Permutation entropy improves fetal behavioural state classification based on heart rate analysis from biomagnetic recordings in near term fetuses

Persistent homology of coarse-grained state-space networks

This work is dedicated to the topological analysis of complex transitional networks for dynamic state detection. Transitional networks are formed from time series data and they leverage graph theory tools to reveal information about the underlying dynamic system. However, traditional tools can fail to summarize the complex topology present in such graphs. In this work, we leverage persistent homology from topological data analysis to study the structure of these networks. We contrast dynamic state detection from time series using a coarse-grained state-space network (CGSSN) and topological data analysis (TDA) to two state of the art approaches: ordinal partition networks (OPNs) combined with TDA and the standard application of persistent homology to the time-delay embedding of the signal. We show that the CGSSN captures rich information about the dynamic state of the underlying dynamical system as evidenced by a significant improvement in dynamic state detection and noise robustness in comparison to OPNs. We also show that because the computational time of CGSSN is not linearly dependent on the signal's length, it is more computationally efficient than applying TDA to the time-delay embedding of the time series.

Information-Based Similarity of Ordinal Pattern Sequences as a Novel Descriptor in Obstructive Sleep Apnea Screening Based on Wearable Photoplethysmography Bracelets

Obstructive sleep apnea (OSA) is a common respiratory disorder associated with autonomic nervous system (ANS) dysfunction, resulting in abnormal heart rate variability (HRV). Capable of acquiring heart rate (HR) information with more convenience, wearable photoplethysmography (PPG) bracelets are proven to be a potential surrogate for electrocardiogram (ECG)-based devices. Meanwhile, bracelet-type PPG has been heavily marketed and widely accepted. This study aims to investigate the algorithm that can identify OSA with wearable devices. The information-based similarity of ordinal pattern sequences (OP_IBS), which is a modified version of the information-based similarity (IBS), has been proposed as a novel index to detect OSA based on wearable PPG signals. A total of 92 PPG recordings (29 normal subjects, 39 mild-moderate OSA subjects and 24 severe OSA subjects) were included in this study. OP_IBS along with classical indices were calculated. For severe OSA detection, the accuracy of OP_IBS was 85.9%, much higher than that of the low-frequency power to high-frequency power ratio (70.7%). The combination of OP_IBS, IBS, CV and LF/HF can achieve 91.3% accuracy, 91.0% sensitivity and 91.5% specificity. The performance of OP_IBS is significantly improved compared with our previous study based on the same database with the IBS method. In the Physionet database, OP_IBS also performed exceptionally well with an accuracy of 91.7%. This research shows that the OP_IBS method can access the HR dynamics of OSA subjects and help diagnose OSA in clinical environments.

Fetal respiratory movements improve reliability of heart rate variability and suggest a coupling between fetal respiratory arrhythmia and vagal activity

Fetal heart rate variability (FHRV) reflects autonomic cardiac regulation. The autonomic nervous system constantly adjusts the heart rate to maintain homeostasis. By providing insight into the fetal autonomic state, FHRV has the potential to become an investigational and clinical instrument. However, the method needs standardization and the influence of fetal movements, including fetal respiratory movements, is not well explored. Therefore, in a highly standardized setting, the aim was to evaluate the association between fetal movements and fetal heart rate variability (FHRV) including their impact on reliability. Fetal heart rate was obtained by noninvasive fetal electrocardiography (NI-FECG) and fetal movements by simultaneous ultrasound scanning in 30 healthy singleton pregnant women on two occasions with a maximum interval of 7 days. The standard deviation of normal-to-normal RR-intervals (SDNN), root mean square of successive RR-interval differences (RMDDS), high-frequency power (HF-power), low-frequency power (LF-power), and LF/HF were measured. A multivariate mixed model was used and reliability was defined as acceptable by a coefficient of variance (CV) ≤15% and an intraclass correlation coefficient (ICC) ≥0.80. During time periods with fetal respiratory movements, the highest reliability was achieved. Intra- and inter-observer reliability measurements were very high (CV: 0-9%; ICC ≧ 0.86). Within the same recording, SDNN and RMSSD achieved acceptable reliability (CV: 14-15%; ICC ≧ 0.80). However, day-to-day reliability displayed high CV's. In time periods with fetal respiratory movements, as compared to periods with quiescence RMSSD and HF-power were higher (Ratio: 1.33-2.03) and LF/HF power lower (Ratio: 0.54). In periods with fetal body movements SDNN, RMSSD and HF-power were higher (Ratio: 1.27-1.65). In conclusion, time periods with fetal respiratory movements were associated with high reliability of FHRV analyses and the highest values of parameters supposed to represent vagal activity.

Non-linear Methods Predominant in Fetal Heart Rate Analysis: A Systematic Review

The analysis of fetal heart rate variability has served as a scientific and diagnostic tool to quantify cardiac activity fluctuations, being good indicators of fetal well-being. Many mathematical analyses were proposed to evaluate fetal heart rate variability. We focused on non-linear analysis based on concepts of chaos, fractality, and complexity: entropies, compression, fractal analysis, and wavelets. These methods have been successfully applied in the signal processing phase and increase knowledge about cardiovascular dynamics in healthy and pathological fetuses. This review summarizes those methods and investigates how non-linear measures are related to each paper's research objectives. Of the 388 articles obtained in the PubMed/Medline database and of the 421 articles in the Web of Science database, 270 articles were included in the review after all exclusion criteria were applied. While approximate entropy is the most used method in classification papers, in signal processing, the most used non-linear method was Daubechies wavelets. The top five primary research objectives covered by the selected papers were detection of signal processing, hypoxia, maturation or gestational age, intrauterine growth restriction, and fetal distress. This review shows that non-linear indices can be used to assess numerous prenatal conditions. However, they are not yet applied in clinical practice due to some critical concerns. Some studies show that the combination of several linear and non-linear indices would be ideal for improving the analysis of the fetus's well-being. Future studies should narrow the research question so a meta-analysis could be performed, probing the indices' performance.

