FIG 5
(a) The CMI as a function of time lag (black) together with the surrogate ranges given by the surrogate mean ±1.96σ (gray) from 30 permutation surrogate realizations. For case (1), Fig. 4, the CMI is applied here with Δ τ í µí¼ 1,2 according to Eq. (18).
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In the natural world, the properties of interacting oscillatory systems are not constant, but evolve or fluctuating continuously in time. Thus, the basic frequencies of the interacting oscillators are time varying, which makes the system analysis complex. For studying their interactions we propose a complementary approach combining wavelet bispectr...
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Context 1
... using the phase increment according to Eq. (18) in CMI I[í µí¼ 1 (t) ; Δ τ í µí¼ 2 (t) | í µí¼ 2 (t)] and I[í µí¼ 2 (t) ; Δ τ í µí¼ 1 (t) | í µí¼ 1 (t)], we have a more sensitive measure. Results for the first 400 s are shown in Fig. 5. The CMI is clearly above the surrogate range in the case when the second oscillator influences the first one, Fig. 5(a), whereas the opposite CMI is within the surrogates, Fig. 5(b). In spite of the presence of phase synchronization and intermittent coupling the causality is inferred ...
Context 2
... the phase increment according to Eq. (18) in CMI I[í µí¼ 1 (t) ; Δ τ í µí¼ 2 (t) | í µí¼ 2 (t)] and I[í µí¼ 2 (t) ; Δ τ í µí¼ 1 (t) | í µí¼ 1 (t)], we have a more sensitive measure. Results for the first 400 s are shown in Fig. 5. The CMI is clearly above the surrogate range in the case when the second oscillator influences the first one, Fig. 5(a), whereas the opposite CMI is within the surrogates, Fig. 5(b). In spite of the presence of phase synchronization and intermittent coupling the causality is inferred ...
Context 3
... (t) ; Δ τ í µí¼ 2 (t) | í µí¼ 2 (t)] and I[í µí¼ 2 (t) ; Δ τ í µí¼ 1 (t) | í µí¼ 1 (t)], we have a more sensitive measure. Results for the first 400 s are shown in Fig. 5. The CMI is clearly above the surrogate range in the case when the second oscillator influences the first one, Fig. 5(a), whereas the opposite CMI is within the surrogates, Fig. 5(b). In spite of the presence of phase synchronization and intermittent coupling the causality is inferred ...
Citations
... Surrogate data testing provides a versatile framework applicable to signals that arise from any physical system, allowing the exploration of fundamental questions about the system. Moreover, incorporating information dynamic measures, such as IS and MIR, into these methods enables the testing of specific hypotheses related to the observed data (Jamšek et al., 2010;Cliff et al., 2021). In typical surrogate analysis methods, a null hypothesis is formulated, assuming the absence of a specific characteristic to be tested in the observed data. ...
The increasing availability of time series data depicting the evolution of physical system properties has prompted the development of methods focused on extracting insights into the system behavior over time, discerning whether it stems from deterministic or stochastic dynamical systems. Surrogate data testing plays a crucial role in this process by facilitating robust statistical assessments. This ensures that the observed results are not mere occurrences by chance, but genuinely reflect the inherent characteristics of the underlying system. The initial process involves formulating a null hypothesis, which is tested using surrogate data in cases where assumptions about the underlying distributions are absent. A discriminating statistic is then computed for both the original data and each surrogate data set. Significantly deviating values between the original data and the surrogate data ensemble lead to the rejection of the null hypothesis. In this work, we present various surrogate methods designed to assess specific statistical properties in random processes. Specifically, we introduce methods for evaluating the presence of autodependencies and nonlinear dynamics within individual processes, using Information Storage as a discriminating statistic. Additionally, methods are introduced for detecting coupling and nonlinearities in bivariate processes, employing the Mutual Information Rate for this purpose. The surrogate methods introduced are first tested through simulations involving univariate and bivariate processes exhibiting both linear and nonlinear dynamics. Then, they are applied to physiological time series of Heart Period (RR intervals) and respiratory flow (RESP) variability measured during spontaneous and paced breathing. Simulations demonstrated that the proposed methods effectively identify essential dynamical features of stochastic systems. The real data application showed that paced breathing, at low breathing rate, increases the predictability of the individual dynamics of RR and RESP and dampens nonlinearity in their coupled dynamics.
