A Fast DFA Algorithm for Multifractal Multiscale Analysis of Physiological Time Series

Multifractal analysis of heart rate variability in pregnancy during sleep

Understanding the complex dynamics of heart rate variability (HRV) during pregnancy is crucial for monitoring both maternal well-being and fetal health. In this study, we use the Multifractal Detrended Fluctuations Analysis approach to investigate HRV patterns in pregnant individuals during sleep based on RR interval maxima (MM fluctuations). In addition, we study the type of multifractality within MM fluctuations, that is, if it arises from a broad probability density function or from varying long-range correlations. Furthermore, to provide a comprehensive view of HRV changes during sleep in pregnancy, classical temporal and spectral HRV indices were calculated at quarterly intervals during sleep. Our study population consists of 21 recordings from nonpregnant women, 18 from the first trimester (early-pregnancy) and 18 from the second trimester (middle-pregnancy) of pregnancy. Results. There are statistically significant differences ( p -value < 0.05) in mean heart rate, rms heart rate, mean MM fluctuations, and standard deviation of MM fluctuations, particularly in the third and fourth quarter of sleep between pregnant and non-pregnant states. In addition, the early-pregnancy group shows significant differences ( p -value < 0.05) in spectral indices during the first and fourth quarter of sleep compared to the non-pregnancy group. Furthermore, the results of our research show striking similarities in the average multifractal structure of MM fluctuations between pregnant and non-pregnant states during normal sleep. These results highlight the influence of different long-range correlations within the MM fluctuations, which could be primarily associated with the emergence of sleep cycles on multifractality during sleep. Finally, we performed a separability analysis between groups using temporal and spectral HRV indices as features per sleep quarter. Employing only three features after Principal Component Analysis (PCA) to the original feature set, achieving complete separability among all groups appears feasible. Using multifractal analysis, our study provides a comprehensive understanding of the complex HRV patterns during pregnancy, which holds promise for maternal and fetal health monitoring. The separability analysis also provides valuable insights into the potential for group differentiation using simple measures such as mean heart rate, rms heart rate, and mean MM fluctuations or in the transformed feature space based on PCA.

Assessing cardiovascular stress based on heart rate variability in female shift workers: a multiscale-multifractal analysis approach

Introduction

Sleep-wake cycle disruption caused by shift work may lead to cardiovascular stress, which is observed as an alteration in the behavior of heart rate variability (HRV). In particular, HRV exhibits complex patterns over different time scales that help to understand the regulatory mechanisms of the autonomic nervous system, and changes in the fractality of HRV may be associated with pathological conditions, including cardiovascular disease, diabetes, or even psychological stress. The main purpose of this study is to evaluate the multifractal-multiscale structure of HRV during sleep in healthy shift and non-shift workers to identify conditions of cardiovascular stress that may be associated with shift work.

Methods

The whole-sleep HRV signal was analyzed from female participants: eleven healthy shift workers and seven non-shift workers. The HRV signal was decomposed into intrinsic mode functions (IMFs) using the empirical mode decomposition method, and then the IMFs were analyzed using the multiscale-multifractal detrended fluctuation analysis (MMF-DFA) method. The MMF-DFA was applied to estimate the self-similarity coefficients, α(q, τ), considering moment orders (q) between -5 and +5 and scales (τ) between 8 and 2,048 s. Additionally, to describe the multifractality at each τ in a simple way, a multifractal index, MFI(τ), was computed.

Results

Compared to non-shift workers, shift workers presented an increase in the scaling exponent, α(q, τ), at short scales (τ < 64 s) with q < 0 in the high-frequency component (IMF1, 0.15-0.4 Hz) and low-frequency components (IMF2-IMF3, 0.04-0.15 Hz), and with q> 0 in the very low frequencies (IMF4, < 0.04 Hz). In addition, at large scales (τ> 1,024 s), a decrease in α(q, τ) was observed in IMF3, suggesting an alteration in the multifractal dynamic. MFI(τ) showed an increase at small scales and a decrease at large scales in IMFs of shift workers.