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.

Machine learning model on heart rate variability metrics identifies asymptomatic toddlers exposed to zika virus during pregnancy

Objective. Although the Zika virus (ZIKV) seems to be prominently neurotropic, there are some reports of involvement of other organs, particularly the heart. Of special concern are those children exposed prenatally to ZIKV and born without microcephaly or other congenital anomalies. Electrocardiogram (ECG)-derived heart rate variability (HRV) metrics represent an attractive, low-cost, widely deployable tool for early identification of developmental functional alterations in exposed children born without such overt clinical symptoms. We hypothesized that HRV in such children would yield a biomarker of fetal ZIKV exposure. Our objective was to test this hypothesis in young children exposed to ZIKV during pregnancy.Approach. We investigated the HRV properties of 21 children aged 4-25 months from Brazil. The infants were divided into two groups, the ZIKV-exposed (n = 13) and controls (n = 8). Single-channel ECG was recorded in each child at ∼15 months of age and HRV was analyzed in 5 min segments to provide a comprehensive characterization of the degree of variability and complexity of the heart rate.Main results.Using a cubic support vector machine classifier we identified babies as Zika cases or controls with a negative predictive value of 92% and a positive predictive value of 86%. Our results show that a machine learning model derived from HRV metrics can help differentiate between ZIKV-affected, yet asymptomatic, and non-ZIKV-exposed babies. We identified the box count as the best HRV metric in this study allowing such differentiation, regardless of the presence of microcephaly.Significance.We show that it is feasible to measure HRV in infants and toddlers using a small non-invasive portable ECG device and that such an approach may uncover the memory ofin uteroexposure to ZIKV. We discuss putative mechanisms. This approach may be useful for future studies and low-cost screening tools involving this challenging to examine population.

Ensemble Empirical Mode Decomposition with Adaptive Noise with Convolution Based Gated Recurrent Neural Network: A New Deep Learning Model for South Asian High Intensity Forecasting

The intensity variation of the South Asian high (SAH) plays an important role in the formation and extinction of many kinds of mesoscale systems, including tropical cyclones, southwest vortices in the Asian summer monsoon (ASM) region, and the precipitation in the whole Asia Europe region, and the SAH has a vortex symmetrical structure; its dynamic field also has the symmetry form. Not enough previous studies focus on the variation of SAH daily intensity. The purpose of this study is to establish a day-to-day prediction model of the SAH intensity, which can accurately predict not only the interannual variation but also the day-to-day variation of the SAH. Focusing on the summer period when the SAH is the strongest, this paper selects the geopotential height data between 1948 and 2020 from NCEP to construct the SAH intensity datasets. Compared with the classical deep learning methods of various kinds of efficient time series prediction model, we ultimately combine the Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) method, which has the ability to deal with the nonlinear and unstable single system, with the Permutation Entropy (PE) method, which can extract the SAH intensity feature of IMF decomposed by CEEMDAN, and the Convolution-based Gated Recurrent Neural Network (ConvGRU) model is used to train, test, and predict the intensity of the SAH. The prediction results show that the combination of CEEMDAN and ConvGRU can have a higher accuracy and more stable prediction ability than the traditional deep learning model. After removing the redundant features in the time series, the prediction accuracy of the SAH intensity is higher than that of the classical model, which proves that the method has good applicability for the prediction of nonlinear systems in the atmosphere.

Information-theoretic characterization of eye-tracking signals with relation to cognitive tasks

Eye tracking is being increasingly used as a more powerful diagnosis instrument when compared with traditional pen-and-paper tests in psychopedagogy and psychology. This technology may significantly improve neurocognitive assessments in gathering indirect latent information about the subjects' performance. However, the meaning and implications of these data are far from being fully understood. In this work, we present a comprehensive study of eye tracking time series in terms of statistical complexity measures. We registered the eye tracking movements of several subjects solving the two parts of the commonly applied Trail Making Test (TMT-A and TMT-B) and studied their Shannon entropy, disequilibrium, statistical complexity, and Fisher information with respect to three different probability distributions. The results show that these quantifiers reveal information about different features of the gaze depending on the distribution considered. As a meaningful result, we found that Fisher information in the position distribution reflects the difficulties encountered by the subject when solving the task. Such a characterization may be of interest to understand the underlying cognitive tasks performed by the subjects, and, additionally, it can serve as a source of valuable parameters to quantitatively assess how and why the subjects budget their attention, providing psychologists and psychopedagogues with more refined neuropsychological evaluation features and tools.

On the Use of Fuzzy and Permutation Entropy in Hand Gesture Characterization from EMG Signals: Parameters Selection and Comparison

The surface electromyography signal (sEMG) is widely used for gesture characterization; its reliability is strongly connected to the features extracted from sEMG recordings. This study aimed to investigate the use of two complexity measures, i.e., fuzzy entropy (FEn) and permutation entropy (PEn) for hand gesture characterization. Fourteen upper limb movements, sorted into three sets, were collected on ten subjects and the performances of FEn and PEn for gesture descriptions were analyzed for different computational parameters. FEn and PEn were able to properly cluster the expected numbers of gestures, but computational parameters were crucial for ensuring clusters’ separability and proper gesture characterization. FEn and PEn were also compared with other eighteen classical time and frequency domain features through the minimum redundancy maximum relevance algorithm and showed the best predictive importance scores in two gesture sets; they also had scores within the subset of the best five features in the remaining one. Further, the classification accuracies of four different feature sets presented remarkable increases when FEn and PEn are included as additional features. Outcomes support the use of FEn and PEn for hand gesture description when computational parameters are properly selected, and they could be useful in supporting the development of robotic arms and prostheses myoelectric control.

