Correlations in magnitude series to assess nonlinearities: Application to multifractal models and heartbeat fluctuations

High-intensity physiological activation disrupts the neural signatures of conflict processing

Physiological activation fluctuates throughout the day. Previous studies have shown that during periods of reduced activation, cognitive control remains resilient due to neural compensatory mechanisms. In this study, we investigate the effects of high physiological activation on both behavioural and neural markers of cognitive control. We hypothesize that while behavioural measures of cognitive control would remain intact during periods of high activation, there would be observable changes in neural correlates. In our electroencephalography study, we manipulate levels of physiological activation through physical exercise. Although we observe no significant impact on behavioural measures of cognitive conflict, both univariate and multivariate time-frequency markers prove unreliable under conditions of high activation. Moreover, we observe no modulation of whole-brain connectivity measures by physiological activation. We suggest that this dissociation between behavioural and neural measures indicates that the human cognitive control system remains resilient even at high activation, possibly due to underlying neural compensatory mechanisms.

Visibility Graph Analysis of Heartbeat Time Series: Comparison of Young vs. Old, Healthy vs. Diseased, Rest vs. Exercise, and Sedentary vs. Active

Using the visibility graph algorithm (VGA), a complex network can be associated with a time series, such that the properties of the time series can be obtained by studying those of the network. Any value of the time series becomes a node of the network, and the number of other nodes that it is connected to can be quantified. The degree of connectivity of a node is positively correlated with its magnitude. The slope of the regression line is denoted by k-M, and, in this work, this parameter was calculated for the cardiac interbeat time series of different contrasting groups, namely: young vs. elderly; healthy subjects vs. patients with congestive heart failure (CHF); young subjects and adults at rest vs. exercising young subjects and adults; and, finally, sedentary young subjects and adults vs. active young subjects and adults. In addition, other network parameters, including the average degree and the average path length, of these time series networks were also analyzed. Significant differences were observed in the k-M parameter, average degree, and average path length for all analyzed groups. This methodology based on the analysis of the three mentioned parameters of complex networks has the advantage that such parameters are very easy to calculate, and it is useful to classify heartbeat time series of subjects with CHF vs. healthy subjects, and also for young vs. elderly subjects and sedentary vs. active subjects.

Complexity and Entropy in Physiological Signals (CEPS): Resonance Breathing Rate Assessed Using Measures of Fractal Dimension, Heart Rate Asymmetry and Permutation Entropy

BACKGROUND

As technology becomes more sophisticated, more accessible methods of interpretating Big Data become essential. We have continued to develop Complexity and Entropy in Physiological Signals (CEPS) as an open access MATLAB® GUI (graphical user interface) providing multiple methods for the modification and analysis of physiological data.

METHODS

To demonstrate the functionality of the software, data were collected from 44 healthy adults for a study investigating the effects on vagal tone of breathing paced at five different rates, as well as self-paced and un-paced. Five-minute 15-s recordings were used. Results were also compared with those from shorter segments of the data. Electrocardiogram (ECG), electrodermal activity (EDA) and Respiration (RSP) data were recorded. Particular attention was paid to COVID risk mitigation, and to parameter tuning for the CEPS measures. For comparison, data were processed using Kubios HRV, RR-APET and DynamicalSystems.jl software. We also compared findings for ECG RR interval (RRi) data resampled at 4 Hz (4R) or 10 Hz (10R), and non-resampled (noR). In total, we used around 190-220 measures from CEPS at various scales, depending on the analysis undertaken, with our investigation focused on three families of measures: 22 fractal dimension (FD) measures, 40 heart rate asymmetries or measures derived from Poincaré plots (HRA), and 8 measures based on permutation entropy (PE).

RESULTS

FDs for the RRi data differentiated strongly between breathing rates, whether data were resampled or not, increasing between 5 and 7 breaths per minute (BrPM). Largest effect sizes for RRi (4R and noR) differentiation between breathing rates were found for the PE-based measures. Measures that both differentiated well between breathing rates and were consistent across different RRi data lengths (1-5 min) included five PE-based (noR) and three FDs (4R). Of the top 12 measures with short-data values consistently within ± 5% of their values for the 5-min data, five were FDs, one was PE-based, and none were HRAs. Effect sizes were usually greater for CEPS measures than for those implemented in DynamicalSystems.jl.

