Cross-sample entropy statistic as a measure of complexity and regularity of renal sympathetic nerve activity in the rat

Use of Intrinsic Entropy to Assess the Instantaneous Complexity of Thoracoabdominal Movement Patterns to Indicate the Effect of the Iso-Volume Maneuver Trial on the Performance of the Step Test

The recent surge in interest surrounds the analysis of physiological signals with a non-linear dynamic approach. The measurement of entropy serves as a renowned method for indicating the complexity of a signal. However, there is a dearth of research concerning the non-linear dynamic analysis of respiratory signals. Therefore, this study employs a novel method known as intrinsic entropy (IE) to assess the short-term dynamic changes in thoracoabdominal movement patterns, as measured by respiratory inductance plethysmography (RIP), during various states such as resting, step test, recovery, and iso-volume maneuver (IVM) trials. The findings reveal a decrease in IE of thoracic wall movement (TWM) and an increase in IE of abdominal wall movement (AWM) following the IVM trial. This suggests that AWM may dominate the breathing exercise after the IVM trial. Moreover, due to the high temporal resolution of IE, it proves to be a suitable measure for assessing the complexity of thoracoabdominal movement patterns under non-stationary states such as the step test and recovery. The results also demonstrate that the instantaneous complexity of TWM and AWM can effectively capture instantaneous changes during non-stationary states, which may prove valuable in understanding the respiratory mechanism for healthcare purposes in daily life.

Detection of intermuscular coordination based on the causality of empirical mode decomposition

Considering the stochastic nature of electromyographic (EMG) signals, nonlinear methods may be a more accurate approach to study intermuscular coordination than the linear approach. The aims of this study were to assess the coordination between two ankle plantar flexors using EMG by applying the causal decomposition approach and assessing whether the intermuscular coordination is affected by the slope of the treadmill. The medial gastrocnemius (MG) and soleus muscles (SOL) were analyzed during the treadmill walking at inclinations of 0°, 5°, and 10°. The coordination was evaluated using ensemble empirical mode decomposition, and the causal interaction was encoded by the instantaneous phase dependence of time series bi-directional causality. To estimate the mutual predictability between MG and SOL, the cross-approximate entropy (XApEn) was assessed. The maximal causal interaction was observed between 40 and 75 Hz independent of inclination. XApEn showed a significant decrease between 0° and 5° (p = 0.028), between 5° and 10° (p = 0.038), and between 0° and 10° (p = 0.014), indicating an increase in coordination. Thus, causal decomposition is an appropriate methodology to study intermuscular coordination. These results indicate that the variation of loading through the change in treadmill inclination increases the interaction of the shared input between MG and SOL, suggesting increased intermuscular coordination.

Automatic diagnosis of late-life depression by 3D convolutional neural networks and cross-sample Entropy analysis from resting-state fMRI

Resting-state fMRI has been widely used in investigating the pathophysiology of late-life depression (LLD). Unlike the conventional linear approach, cross-sample entropy (CSE) analysis shows the nonlinear property in fMRI signals between brain regions. Moreover, recent advances in deep learning, such as convolutional neural networks (CNNs), provide a timely application for understanding LLD. Accurate and prompt diagnosis is essential in LLD; hence, this study aimed to combine CNN and CSE analysis to discriminate LLD patients and non-depressed comparison older adults based on brain resting-state fMRI signals. Seventy-seven older adults, including 49 patients and 28 comparison older adults, were included for fMRI scans. Three-dimensional CSEs with volumes corresponding to 90 seed regions of interest of each participant were developed and fed into models for disease classification and depression severity prediction. We obtained a diagnostic accuracy > 85% in the superior frontal gyrus (left dorsolateral and right orbital parts), left insula, and right middle occipital gyrus. With a mean root-mean-square error (RMSE) of 2.41, three separate models were required to predict depressive symptoms in the severe, moderate, and mild depression groups. The CSE volumes in the left inferior parietal lobule, left parahippocampal gyrus, and left postcentral gyrus performed best in each respective model. Combined complexity analysis and deep learning algorithms can classify patients with LLD from comparison older adults and predict symptom severity based on fMRI data. Such application can be utilized in precision medicine for disease detection and symptom monitoring in LLD.