Fetal Heart Rate Variability Is Affected by Fetal Movements: A Systematic Review

Introduction: Fetal heart rate variability (FHRV) evaluates the fetal neurological state, which is poorly assessed by conventional prenatal surveillance including cardiotocography (CTG). Accurate FHRV on a beat-to-beat basis, assessed by time domain and spectral domain analyses, has shown promising results in the scope of fetal surveillance. However, accepted standards for these techniques are lacking, and the influence of fetal breathing movements and gross movements may be especially challenging. Thus, current standards for equivalent assessments in adults prescribe rest and controlled respiration. The aim of this review is to clarify the importance of fetal movements on FHRV. Methods: A systematic review in accordance with the PRISMA guidelines based on publications in the EMBASE, the MEDLINE, and the Cochrane Library databases was performed. Studies describing the impact of fetal movements on time domain, spectral domain and entropy analyses in healthy human fetuses were reviewed. Only studies based on fetal electrocardiography or fetal magnetocardiography were included. PROSPERO registration number: CRD42018068806. Results: In total, 14 observational studies were included. Fetal movement detection, signal processing, length, and selection of appropriate time series varied across studies. Despite these divergences, all studies showed an increase in overall FHRV in the moving fetus compared to the resting fetus. Especially short-term, vagal mediated indexes showed an increase during fetal breathing movements including an increase in Root Mean Square of the Successive Differences (RMSSD) and High Frequency power (HF) and a decrease in Low Frequency power/High Frequency power (LF/HF). These findings were present even in analyses restricted to one specific fetal behavioral state defined by Nijhuis. On the other hand, fetal body movements seemed to increase parameters supposed to represent the sympathetic response [LF and Standard Deviation of RR-intervals from normal sinus beats (SDNN)] proportionally more than parameters representing the parasympathetic response (RMSSD, HF). Results regarding entropy analyses were inconclusive. Conclusion: Time domain analyses as well as spectral domain analyses are affected by fetal movements. Fetal movements and especially breathing movements should be considered in these analyses of FHRV.

A pilot study: Auditory steady-state responses (ASSR) can be measured in human fetuses using fetal magnetoencephalography (fMEG)

Background

Auditory steady-state responses (ASSRs) are ongoing evoked brain responses to continuous auditory stimuli that play a role for auditory processing of complex sounds and speech perception. Transient auditory event-related responses (AERRs) have previously been recorded using fetal magnetoencephalography (fMEG) but involve different neurological pathways. Previous studies in children and adults demonstrated that the cortical components of the ASSR are significantly affected by state of consciousness and by maturational changes in neonates and young infants. To our knowledge, this is the first study to investigate ASSRs in human fetuses.

Methods

47 fMEG sessions were conducted with 24 healthy pregnant women in three gestational age groups (30–32 weeks, 33–35 weeks and 36–39 weeks). The stimulation consisted of amplitude-modulated (AM) tones with a duration of one second, a carrier frequency (CF) of 500 Hz and a modulation frequency (MF) of 27 Hz or 42 Hz. Both tones were presented in a random order with equal probability adding up to 80–100 repetitions per tone. The ASSR across trials was quantified by assessing phase synchrony in the cortical signals at the stimulation frequency.

Results and conclusion

Ten out of 47 recordings were excluded due to technical problems or maternal movements. Analysis of the included 37 fetal recordings revealed a statistically significant response for the phase coherence between trials for the MF of 27 Hz but not for 42 Hz. An exploratory subgroup analysis moreover suggested an advantage in detectability for fetal behavioral state 2F (active asleep) compared to 1F (quiet asleep) detected using fetal heart rate. In conclusion, this pilot study is the first description of a method to detect human ASSRs in fetuses. The findings warrant further investigations of the developing fetal brain.

Nonlinear Methods Most Applied to Heart-Rate Time Series: A Review

The heart-rate dynamics are one of the most analyzed physiological interactions. Many mathematical methods were proposed to evaluate heart-rate variability. These methods have been successfully applied in research to expand knowledge concerning the cardiovascular dynamics in healthy as well as in pathological conditions. Notwithstanding, they are still far from clinical practice. In this paper, we aim to review the nonlinear methods most used to assess heart-rate dynamics. We focused on methods based on concepts of chaos, fractality, and complexity: Poincaré plot, recurrence plot analysis, fractal dimension (and the correlation dimension), detrended fluctuation analysis, Hurst exponent, Lyapunov exponent entropies (Shannon, conditional, approximate, sample entropy, and multiscale entropy), and symbolic dynamics. We present the description of the methods along with their most notable applications.

On the automatic parameter selection for permutation entropy

Permutation Entropy (PE) is a cost effective tool for summarizing the complexity of a time series. It has been used in many applications including damage detection, disease forecasting, detection of dynamical changes, and financial volatility analysis. However, to successfully use PE, an accurate selection of two parameters is needed: the permutation dimension n and embedding delay τ. These parameters are often suggested by experts based on a heuristic or by a trial and error approach. Both of these methods can be time-consuming and lead to inaccurate results. In this work, we investigate multiple schemes for automatically selecting these parameters with only the corresponding time series as the input. Specifically, we develop a frequency-domain approach based on the least median of squares and the Fourier spectrum, as well as extend two existing methods: Permutation Auto-Mutual Information Function and Multi-scale Permutation Entropy (MPE) for determining τ. We then compare our methods as well as current methods in the literature for obtaining both τ and n against expert-suggested values in published works. We show that the success of any method in automatically generating the correct PE parameters depends on the category of the studied system. Specifically, for the delay parameter τ, we show that our frequency approach provides accurate suggestions for periodic systems, nonlinear difference equations, and electrocardiogram/electroencephalogram data, while the mutual information function computed using adaptive partitions provides the most accurate results for chaotic differential equations. For the permutation dimension n, both False Nearest Neighbors and MPE provide accurate values for n for most of the systems with a value of n=5 being suitable in most cases.

