Evaluation of scale invariance in physiological signals by means of balanced estimation of diffusion entropy

Cellular coupling determines scale-invariant behavior of neurons in suprachiasmatic nucleus

The main clock in mammals, located in the suprachiasmatic nucleus (SCN) of hypothalamus, not only regulates the daily rhythms in physiological and behavioral activities, but also plays a key role as one of the control nodes in the brain regulating behavioral activity. As such, it induces scale-invariance in the temporal patterns of behavioral activity and of multi-unit neural activity of the SCN network. In particular, the scale-invariant patterns maintain across multiple time scales from 3 minutes to 10 hours, characterized by a scaling exponent around 1. Thus far, no study found the origin of the scale-invariance of the SCN network. Using the method of correlation-dependent balance estimation of diffusion entropy (cBEDE), we found that scale-invariance also exists in the individual neurons of the SCN, and the scale invariance properties are significantly increased when the neurons are coupled in a network of neurons. Improved scale invariance in the single neurons is, therefore, imposed by the emergent network properties of the SCN network. Our findings show that the scale-invariance of the SCN can already be found at the level of the individual neurons and that the application of a scale invariance measure, such as cBEDE, can help in determining the network status of the SCN.

Brain Network Constancy and Participant Recognition: an Integrated Approach to Big Data and Complex Network Analysis

With the development of big data sharing and data standardization, electroencephalogram (EEG) data are increasingly used in the exploration of human cognitive behavior. Most of the existing studies focus on the changes of human brain network topology (the number of connections, degree distribution, clustering coefficient phantom) in various cognitive behaviors. However, there has been little exploration into the steady state of multi-cognitive behaviors and the recognition of multi-participant brain networks. To solve these two problems, we used EEG data of 99 healthy participants from the PhysioBank to study multi-cognitive behaviors. Specifically, we calculated the symbolic transfer entropy (STE) between 64 electrode sequences of EEG data and constructed the brain networks of various cognitive behaviors of each participant using the directed minimum spanning tree (DMST) algorithm. We then investigated the eigenvalue spectrum of the STE matrix of each individual's cognitive behavior. The results also showed that the spectrum distributions of different cognitive states of the same participant remained relatively stable, but those of the same cognitive state of different participants varied considerably, verifying the relative stability and uniqueness of the human brain network similar to a human's fingerprint. Based on these features, we used the spectral distribution set of 99 participants of various cognitive states as the original data set and developed a spectral distribution set scoring (SDSS) method to identify the brain network participants. It was found that most labels (69.35%) of the test participant with the highest score were identical to the labeled participant. This study provided further evidence for the existence of human brain fingerprints, and furnished a new approach for dynamic identification of brain fingerprints.

Singular value decomposition entropy as a measure of ankle dynamics efficacy in a Y-balance test following supportive lower limb taping

Complexity versus regularity is an important component of appropriate joint position to retain balance but has not received much attention. The Singular value decomposition entropy (SvdEn) characterizes information content or regularity of a signal depending on the number of vectors attributed to the process. The current study aimed to investigate the effect of kinesiology tape compared to static strapping tape and no tape on ankle joint dynamics during the Y balance test. Forty-one participants (21 males; 20 females) aged between 18 and 34 years of age completed the Y-balance test with kinesiology tape, with strapping tape and without tape applied to the dominant leg. SvdEn was obtained from center of pressure values, as well as ankle and knee movement variability during the Y balance test. Center of pressure and knee joint dynamics did not change significantly between the two taped and no tape conditions during the YBT. Ankle joint SvdEn was significantly lower in the anterior-posterior (p<; .05) and superior-inferior (p<; .001) direction for both tape conditions compared to no tape. Greater regularity in the ankle joint dynamics indicates less vectors are required to describe the signal, which can be interpreted from a neurophysiological perspective as a decrease in feedforward and/or feedback input along the hierarchical sensorimotor processing pathway as an adjustment to taping and a possibly more reflex oriented response localised at the spinal cord level.

The Effect of Ankle Support on Lower Limb Kinematics During the Y-Balance Test Using Non-linear Dynamic Measures

Background: According to dynamical systems theory, an increase in movement variability leads to greater adaptability, which may be related to the number of feedforward and feedback mechanisms associated with movement and postural control. Using Higuchi dimension (HDf) to measure complexity of the signal and Singular Value Decomposition Entropy (SvdEn) to measure the number of attributes required to describe the biosignal, the purpose of this study was to determine the effect of kinesiology and strapping tape on center of pressure dynamics, myoelectric muscle activity, and joint angle during the Y balance test.

Method: Forty-one participants between 18 and 34 years of age completed five trials of the Y balance test without tape, with strapping tape (ST), and with kinesiology tape (KT) in a cross-sectional study. The mean and standard errors were calculated for the center of pressure, joint angles, and muscle activities with no tape, ST, and KT. The results were analyzed with a repeated measures ANOVA model (P_A < 0.05) fit and followed by Tukey post hoc analysis from the R package with probability set at P < 0.05.