Conclusion

This study helps to recognize the multifractality of HRV during sleep, beyond simply looking at indices based on means and variances. This analysis helps to identify that shift workers show alterations in fractal properties, mainly on short scales. These findings suggest a disturbance in the autonomic nervous system induced by the cardiovascular stress of shift work.

Persistence of Basal Ganglia Oscillatory Activity During Tremor Attenuation by Movement in Parkinson's Disease Patients

BACKGROUND

One of the characteristics of parkinsonian tremor is that its amplitude decreases with movement. Current models suggest an interaction between basal ganglia (BG) and cerebello-thalamo-cortical circuits in parkinsonian tremor pathophysiology.

OBJECTIVE

We aimed to correlate central oscillation in the BG with electromyographic activity during re-emergent tremor in order to detect changes in BG oscillatory activity when tremor is attenuated by movement.

METHODS

We performed a prospective, observational study on consecutive parkinsonian patients who underwent deep brain stimulation surgery and presented re-emergent tremor. Coherence analysis between subthalamic nucleus/globus pallidus internus (STN/GPi) tremorous activity measured by microrecording (MER) and electromyogram (EMG) from flexor and extensor wrist muscles during rest, posture, and re-emergent tremor pause was performed during surgery. The statistical significance level of the MER-EMG coherence was determined using surrogate data analysis, and the directionality of information transfer between BG and muscle was performed using entropy transfer analysis.

RESULTS

We analyzed 148 MERs with tremor-like activity from 6 patients which were evaluated against the simultaneous EMGs, resulting in 296 correlations. Of these, 26 presented a significant level of coherence at tremor frequency, throughout rest and posture, with a complete EMG stop in between. During the pause, all recordings showed sustained MER peaks at tremor frequency (±1.5 Hz). Information flows preferentially from BG to muscle during rest and posture, with a loss of directionality during the pause.

CONCLUSIONS

Our results suggest that oscillatory activity in STN/GPi functionally linked to tremor sustains firing frequency during re-emergent tremor pause, thus suggesting no direct role of the BG circuit on tremor attenuation due to voluntary movements. © 2024 International Parkinson and Movement Disorder Society.

Multifractal Multiscale Analysis of Human Movements during Cognitive Tasks

Continuous adaptations of the movement system to changing environments or task demands rely on superposed fractal processes exhibiting power laws, that is, multifractality. The estimators of the multifractal spectrum potentially reflect the adaptive use of perception, cognition, and action. To observe time-specific behavior in multifractal dynamics, a multiscale multifractal analysis based on DFA (MFMS-DFA) has been recently proposed and applied to cardiovascular dynamics. Here we aimed at evaluating whether MFMS-DFA allows identifying multiscale structures in the dynamics of human movements. Thirty-six (12 females) participants pedaled freely, after a metronomic initiation of the cadence at 60 rpm, against a light workload for 10 min: in reference to cycling (C), cycling while playing "Tetris" on a computer, alone (CT) or collaboratively (CTC) with another pedaling participant. Pedal revolution periods (PRP) series were examined with MFMS-DFA and compared to linearized surrogates, which attested to a presence of multifractality at almost all scales. A marked alteration in multifractality when playing Tetris was evidenced at two scales, τ ≈ 16 and τ ≈ 64 s, yet less marked at τ ≈ 16 s when playing collaboratively. Playing Tetris in collaboration attenuated these alterations, especially in the best Tetris players. This observation suggests the high sensitivity to cognitive demand of MFMS-DFA estimators, extending to the assessment of skill/demand interplay from individual behavior. So, by identifying scale-dependent multifractal structures in movement dynamics, MFMS-DFA has obvious potential for examining brain-movement coordinative structures, likely with sufficient sensitivity to find echo in diagnosing disorders and monitoring the progress of diseases that affect cognition and movement control.