... The primary aim is to produce a large number of surrogate datasets using the observed data under this null hypothesis. Previous methodologies, such as Fourier transform-based surrogates and cycle phase permutation,40 have been ineffective in eliminating the property under investigation in surrogates for oscillatory data. This inadequacy can be attributed to periodicity in the data, as similar patterns repeat themselves at regular intervals. ...
Identification of the source of plantwide oscillations is a challenging problem, even with the availability of big data. Causality analysis is often used to construct causal maps and obtain a sequence of fault propagation for an in‐depth investigation. The reliability of the widely used Granger causality depends on the quality of the observed data. But since real‐world industrial data are prone to sensor errors, their accuracy is significantly compromised. Experienced engineers possess years of valuable process knowledge which when introduced into the modeling can significantly reduce the over‐dependence on data. In this article, we propose a novel approach to efficiently amalgamate expert information with the observed data to reconstruct causal maps. A new surrogate‐data‐based approach to test the significance of the causal relations obtained for oscillatory data is also proposed in this article. The efficiency of the proposed methodology is demonstrated using a simulation and an industrial case study.
... It characterises the complexity of a signal and estimates how much of this complexity originates from other input signals, thereby determining the strength and direction of the information transfer between the signals. When applied to biomedical data this can be used to identify couplings and their associated physiological functions (Jamšek et al., 2010). ...
For decades the role of autonomic regulation and the baroreflex in the generation of the respiratory sinus arrhythmia (RSA) - modulation of heart rate by the frequency of breathing - has been under dispute. We hypothesized that by using autonomic blockers we can reveal which oscillations and their interactions are suppressed, elucidating their involvement in RSA as well as in cardiovascular regulation more generally. R-R intervals, end tidal CO2, finger arterial pressure, and muscle sympathetic nerve activity (MSNA) were measured simultaneously in 7 subjects during saline, atropine and propranolol infusion. The measurements were repeated during spontaneous and fixed-frequency breathing, and apnea. The power spectra, phase coherence and couplings were calculated to characterise the variability and interactions within the cardiovascular system. Atropine reduced R-R interval variability (p < 0.05) in all three breathing conditions, reduced MSNA power during apnea and removed much of the significant coherence and couplings. Propranolol had smaller effect on the power of oscillations and did not change the number of significant interactions. Most notably, atropine reduced R-R interval power in the 0.145–0.6 Hz interval during apnea, which supports the hypothesis that the RSA is modulated by a mechanism other than the baroreflex. Atropine also reduced or made negative the phase shift between the systolic and diastolic pressure, indicating the cessation of baroreflex-dependent blood pressure variability. This result suggests that coherent respiratory oscillations in the blood pressure can be used for the non-invasive assessment of autonomic regulation.
... Where s, t, and are scale variable, time variable, and mother wavelet function respectively. Wavelet bispectrum is defined as follows [58], [59]: ...
... The following equation (9) represents the instantaneous biphase. Wavelet bispectrum has been used in different approaches in previous studies (see [51], [59] and [60]). For example in [51] authors eliminate the s1 and s2 effect in instantaneous biphase by averaging and create a time signal as wavelet-based bicoherence. ...