Decreased sample entropy during sleep-to-wake transition in sleep apnea patients

Objective. This study aimed to prove that there is a sudden change in the human physiology system when switching from one sleep stage to another and physical threshold-based sample entropy (SampEn) is able to capture this transition in an RR interval time series from patients with disorders such as sleep apnea.Approach. Physical threshold-based SampEn was used to analyze different sleep-stage RR segments from sleep apnea subjects in the St. Vincents University Hospital/University College Dublin Sleep Apnea Database, and SampEn differences were compared between two consecutive sleep stages. Additionally, other standard heart rate variability (HRV) measures were also analyzed to make comparisons.Main results. The findings suggested that the sleep-to-wake transitions presented a SampEn decrease significantly larger than intra-sleep ones (P < 0.01), which outperformed other standard HRV measures. Moreover, significant entropy differences between sleep and subsequent wakefulness appeared when the previous sleep stage was either S1 (P < 0.05), S2 (P < 0.01) or S4 (P < 0.05).Significance. The results demonstrated that physical threshold-based SampEn has the capability of depicting physiological changes in the cardiovascular system during the sleep-to-wake transition in sleep apnea patients and it is more reliable than the other analyzed HRV measures. This noninvasive HRV measure is a potential tool for further evaluation of sleep physiological time series.

Exercise intervention protocol in children and young adults with cerebral palsy: the effects of strength, flexibility and gait training on physical performance, neuromuscular mechanisms and cardiometabolic risk factors (EXECP)

Background

Individuals with cerebral palsy (CP) have problems in everyday tasks such as walking and climbing stairs due to a combination of neuromuscular impairments such as spasticity, muscle weakness, reduced joint flexibility and poor coordination. Development of evidence-based interventions are in pivotal role in the development of better targeted rehabilitation of CP, and thus in maintaining their motor function and wellbeing. Our aim is to investigate the efficacy of an individually tailored, multifaceted exercise intervention (EXECP) in children and young adults with CP. EXECP is composed of strength, flexibility and gait training. Furthermore, this study aims to verify the short-term retention of the adaptations three months after the end of the EXECP intervention.

Methods

Twenty-four children and young adults with spastic CP will be recruited to participate in a 9-month research project with a 3-month training intervention, consisting of two to three 90-min sessions per week. In each session, strength training for the lower limbs and trunk muscles, flexibility training for the lower limbs and inclined treadmill gait training will be performed. We will evaluate muscle strength, joint flexibility, neuromuscular and cardiometabolic parameters. A nonconcurrent multiple baseline design with two pre-tests and two post-tests all interspaced by three months is used. In addition to the CP participants, 24 typically developing age and sex-matched participants will perform the two pre-tests (i.e. no intervention) to provide normative data.

Discussion

This study has a comprehensive approach examining longitudinal effects of wide variety of variables ranging from physical activity and gross motor function to sensorimotor functions of the brain and neuromuscular and cardiometabolic parameters, providing novel information about the adaptation mechanisms in cerebral palsy. To the best of our knowledge, this is the first intervention study providing supervised combined strength, flexibility and gait training for young individuals with CP.

Trial registration number

ISRCTN69044459, prospectively registered (21/04/2017).

Depression Scale Prediction with Cross-Sample Entropy and Deep Learning

A two-stage deep learning-based scheme is presented to predict the Hamilton Depression Scale (HAM-D) in this study. First, the cross-sample entropy (CSE) that allows assessing the degree of similarity of two data series are evaluated for the 90 brain regions of interest partitioned according to Automated Anatomical Labeling. The obtained CSE maps are then converted to 3D CSE volumes to serve as the inputs to the deep learning network models for the HAM-D scale level classification and prediction. The efficacy of the proposed scheme was illustrated by the resting-state functional magnetic resonance imaging data from 38 patients. From the results, the root mean square errors for the HAM-D scale prediction obtained during training, validation, and testing were 2.73, 2.66, and 2.18, which were less than those of a scheme having only a regression stage.