First evidence that intrinsic fetal heart rate variability exists and is affected by hypoxic pregnancy

KEY POINTS

We introduce a technique to test whether intrinsic fetal heart rate variability (iFHRV) exists and we show the utility of the technique by testing the hypothesis that iFHRV is affected by chronic fetal hypoxia, one of the most common adverse outcomes of human pregnancy complicated by fetal growth restriction. Using an established late gestation ovine model of fetal development under chronic hypoxic conditions, we identify iFHRV in isolated fetal hearts and show that it is markedly affected by hypoxic pregnancy. Therefore, the isolated fetal heart has intrinsic variability and carries a memory of adverse intrauterine conditions experienced during the last third of pregnancy.

ABSTRACT

Fetal heart rate variability (FHRV) emerges from influences of the autonomic nervous system, fetal body and breathing movements, and from baroreflex and circadian processes. We tested whether intrinsic heart rate variability (iHRV), devoid of any external influences, exists in the fetal period and whether it is affected by chronic fetal hypoxia. Chronically catheterized ewes carrying male singleton fetuses were exposed to normoxia (n = 6) or hypoxia (10% inspired O₂ , n = 9) for the last third of gestation (105-138 days of gestation (dG); term ∼145 dG) in isobaric chambers. At 138 dG, isolated hearts were studied using a Langendorff preparation. We calculated basal intrinsic FHRV (iFHRV) indices reflecting iFHRV's variability, predictability, temporal symmetry, fractality and chaotic behaviour, from the systolic peaks within 15 min segments in each heart. Significance was assumed at P < 0.05. Hearts of fetuses isolated from hypoxic pregnancy showed approximately 4-fold increases in the Grid transformation as well as the AND similarity index (sgridAND) and a 4-fold reduction in the scale-dependent Lyapunov exponent slope. We also detected a 2-fold reduction in the Recurrence quantification analysis, percentage of laminarity (pL) and recurrences, maximum and average diagonal line (dlmax, dlmean) and the Multiscale time irreversibility asymmetry index. The iHRV measures dlmax, dlmean, pL and sgridAND correlated with left ventricular end-diastolic pressure across both groups (average R²  = 0.38 ± 0.03). This is the first evidence that iHRV originates in fetal life and that chronic fetal hypoxia significantly alters it. Isolated fetal hearts from hypoxic pregnancy exhibit a time scale-dependent higher complexity in iFHRV.

Teaching Ordinal Patterns to a Computer: Efficient Encoding Algorithms Based on the Lehmer Code

Ordinal patterns are the common basis of various techniques used in the study of dynamical systems and nonlinear time series analysis. The present article focusses on the computational problem of turning time series into sequences of ordinal patterns. In a first step, a numerical encoding scheme for ordinal patterns is proposed. Utilising the classical Lehmer code, it enumerates ordinal patterns by consecutive non-negative integers, starting from zero. This compact representation considerably simplifies working with ordinal patterns in the digital domain. Subsequently, three algorithms for the efficient extraction of ordinal patterns from time series are discussed, including previously published approaches that can be adapted to the Lehmer code. The respective strengths and weaknesses of those algorithms are discussed, and further substantiated by benchmark results. One of the algorithms stands out in terms of scalability: its run-time increases linearly with both the pattern order and the sequence length, while its memory footprint is practically negligible. These properties enable the study of high-dimensional pattern spaces at low computational cost. In summary, the tools described herein may improve the efficiency of virtually any ordinal pattern-based analysis method, among them quantitative measures like permutation entropy and symbolic transfer entropy, but also techniques like forbidden pattern identification. Moreover, the concepts presented may allow for putting ideas into practice that up to now had been hindered by computational burden. To enable smooth evaluation, a function library written in the C programming language, as well as language bindings and native implementations for various numerical computation environments are provided in the supplements.

A Method for Detecting Dynamic Mutation of Complex Systems Using Improved Information Entropy

In this study, a nonlinear analysis method called improved information entropy (IIE) is proposed on the basis of constructing a special probability mass function for the normalized analysis of Shannon entropy for a time series. The definition is directly applied to several typical time series, and the characteristic of IIE is analyzed. This method can distinguish different kinds of signals and reflects the complexity of one-dimensional time series of high sensitivity to the changes in signal. Thus, the method is applied to the fault diagnosis of a rolling bearing. Experimental results show that the method can effectively extract the sensitive characteristics of the bearing running state and has fast operation time and minimal parameter requirements.

Assessing sustainability in North America's ecosystems using criticality and information theory

Sustainability is a key concept in economic and policy debates. Nevertheless, it is usually treated only in a qualitative way and has eluded quantitative analysis. Here, we propose a sustainability index based on the premise that sustainable systems do not lose or gain Fisher Information over time. We test this approach using time series data from the AmeriFlux network that measures ecosystem respiration, water and energy fluxes in order to elucidate two key sustainability features: ecosystem health and stability. A novel definition of ecosystem health is developed based on the concept of criticality, which implies that if a system's fluctuations are scale invariant then the system is in a balance between robustness and adaptability. We define ecosystem stability by taking an information theory approach that measures its entropy and Fisher information. Analysis of the Ameriflux consortium big data set of ecosystem respiration time series is contrasted with land condition data. In general we find a good agreement between the sustainability index and land condition data. However, we acknowledge that the results are a preliminary test of the approach and further verification will require a multi-signal analysis. For example, high values of the sustainability index for some croplands are counter-intuitive and we interpret these results as ecosystems maintained in artificial health due to continuous human-induced inflows of matter and energy in the form of soil nutrients and control of competition, pests and disease.