Scale-Fractal Detrended Fluctuation Analysis for Fault Diagnosis of a Centrifugal Pump and a Reciprocating Compressor

Reciprocating compressors and centrifugal pumps are rotating machines used in industry, where fault detection is crucial for avoiding unnecessary and costly downtime. A novel method for fault classification in reciprocating compressors and multi-stage centrifugal pumps is proposed. In the feature extraction stage, raw vibration signals are processed using multi-fractal detrended fluctuation analysis (MFDFA) to extract features indicative of different types of faults. Such MFDFA features enable the training of machine learning models for classifying faults. Several classical machine learning models and a deep learning model corresponding to the convolutional neural network (CNN) are compared with respect to their classification accuracy. The cross-validation results show that all models are highly accurate for classifying the 13 types of faults in the centrifugal pump, the 17 valve faults, and the 13 multi-faults in the reciprocating compressor. The random forest subspace discriminant (RFSD) and the CNN model achieved the best results using MFDFA features calculated with quadratic approximations. The proposed method is a promising approach for fault classification in reciprocating compressors and multi-stage centrifugal pumps.

Entropy-Based Multifractal Testing of Heart Rate Variability during Cognitive-Autonomic Interplay

Entropy-based and fractal-based metrics derived from heart rate variability (HRV) have enriched the way cardiovascular dynamics can be described in terms of complexity. The most commonly used multifractal testing, a method using q moments to explore a range of fractal scaling in small-sized and large-sized fluctuations, is based on detrended fluctuation analysis, which examines the power-law relationship of standard deviation with the timescale in the measured signal. A more direct testing of a multifractal structure exists based on the Shannon entropy of bin (signal subparts) proportion. This work aims to reanalyze HRV during cognitive tasks to obtain new markers of HRV complexity provided by entropy-based multifractal spectra using the method proposed by Chhabra and Jensen in 1989. Inter-beat interval durations (RR) time series were obtained in 28 students comparatively in baseline (viewing a video) and during three cognitive tasks: Stroop color and word task, stop-signal, and go/no-go. The new HRV estimators were extracted from the f/α singularity spectrum of the RR magnitude increment series, established from q-weighted stable (log-log linear) power laws, namely: (i) the whole spectrum width (MF) calculated as αmax - αmin; the specific width representing large-sized fluctuations (MFlarge) calculated as α0 - αq+; and small-sized fluctuations (MFsmall) calculated as αq- - α0. As the main results, cardiovascular dynamics during Stroop had a specific MF signature while MFlarge was rather specific to go/no-go. The way these new HRV markers could represent different aspects of a complete picture of the cognitive-autonomic interplay is discussed, based on previously used entropy- and fractal-based markers, and the introduction of distribution entropy (DistEn), as a marker recently associated specifically with complexity in the cardiovascular control.

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.

Better than DFA? A Bayesian Method for Estimating the Hurst Exponent in Behavioral Sciences

Detrended Fluctuation Analysis (DFA) is the most popular fractal analytical technique used to evaluate the strength of long-range correlations in empirical time series in terms of the Hurst exponent, H. Specifically, DFA quantifies the linear regression slope in log-log coordinates representing the relationship between the time series' variability and the number of timescales over which this variability is computed. We compared the performance of two methods of fractal analysis-the current gold standard, DFA, and a Bayesian method that is not currently well-known in behavioral sciences: the Hurst-Kolmogorov (HK) method-in estimating the Hurst exponent of synthetic and empirical time series. Simulations demonstrate that the HK method consistently outperforms DFA in three important ways. The HK method: (i) accurately assesses long-range correlations when the measurement time series is short, (ii) shows minimal dispersion about the central tendency, and (iii) yields a point estimate that does not depend on the length of the measurement time series or its underlying Hurst exponent. Comparing the two methods using empirical time series from multiple settings further supports these findings. We conclude that applying DFA to synthetic time series and empirical time series during brief trials is unreliable and encourage the systematic application of the HK method to assess the Hurst exponent of empirical time series in behavioral sciences.