Improving Accuracy of Heart Failure Detection Using Data Refinement

Due to the wide inter- and intra-individual variability, short-term heart rate variability (HRV) analysis (usually 5 min) might lead to inaccuracy in detecting heart failure. Therefore, RR interval segmentation, which can reflect the individual heart condition, has been a key research challenge for accurate detection of heart failure. Previous studies mainly focus on analyzing the entire 24-h ECG recordings from all individuals in the database which often led to poor detection rate. In this study, we propose a set of data refinement procedures, which can automatically extract heart failure segments and yield better detection of heart failure. The procedures roughly contain three steps: (1) select fast heart rate sequences, (2) apply dynamic time warping (DTW) measure to filter out dissimilar segments, and (3) pick out individuals with large numbers of segments preserved. A physical threshold-based Sample Entropy (SampEn) was applied to distinguish congestive heart failure (CHF) subjects from normal sinus rhythm (NSR) ones, and results using the traditional threshold were also discussed. Experiment on the PhysioNet/MIT RR Interval Databases showed that in SampEn analysis (embedding dimension m = 1, tolerance threshold r = 12 ms and time series length N = 300), the accuracy value after data refinement has increased to 90.46% from 75.07%. Meanwhile, for the proposed procedures, the area under receiver operating characteristic curve (AUC) value has reached 95.73%, which outperforms the original method (i.e., without applying the proposed data refinement procedures) with AUC of 76.83%. The results have shown that our proposed data refinement procedures can significantly improve the accuracy in heart failure detection.

A New Physically Meaningful Threshold of Sample Entropy for Detecting Cardiovascular Diseases

Sample Entropy (SampEn) is a popular method for assessing the regularity of physiological signals. Prior to the entropy calculation, certain common parameters need to be initialized: Embedding dimension m, tolerance threshold r and time series length N. Nevertheless, the determination of these parameters is usually based on expert experience. Improper assignments of these parameters tend to bring invalid values, inconsistency and low statistical significance in entropy calculation. In this study, we proposed a new tolerance threshold with physical meaning (rp), which was based on the sampling resolution of physiological signals. Statistical significance, percentage of invalid entropy values and ROC curve were used to evaluate the proposed rp against the traditional threshold (rt). Normal sinus rhythm (NSR), congestive heart failure (CHF) as well as atrial fibrillation (AF) RR interval recordings from Physionet were used as the test data. The results demonstrated that the proposed rp had better stability than rt, hence more adaptive to detect cardiovascular diseases of CHF and AF.

Measuring Coupling of Rhythmical Time Series Using Cross Sample Entropy and Cross Recurrence Quantification Analysis

The aim of this investigation was to compare and contrast the use of cross sample entropy (xSE) and cross recurrence quantification analysis (cRQA) measures for the assessment of coupling of rhythmical patterns. Measures were assessed using simulated signals with regular, chaotic, and random fluctuations in frequency, amplitude, and a combination of both. Biological data were studied as models of normal and abnormal locomotor-respiratory coupling. Nine signal types were generated for seven frequency ratios. Fifteen patients with COPD (abnormal coupling) and twenty-one healthy controls (normal coupling) walked on a treadmill at three speeds while breathing and walking were recorded. xSE and the cRQA measures of percent determinism, maximum line, mean line, and entropy were quantified for both the simulated and experimental data. In the simulated data, xSE, percent determinism, and entropy were influenced by the frequency manipulation. The 1 : 1 frequency ratio was different than other frequency ratios for almost all measures and/or manipulations. The patients with COPD used a 2 : 3 ratio more often and xSE, percent determinism, maximum line, mean line, and cRQA entropy were able to discriminate between the groups. Analysis of the effects of walking speed indicated that all measures were able to discriminate between speeds.

Short-term pulse rate variability is better characterized by functional near-infrared spectroscopy than by photoplethysmography

Pulse rate variability (PRV) can be extracted from functional near-infrared spectroscopy (fNIRS) (PRV(NIRS)) and photoplethysmography (PPG) (PRV(PPG)) signals. The present study compared the accuracy of simultaneously acquired PRV(NIRS) and PRV(PPG), and evaluated their different characterizations of the sympathetic (SNS) and parasympathetic (PSNS) autonomous nervous system activity. Ten healthy subjects were recorded during resting-state (RS) and respiratory challenges in two temperature conditions, i.e., room temperature (23°C) and cold temperature (4°C). PRV(NIRS) was recorded based on fNIRS measurement on the head, whereas PRV(PPG) was determined based on PPG measured at the finger. Accuracy between PRV(NIRS) and PRV(PPG), as assessed by cross-covariance and cross-sample entropy, demonstrated a high degree of correlation (r > 0.9), which was significantly reduced by respiration and cold temperature. Characterization of SNS and PSNS using frequency-domain, time-domain, and nonlinear methods showed that PRV(NIRS) provided significantly better information on increasing PSNS activity in response to respiration and cold temperature than PRV(PPG). The findings show that PRV(NIRS) may outperform PRV(PPG) under conditions in which respiration and temperature changes are present, and may, therefore, be advantageous in research and clinical settings, especially if characterization of the autonomous nervous system is desired.