Linear and nonlinear features of fetal heart rate on the assessment of fetal development in the course of pregnancy and the impact of fetal gender

OBJECTIVE

This work aims to investigate the impact of gestational age and fetal gender on fetal heart rate (FHR) tracings.

APPROACH

Different linear and nonlinear parameters indicating correlation or complexity were used to study the influence of fetal age and gender on FHR tracings. The signals were recorded from 99 normal pregnant women in a singleton pregnancy at gestational ages from 28 to 40 weeks, before the onset of labor. There were 56 female fetuses and 43 male.

MAIN RESULTS

Analysis of FHR shows that the means as well as measures of irregularity of FHR, such as approximate entropy and algorithmic complexity, decrease as gestation progresses. There were also indications that mutual information and multiscale entropy were lower in male fetuses in early pregnancy.

SIGNIFICANCE

Fetal age and gender seem to influence FHR tracings. Taking this into consideration would improve the interpretation of FHR monitoring.

Permutation entropy with vector embedding delays

Permutation entropy (PE) is a statistic used widely for the detection of structure within a time series. Embedding delay times at which the PE is reduced are characteristic timescales for which such structure exists. Here, a generalized scheme is investigated where embedding delays are represented by vectors rather than scalars, permitting PE to be calculated over a (D-1)-dimensional space, where D is the embedding dimension. This scheme is applied to numerically generated noise, sine wave and logistic map series, and experimental data sets taken from a vertical-cavity surface emitting laser exhibiting temporally localized pulse structures within the round-trip time of the laser cavity. Results are visualized as PE maps as a function of embedding delay, with low PE values indicating combinations of embedding delays where correlation structure is present. It is demonstrated that vector embedding delays enable identification of structure that is ambiguous or masked, when the embedding delay is constrained to scalar form.

Variance of permutation entropy and the influence of ordinal pattern selection

Permutation entropy (PE) is a widely used measure for complexity, often used to distinguish between complex systems (or complex systems in different states). Here, the PE variance for a stationary time series is derived, and the influence of ordinal pattern selection, specifically whether the ordinal patterns are permitted to overlap or not, is examined. It was found that permitting ordinal patterns to overlap reduces the PE variance, improving the ability of this statistic to distinguish between complex system states for both numeric (fractional Gaussian noise) and experimental (semiconductor laser with optical feedback) systems. However, with overlapping ordinal patterns, the precision to which the PE variance can be estimated becomes diminished, which can manifest as increased incidences of false positive and false negative errors when applying PE to statistical inference problems.

Permutation entropy of finite-length white-noise time series

Permutation entropy (PE) is commonly used to discriminate complex structure from white noise in a time series. While the PE of white noise is well understood in the long time-series limit, analysis in the general case is currently lacking. Here the expectation value and variance of white-noise PE are derived as functions of the number of ordinal pattern trials, N, and the embedding dimension, D. It is demonstrated that the probability distribution of the white-noise PE converges to a χ^{2} distribution with D!-1 degrees of freedom as N becomes large. It is further demonstrated that the PE variance for an arbitrary time series can be estimated as the variance of a related metric, the Kullback-Leibler entropy (KLE), allowing the qualitative N≫D! condition to be recast as a quantitative estimate of the N required to achieve a desired PE calculation precision. Application of this theory to statistical inference is demonstrated in the case of an experimentally obtained noise series, where the probability of obtaining the observed PE value was calculated assuming a white-noise time series. Standard statistical inference can be used to draw conclusions whether the white-noise null hypothesis can be accepted or rejected. This methodology can be applied to other null hypotheses, such as discriminating whether two time series are generated from different complex system states.

Assessment of fetal maturation age by heart rate variability measures using random forest methodology

Fetal maturation age assessment based on heart rate variability (HRV) is a predestinated tool in prenatal diagnosis. To date, almost linear maturation characteristic curves are used in univariate and multivariate models. Models using complex multivariate maturation characteristic curves are pending. To address this problem, we use Random Forest (RF) to assess fetal maturation age and compare RF with linear, multivariate age regression. We include previously developed HRV indices such as traditional time and frequency domain indices and complexity indices of multiple scales. We found that fetal maturation was best assessed by complexity indices of short scales and skewness in state-dependent datasets (quiet sleep, active sleep) as well as in state-independent recordings. Additionally, increasing fluctuation amplitude contributed to the model in the active sleep state. None of the traditional linear HRV parameters contributed to the RF models. Compared to linear, multivariate regression, the mean prediction of gestational age (GA) is more accurate with RF than in linear, multivariate regression (quiet state: R(2)=0,617 vs. R(2)=0,461, active state: R(2)=0,521 vs. R(2)=0,436, state independent: R(2)=0,583 vs. R(2)=0,548). We conclude that classification and regression tree models such as RF methodology are appropriate for the evaluation of fetal maturation age. The decisive role of adjustments between different time scales of complexity may essentially extend previous analysis concepts mainly based on rhythms and univariate complexity indices. Those system characteristics may have implication for better understanding and accessibility of the maturating complex autonomic control and its disturbance.