Evaluation of non-linear heart rate variability using multi-scale multi-fractal detrended fluctuation analysis in mice: Roles of the autonomic nervous system and sinoatrial node

Nonlinear analyses of heart rate variability (HRV) can be used to quantify the unpredictability, fractal properties and complexity of heart rate. Fractality and its analysis provides valuable information about cardiovascular health. Multi-Scale Multi-Fractal Detrended Fluctuation Analysis (MSMFDFA) is a complexity-based algorithm that can be used to quantify the multi-fractal dynamics of the HRV time series through investigating characteristic exponents at different time scales. This method is applicable to short time series and it is robust to noise and nonstationarity. We have used MSMFDFA, which enables assessment of HRV in the frequency ranges encompassing the very-low frequency and ultra-low frequency bands, to jointly assess multi-scale and multi-fractal dynamics of HRV signals obtained from telemetric ECG recordings in wildtype mice at baseline and after autonomic nervous system (ANS) blockade, from electrograms recorded from isolated atrial preparations and from spontaneous action potential recordings in isolated sinoatrial node myocytes. Data demonstrate that the fractal profile of the intrinsic heart rate is significantly different from the baseline heart rate in vivo, and it is also altered after ANS blockade at specific scales and fractal order domains. For beating rate in isolated atrial preparations and intrinsic heart rate in vivo, the average fractal structure of the HRV increased and multi-fractality strength decreased. These data demonstrate that fractal properties of the HRV depend on both ANS activity and intrinsic sinoatrial node function and that assessing multi-fractality at different time scales is an effective approach for HRV assessment.

A Single Session of SMR-Neurofeedback Training Improves Selective Attention Emerging from a Dynamic Structuring of Brain-Heart Interplay

Research on sensorimotor rhythms (SMR) based on neurofeedback (NFb) emphasizes improvements in selective attention associated with SMR amplification. However, the long-term training proposed in most studies posed the question of acceptability, which led to the evaluation of the potential of a single NFb session. Based on cognitive and autonomic controls interfering with attention processes, we hypothesized changes in selective attention after a single SMR-NFb session, along with changes in brain-heart interplay, which are reflected in the multifractality of heartbeat dynamics. Here, young healthy participants (n = 35, 20 females, 21 ± 3 years) were randomly assigned either to a control group (Ctrl) watching a movie or to a neurofeedback (NFb) group performing a single session of SMR-NFb. A headset with EEG electrodes (positioned on C3 and C4) connected to a smartphone app served to guide and to evaluate NFb training efficacy. A Stroop task was performed for 8 min by each group before and after the intervention (movie vs. SMR-NFb) while collecting heart rate variability and C4-EEG for 20 min. When compared to Ctrl, the NFb group exhibited better Stroop performance, especially when facing incongruent trials. The multifractality and NFb training efficacy were identified as strong predictors of the gain in global Stroop performance, while multifractality was the only predictor regarding incongruent trials. We conclude that a single session of SMR-NFb improves selective attention in healthy individuals through the specific reorganization of brain-heart interplay, which is reflected in multifractal heartbeat dynamics.

Heart Rate Variability from Wearable Photoplethysmography Systems: Implications in Sleep Studies at High Altitude

The interest in photoplethysmography (PPG) for sleep monitoring is increasing because PPG may allow assessing heart rate variability (HRV), which is particularly important in breathing disorders. Thus, we aimed to evaluate how PPG wearable systems measure HRV during sleep at high altitudes, where hypobaric hypoxia induces respiratory disturbances. We considered PPG and electrocardiographic recordings in 21 volunteers sleeping at 4554 m a.s.l. (as a model of sleep breathing disorder), and five alpine guides sleeping at sea level, 6000 m and 6800 m a.s.l. Power spectra, multiscale entropy, and self-similarity were calculated for PPG tachograms and electrocardiography R–R intervals (RRI). Results demonstrated that wearable PPG devices provide HRV measures even at extremely high altitudes. However, the comparison between PPG tachograms and RRI showed discrepancies in the faster spectral components and at the shorter scales of self-similarity and entropy. Furthermore, the changes in sleep HRV from sea level to extremely high altitudes quantified by RRI and PPG tachograms in the five alpine guides tended to be different at the faster frequencies and shorter scales. Discrepancies may be explained by modulations of pulse wave velocity and should be considered to interpret correctly autonomic alterations during sleep from HRV analysis.

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.