Effect of salt intake on beat-to-beat blood pressure nonlinear dynamics and entropy in salt-sensitive versus salt-protected rats

Blood pressure exhibits substantial short- and long-term variability (BPV). We assessed the hypothesis that the complexity of beat-to-beat BPV will be differentially altered in salt-sensitive hypertensive Dahl rats (SS) versus rats protected from salt-induced hypertension (SSBN13) maintained on high-salt versus low-salt diet. Beat-to-beat systolic and diastolic BP series from nine SS and six SSBN13 rats (http://www.physionet.org) were analyzed following 9 weeks on low salt and repeated after 2 weeks on high salt. BP complexity was quantified by detrended fluctuation analysis (DFA), short- and long-range scaling exponents (αS and αL), sample entropy (SampEn), and traditional standard deviation (SD) and coefficient of variation (CV(%)). Mean systolic and diastolic BP increased on high-salt diet (P < 0.01) particularly for SS rats. SD and CV(%) were similar across groups irrespective of diet. Salt-sensitive and -protected rats exhibited similar complexity indices on low-salt diet. On high salt, (1) SS rats showed increased scaling exponents or smoother, systolic (P = 0.007 [αL]) and diastolic (P = 0.008 [αL]) BP series; (2) salt-protected rats showed lower SampEn (less complex) systolic and diastolic BP (P = 0.046); and (3) compared to protected SSBN13 rats, SS showed higher αL for systolic (P = 0.01) and diastolic (P = 0.005) BP Hypertensive SS rats are more susceptible to high salt with a greater rise in mean BP and reduced complexity. Comparable mean pressures in sensitive and protective rats when on low-salt diet coupled with similar BPV dynamics suggest a protective role of low-salt intake in hypertensive rats. This effect likely reflects better coupling of biologic oscillators.

Classification of Asthma Based on Nonlinear Analysis of Breathing Pattern

Normal human breathing exhibits complex variability in both respiratory rhythm and volume. Analyzing such nonlinear fluctuations may provide clinically relevant information in patients with complex illnesses such as asthma. We compared the cycle-by-cycle fluctuations of inter-breath interval (IBI) and lung volume (LV) among healthy volunteers and patients with various types of asthma. Continuous respiratory datasets were collected from forty age-matched men including 10 healthy volunteers, 10 patients with controlled atopic asthma, 10 patients with uncontrolled atopic asthma, and 10 patients with uncontrolled non-atopic asthma during 60 min spontaneous breathing. Complexity of breathing pattern was quantified by calculating detrended fluctuation analysis, largest Lyapunov exponents, sample entropy, and cross-sample entropy. The IBI as well as LV fluctuations showed decreased long-range correlation, increased regularity and reduced sensitivity to initial conditions in patients with asthma, particularly in uncontrolled state. Our results also showed a strong synchronization between the IBI and LV in patients with uncontrolled asthma. Receiver operating characteristic (ROC) curve analysis showed that nonlinear analysis of breathing pattern has a diagnostic value in asthma and can be used in differentiating uncontrolled from controlled and non-atopic from atopic asthma. We suggest that complexity analysis of breathing dynamics may represent a novel physiologic marker to facilitate diagnosis and management of patients with asthma. However, future studies are needed to increase the validity of the study and to improve these novel methods for better patient management.

Detection of Coupling in Short Physiological Series by a Joint Distribution Entropy Method

OBJECTIVE

In this study, we developed a joint distribution entropy (JDistEn) method to robustly estimate the coupling in short physiological series.

