particular, this review provides a framework to tackle inverse problems in these areas and clearly distinguish non-autonomous dynamics from chaos or stochasticity.
The purpose of the present study was to compare the effects of endothelium-dependent acetylcholine (ACh) and endothelium-independent sodium nitroprusside (SNP) vasodilators on the oscillatory ...components of the cutaneous blood perfusion signals in humans. The unstimulated basal blood perfusion and the blood perfusion during iontophoretically delivered ACh and SNP were measured using laser Doppler flowmetry (LDF). The wavelet transform was calculated before spectral analysis of the measured signals. In the frequency interval from 0.0095 to 1.6 Hz the LDF signal consists of oscillations with five different characteristic frequencies. In addition to the cardiac (1 Hz) and respiratory (0.3 Hz) rhythms, three other oscillations in the regions around 0.1, 0.04, and 0.01 Hz were detected. The oscillations with the different frequencies were observed in unstimulated blood flow and also during stimulation with ACh and SNP. Compared to the unstimulated blood flow, both ACh and SNP increased the mean amplitude of the total spectrum (P< 0.005 for both substances). The only significant difference between the effects of ACh and SNP was observed in the amplitude of oscillations with the frequency of around 0.01 Hz. ACh increased the absolute amplitude of this frequency to a greater extent than SNP in athletes (P= 0.03), whereas only a trend was observed in controls (P= 0.2). The relative amplitude, defined as the ratio between the absolute amplitude of a particular frequency interval and the mean amplitude of the total spectrum, was also higher for ACh compared to SNP both in controls (P= 0.008) and in athletes (P= 0.004), only for oscillations with the frequency of around 0.01 Hz. We conclude that ACh selectively influences the oscillatory component of around 0.01 Hz in the cutaneous blood perfusion signal to a greater extent than SNP. This finding indicates that endothelium-mediated vasodilatation is manifested as oscillations with a repetition time of approximately 1 min. The mechanisms for the endothelial dependency of this frequency remain to be elucidated. Our data indicate that spectral analysis based on wavelet transform of the cutaneous perfusion signal can be used clinically to investigate endothelial function. The described noninvasive method might be used to evaluate endothelial function for research, for diagnostic purposes, and maybe also to assess effects of therapy in cardiovascular diseases.
A directionality index based on conditional mutual information is proposed for application to the instantaneous phases of weakly coupled oscillators. Its abilities to distinguish unidirectional from ...bidirectional coupling, as well as to reveal and quantify asymmetry in bidirectional coupling, are demonstrated using numerical examples of quasiperiodic, chaotic, and noisy oscillators, as well as real human cardiorespiratory data.
Reconstructing Time-Dependent Dynamics Clemson, Philip; Lancaster, Gemma; Stefanovska, Aneta
Proceedings of the IEEE,
02/2016, Volume:
104, Issue:
2
Journal Article
Peer reviewed
Open access
The usefulness of the information contained in biomedical data relies heavily on the reliability and accuracy of the methods used for its extraction. The conventional assumptions of stationarity and ...autonomicity break down in the case of living systems because they are thermodynamically open, and thus constantly interacting with their environments. This leads to an inherent time-variability and results in highly nonlinear, time-dependent dynamics. The aim of signal analysis usually is to gain insight into the behavior of the system from which the signal originated. Here, a range of signal analysis methods is presented and applied to extract information about time-varying oscillatory modes and their interactions. Methods are discussed for the characterization of signals and their underlying nonautonomous dynamics, including time-frequency analysis, decomposition, coherence analysis and dynamical Bayesian inference to study interactions and coupling functions. They are illustrated by being applied to cardiovascular and EEG data. The recent introduction of chronotaxic systems provides a theoretical framework within which dynamical systems can have amplitudes and frequencies which are time-varying, yet remain stable, matching well the characteristics of life. We demonstrate that, when applied in the context of chronotaxic systems, the methods presented facilitate the accurate extraction of the system dynamics over many scales of time and space.
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.
Defining the wavelet bispectrum Newman, Julian; Pidde, Aleksandra; Stefanovska, Aneta
Applied and computational harmonic analysis,
March 2021, 2021-03-00, Volume:
51
Journal Article
Peer reviewed
Open access
Bispectral analysis is an effective signal processing tool for investigating interactions between oscillations, and has been adapted to the continuous wavelet transform for time-evolving analysis of ...open systems. However, one unaddressed question for the wavelet bispectrum is quantification of the bispectral content of an area of scale-scale space. Without this, interpretation of wavelet bispectrum computations is merely qualitative. We now overcome this limitation by providing suitable normalisations of the wavelet bispectrum formula that enable it to be treated as a density to be integrated. These are roughly analogous to the normalisation for second-order wavelet spectral densities. We prove that our definition of the wavelet bispectrum matches the traditional bispectrum of sums of sinusoids, in the limit as the frequency resolution tends to infinity. We illustrate the improved quantitative power of our definition with numerical and experimental data. We also discuss notions of bicoherence and its practical implementation.
Spectral components of heart rate variability (HRV) are determined in the time-frequency domain using a wavelet transform. Based on the finer estimation of low-frequency content enabled by the ...logarithmic resolution of the wavelet transform, corrections of spectral intervals, already defined by Fourier and model based methods, are proposed. The characteristic peaks between 0.0095 and 0.6 Hz are traced in time and four spectral intervals are defined, I (0.0095-0.021 Hz), II (0.021-0.052 Hz), III (0.052-0.145 Hz) and IV (0.145-0.6 Hz), within which peaks are located for all subjects included. These intervals are shown to be invariant regardless of the age and the state of the system. We also show that the frequency and power of the spectral components are related to age, AMI and particularly to type II diabetes mellitus.
Time–frequency representations (TFRs) of signals, such as the windowed Fourier transform (WFT), wavelet transform (WT) and their synchrosqueezed versions (SWFT, SWT), provide powerful analysis tools. ...Here we present a thorough review of these TFRs, summarizing all practically relevant aspects of their use, reconsidering some conventions and introducing new concepts and procedures to advance their applicability and value. Furthermore, a detailed numerical and theoretical study of three specific questions is provided, relevant to the application of these methods, namely: the effects of the window/wavelet parameters on the resultant TFR; the relative performance of different approaches for estimating parameters of the components present in the signal from its TFR; and the advantages/drawbacks of synchrosqueezing. In particular, we show that the higher concentration of the synchrosqueezed transforms does not seem to imply better resolution properties, so that the SWFT and SWT do not appear to provide any significant advantages over the original WFT and WT apart from a more visually appealing pictures. The algorithms and Matlab codes used in this work, e.g. those for calculating (S)WFT and (S)WT, are freely available for download.
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