Multiscale assessment of the degree of multifractality for physiological time series

Recent advancements in detrended fluctuation analysis (DFA) allow evaluating multifractal coefficients scale-by-scale, a promising approach for assessing the complexity of biomedical signals. The multifractality degree is typically quantified by the singularity spectrum width (W_SS), a method that is critically unstable in multiscale applications. Thus, we aim to propose a robust multiscale index of multifractality, compare it with W_SS and illustrate its performance on real biosignals. The proposed index is the cumulative function of squared increments between consecutive DFA coefficients at each scale n: α_CF(n). We compared it with W_SS calculated scale-by-scale considering monofractal/monoscale, monofractal/multiscale, multifractal/monoscale and multifractal/multiscale random processes. The two indices provided qualitatively similar descriptions of multifractality, but α_CF(n) differentiated better the multifractal components from artefacts due to crossovers or detrending overfitting. Applied on 24 h heart rate recordings of 14 participants, the singularity spectrum failed to always satisfy the concavity requirement for providing meaningful W_SS, while α_CF(n) demonstrated a statistically significant heart rate multifractality at night in the scale ranges 16-100 and 256-680 s. Furthermore, α_CF(n) did not reject the hypothesis of monofractality at daytime, coherently with previous reports of lower nonlinearity and monoscale multifractality during the day. Thus, α_CF(n) appears a robust index of multiscale multifractality that is useful for quantifying complexity alterations of physiological series. This article is part of the theme issue 'Advanced computation in cardiovascular physiology: new challenges and opportunities'.

Multifractal and Multiscale Detrended Fluctuation Analysis of Cardiovascular Signals: how the Estimation Bias Affects ShortTerm Coefficients and a Way to mitigate this Error

The Detrended Fluctuation Analysis (DFA) is a popular method for quantifying the self-similarity of the heart rate that may reveal complexity aspects in cardiovascular regulation. However, the self-similarity coefficients provided by DFA may be affected by an overestimation error associated with the shortest scales. Recently, the DFA has been extended to calculate the multifractal-multiscale self-similarity and some evidence suggests that overestimation errors may affect different multifractal orders. If this is the case, the error might alter substantially the multifractal-multiscale representation of the cardiovascular self-similarity. The aim of this work is 1) to describe how this error depends on the multifractal orders and scales and 2) to propose a way to mitigate this error applicable to real cardiovascular series.Clinical Relevance- The proposed correction method may extend the multifractal analysis at the shortest scales, thus allowing to better assess complexity alterations in the cardiac autonomic regulation and to increase the clinical value of DFA.

A Comprehensive Review of Techniques for Processing and Analyzing Fetal Heart Rate Signals

The availability of standardized guidelines regarding the use of electronic fetal monitoring (EFM) in clinical practice has not effectively helped to solve the main drawbacks of fetal heart rate (FHR) surveillance methodology, which still presents inter- and intra-observer variability as well as uncertainty in the classification of unreassuring or risky FHR recordings. Given the clinical relevance of the interpretation of FHR traces as well as the role of FHR as a marker of fetal wellbeing autonomous nervous system development, many different approaches for computerized processing and analysis of FHR patterns have been proposed in the literature. The objective of this review is to describe the techniques, methodologies, and algorithms proposed in this field so far, reporting their main achievements and discussing the value they brought to the scientific and clinical community. The review explores the following two main approaches to the processing and analysis of FHR signals: traditional (or linear) methodologies, namely, time and frequency domain analysis, and less conventional (or nonlinear) techniques. In this scenario, the emerging role and the opportunities offered by Artificial Intelligence tools, representing the future direction of EFM, are also discussed with a specific focus on the use of Artificial Neural Networks, whose application to the analysis of accelerations in FHR signals is also examined in a case study conducted by the authors.

