Determining embedding dimension for phase-space reconstruction using a geometrical construction

Synchronized arousal between performers and related spectators in a fire-walking ritual

Collective rituals are present in all known societies, but their function is a matter of long-standing debates. Field observations suggest that they may enhance social cohesion and that their effects are not limited to those actively performing but affect the audience as well. Here we show physiological effects of synchronized arousal in a Spanish fire-walking ritual, between active participants and related spectators, but not participants and other members of the audience. We assessed arousal by heart rate dynamics and applied nonlinear mathematical analysis to heart rate data obtained from 38 participants. We compared synchronized arousal between fire-walkers and spectators. For this comparison, we used recurrence quantification analysis on individual data and cross-recurrence quantification analysis on pairs of participants' data. These methods identified fine-grained commonalities of arousal during the 30-min ritual between fire-walkers and related spectators but not unrelated spectators. This indicates that the mediating mechanism may be informational, because participants and related observers had very different bodily behavior. This study demonstrates that a collective ritual may evoke synchronized arousal over time between active participants and bystanders. It links field observations to a physiological basis and offers a unique approach for the quantification of social effects on human physiology during real-world interactions.

Noise and complexity in human postural control: interpreting the different estimations of entropy

Background

Over the last two decades, various measures of entropy have been used to examine the complexity of human postural control. In general, entropy measures provide information regarding the health, stability and adaptability of the postural system that is not captured when using more traditional analytical techniques. The purpose of this study was to examine how noise, sampling frequency and time series length influence various measures of entropy when applied to human center of pressure (CoP) data, as well as in synthetic signals with known properties. Such a comparison is necessary to interpret data between and within studies that use different entropy measures, equipment, sampling frequencies or data collection durations.

Methods and Findings

The complexity of synthetic signals with known properties and standing CoP data was calculated using Approximate Entropy (ApEn), Sample Entropy (SampEn) and Recurrence Quantification Analysis Entropy (RQAEn). All signals were examined at varying sampling frequencies and with varying amounts of added noise. Additionally, an increment time series of the original CoP data was examined to remove long-range correlations. Of the three measures examined, ApEn was the least robust to sampling frequency and noise manipulations. Additionally, increased noise led to an increase in SampEn, but a decrease in RQAEn. Thus, noise can yield inconsistent results between the various entropy measures. Finally, the differences between the entropy measures were minimized in the increment CoP data, suggesting that long-range correlations should be removed from CoP data prior to calculating entropy.

Conclusions

The various algorithms typically used to quantify the complexity (entropy) of CoP may yield very different results, particularly when sampling frequency and noise are different. The results of this study are discussed within the context of the neural noise and loss of complexity hypotheses.

BrainNetVis: an open-access tool to effectively quantify and visualize brain networks

This paper presents BrainNetVis, a tool which serves brain network modelling and visualization, by providing both quantitative and qualitative network measures of brain interconnectivity. It emphasizes the needs that led to the creation of this tool by presenting similar works in the field and by describing how our tool contributes to the existing scenery. It also describes the methods used for the calculation of the graph metrics (global network metrics and vertex metrics), which carry the brain network information. To make the methods clear and understandable, we use an exemplar dataset throughout the paper, on which the calculations and the visualizations are performed. This dataset consists of an alcoholic and a control group of subjects.

Nonlinear smooth orthogonal decomposition of kinematic features of sawing reconstructs muscle fatigue evolution as indicated by electromyography

Tracking or predicting physiological fatigue is important for developing more robust training protocols and better energy supplements and/or reducing muscle injuries. Current methodologies are usually impractical and/or invasive and may not be realizable outside of laboratory settings. It was recently demonstrated that smooth orthogonal decomposition (SOD) of phase space warping (PSW) features of motion kinematics can identify fatigue in individual muscle groups. We hypothesize that a nonlinear extension of SOD will identify more optimal fatigue coordinates and provide a lower-dimensional reconstruction of local fatigue dynamics than the linear SOD. Both linear and nonlinear SODs were applied to PSW features estimated from measured kinematics to reconstruct muscle fatigue dynamics in subjects performing a sawing motion. Ten healthy young right-handed subjects pushed a weighted handle back and forth until voluntary exhaustion. Three sets of joint kinematic angles were measured from the right upper extremity in addition to surface electromyography (EMG) recordings. The SOD coordinates of kinematic PSW features were compared against independently measured fatigue markers (i.e., mean and median EMG spectrum frequencies of individual muscle groups). This comparison was based on a least-squares linear fit of a fixed number of the dominant SOD coordinates to the appropriate local fatigue markers. Between subject variability showed that at most four to five nonlinear SOD coordinates were needed to reconstruct fatigue in local muscle groups, while on average 15 coordinates were needed for the linear SOD. Thus, the nonlinear coordinates provided a one-order-of-magnitude improvement over the linear ones.