METHODS

The JDistEn method is derived from a joint distance matrix which is constructed from a combination of the distance matrix corresponding to each individual data channel using a geometric mean calculation. A coupled Rössler system and a coupled dual-kinetics neural mass model were used to examine how well JDistEn performed, specifically, its sensitivity for detecting weak coupling, its consistency in gauging coupling strength, and its reliability in processing input of decreased data length. Performance of JDistEn in estimating physiological coupling was further examined with bivariate electroencephalography data from rats and RR interval and diastolic time interval series from human beings. Cross-sample entropy (XSampEn), cross-conditional entropy (XCE), and Shannon entropy of diagonal lines in the joint recurrence plots (JENT) were applied for purposes of comparison.

RESULTS

Simulation results suggest that JDistEn showed markedly higher sensitivity than XSampEn, XCE, and JENT for dynamics in weak coupling, although as the simulation models were more intensively coupled, JDistEn performance was comparable to the three others. In addition, this improved sensitivity was much more pronounced for short datasets. Experimental results further confirmed that JDistEn outperformed XSampEn, XCE, and JENT for detecting weak coupling, especially for short physiological data.

CONCLUSION

This study suggested that our proposed JDistEn could be useful for continuous and even real-time coupling analysis for physiological signals in clinical practice.

Functional brain networks formed using cross-sample entropy are scale free

Over the previous decade, there has been an explosion of interest in network science, in general, and its application to the human brain, in particular. Most brain network investigations to date have used linear correlations (LinCorr) between brain areas to construct and then interpret brain networks. In this study, we applied an entropy-based method to establish functional connectivity between brain areas. This method is sensitive to both nonlinear and linear associations. The LinCorr-based and entropy-based techniques were applied to resting-state functional magnetic resonance imaging data from 10 subjects, and the resulting networks were compared. The networks derived from the entropy-based method exhibited power-law degree distributions. Moreover, the entropy-based networks had a higher clustering coefficient and a shorter path length compared with that of the LinCorr-based networks. While the LinCorr-based networks were assortative, with nodes with similar degrees preferentially connected, the entropy-based networks were disassortative, with high-degree hubs directly connected to low-degree nodes. It is likely that the differences in clustering and assortativity are due to "mega-hubs" in the entropy-based networks. These mega-hubs connect to a large majority of the nodes in the network. This is the first work clearly demonstrating differences between functional brain networks using linear and nonlinear techniques. The key finding is that the nonlinear technique produced networks with scale-free degree distributions. There remains debate among the neuroscience community as to whether human brains are scale free. These data support the argument that at least some aspects of the human brain are perhaps scale free.

Perceptual suppression revealed by adaptive multi-scale entropy analysis of local field potential in monkey visual cortex

Generalized flash suppression (GFS), in which a salient visual stimulus can be rendered invisible despite continuous retinal input, provides a rare opportunity to directly study the neural mechanism of visual perception. Previous work based on linear methods, such as spectral analysis, on local field potential (LFP) during GFS has shown that the LFP power at distinctive frequency bands are differentially modulated by perceptual suppression. Yet, the linear method alone may be insufficient for the full assessment of neural dynamic due to the fundamentally nonlinear nature of neural signals. In this study, we set forth to analyze the LFP data collected from multiple visual areas in V1, V2 and V4 of macaque monkeys while performing the GFS task using a nonlinear method - adaptive multi-scale entropy (AME) - to reveal the neural dynamic of perceptual suppression. In addition, we propose a new cross-entropy measure at multiple scales, namely adaptive multi-scale cross-entropy (AMCE), to assess the nonlinear functional connectivity between two cortical areas. We show that: (1) multi-scale entropy exhibits percept-related changes in all three areas, with higher entropy observed during perceptual suppression; (2) the magnitude of the perception-related entropy changes increases systematically over successive hierarchical stages (i.e. from lower areas V1 to V2, up to higher area V4); and (3) cross-entropy between any two cortical areas reveals higher degree of asynchrony or dissimilarity during perceptual suppression, indicating a decreased functional connectivity between cortical areas. These results, taken together, suggest that perceptual suppression is related to a reduced functional connectivity and increased uncertainty of neural responses, and the modulation of perceptual suppression is more effective at higher visual cortical areas. AME is demonstrated to be a useful technique in revealing the underlying dynamic of nonlinear/nonstationary neural signal.