Formation of functional associations across time scales in the fetal autonomic control system--a multifractal analysis

During fetal development, different control systems mediated by the autonomic nervous system form functional connections over a wide range of time scales. Using multiscale multifractal analysis (MMA) of fetal heart rate variability (HRV), we describe fundamental relationships in the developing scale-wide adjustments within fetal behavior states as well as across state changes. MMA yields the local Hurst exponent surface h(q,s) with q as the multifractal parameter and s as the scale. In 30-minute recordings of healthy fetuses between 24 and 36weeks of gestation (n=25 in quiet sleep, n=29 in active sleep, n=30 changing sleep state) we investigated the dependency of h(q,s) on gestation age. In univariate models, we found a decreasing persistence for short scales and small amplitudes in the quiet (s1=39, q1=-0.7, R(2)=0.52) and in the active (s1=69, q1=-1.4, R(2)=0.23) sleep in contrast to an increasing persistency for long scales and large amplitudes (s1=147, q1=2.4, R(2)=0.29) in the mixed state. Bivariate models (additional scales considered) presented increased coefficients of determination R(2)=0.56, 0.4, and 0.43, respectively. Persistency increasing with age in connection with the sleep state changes (independent of the age related short range dependencies within the separate homogeneous sleep states) is reported here for the first time. The MMA indices obtained for the fetal HRV represent characteristics of the maturating scale-wide cardiovascular control in the context of the evolving sleep state dynamics, which have so far not been considered. They should be incorporated in the search for HRV indices for prenatal diagnosis of developmental disorders and risk assessment.

Heart rate variability parameters and fetal movement complement fetal behavioral states detection via magnetography to monitor neurovegetative development

Fetal behavioral states are defined by fetal movement and heart rate variability (HRV). At 32 weeks of gestational age (GA) the distinction of four fetal behavioral states represented by combinations of quiet or active sleep or awakeness is possible. Prior to 32 weeks, only periods of fetal activity and quiesence can be distinguished. The increasing synchronization of fetal movement and HRV reflects the development of the autonomic nervous system (ANS) control. Fetal magnetocardiography (fMCG) detects fetal heart activity at high temporal resolution, enabling the calculation of HRV parameters. This study combined the criteria of fetal movement with the HRV analysis to complete the criteria for fetal state detection. HRV parameters were calculated including the standard deviation of the normal-to-normal R–R interval (SDNN), the mean square of successive differences of the R–R intervals (RMSSD, SDNN/RMSSD ratio, and permutation entropy (PE) to gain information about the developing influence of the ANS within each fetal state. In this study, 55 magnetocardiograms from healthy fetuses of 24–41 weeks’ GA were recorded for up to 45 min using a fetal biomagnetometer. Fetal states were classified based on HRV and movement detection. HRV parameters were calculated for each state. Before GA 32 weeks, 58.4% quiescence and 41.6% activity cycles were observed. Later, 24% quiet sleep state (1F), 65.4% active sleep state (2F), and 10.6% active awake state (4F) were observed. SDNN increased over gestation. Changes of HRV parameters between the fetal behavioral states, especially between 1F and 4F, were statistically significant. Increasing fetal activity was confirmed by a decrease in PE complexity measures. The fHRV parameters support the differentiation between states and indicate the development of autonomous nervous control of heart rate function.

Ordinal symbolic analysis and its application to biomedical recordings

Ordinal symbolic analysis opens an interesting and powerful perspective on time-series analysis. Here, we review this relatively new approach and highlight its relation to symbolic dynamics and representations. Our exposition reaches from the general ideas up to recent developments, with special emphasis on its applications to biomedical recordings. The latter will be illustrated with epilepsy data.

Visual availability, balance performance and movement complexity in dancers

Research regarding the complex fluctuations of postural sway in an upright standing posture has yielded controversial results about the relationship between complexity and the capacity of the system to generate adaptive responses. The aim of this study is to compare the performance and complexity of two groups with different levels of expertise in postural control during a balance task. We examined the balance ability and time varying (dynamic) characteristics in a group of 18 contemporary dancers and 30 non-dancers in different visual conditions. The task involved maintaining balance for 30s on a stability platform with opened or closed eyes. The results showed that dancers exhibited greater balance ability only in open eyes task than non-dancers. We also observed a lower performance in both groups during the test with closed eyes, but only dancers reduced their complexity in closed eyes task. The main conclusion is that the greater postural control exhibited by dancers depends on the availability of visual information.

Fetal behavioral states are stable over daytime - evidence by longitudinal and cross-sectional fetal biomagnetic recordings

AIMS

Fetal behavioral states can be distinguished by biomagnetic recordings. We performed a longitudinal and a cross-sectional study to address the question whether the distribution of fetal behavioral states changes during the daytime.

METHODS

For the longitudinal study, 32 magnetocardiographic recordings were performed on a singleton pregnancy on a weekly basis. On each examination day, two recordings were performed at different times between 25 and 40 weeks of gestation. For the cross-sectional study, fetal magnetocardiograms (fMCG) were recorded in a group of 32 singleton pregnancies matched for gestational age and daytime to the longitudinal study. The recordings were separated into two gestational age groups (less and more than 32 weeks). Fetal behavioral states were extracted from actocardiograms generated from MCG.

RESULTS

No significant differences in fetal behavioral state distribution were found between morning and afternoon recordings in either the longitudinal or the cross-sectional study.

CONCLUSION

This is the first magnetographic approach to show that daytime does not influence the distribution of fetal behavioral states in standardized recordings of 30 min length. This result implies that fetal magnetography recordings at normal daytimes can be combined without a bias and future recordings can be conducted independently of daytime as long as the varying behavioral states are generally taken into account during analysis.

New estimators and guidelines for better use of fetal heart rate estimators with Doppler ultrasound devices

Characterizing fetal wellbeing with a Doppler ultrasound device requires computation of a score based on fetal parameters. In order to analyze the parameters derived from the fetal heart rate correctly, an accuracy of 0.25 beats per minute is needed. Simultaneously with the lowest false negative rate and the highest sensitivity, we investigated whether various Doppler techniques ensure this accuracy. We found that the accuracy was ensured if directional Doppler signals and autocorrelation estimation were used. Our best estimator provided sensitivity of 95.5%, corresponding to an improvement of 14% compared to the standard estimator.

