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In the present work, we present a new method for assessing causality in uni-directionally coupled chaotic oscillators with small frequency mismatch. This method is based on the correlation between changes in the phase dynamics of the slave oscillator and the dynamics of the phase difference between the oscillators. Application of the proposed approach to master-slave R¨ossler systems showed that the new algorithm is well-suited for assessing the presence and direction of coupling, especially in the case of weak coupling.

In the present work, we present a new algorithm for assessing causality in uni-directionally coupled chaotic oscillators with small frequency mismatch embedded in heavy white Gaussian noise. This method is based on the correlation between changes in the phase dynamics of the slave oscillator and the dynamics of the phase difference between the oscillators. To recover the phase at low signal-to-noise ratio, a nonlinear adaptive denoising algorithm based on finding sinusoidal fits to the local neighbourhood of the reconstructed phase space is used. Application of the proposed approach to master-slave Rössler systems showed that the new algorithm is well-suited for assessing the presence and direction of coupling in highly noisy uni-directionally coupled chaotic oscillators, especially in the case of weak and moderate coupling.

In this study, we present a new method for assessing the functional connectivity between the drive and the noisy response chaotic oscillators with significant frequency mismatch. This method is based on the footprint of the drive oscillator on the response oscillator. Specifically, we calculate the magnitude squared coherence between the intrinsic frequency of the drive oscillator and the oscillation frequency of the dynamics of the spectral energy of the response oscillator. The spectral energy of the response oscillator is estimated using a modified S-transform algorithm with a short sliding window. We apply the proposed approach to master–slave Rossler systems and coupled Van der Pol oscillators with a frequency ratio of $$\sim $$ 1:4 and show that when the slave signal is contaminated with white Gaussian noise at different signal-to-noise ratios (SNR), the new algorithm is well suited to assess the presence of coupling with a priori known direction in noisy, unidirectionally coupled chaotic oscillators, especially in the case of weak and moderate coupling.

In this paper, we present some improvements to the convergent cross mapping (CCM) algorithm for detecting causality in uni-directionally connected chaotic systems. The basic concept of the CCM algorithm is that the causal influence of system X on system Y appears as mapping of the neighbouring states in the reconstructed d-dimensional manifold, My, to the neighbouring states in the reconstructed d-dimensional manifold, Mx, and this effect is evaluated using the correlation coefficient between the estimated and observed values of Mx. We proposed a composite indicator of causality as the ratio between the correlation coefficient and the Shannon entropy of the distribution of the residuals between the estimated and observed values of Mx. Application of the proposed approach to four master-slave Rössler and Lorenz systems and real-world data showed that the new algorithm allowed a slight increase in capability to reveal the presence and direction of couplings.

In this paper, we propose a novel algorithm for the detection changes in the spatio-spectral dynamics of a multivariate time series based on a sliced cross-bispectrum. The singular value decomposition is performed on the matrix of the sliced cross-bispectrum at every frequency of the bandwidth, and a 2D spectrum of the eigenvalues is obtained. The changes in the dynamics are translated into differences between the reference and current values of the 2D eigenvalue spectrum. Hellinger divergence as the measure of the distance between eigenvalue spectrum matrices is used. The performance of the proposed approach is tested on model data and applied to multi-channel electroencephalogram recordings from epilepsy patients to detect ictal states.

In this paper, a novel algorithm based on the local projection noise reduction approach is applied to smooth noise for strongly contaminated chaotic oscillators. Specifically, one-dimensional time series are embedded into a high dimensional phase space and the noise level is defined through orthogonal projections of the data points within the neighbourhood of the reference point onto linear subspaces. The current vector of the phase space is denoised by performing two-dimensional discrete stationary wavelet transform (SWT)-based filtering in the neighbourhood of the phase point. Numerical results show that our algorithm effectively recovers continuous-time chaotic signals in heavy-noise environments and outperforms the classical local projection noise reduction approach for simulated data from the Rössler system and Duffing oscillator at signal-to-noise ratios (SNRs) from 15 to 0 dB, either for the real world data – human breath time series.

In this paper, a novel algorithm based on the moving block-bootstrap (MBB) technique for the detection of coherence between oscillators in heavy-noise environments is proposed. To verify the null hypothesis of absent significant coherence against the alternative, the average coherence of MBB resampled data is compared with a certain threshold over all frequencies. The threshold is defined as the upper bound of a 95 % confidence interval for bootstrapping coherence performed with independent resampling indexes for both time series, so the dependence between the time series under consideration is destroyed. The benefits of the proposed method are illustrated by simulations of the phase synchronization effects of two linearly coupled chaotic Rössler oscillators.

In this paper, it is shown that the block circulant matrix decomposition technique makes the multivariate singular spectrum analysis (M-SSA) a well suited tool for detecting of changes of the correlation structure in non-stationary multivariate time series in the presence of high observational noise levels. The major drawback of M-SSA, that it operates on a large covariance matrix and becomes computationally expensive, can be avoided by reordering the Toeplitz-block covariance matrix into a block Toeplitz matrix, embedding this into a block circulant matrix and efficiently block-diagonalizing this by the means of the Fast Fourier Transform (FFT) using the well known algorithm. The overall degree of synchronization among multiple-channel signals is defined by the synchronization index (the S-estimator) of the rearranged and truncated eigenvalue spectrum. Throughout the experiment, the high capability of the proposed algorithm to detect the lag-synchronized state under the influence of strong noise is validated with simulated data—a network of time series generated by autoregressive models (AR) and a network of coupled chaotic Roessler oscillators.

The purpose of this investigation was to estimate the test-retest cross-reliability of peripheral and central changes with respect to nonlinear and linear measures of a surface electromyography (EMG) signal measured during isometric maximal voluntary contraction (MVC) combined with superimposed electrical stimulation during a brief and fatiguing task involving the ankle plantar flexors over 2 follicular phases of menstrual cycle. Ten healthy female adults underwent 1 familiarization session and 5 identical test-retest sessions. The results showed that the decrease in plantar flexor EMG components (root mean square [RMS], mean frequency [MnF], wavelet packet entropy [WPE]) for soleus and gastrocnemius muscles, central activation ratio (CAR) and MVC, and contractile properties (P20, P100, PTT-100, and half-relaxation time) of the plantar flexor muscles at the end of 2-minute MVC were similar (time effect; p0.001, η(p)²0.7, statistical power [SP]99%) and exhibited high stability over 5 trials (trial effect; p0.05; η(p)²0.2, SP30%). High reliability between trials was found for 5-second MVC (intraclass correlation coefficient [ICC]0.82, p0.001) and meaningful reliability for 2-minute MVC (ICC0.66, p0.01). In conclusion, in young healthy women, measurements of neuromuscular function, such as RMS, MnF, and WPE of a surface EMG signal, MVC, and CAR from a brief and sustained MVC of the ankle plantar flexors, are reliable, and multidimensional stability was found with respect to both high and low correlation outcomes across the 5 identical test-retest trials of any 2 properties measured during brief and sustained MVC.