Results: SvdEn indicated significantly decreased complexity in the anterior-posterior (p < 0.05) and internal-external rotation (p < 0.001) direction of the ankle, whilst HDf for both ST and KT identified a significant increase in ankle dynamics when compared to no tape (p < 0.0001) in the mediolateral direction. Taping also resulted in a significant difference in gastrocnemius muscle myoelectric muscle activity between ST and KT (p = 0.047).

Conclusion: Complexity of ankle joint dynamics increased in the sagittal plane of movement with no significant changes in the possible number of physiological attributes. In contrast, the number of possible physiological attributes contributing to ankle movement was significantly lower in the frontal and transverse planes. Simply adhering tape to the skin is sufficient to influence neurological control and adaptability of movement. In addition, adaptation of ankle joint dynamics to retain postural stability during a Y Balance test is achieved differently depending on the direction of movement.

Sampling frequency dependent visibility graphlet approach to time series

Recent years have witnessed special attention on complex network based time series analysis. To extract evolutionary behaviors of a complex system, an interesting strategy is to separate the time series into successive segments, map them further to graphlets as representatives of states, and extract from the state (graphlet) chain transition properties, called graphlet based time series analysis. Generally speaking, properties of time series depend on the time scale. In reality, a time series consists of records that are sampled usually with a specific frequency. A natural question is how the evolutionary behaviors obtained with the graphlet approach depend on the sampling frequency? In the present paper, a new concept called the sampling frequency dependent visibility graphlet is proposed to answer this problem. The key idea is to extract a new set of series in which the successive elements have a specified delay and obtain the state transition network with the graphlet based approach. The dependence of the state transition network on the sampling period (delay) can show us the characteristics of the time series at different time scales. Detailed calculations are conducted with time series produced by the fractional Brownian motion, logistic map and Rössler system, and the empirical sentence length series for the famous Chinese novel entitled A Story of the Stone. It is found that the transition networks for fractional Brownian motions with different Hurst exponents all share a backbone pattern. The linkage strengths in the backbones for the motions with different Hurst exponents have small but distinguishable differences in quantity. The pattern also occurs in the sentence length series; however, the linkage strengths in the pattern have significant differences with that for the fractional Brownian motions. For the period-eight trajectory generated with the logistic map, there appear three different patterns corresponding to the conditions of the sampling period being odd/even-fold of eight or not both. For the chaotic trajectory of the logistic map, the backbone pattern of the transition network for sampling 1 saturates rapidly to a new structure when the sampling period is larger than 2. For the chaotic trajectory of the Rössler system, the backbone structure of the transition network is initially formed with two self-loops, the linkage strengths of which decrease monotonically with the increase of the sampling period. When the sampling period reaches 9, a new large loop appears. The pattern saturates to a complex structure when the sampling period is larger than 11. Hence, the new concept can tell us new information on the trajectories. It can be extended to analyze other series produced by brains, stock markets, and so on.

A Patient Suffering From Neurodegenerative Disease May Have a Strengthened Fractal Gait Rhythm

Scale invariance in stride series, namely, the series shows similar patterns across multiple time scales, is used widely as a non-invasive identifier of health assessment. Detailed calculations in the literature with standard tools, such as de-trended fluctuation analysis and wavelet transform modulus maxima seem to lead a conclusion that patients suffering from neurodegenerative diseases have weakened fractal gait rhythm compared with healthy persons. These variance-based methods are dynamical mechanism dependent, namely, for some dynamical process the scale invariance cannot be detected qualitatively, while for some others the scale invariance can be detected correctly, but the estimated value of scaling exponent is not correct. Generally, we have limited knowledge on the dynamical mechanism. What is more, the stride series for the patients have a typical finite length of ~300, which may lead to unreasonable statistical fluctuations to the evaluation procedure. Hence, how a neurodegenerative disorder disease affects the scale invariance is still an open problem. In this paper the balanced estimation of diffusion entropy (cBEDE) is used to overcome the limitations. The volunteers include healthy individuals and patients with/without freezing of gait (FOG). It is found that scale invariance exists widely in the gait time series for all the individuals. The average of scaling exponents for patients suffering from FOG is similar with or larger than that for healthy individuals, and similar with that for patients without FOG. The patients not suffering from FOG have an average of scaling exponent significantly larger than that for healthy people. From the results estimated by cBEDE, we can conclude that a patient may have an increased scaling exponent, which is contradictory qualitatively with that in the literatures.