Asymmetric multiscale multifractal analysis (AMMA) of heart rate variability

Objective.The physiological activity of the heart is controlled and modulated mostly by the parasympathetic and sympathetic nervous systems. Heart rate variability (HRV) analysis is therefore used to observe fluctuations that reflect changes in the activity in these two branches. Knowing that acceleration and deceleration patterns in heart rate fluctuations are asymmetrically distributed, the ability to analyze HRV asymmetry was introduced into MMA.Approach. The new method is called asymmetric multiscale multifractal analysis (AMMA) and the analysis involved six groups: 36 healthy persons, 103 cases with aortic valve stenosis, 36 with hypertrophic cardiomyopathy, 32 with atrial fibrillation, 59 patients with coronary artery disease (CAD) and 13 with congestive heart failure.Main results. Analyzing the results obtained for the 6 groups of patients based on the AMMA method, i.e. comparing the Hurst surfaces for heart rate decelerations and accelerations, it was noticed that these surfaces differ significantly. And the differences occur in most groups for large fluctuations (multifractal parameterq > 0). In addition, a similarity was found for all groups for the AMMA Hurst surface for decelerations to the MMA Hurst surface-heart rate decelerations (lengthening of the RR intervals) appears to be the main factor determining the shape of the complete Hurst surface and so the multifractal properties of HRV. The differences between the groups, especially for CAD, hypertrophic cardiomyopathy and aortic valve stenosis, are more visible if the Hurst surfaces are analyzed separately for accelerations and decelerations.Significance. The AMMA results presented here may provide additional input for HRV analysis and create a new paradigm for future medical screening. Note that the HRV analysis using MMA (without distinguishing accelerations from decelerations) gave satisfactory screening statistics in our previous studies.

Entropy and Multifractal-Multiscale Indices of Heart Rate Time Series to Evaluate Intricate Cognitive-Autonomic Interactions

Recent research has clarified the existence of a networked system involving a cortical and subcortical circuitry regulating both cognition and cardiac autonomic control, which is dynamically organized as a function of cognitive demand. The main interactions span multiple temporal and spatial scales and are extensively governed by nonlinear processes. Hence, entropy and (multi)fractality in heart period time series are suitable to capture emergent behavior of the cognitive-autonomic network coordination. This study investigated how entropy and multifractal-multiscale analyses could depict specific cognitive-autonomic architectures reflected in the heart rate dynamics when students performed selective inhibition tasks. The participants (N=37) completed cognitive interference (Stroop color and word task), action cancellation (stop-signal) and action restraint (go/no-go) tasks, compared to watching a neutral movie as baseline. Entropy and fractal markers (respectively, the refined composite multiscale entropy and multifractal-multiscale detrended fluctuation analysis) outperformed other time-domain and frequency-domain markers of the heart rate variability in distinguishing cognitive tasks. Crucially, the entropy increased selectively during cognitive interference and the multifractality increased during action cancellation. An interpretative hypothesis is that cognitive interference elicited a greater richness in interactive processes that form the central autonomic network while action cancellation, which is achieved via biasing a sensorimotor network, could lead to a scale-specific heightening of multifractal behavior.

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.

Sex Differences in Heart Rate Nonlinearity by Multifractal Multiscale Detrended Fluctuation Analysis

Recent developments of detrended fluctuation analysis (DFA) provide multifractal/multiscale (MFMS) descriptions of the heart rate self-similarity, a promising approach to cardiovascular complexity. However, it is unclear whether the MFMS DFA may also describe the nonlinear components of heart rate variability. Our aim is to define MFMS DFA indices for quantifying the short-term and long-term degree of the heart-rate nonlinearity and to apply these indices to detect possible sex-related differences.We recorded the inter-beat-interval (IBI) series in 42 male and in 42 female healthy participants sitting at rest for about 2 hours. For each series j, we generated 100 phase-randomized surrogate series. We applied the MFMS DFA to estimate the self-similarity coefficients α over scales τ between 8 and 512 s and moment orders q between -5 and +5, obtaining coefficients for the original series, α^O,j (q, τ), and for each surrogate, α^i,j (q, τ) with 1≤i≤100. We first evaluated π^j(q, τ), percentile of α^i,j (q, τ) distribution in which was α^O,j (q, τ). Then we calculated the percentages of scales where π^j(q, τ) was <5% for 8≤τ≤16 s (short-term nonlinearity index NL₁(q)) and for 16≤τ≤512 s (long-term nonlinearity index NL₂(q)). We found that NL₁(q) was generally greater than 50% at all q≥0 but q=2 (i.e., moment order of the monofractal DFA), while at q<0 it was high in males only, with significant sex differences at q=-1 and q=-2. Results indicate that the multifractal DFA may highlight nonlinear heart-rate components at the short scales that are not revealed by the traditional monofractal DFA and that appear related to gender differences.Clinical Relevance- This supports the use of MFMS DFA to integrate the linear information from traditional spectral methods of heart rate variability in clinical studies aimed at improving the stratification of the cardiovascular risk.