Patterns of cardiorespiratory coordination in young women with recurrent major depressive disorder treated with escitalopram or venlafaxine

Evidence from previous studies suggests autonomic dysregulation in patients with major depressive disorder (MDD). Antidepressant treatment may also affect central autonomic function. We investigated whether the type of antidepressant might be associated with the pattern of cardiorespiratory coordination in non-depressed women with recurrent MDD. Resting electrocardiograms and respiratory signals were simultaneously recorded from 38 euthymic women with recurrent MDD who were treated with either escitalopram (n=19) or venlafaxine (n=19) monotherapy and from 38 healthy women. Linear measures of heart rate variability were extracted to assess cardiac autonomic control. Sample entropy (SampEn) was computed to assess the complexity of heart rate and respiratory signals, and cross-SampEn was calculated to measure the nonlinear interaction of both signals. Significant decreases in the cardiovagal tone and cardiorespiratory coupling of women with recurrent MDD receiving venlafaxine, and tendencies toward lower cardiovagal tone and cardiorespiratory coupling in women with recurrent MDD receiving escitalopram were observed when compared with healthy controls. Effect sizes for these differences were large between women receiving venlafaxine and healthy controls. We found a positive association between cardiorespiratory decoupling and venlafaxine dose. Norepinephrine-enhancement, within a therapeutic dose range, seems to be closely associated with decreased vagal tone and reduced nonlinear coupling between heart rate and respiration in euthymic women with recurrent MDD. However, the effects of serotonin enhancement on cardiovagal tone should be considered. Our results suggest that the pharmacodynamic properties of antidepressants may affect autonomic regulation of women with recurrent MDD even in euthymic state.

Cross-conditional entropy and coherence analysis of pharmaco-EEG changes induced by alprazolam

RATIONALE

Quantitative analysis of electroencephalographic signals (EEG) and their interpretation constitute a helpful tool in the assessment of the bioavailability of psychoactive drugs in the brain. Furthermore, psychotropic drug groups have typical signatures which relate biochemical mechanisms with specific EEG changes.

OBJECTIVES

To analyze the pharmacological effect of a dose of alprazolam on the connectivity of the brain during wakefulness by means of linear and nonlinear approaches.

METHODS

EEG signals were recorded after alprazolam administration in a placebo-controlled crossover clinical trial. Nonlinear couplings assessed by means of corrected cross-conditional entropy were compared to linear couplings measured with the classical magnitude squared coherence.

RESULTS

Linear variables evidenced a statistically significant drug-induced decrease, whereas nonlinear variables showed significant increases. All changes were highly correlated to drug plasma concentrations. The spatial distribution of the observed connectivity changes clearly differed from a previous study: changes before and after the maximum drug effect were mainly observed over the anterior half of the scalp. Additionally, a new variable with very low computational cost was defined to evaluate nonlinear coupling. This is particularly interesting when all pairs of EEG channels are assessed as in this study.

CONCLUSIONS

Results showed that alprazolam induced changes in terms of uncoupling between regions of the scalp, with opposite trends depending on the variables: decrease in linear ones and increase in nonlinear features. Maps provided consistent information about the way brain changed in terms of connectivity being definitely necessary to evaluate separately linear and nonlinear interactions.

Decreased thalamo-cortical connectivity by alteration of neural information flow in theta oscillation in depression-model rats

Alterations in oscillatory brain activity are strongly correlated with cognitive performance in various physiological rhythms. The present study investigated whether the directionality of neural information flow (NIF) could be used to characterize the synaptic plasticity in thalamocortical (TC) pathway, and examined which frequency field oscillations were mostly related to the cognitive deficiency in depression. Two novel algorithms were employed to determine the coupling interaction between the LD thalamus and medial prefrontal cortex (mPFC) in five frequency bands, using the phase signals of local field potentials (LFP) in these two regions. The results showed that the power of neural activity in mPFC was increased in delta, theta and beta frequency bands in depression. However, the nonlinear characteristics of LFP activity were weakened in depression by means of sample entropy measurements. In the analysis of phase dynamics, the phase synchronization values were reduced in theta rhythm in stressed rats. Importantly, the coupling direction index d and the unidirectional influence from LD thalamus to mPFC were significantly reduced at the theta rhythm in rats in depression, and increased after memantine treatment, which were associated with the LTP alterations and cognitive impairment in our previous report. Moreover, the fact that the reduced entropy value was only found in mPFC might implicate postsynaptic effect involved in synaptic plasticity alteration in the depression model. The results suggest that the effects of depression on cognitive deficits are mediated via profound alterations in information flow in the TC pathway, and the directional index at theta rhythm could be used as a measurement of synaptic plasticity.