CONCLUSION

The updated CEPS software enables visualisation and analysis of multichannel physiological data using a variety of established and recently introduced complexity entropy measures. Although equal resampling is theoretically important for FD estimation, it appears that FD measures may also be usefully applied to non-resampled data.

Decrease of heart rate variability during exercise: An index of cardiorespiratory fitness

The present study proposes to measure and quantify the heart rate variability (HRV) changes during effort as a function of the heart rate and to test the capacity of the produced indices to predict cardiorespiratory fitness measures. Therefore, the beat-to-beat cardiac time interval series of 18 adolescent athletes (15.2 ± 2.0 years) measured during maximal graded effort test were detrended using a dynamical first-order differential equation model. HRV was then calculated as the standard deviation of the detrended RR intervals (SDRR) within successive windows of one minute. The variation of this measure of HRV during exercise is properly fitted by an exponential decrease of the heart rate: the SDRR is divided by 2 every increase of heart rate of 20 beats/min. The HR increase necessary to divide by 2 the HRV is linearly inversely correlated with the maximum oxygen consumption (r = -0.60, p = 0.006), the maximal aerobic power (r = -0.62, p = 0.006), and, to a lesser extent, to the power at the ventilatory thresholds (r = -0.53, p = 0.02 and r = -0.47, p = 0.05 for the first and second threshold). It indicates that the decrease of the HRV when the heart rate increases is faster among athletes with better fitness. This analysis, based only on cardiac measurements, provides a promising tool for the study of cardiac measurements generated by portable devices.

Permutation Jensen-Shannon distance: A versatile and fast symbolic tool for complex time-series analysis

The main motivation of this paper is to introduce the permutation Jensen-Shannon distance, a symbolic tool able to quantify the degree of similarity between two arbitrary time series. This quantifier results from the fusion of two concepts, the Jensen-Shannon divergence and the encoding scheme based on the sequential ordering of the elements in the data series. The versatility and robustness of this ordinal symbolic distance for characterizing and discriminating different dynamics are illustrated through several numerical and experimental applications. Results obtained allow us to be optimistic about its usefulness in the field of complex time-series analysis. Moreover, thanks to its simplicity, low computational cost, wide applicability, and less susceptibility to outliers and artifacts, this ordinal measure can efficiently handle large amounts of data and help to tackle the current big data challenges.

On the Validity of Detrended Fluctuation Analysis at Short Scales

Detrended Fluctuation Analysis (DFA) has become a standard method to quantify the correlations and scaling properties of real-world complex time series. For a given scale ℓ of observation, DFA provides the function F(ℓ), which quantifies the fluctuations of the time series around the local trend, which is substracted (detrended). If the time series exhibits scaling properties, then F(ℓ)∼ℓα asymptotically, and the scaling exponent α is typically estimated as the slope of a linear fitting in the logF(ℓ) vs. log(ℓ) plot. In this way, α measures the strength of the correlations and characterizes the underlying dynamical system. However, in many cases, and especially in a physiological time series, the scaling behavior is different at short and long scales, resulting in logF(ℓ) vs. log(ℓ) plots with two different slopes, α1 at short scales and α2 at large scales of observation. These two exponents are usually associated with the existence of different mechanisms that work at distinct time scales acting on the underlying dynamical system. Here, however, and since the power-law behavior of F(ℓ) is asymptotic, we question the use of α1 to characterize the correlations at short scales. To this end, we show first that, even for artificial time series with perfect scaling, i.e., with a single exponent α valid for all scales, DFA provides an α1 value that systematically overestimates the true exponent α. In addition, second, when artificial time series with two different scaling exponents at short and large scales are considered, the α1 value provided by DFA not only can severely underestimate or overestimate the true short-scale exponent, but also depends on the value of the large scale exponent. This behavior should prevent the use of α1 to describe the scaling properties at short scales: if DFA is used in two time series with the same scaling behavior at short scales but very different scaling properties at large scales, very different values of α1 will be obtained, although the short scale properties are identical. These artifacts may lead to wrong interpretations when analyzing real-world time series: on the one hand, for time series with truly perfect scaling, the spurious value of α1 could lead to wrongly thinking that there exists some specific mechanism acting only at short time scales in the dynamical system. On the other hand, for time series with true different scaling at short and large scales, the incorrect α1 value would not characterize properly the short scale behavior of the dynamical system.