Many studies in the field of sleep have focused on connectivity and coherence. Still, the nonstationary nature of electroencephalography (EEG) makes many of the previous methods unsuitable for automatic sleep detection. Time-frequency representations and high-order spectra are applied to nonstationary signal analysis and nonlinearity investigation, respectively. Therefore, combining wavelet and bispectrum, wavelet-based bi-phase (Wbiph) was proposed and used as a novel feature for sleep–wake classification. The results of the statistical analysis with emphasis on the importance of the gamma rhythm in sleep detection show that the Wbiph is more potent than coherence in the wake–sleep classification. The Wbiph has not been used in sleep studies before. However, the results and inherent advantages, such as the use of wavelet and bispectrum in its definition, suggest it as an excellent alternative to coherence. In the next part of this paper, a convolutional neural network (CNN) classifier was applied for the sleep–wake classification by Wbiph. The classification accuracy was 97.17% in nonLOSO and 95.48% in LOSO cross-validation, which is the best among previous studies on sleep–wake classification.
... The CWT method has the application in various branches of applied mathematics [13] - [17]. In [18] - [19], CWT is used to analyze the phase synchronization of chaotic signals, and in [20] − for the analysis of regular and chaotic oscillations generated by the Rössler system of differential equations. The Wavelet transform is used in [21] to process brain electroencephalogram (EEG) signals in order to identify interactions between different parts of the brain. ...
... In particular, we introduce a new definition of "wavelet-phase bicoherence". We also briefly describe how wavelet bispectral analysis can be used to suggest unidirectionality of coupling [31]. ...
... Again we can extend p ψ,κ,xy to all sets A ∈ B(R 2 ) with A \({0} ×R) ∈ Bp ψ,κ,xy using (31). We refer to p ψ,κ,xy as the wavelet cross-energy spectrum of x with y. ...
... A remark about unidirectional coupling Sometimes unidirectionality of coupling of oscillators can be suggested by analysis of bicoherences and bispectral densities. Slightly generalising the description in [31]: Suppose we have signals x, y and z, where y and z could be the same as each other or different, recorded from a system that contains an oscillatory process consisting of two phase-coupled oscillators. Suppose that • analysis of properties of the bispectrum b ψ,κ,xyz around (f 1 , f 2 , t) detects the presence of this oscillatory process; • analysis of properties of the bispectrum b ψ,κ,xyx around (f 1 , f 2 , t) does not detect the presence of this oscillatory process; ...
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.
... To learn about interactions between oscillators, many methods can be applied [67][68][69][70][71][72]. To find the underlying mechanisms responsible for these interactions, the dynamical Bayesian inference (DBI) [20,60,73] can be used. ...
The precise mechanisms connecting the cardiovascular system and the cerebrospinal fluid (CSF) are not well understood in detail. This paper investigates the couplings between the cardiac and respiratory components, as extracted from blood pressure (BP) signals and oscillations of the subarachnoid space width (SAS), collected during slow ventilation and ventilation against inspiration resistance. The experiment was performed on a group of 20 healthy volunteers (12 females and 8 males; BMI=22.1±3.2 kg/m2; age 25.3±7.9 years). We analysed the recorded signals with a wavelet transform. For the first time, a method based on dynamical Bayesian inference was used to detect the effective phase connectivity and the underlying coupling functions between the SAS and BP signals. There are several new findings. Slow breathing with or without resistance increases the strength of the coupling between the respiratory and cardiac components of both measured signals. We also observed increases in the strength of the coupling between the respiratory component of the BP and the cardiac component of the SAS and vice versa. Slow breathing synchronises the SAS oscillations, between the brain hemispheres. It also diminishes the similarity of the coupling between all analysed pairs of oscillators, while inspiratory resistance partially reverses this phenomenon. BP–SAS and SAS–BP interactions may reflect changes in the overall biomechanical characteristics of the brain.
... After calculating the directional index of the original sequences, we repeat the steps of shuffling the original sequences and calculating the directional index 30 times to get the distribution of the direction index. The deviation of the DCMI, obtained from the studied data, from the DCMI generated by the shuffled data means that the DCMI reflects the causal relationship of the data [35]. ...