Can the Detrended Fluctuation Analysis Reveal Nonlinear Components of Heart Rate Variabilityƒ

The Detrended Fluctuation Analysis (DFA) is widely employed to quantify the fractal dynamics of R-R intervals (RRI). This is usually done by estimating a short- and a long-term coefficient, but it is still unclear how much the information provided by such a bi-scale DFA is independent of that of traditional spectral indices. However, more sophisticated DFA approaches have been recently proposed, including the multifractal-multiscale DFA and the DFA for magnitude and sign of RRI changes. The aim of our work is to investigate whether novel DFA approaches allow extracting the information on the nonlinear RRI dynamics that traditional spectral methods cannot retrieve.We selected 4-hour segments of beat-by-beat RRI series from a 24-hour Holter recording, one during daytime (wake), one at night (sleep) in a healthy volunteer. From the wake segment, we generated 100 surrogate series shuffling the phases but preserving the power spectrum, and then from each of the resulting RRI series, we generated the series of the sign and the series of the magnitude of successive RRI changes. We generated similar series from the sleep recording. Thus, we finally obtained 6 original beat-to-beat series to be compared with 600 surrogate series, each of 4-hour duration.The comparison between original and surrogate series showed that for this experimental setting, the traditional monofractal DFA provides the same information retrievable by the power spectrum. However, specific components of the multifractal DFA reveal information not detectable by the power spectrum, particularly in the sleep condition. Furthermore, the DFA of the magnitude of RRI changes reflects important nonlinear components. Therefore, these more sophisticated DFA approaches might effectively improve the clinical value of RRI variability analysis.

Complexity Analysis of Surface Electromyography for Assessing the Myoelectric Manifestation of Muscle Fatigue: A Review

The surface electromyography (sEMG) records the electrical activity of muscle fibers during contraction: one of its uses is to assess changes taking place within muscles in the course of a fatiguing contraction to provide insights into our understanding of muscle fatigue in training protocols and rehabilitation medicine. Until recently, these myoelectric manifestations of muscle fatigue (MMF) have been assessed essentially by linear sEMG analyses. However, sEMG shows a complex behavior, due to many concurrent factors. Therefore, in the last years, complexity-based methods have been tentatively applied to the sEMG signal to better individuate the MMF onset during sustained contractions. In this review, after describing concisely the traditional linear methods employed to assess MMF we present the complexity methods used for sEMG analysis based on an extensive literature search. We show that some of these indices, like those derived from recurrence plots, from entropy or fractal analysis, can detect MMF efficiently. However, we also show that more work remains to be done to compare the complexity indices in terms of reliability and sensibility; to optimize the choice of embedding dimension, time delay and threshold distance in reconstructing the phase space; and to elucidate the relationship between complexity estimators and the physiologic phenomena underlying the onset of MMF in exercising muscles.

Day and Night Changes of Cardiovascular Complexity: A Multi-Fractal Multi-Scale Analysis

Recently, a multifractal-multiscale approach to detrended fluctuation analysis (DFA) was proposed to evaluate the cardiovascular fractal dynamics providing a surface of self-similarity coefficients α(q,τ), function of the scale τ, and moment order q. We hypothesize that this versatile DFA approach may reflect the cardiocirculatory adaptations in complexity and nonlinearity occurring during the day/night cycle. Our aim is, therefore, to quantify how α(q, τ) surfaces of cardiovascular series differ between daytime and night-time. We estimated α(q,τ) with -5 ≤ q ≤ 5 and 8 ≤ τ ≤ 2048 s for heart rate and blood pressure beat-to-beat series over periods of few hours during daytime wake and night-time sleep in 14 healthy participants. From the α(q,τ) surfaces, we estimated short-term (<16 s) and long-term (from 16 to 512 s) multifractal coefficients. Generating phase-shuffled surrogate series, we evaluated short-term and long-term indices of nonlinearity for each q. We found a long-term night/day modulation of α(q,τ) between 128 and 256 s affecting heart rate and blood pressure similarly, and multifractal short-term modulations at q < 0 for the heart rate and at q > 0 for the blood pressure. Consistent nonlinearity appeared at the shorter scales at night excluding q = 2. Long-term circadian modulations of the heart rate DFA were previously associated with the cardiac vulnerability period and our results may improve the risk stratification indicating the more relevant α(q,τ) area reflecting this rhythm. Furthermore, nonlinear components in the nocturnal α(q,τ) at q ≠ 2 suggest that DFA may effectively integrate the linear spectral information with complexity-domain information, possibly improving the monitoring of cardiac interventions and protocols of rehabilitation medicine.