CEPS: An Open Access MATLAB Graphical User Interface (GUI) for the Analysis of Complexity and Entropy in Physiological Signals

BACKGROUND

We developed CEPS as an open access MATLAB^® GUI (graphical user interface) for the analysis of Complexity and Entropy in Physiological Signals (CEPS), and demonstrate its use with an example data set that shows the effects of paced breathing (PB) on variability of heart, pulse and respiration rates. CEPS is also sufficiently adaptable to be used for other time series physiological data such as EEG (electroencephalography), postural sway or temperature measurements.

METHODS

Data were collected from a convenience sample of nine healthy adults in a pilot for a larger study investigating the effects on vagal tone of breathing paced at various different rates, part of a development programme for a home training stress reduction system.

RESULTS

The current version of CEPS focuses on those complexity and entropy measures that appear most frequently in the literature, together with some recently introduced entropy measures which may have advantages over those that are more established. Ten methods of estimating data complexity are currently included, and some 28 entropy measures. The GUI also includes a section for data pre-processing and standard ancillary methods to enable parameter estimation of embedding dimension m and time delay τ ('tau') where required. The software is freely available under version 3 of the GNU Lesser General Public License (LGPLv3) for non-commercial users. CEPS can be downloaded from Bitbucket. In our illustration on PB, most complexity and entropy measures decreased significantly in response to breathing at 7 breaths per minute, differentiating more clearly than conventional linear, time- and frequency-domain measures between breathing states. In contrast, Higuchi fractal dimension increased during paced breathing.

CONCLUSIONS

We have developed CEPS software as a physiological data visualiser able to integrate state of the art techniques. The interface is designed for clinical research and has a structure designed for integrating new tools. The aim is to strengthen collaboration between clinicians and the biomedical community, as demonstrated here by using CEPS to analyse various physiological responses to paced breathing.

Entropy Analysis of RR-Time Series From Stress Tests

The RR-interval time series or tachograms obtained from electrocardiograms have been widely studied since they reflect the cardiac variability, and this is an indicative of the health status of a person. The tachogram can be seen as a highly non-linear and complex time series, and therefore, should be analyzed with non-linear techniques. In this work, several entropy measures, Sample Entropy (SampEn), Approximate Entropy (ApEn), and Fuzzy Entropy (FuzzyEn) are used as a measure of heart rate variability (HRV). Tachograms belonging to thirty-nine subjects were obtained from a cardiac stress test consisting of a rest period followed by a period of moderate physical activity. Subjects are grouped according to their physical activity using the IPAQ sedentary and active questionnaire, we work with youth and middle-aged adults. The entropy measures for each group show that for the sedentary subjects the values are high at rest and decrease appreciably with moderate physical activity, This happens for both young and middle-aged adults. These results are highly reproducible. In the case of the subjects that exercise regularly, an increase in entropy is observed or they tend to retain the entropy value that they had at rest. It seems that there is a possible correlation between the physical condition of a person with the increase or decrease in entropy during moderate physical activity with respect to the entropy at rest. It was also observed that entropy during longer physical activity tests tends to decrease as fatigue accumulates, but this decrease is small compared to the change that occurs when going from rest to physical activity.

Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution

The observable outputs of many complex dynamical systems consist of time series exhibiting autocorrelation functions of great diversity of behaviors, including long-range power-law autocorrelation functions, as a signature of interactions operating at many temporal or spatial scales. Often, numerical algorithms able to generate correlated noises reproducing the properties of real time series are used to study and characterize such systems. Typically, many of those algorithms produce a Gaussian time series. However, the real, experimentally observed time series are often non-Gaussian and may follow distributions with a diversity of behaviors concerning the support, the symmetry, or the tail properties. It is always possible to transform a correlated Gaussian time series into a time series with a different marginal distribution, but the question is how this transformation affects the behavior of the autocorrelation function. Here, we study analytically and numerically how the Pearson's correlation of two Gaussian variables changes when the variables are transformed to follow a different destination distribution. Specifically, we consider bounded and unbounded distributions, symmetric and non-symmetric distributions, and distributions with different tail properties from decays faster than exponential to heavy-tail cases including power laws, and we find how these properties affect the correlation of the final variables. We extend these results to a Gaussian time series, which are transformed to have a different marginal distribution, and show how the autocorrelation function of the final non-Gaussian time series depends on the Gaussian correlations and on the final marginal distribution. As an application of our results, we propose how to generalize standard algorithms producing a Gaussian power-law correlated time series in order to create a synthetic time series with an arbitrary distribution and controlled power-law correlations. Finally, we show a practical example of this algorithm by generating time series mimicking the marginal distribution and the power-law tail of the autocorrelation function of real time series: the absolute returns of stock prices.

Nonlinear correlations in multifractals: Visibility graphs of magnitude and sign series

Correlations in a multifractal series have been investigated extensively. Almost all approaches try to find scaling features of a given time series. However, the scaling analysis has always been encountered with some difficulties. Of particular importance is finding a proper scaling region and removing the impact of the probability distribution function of the series on the correlation extraction methods. In this article, we apply the horizontal visibility graph algorithm to map a stochastic time series into networks. By investigating the magnitude and sign of a multifractal time series, we show that one can detect linear as well as nonlinear correlations, even for situations that have been considered as uncorrelated noises by typical approaches such as the multifractal detrended fluctuation analysis. Furthermore, we introduce a topological parameter that can well measure the strength of nonlinear correlations. This parameter is independent of the probability distribution function and calculated without the need to find any scaling region. Our findings may provide new insights about the multifractal analysis of a time series in a variety of complex systems.

Comparison of methods for the assessment of nonlinearity in short-term heart rate variability under different physiopathological states

Despite the widespread diffusion of nonlinear methods for heart rate variability (HRV) analysis, the presence and the extent to which nonlinear dynamics contribute to short-term HRV are still controversial. This work aims at testing the hypothesis that different types of nonlinearity can be observed in HRV depending on the method adopted and on the physiopathological state. Two entropy-based measures of time series complexity (normalized complexity index, NCI) and regularity (information storage, IS), and a measure quantifying deviations from linear correlations in a time series (Gaussian linear contrast, GLC), are applied to short HRV recordings obtained in young (Y) and old (O) healthy subjects and in myocardial infarction (MI) patients monitored in the resting supine position and in the upright position reached through head-up tilt. The method of surrogate data is employed to detect the presence and quantify the contribution of nonlinear dynamics to HRV. We find that the three measures differ both in their variations across groups and conditions and in the percentage and strength of nonlinear HRV dynamics. NCI and IS displayed opposite variations, suggesting more complex dynamics in O and MI compared to Y and less complex dynamics during tilt. The strength of nonlinear dynamics is reduced by tilt using all measures in Y, while only GLC detects a significant strengthening of such dynamics in MI. A large percentage of detected nonlinear dynamics is revealed only by the IS measure in the Y group at rest, with a decrease in O and MI and during T, while NCI and GLC detect lower percentages in all groups and conditions. While these results suggest that distinct dynamic structures may lie beneath short-term HRV in different physiological states and pathological conditions, the strong dependence on the measure adopted and on their implementation suggests that physiological interpretations should be provided with caution.