Conditional mutual information (CMI) is the basis of many coupling direction metrics and plays an important role in revealing the causal relationship between different signals. In this paper, we propose dispersion conditional mutual information (DCMI) which uses dispersion patterns to calculate the probability distribution. This method is computationally fast and can accurately extract the dynamical characteristics of signals even in the presence of noise. The effects of time lag between signals, coupling coefficient, noise, data length and sudden change in coupling direction on the performance of DCMI are evaluated by simulation experiments. Moreover, we extend DCMI to multiscale DCMI (MDCMI) through a modified multiscale method. MDCMI is adopted on the coupled chaotic model and coupled stochastic process to research the properties. The results show that both DCMI and MDCMI have excellent properties in detecting the coupling relationship and are easy to calculate. Finally, we apply the MDCMI on stock indexes of USA and the results show that there is a cross-correlation between stock price and trading volume.
... To test the null hypothesis that the power spectra time series of two different frequencies, and , are not coupled in the data, non-parametric surrogate data method is supported as it makes minimum assumptions. It preserves the original data's statistical properties while generating time series that are randomized such that any possible nonlinear coupling is removed [54]. It has also been actively used to test coupling in nonlinear systems in many EEG studies [55,56]. ...
... To test the null hypothesis that the power spectra time series of two different frequencies, f i and f j , are not coupled in the data, non-parametric surrogate data method is supported as it makes minimum assumptions. It preserves the original data's statistical properties while generating time series that are randomized such that any possible nonlinear coupling is removed [54]. It has also been actively used to test coupling in nonlinear systems in many EEG studies [55,56]. ...
Magneto-/Electro-encephalography (M/EEG) commonly uses (fast) Fourier transformation to compute power spectral density (PSD). However, the resulting PSD plot lacks temporal information, making interpretation sometimes equivocal. For example, consider two different PSDs: a central parietal EEG PSD with twin peaks at 10 Hz and 20 Hz and a central parietal PSD with twin peaks at 10 Hz and 50 Hz. We can assume the first PSD shows a mu rhythm and the second harmonic; however, the latter PSD likely shows an alpha peak and an independent line noise. Without prior knowledge, however, the PSD alone cannot distinguish between the two cases. To address this limitation of PSD, we propose using cross-frequency power–power coupling (PPC) as a post-processing of independent component (IC) analysis (ICA) to distinguish brain components from muscle and environmental artifact sources. We conclude that post-ICA PPC analysis could serve as a new data-driven EEG classifier in M/EEG studies. For the reader’s convenience, we offer a brief literature overview on the disparate use of PPC. The proposed cross-frequency power–power coupling analysis toolbox (PowPowCAT) is a free, open-source toolbox, which works as an EEGLAB extension.
... Mutual interactions among subsystems, their frequencies, and amplitudes are all time-varying. 22 The time-dependent interaction is quite common in biological, 23 epidemiological, 24 and social network. 25 In biological systems, timevarying dynamic Bayesian networks (TV-DBNs) for modeling the varying network structures underlying non-stationary biological time series have been studied. ...
Many systems exhibit both attractive and repulsive types of interactions, which may be dynamic or static. A detailed understanding of the dynamical properties of a system under the influence of dynamically switching attractive or repulsive interactions is of practical significance. However, it can also be effectively modeled with two coexisting competing interactions. In this work, we investigate the effect of time-varying attractive–repulsive interactions as well as the hybrid model of coexisting attractive–repulsive interactions in two coupled nonlinear oscillators. The dynamics of two coupled nonlinear oscillators, specifically limit cycles as well as chaotic oscillators, are studied in detail for various dynamical transitions for both cases. Here, we show that dynamic or static attractive–repulsive interactions can induce an important transition from the oscillatory to steady state in identical nonlinear oscillators due to competitive effects. The analytical condition for the stable steady state in dynamic interactions at the low switching time period and static coexisting interactions are calculated using linear stability analysis, which is found to be in good agreement with the numerical results. In the case of a high switching time period, oscillations are revived for higher interaction strength.