Entropy in Heart Rate Dynamics Reflects How HRV-Biofeedback Training Improves Neurovisceral Complexity during Stress-Cognition Interactions

Despite considerable appeal, the growing appreciation of biosignals complexity reflects that system complexity needs additional support. A dynamically coordinated network of neurovisceral integration has been described that links prefrontal-subcortical inhibitory circuits to vagally-mediated heart rate variability. Chronic stress is known to alter network interactions by impairing amygdala functional connectivity. HRV-biofeedback training can counteract stress defects. We hypothesized the great value of an entropy-based approach of beat-to-beat biosignals to illustrate how HRVB training restores neurovisceral complexity, which should be reflected in signal complexity. In thirteen moderately-stressed participants, we obtained vagal tone markers and psychological indexes (state anxiety, cognitive workload, and Perceived Stress Scale) before and after five-weeks of daily HRVB training, at rest and during stressful cognitive tasking. Refined Composite Multiscale Entropy (RCMSE) was computed over short time scales as a marker of signal complexity. Heightened vagal tone at rest and during stressful tasking illustrates training benefits in the brain-to-heart circuitry. The entropy index reached the highest significance levels in both variance and ROC curves analyses. Restored vagal activity at rest correlated with gain in entropy. We conclude that HRVB training is efficient in restoring healthy neurovisceral complexity and stress defense, which is reflected in HRV signal complexity. The very mechanisms that are involved in system complexity remain to be elucidated, despite abundant literature existing on the role played by amygdala in brain interconnections.

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.

Decomposing the complexity of heart-rate variability by the multifractal-multiscale approach to detrended fluctuation analysis: an application to low-level spinal cord injury

OBJECTIVE

While several studies have assessed autonomic cardiovascular control after a spinal cord lesion using heart-rate variability (HRV) indices in the frequency and time domains, complexity measures have rarely been used, even if detrended fluctuation analysis (DFA) appeared promising. Recent developments in DFA decompose the multifractal contributions using temporal scales. Our aim is to evaluate the potential of these new DFA tools, considering as an example application the decomposition of HRV complexity in individuals with spinal cord injury (SCI) at a low lesion level, for whom alterations in traditional indices are not expected.

APPROACH

We enrolled 14 subjects with SCI with a lesion below the eleventh thoracic vertebra and 34 able-bodied (AB) controls. We recorded the R-R intervals (RRI) for 10 min in supine and sitting postures. We applied the multifractal-multiscale (MFMS) DFA to derive scale coefficients, α(q,τ), with function of the multifractal order q and scale τ, and evaluated a scale-coefficient dispersion index, α _SD(τ), as the standard deviation of α(q,τ) over q. We calculated the RRI increments, their magnitude and sign, estimating the MFMS DFA coefficients for the series of magnitude α _m(q,τ) and sign α _s(q,τ).

MAIN RESULTS

While sitting, differences between SCI and AB groups depended on q for coefficients 16  <  τ  <  32 s, so that α _SD(τ) was lower in individuals with SCI at τ  =  25 s. In the supine condition, short-term scales were greater in individuals with SCI for all q, and α _SD(τ) did not differ between groups. Group differences were found in α _s(q,τ) and not in α _m(q,τ) or in traditional HRV indices. The surrogate analysis showed AB-SCI differences in linear HRV components at scales τ  <  16 s and nonlinear components at larger scales.

SIGNIFICANCE

Complexity decomposition by DFA describes autonomic alterations in HRV in low-level paraplegia better than traditional indices, probably pointing out a loss of system complexity in the sitting posture and an impaired sympatho/vagal modulation in the supine position.