Multifractal and entropy analysis of resting-state electroencephalography reveals spatial organization in local dynamic functional connectivity

Functional connectivity of the brain fluctuates even in resting-state condition. It has been reported recently that fluctuations of global functional network topology and those of individual connections between brain regions expressed multifractal scaling. To expand on these findings, in this study we investigated if multifractality was indeed an inherent property of dynamic functional connectivity (DFC) on the regional level as well. Furthermore, we explored if local DFC showed region-specific differences in its multifractal and entropy-related features. DFC analyses were performed on 62-channel, resting-state electroencephalography recordings of twelve young, healthy subjects. Surrogate data testing verified the true multifractal nature of regional DFC that could be attributed to the presumed nonlinear nature of the underlying processes. Moreover, we found a characteristic spatial distribution of local connectivity dynamics, in that frontal and occipital regions showed stronger long-range correlation and higher degree of multifractality, whereas the highest values of entropy were found over the central and temporal regions. The revealed topology reflected well the underlying resting-state network organization of the brain. The presented results and the proposed analysis framework could improve our understanding on how resting-state brain activity is spatio-temporally organized and may provide potential biomarkers for future clinical research.

Correlation properties of heart rate variability during endurance exercise: A systematic review

Background

Non‐linear measures of heart rate variability (HRV) may provide new opportunities to monitor cardiac autonomic regulation during exercise. In healthy individuals, the HRV signal is mainly composed of quasi‐periodic oscillations, but it also possesses random fluctuations and so‐called fractal structures. One widely applied approach to investigate fractal correlation properties of heart rate (HR) time series is the detrended fluctuation analysis (DFA). DFA is a non‐linear method to quantify the fractal scale and the degree of correlation of a time series. Regarding the HRV analysis, it should be noted that the short‐term scaling exponent alpha1 of DFA has been used not only to assess cardiovascular risk but also to assess prognosis and predict mortality in clinical settings. It has also been proven to be useful for application in exercise settings including higher exercise intensities, non‐stationary data segments, and relatively short recording times.

Method

Therefore, the purpose of this systematic review was to analyze studies that investigated the effects of acute dynamic endurance exercise on DFA‐alpha1 as a proxy of correlation properties in the HR time series.

Results

The initial search identified 442 articles (351 in PubMed, 91 in Scopus), of which 11 met all inclusion criteria.

Conclusions

The included studies show that DFA‐alpha1 of HRV is suitable for distinguishing between different organismic demands during endurance exercise and may prove helpful to monitor responses to different exercise intensities, movement frequencies, and exercise durations. Additionally, non‐linear DFA of HRV is a suitable analytical approach, providing a differentiated and qualitative view of exercise physiology.

Multifractal Dynamic Functional Connectivity in the Resting-State Brain

Assessing the functional connectivity (FC) of the brain has proven valuable in enhancing our understanding of brain function. Recent developments in the field demonstrated that FC fluctuates even in the resting state, which has not been taken into account by the widely applied static approaches introduced earlier. In a recent study using functional near-infrared spectroscopy (fNIRS) global dynamic functional connectivity (DFC) has also been found to fluctuate according to scale-free i.e., fractal dynamics evidencing the true multifractal (MF) nature of DFC in the human prefrontal cortex. Expanding on these findings, we performed electroencephalography (EEG) measurements in 14 regions over the whole cortex of 24 healthy, young adult subjects in eyes open (EO) and eyes closed (EC) states. We applied dynamic graph theoretical analysis to capture DFC by computing the pairwise time-dependent synchronization between brain regions and subsequently calculating the following dynamic graph topological measures: Density, Clustering Coefficient, and Efficiency. We characterized the dynamic nature of these global network metrics as well as local individual connections in the networks using focus-based multifractal time series analysis in all traditional EEG frequency bands. Global network topological measures were found fluctuating–albeit at different extent–according to true multifractal nature in all frequency bands. Moreover, the monofractal Hurst exponent was found higher during EC than EO in the alpha and beta bands. Individual connections showed a characteristic topology in their fractal properties, with higher autocorrelation owing to short-distance connections–especially those in the frontal and pre-frontal cortex–while long-distance connections linking the occipital to the frontal and pre-frontal areas expressed lower values. The same topology was found with connection-wise multifractality in all but delta band connections, where the very opposite pattern appeared. This resulted in a positive correlation between global autocorrelation and connection-wise multifractality in the higher frequency bands, while a strong anticorrelation in the delta band. The proposed analytical tools allow for capturing the fine details of functional connectivity dynamics that are evidently present in DFC, with the presented results implying that multifractality is indeed an inherent property of both global and local DFC.