Evolution of scaling behaviors embedded in sentence series from A Story of the Stone

The novel entitled A Story of the Stone provides us precise details of life and social structure of the 18th century China. Its writing lasted a long duration of about 10 years, in which the author’s habit may change significantly. It had been published anonymously up to the beginning of the 20th century, which left a mystery of the author’s attribution. In the present work we focus our attention on scaling behavior embedded in the sentence series from this novel, hope to find how the ideas are organized from single sentences to the whole text. Especially we are interested in the evolution of scale invariance to monitor the changes of the author’s language habit and to find some clues on the author’s attribution. The sentence series are separated into a total of 69 non-overlapping segments with a length of 500 sentences each. The correlation dependent balanced estimation of diffusion entropy (cBEDE) is employed to evaluate the scaling behaviors embedded in the short segments. It is found that the total, the part attributed currently to Xueqin Cao (X-part), and the other part attributed to E Gao (E-part), display scale invariance in a large scale up to 10³ sentences, while their scaling exponents are almost identical. All the segments behave scale invariant in considerable wide scales, most of which reach one third of the length. In the curve of scaling exponent versus segment number, the X-part has rich patterns with averagely larger values, while the E-part has a U-shape with a significant low bottom. This finding is a new clue to support the attribution of the E-part to E Gao.

Multifractals embedded in short time series: An unbiased estimation of probability moment

An exact estimation of probability moments is the base for several essential concepts, such as the multifractals, the Tsallis entropy, and the transfer entropy. By means of approximation theory we propose a new method called factorial-moment-based estimation of probability moments. Theoretical prediction and computational results show that it can provide us an unbiased estimation of the probability moments of continuous order. Calculations on probability redistribution model verify that it can extract exactly multifractal behaviors from several hundred recordings. Its powerfulness in monitoring evolution of scaling behaviors is exemplified by two empirical cases, i.e., the gait time series for fast, normal, and slow trials of a healthy volunteer, and the closing price series for Shanghai stock market. By using short time series with several hundred lengths, a comparison with the well-established tools displays significant advantages of its performance over the other methods. The factorial-moment-based estimation can evaluate correctly the scaling behaviors in a scale range about three generations wider than the multifractal detrended fluctuation analysis and the basic estimation. The estimation of partition function given by the wavelet transform modulus maxima has unacceptable fluctuations. Besides the scaling invariance focused in the present paper, the proposed factorial moment of continuous order can find its various uses, such as finding nonextensive behaviors of a complex system and reconstructing the causality relationship network between elements of a complex system.

Long-Range Correlations in Sentence Series from A Story of the Stone

A sentence is the natural unit of language. Patterns embedded in series of sentences can be used to model the formation and evolution of languages, and to solve practical problems such as evaluating linguistic ability. In this paper, we apply de-trended fluctuation analysis to detect long-range correlations embedded in sentence series from A Story of the Stone, one of the greatest masterpieces of Chinese literature. We identified a weak long-range correlation, with a Hurst exponent of 0.575±0.002 up to a scale of 10⁴. We used the structural stability to confirm the behavior of the long-range correlation, and found that different parts of the series had almost identical Hurst exponents. We found that noisy records can lead to false results and conclusions, even if the noise covers a limited proportion of the total records (e.g., less than 1%). Thus, the structural stability test is an essential procedure for confirming the existence of long-range correlations, which has been widely neglected in previous studies. Furthermore, a combination of de-trended fluctuation analysis and diffusion entropy analysis demonstrated that the sentence series was generated by a fractional Brownian motion.

In search of coding and non-coding regions of DNA sequences based on balanced estimation of diffusion entropy

Identification of coding regions in DNA sequences remains challenging. Various methods have been proposed, but these are limited by species-dependence and the need for adequate training sets. The elements in DNA coding regions are known to be distributed in a quasi-random way, while those in non-coding regions have typical similar structures. For short sequences, these statistical characteristics cannot be extracted correctly and cannot even be detected. This paper introduces a new way to solve the problem: balanced estimation of diffusion entropy (BEDE).

Evaluation of scaling invariance embedded in short time series

Scaling invariance of time series has been making great contributions in diverse research fields. But how to evaluate scaling exponent from a real-world series is still an open problem. Finite length of time series may induce unacceptable fluctuation and bias to statistical quantities and consequent invalidation of currently used standard methods. In this paper a new concept called correlation-dependent balanced estimation of diffusion entropy is developed to evaluate scale-invariance in very short time series with length ~10(2). Calculations with specified Hurst exponent values of 0.2,0.3,...,0.9 show that by using the standard central moving average de-trending procedure this method can evaluate the scaling exponents for short time series with ignorable bias (≤0.03) and sharp confidential interval (standard deviation ≤0.05). Considering the stride series from ten volunteers along an approximate oval path of a specified length, we observe that though the averages and deviations of scaling exponents are close, their evolutionary behaviors display rich patterns. It has potential use in analyzing physiological signals, detecting early warning signals, and so on. As an emphasis, the our core contribution is that by means of the proposed method one can estimate precisely shannon entropy from limited records.