Development and application of an automated extraction algorithm for fetal magnetocardiography - normal data and arrhythmia detection

INTRODUCTION

Current standard methods of monitoring fetal heart function are mainly based on echocardiography, which provides indirect information (through mechanical assessment) of the fetal heart rhythm. Fetal magnetocardiography (fMCG) allows a reliable quantification of the temporal structure of fetal heart signals. However, its application in clinical studies is difficult because extracting the fetal heart signal for most current applications requires user intervention. To overcome this limitation, we developed a completely automated extraction algorithm.

PATIENTS AND METHODS

The fMCG recordings were acquired using a 156-channel biomagnetic system. To perform an automated analysis, a combination of orthogonal projection and independent component analysis was used. fMCG recordings from 69 healthy uncomplicated singleton pregnancies with normally developing fetuses were included in the study.

RESULTS

The normal values achieved by the automated algorithm were comparable to previously published data. The majority of the cardiac time intervals were positively correlated with gestational age (GA). The ST segment, T wave and QT interval did not show any correlation.

CONCLUSIONS

The automated detection of fetal heart signals was possible beginning at a GA of 19 weeks. This automated analysis of fMCG recordings might be an objective and easily applicable approach for clinicians to analyze fetal heart signals.

Fetal functional brain age assessed from universal developmental indices obtained from neuro-vegetative activity patterns

Fetal brain development involves the development of the neuro-vegetative (autonomic) control that is mediated by the autonomic nervous system (ANS). Disturbances of the fetal brain development have implications for diseases in later postnatal life. In that context, the fetal functional brain age can be altered. Universal principles of developmental biology applied to patterns of autonomic control may allow a functional age assessment. The work aims at the development of a fetal autonomic brain age score (fABAS) based on heart rate patterns. We analysed n = 113 recordings in quiet sleep, n = 286 in active sleep, and n = 29 in active awakeness from normals. We estimated fABAS from magnetocardiographic recordings (21.4–40.3 weeks of gestation) preclassified in quiet sleep (n = 113, 63 females) and active sleep (n = 286, 145 females) state by cross-validated multivariate linear regression models in a cross-sectional study. According to universal system developmental principles, we included indices that address increasing fluctuation range, increasing complexity, and pattern formation (skewness, power spectral ratio VLF/LF, pNN5). The resulting models constituted fABAS. fABAS explained 66/63% (coefficient of determination R² of training and validation set) of the variance by age in quiet, while 51/50% in active sleep. By means of a logistic regression model using fluctuation range and fetal age, quiet and active sleep were automatically reclassified (94.3/93.1% correct classifications). We did not find relevant gender differences. We conclude that functional brain age can be assessed based on universal developmental indices obtained from autonomic control patterns. fABAS reflect normal complex functional brain maturation. The presented normative data are supplemented by an explorative study of 19 fetuses compromised by intrauterine growth restriction. We observed a shift in the state distribution towards active awakeness. The lower WGA dependent fABAS values found in active sleep may reflect alterations in the universal developmental indices, namely fluctuation amplitude, complexity, and pattern formation that constitute fABAS.

Is there a relationship between fetal brain function and the fetal behavioral state? A fetal MEG-study

AIM

Fetal magnetography enables the recording of biomagnetic fetal signals, including fetal heart and fetal brain signals. These signals allow the determination of fetal behavioral states and functional brain signals with auditory evoked responses (AER). In the current study, we investigated how the behavioral state influences the AER and how stimulation affects fetal state.

METHODS

One hundred and four fetuses in three age groups (28-31 weeks, 32-35 weeks and 36-41 weeks) were recorded with and without auditory stimulation. Both sessions were scored for fetal states. The AER latency was determined for each state separately. Forty-five additional subjects were recorded with two sessions of 10 min without stimulation to investigate a possible time effect on state changes.

RESULTS

The state distribution was significantly different between stimulation and no stimulation conditions. The AER latencies were significantly shorter in active sleep (P=0.013) and active wakefulness (P=0.004) compared to quiet sleep.

CONCLUSION

Auditory stimulation has an effect on fetal states. The state information should be taken into account for the analysis of AER latencies.

Development of multiscale complexity and multifractality of fetal heart rate variability

During fetal development a complex system grows and coordination over multiple time scales is formed towards an integrated behavior of the organism. Since essential cardiovascular and associated coordination is mediated by the autonomic nervous system (ANS) and the ANS activity is reflected in recordable heart rate patterns, multiscale heart rate analysis is a tool predestined for the diagnosis of prenatal maturation. The analyses over multiple time scales requires sufficiently long data sets while the recordings of fetal heart rate as well as the behavioral states studied are themselves short. Care must be taken that the analysis methods used are appropriate for short data lengths. We investigated multiscale entropy and multifractal scaling exponents from 30 minute recordings of 27 normal fetuses, aged between 23 and 38 weeks of gestational age (WGA) during the quiet state. In multiscale entropy, we found complexity lower than that of non-correlated white noise over all 20 coarse graining time scales investigated. Significant maturation age related complexity increase was strongest expressed at scale 2, both using sample entropy and generalized mutual information as complexity estimates. Multiscale multifractal analysis (MMA) in which the Hurst surface h(q,s) is calculated, where q is the multifractal parameter and s is the scale, was applied to the fetal heart rate data. MMA is a method derived from detrended fluctuation analysis (DFA). We modified the base algorithm of MMA to be applicable for short time series analysis using overlapping data windows and a reduction of the scale range. We looked for such q and s for which the Hurst exponent h(q,s) is most correlated with gestational age. We used this value of the Hurst exponent to predict the gestational age based only on fetal heart rate variability properties. Comparison with the true age of the fetus gave satisfying results (error 2.17±3.29 weeks; p<0.001; R(2)=0.52). In addition, we found that the normally used DFA scale range is non-optimal for fetal age evaluation. We conclude that 30 min recordings are appropriate and sufficient for assessing fetal age by multiscale entropy and multiscale multifractal analysis. The predominant prognostic role of scale 2 heart beats for MSE and scale 39 heart beats (at q=-0.7) for MMA cannot be explored neither by single scale complexity measures nor by standard detrended fluctuation analysis.