Differences in nonlinear heart dynamics during rest and exercise and for different training

OBJECTIVE

In this work we want to analyze differences in nonlinear properties between rest and exercise and also to study the permanent effects of physical exercise on heart rate dynamics.

APPROACH

It has been shown that physical exercise alters heart dynamics by increasing heart rate and decreasing variability, modifying spectral power and linear correlations, etc. We hypothesize that physical exercise should also reduce nonlinearity in the heartbeat time series. To quantify nonlinearity in the heartbeat time series, we use an index of nonlinearity recently proposed by Bernaola et al based on correlations of the magnitude time series.

MAIN RESULTS

Our results confirm our initial hypothesis of loss of nonlinearity during physical exercise. Moreover, regarding the permanent effects of physical exercise on heart rate dynamics, we also obtain that aerobic physical training tends to increase nonlinearity in heart dynamics during rest.

SIGNIFICANCE

It is well-known that heart dynamics are controlled by complex interactions between the sympathetic and parasympathetic branches of the autonomic nervous system. Moreover, these two branches act in a competing way, resulting in a clear parasympathetic withdrawal and sympathetic activation during physical exercise. We associate these interactions during physical exercise with a drastic loss of nonlinear properties in the heartbeat time series, revealing the importance of nonlinearity measures in the study of complex systems.

Impact of Healthy Aging on Multifractal Hemodynamic Fluctuations in the Human Prefrontal Cortex

Fluctuations in resting-state cerebral hemodynamics show scale-free behavior over two distinct scaling ranges. Changes in such bimodal (multi) fractal pattern give insight to altered cerebrovascular or neural function. Our main goal was to assess the distribution of local scale-free properties characterizing cerebral hemodynamics and to disentangle the influence of aging on these multifractal parameters. To this end, we obtained extended resting-state records (N = 2¹⁴) of oxyhemoglobin (HbO), deoxyhemoglobin (HbR) and total hemoglobin (HbT) concentration time series with continuous-wave near-infrared spectroscopy technology from the brain cortex. 52 healthy volunteers were enrolled in this study: 24 young (30.6 ± 8.2 years), and 28 elderly (60.5 ± 12.0 years) subjects. Using screening tests on power-law, multifractal noise, and shuffled data sets we evaluated the presence of true multifractal hemodynamics reflecting long-range correlation (LRC). Subsequently, scaling-range adaptive bimodal signal summation conversion (SSC) was performed based on standard deviation (σ) of signal windows across a range of temporal scales (s). Building on moments of different order (q) of the measure, σ(s), multifractal SSC yielded generalized Hurst exponent function, H(q), and singularity spectrum, D(h) separately for a fast and slow component (the latter dominating the highest temporal scales). Parameters were calculated reflecting the estimated measure at s = N (focus), degree of LRC [Hurst exponent, H(2) and maximal Hölder exponent, h_max] and measuring strength of multifractality [full-width-half-maximum of D(h) and ΔH₁₅ = H(−15)−H(15)]. Correlation-based signal improvement (CBSI) enhanced our signal in terms of interpreting changes due to neural activity or local/systemic hemodynamic influences. We characterized the HbO-HbR relationship with the aid of fractal scale-wise correlation coefficient, r_σ(s) and SSC-based multifractal covariance analysis. In the majority of subjects, cerebral hemodynamic fluctuations proved bimodal multifractal. In case of slow component of raw HbT, h_max, and Ĥ(2) were lower in the young group explained by a significantly increased r_σ(s) among elderly at high temporal scales. Regarding the fast component of CBSI-pretreated HbT and that of HbO-HbR covariance, h_max, and focus were decreased in the elderly group. These observations suggest an attenuation of neurovascular coupling reflected by a decreased autocorrelation of the neuronal component concomitant with an accompanying increased autocorrelation of the non-neuronal component in the elderly group.