A FAST ALGORITHM FOR COMPUTING SAMPLE ENTROPY

Sample entropy is a widely used tool for quantifying complexity of a biological system. Computing sample entropy directly using its definition requires large computational costs. We propose a fast algorithm based on a k-d tree data structure for computing sample entropy. We prove that the time complexity of the proposed algorithm is [Formula: see text] and its space complexity is O(N log N), where N is the length of the input time series and m is the length of its pattern templates. We present a numerical experiment that demonstrates significant improvement of the proposed algorithm in computing time.

Network dynamics of the epileptic brain at rest

Baseline neurodynamics are believed to play an important role in normal brain function. A potentially intrinsic property of the brain is the weak coupling between networks at rest, which enables it to be flexible, adapt, process novel stimuli, and learn. Brain regions become differentially coordinated in response to cognitive task and behavior demands and external stimuli. However, abnormally synchronized resting brain networks may also be associated with different pathologies. We investigated baseline network dynamics in the epileptic brain using information theoretic parameters to quantify coupling and directionality of information flow between different cortical regions. We estimated relative entropy, conditional mutual information and a related measure of directional coupling, from EEGs of patients with epilepsy and healthy subjects. At rest, the healthy brain appears to be characterized by low and non-directional network coupling, whereas the epileptic brain appears to be transiently and directionally synchronized.

A comparative study of pattern synchronization detection between neural signals using different cross-entropy measures

Cross-approximate entropy (X-ApEn) and cross-sample entropy (X-SampEn) have been employed as bivariate pattern synchronization measures for characterizing interdependencies between neural signals. In this study, we proposed a new measure, cross-fuzzy entropy (X-FuzzyEn), to describe the synchronicity of patterns. The performances of three statistics were first quantitatively tested using five different coupled systems including both deterministic and stochastic models, i.e., coupled broadband noises, Lorenz-Lorenz, Rossler-Rossler, Rossler-Lorenz, and neural mass model. All the measures were compared with each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise. The three measures were then applied to a real-life problem, pattern synchronization analysis of left and right hemisphere rat electroencephalographic (EEG) signals. Both simulated and real EEG data analysis results showed that the X-FuzzyEn provided an improved evaluation of bivariate series pattern synchronization and could be more conveniently and powerfully applied to different neural dynamical systems contaminated by noise.

Detection of synchrony in biosignals using cross fuzzy entropy

A new method, namely Cross fuzzy entropy (C-FuzzyEn) analysis, that could enable the measurement of the synchrony or similarity of patterns between two distinct signals, was presented in this study. Tests on simulated data sets showed that C-FuzzyEn was superior to the conventional cross sample entropy (C-SampEn) in several aspects, including giving entropy definition in case of small parameters, better relative consistency, and less dependence on record length. The proposed C-FuzzyEn was then applied for the analysis of simultaneously recorded electromyography (EMG) and mechanomyography (MMG) signals during sustained isometric contraction for monitoring local muscle fatigue. The results showed that the C-FuzzyEn of EMG-MMG decreased significantly during the development of muscle fatigue. The time-decrease trend of C-FuzzyEn is similar to the mean frequency (MNF) of EMG, the commonly used muscle fatigue indicator.

Long-range correlation of renal sympathetic nerve activity in both conscious and anesthetized rats

In this study we employed both detrended fluctuation analysis (DFA) and multiscale entropy (MSE) measurements to compare the long-range temporal correlation (LRTC) of multifibre renal sympathetic nerve activity (RSNA) between conscious and anesthetized Wistar rats. It was found that both methods showed the obvious LRTC properties in conscious state. Moreover, the scaling exponent of the RSNA in conscious rats was significantly higher than that in anesthetized rats. The results of MSE analysis showed that the entropy values, derived from the conscious group, increased on small time scales and then stabilized to a relatively constant value whereas the entropy measure, derived from anesthetized animals, almost monotonically decreased. This suggests that the fractal properties of underlying dynamics of the system have been reduced by anesthesia. The results demonstrate that apparently random fluctuations in multifibre RSNA are dictated by a complex deterministic process that imparts "long-term" memory to the dynamic system. However, this memory is significantly weakened by anesthesia.