Fetal development of complex autonomic control evaluated from multiscale heart rate patterns

Development of the fetal autonomic nervous system's integrative capacity in relation to gestational age and emerging behavioral pattern is reflected in fetal heart rate patterns. Conventional indices of vagal and sympathetic rhythms cannot sufficiently reflect their complex interrelationship. Universal behavioral indices of developing complex systems may provide additional information regarding the maturating complex autonomic control. We investigated fetal magnetocardiographic recordings undertaken at 10-min intervals in active (n = 248) and quiet (n = 111) states between 22 and 39 wk gestational age. Standard deviation of heartbeat intervals, skewness, contribution of particular rhythms to the total power, and multiscale entropy were analyzed. The multiscale entropy methodology was validated for 10-min data sets. Age dependence was analyzed by linear regression. In the quiet state, contribution of sympathovagal rhythms and their complexity over a range of corresponding short scales increased with rising age, and skewness shifted from negative to positive values. In the active state, age dependencies were weaker. Skewness as the strongest parameter shifted in the same direction. Fluctuation amplitude and the complexity of scales associated with sympathovagal rhythms increased. We conclude that in the quiet state, stable complex organized rhythms develop. In the active state, however, increasing behavioral variability due to multiple internal coordinations, such as movement-related heart rate accelerations, and external influences develop. Hence, the state-selective assessment in association with developmental indices used herein may substantially improve evaluation of maturation age and early detection and interpretation of developmental problems in prenatal diagnosis.

Linear and nonlinear measures of fetal heart rate patterns evaluated on very short fetal magnetocardiograms

We analyzed the effectiveness of linear short- and long-term variability time domain parameters, an index of sympatho-vagal balance (SDNN/RMSSD) and entropy in differentiating fetal heart rate patterns (fHRPs) on the fetal heart rate (fHR) series of 5, 3 and 2 min duration reconstructed from 46 fetal magnetocardiograms. Gestational age (GA) varied from 21 to 38 weeks. FHRPs were classified based on the fHR standard deviation. In sleep states, we observed that vagal influence increased with GA, and entropy significantly increased (decreased) with GA (SDNN/RMSSD), demonstrating that a prevalence of vagal activity with autonomous nervous system maturation may be associated with increased sleep state complexity. In active wakefulness, we observed a significant negative (positive) correlation of short-term (long-term) variability parameters with SDNN/RMSSD. ANOVA statistics demonstrated that long-term irregularity and standard deviation of normal-to-normal beat intervals (SDNN) best differentiated among fHRPs. Our results confirm that short- and long-term variability parameters are useful to differentiate between quiet and active states, and that entropy improves the characterization of sleep states. All measures differentiated fHRPs more effectively on very short HR series, as a result of the fMCG high temporal resolution and of the intrinsic timescales of the events that originate the different fHRPs.

Modified permutation-entropy analysis of heartbeat dynamics

Heart rate variability (HRV) contains important information about the modulation of the cardiovascular system. Various methods of nonlinear dynamics (e.g., estimating Lyapunov exponents) and complexity measures (e.g., correlation dimension or entropies) have been applied to HRV analysis. Permutation entropy, which was proposed recently, has been widely used in many fields due to its conceptual and computational simplicity. It maps a time series onto a symbolic sequence of permutation ranks. The original permutation entropy assumes the time series under study has a continuous distribution, thus equal values are rare and can be ignored by ranking them according to their order of emergence, or broken by adding small random perturbations to ensure every symbol in a sequence is different. However, when the observed time series is digitized with lower resolution leading to a greater number of equal values, or the equalities represent certain characteristic sequential patterns of the system, it may not be rational to simply ignore or break them. In the present paper, a modified permutation entropy is proposed that, by mapping the equal value onto the same symbol (rank), allows for a more accurate characterization of system states. The application of the modified permutation entropy to the analysis of HRV is investigated using clinically collected data. Results show that modified permutation entropy can greatly improve the ability to distinguish the HRV signals under different physiological and pathological conditions. It can characterize the complexity of HRV more effectively than the original permutation entropy.

Review and classification of variability analysis techniques with clinical applications

Analysis of patterns of variation of time-series, termed variability analysis, represents a rapidly evolving discipline with increasing applications in different fields of science. In medicine and in particular critical care, efforts have focussed on evaluating the clinical utility of variability. However, the growth and complexity of techniques applicable to this field have made interpretation and understanding of variability more challenging. Our objective is to provide an updated review of variability analysis techniques suitable for clinical applications. We review more than 70 variability techniques, providing for each technique a brief description of the underlying theory and assumptions, together with a summary of clinical applications. We propose a revised classification for the domains of variability techniques, which include statistical, geometric, energetic, informational, and invariant. We discuss the process of calculation, often necessitating a mathematical transform of the time-series. Our aims are to summarize a broad literature, promote a shared vocabulary that would improve the exchange of ideas, and the analyses of the results between different studies. We conclude with challenges for the evolving science of variability analysis.

Fetal development assessed by heart rate patterns--time scales of complex autonomic control

The increasing functional integrity of the organism during fetal maturation is connected with increasing complex internal coordination. We hypothesize that time scales of complexity and dynamics of heart rate patterns reflect the increasing inter-dependencies within the fetal organism during its prenatal development. We investigated multi-scale complexity, time irreversibility and fractal scaling from 73 fetal magnetocardiographic 30min recordings over the third trimester. We found different scale dependent complexity changes, increasing medium scale time irreversibility, and increasing long scale fractal correlations (all changes p<0.05). The results confirm the importance of time scales to be considered in fetal heart rate based developmental indices.