Kinematic variability, fractal dynamics and local dynamic stability of treadmill walking

Background

Motorized treadmills are widely used in research or in clinical therapy. Small kinematics, kinetics and energetics changes induced by Treadmill Walking (TW) as compared to Overground Walking (OW) have been reported in literature. The purpose of the present study was to characterize the differences between OW and TW in terms of stride-to-stride variability. Classical (Standard Deviation, SD) and non-linear (fractal dynamics, local dynamic stability) methods were used. In addition, the correlations between the different variability indexes were analyzed.

Methods

Twenty healthy subjects performed 10 min TW and OW in a random sequence. A triaxial accelerometer recorded trunk accelerations. Kinematic variability was computed as the average SD (MeanSD) of acceleration patterns among standardized strides. Fractal dynamics (scaling exponent α) was assessed by Detrended Fluctuation Analysis (DFA) of stride intervals. Short-term and long-term dynamic stability were estimated by computing the maximal Lyapunov exponents of acceleration signals.

Results

TW did not modify kinematic gait variability as compared to OW (multivariate T², p = 0.87). Conversely, TW significantly modified fractal dynamics (t-test, p = 0.01), and both short and long term local dynamic stability (T² p = 0.0002). No relationship was observed between variability indexes with the exception of significant negative correlation between MeanSD and dynamic stability in TW (3 × 6 canonical correlation, r = 0.94).

Conclusions

Treadmill induced a less correlated pattern in the stride intervals and increased gait stability, but did not modify kinematic variability in healthy subjects. This could be due to changes in perceptual information induced by treadmill walking that would affect locomotor control of the gait and hence specifically alter non-linear dependencies among consecutive strides. Consequently, the type of walking (i.e. treadmill or overground) is important to consider in each protocol design.

Reliability of EMG determinism to detect changes in motor unit synchrony and coherence during submaximal contraction

The determinism (DET) is a parameter used in nonlinear analysis to quantify the occurrence of recurrent patterns in a signal. Applied to the electromyographic activity (EMG), DET has been proposed as an index of motor unit synchrony in human. We have recently shown that the amount of motor unit synchronous firings above chance level was enhanced with stronger submaximal muscle contraction. Using these data, we aimed at determining if (1) EMG DET and motor unit synchrony varied in the same way and (2) EMG DET was more specifically related to the degree of oscillatory coupling between motor unit discharges. Cross-correlation and coherence analyses were applied to the discharges of 30 motor unit pairs tested at various force levels to assess the amount of synchronous impulses and the strength of oscillatory coupling in the time and frequency domains, respectively. Recurrent quantification analysis was applied to EMG activity to extract its DET. Overall, changes in EMG DET were poorly explained by changes in motor unit synchronous impulse probability (6%) and frequency (5%), and by changes in motor unit coherence in the 6-12Hz (5%) and 25-40Hz (8%) bands. Moreover, the comparison of the data obtained at the weakest and the strongest contraction levels tested with each motor unit pair showed that EMG DET remained unaltered with stronger contraction despite the occurrence of consistent changes in motor unit synchrony in both time and frequency domains. This speaks strongly against the reliability of DET in evaluating changes in motor unit synchronization during submaximal muscle contraction.

k-Nearest neighbour local linear prediction of scalp EEG activity during intermittent photic stimulation

The characterization of the EEG response to photic stimulation (PS) is an important issue with significant clinical relevance. This study aims to quantify and map the complexity of the EEG during PS, where complexity is measured as the degree of unpredictability resulting from local linear prediction. EEG activity was recorded with eyes closed (EC) and eyes open (EO) during resting and PS at 5, 10, and 15 Hz in a group of 30 healthy subjects and in a case-report of a patient suffering from cerebral ischemia. The mean squared prediction error (MSPE) resulting from k-nearest neighbour local linear prediction was calculated in each condition as an index of EEG unpredictability. The linear or nonlinear nature of the system underlying EEG activity was evaluated quantifying MSPE as a function of the neighbourhood size during local linear prediction, and by surrogate data analysis as well. Unpredictability maps were obtained for each subject interpolating MSPE values over a schematic head representation. Results on healthy subjects evidenced: (i) the prevalence of linear mechanisms in the generation of EEG dynamics, (ii) the lower predictability of EO EEG, (iii) the desynchronization of oscillatory mechanisms during PS leading to increased EEG complexity, (iv) the entrainment of alpha rhythm during EC obtained by 10 Hz PS, and (v) differences of EEG predictability among different scalp regions. Ischemic patient showed different MSPE values in healthy and damaged regions. The EEG predictability decreased moving from the early acute stage to a stage of partial recovery. These results suggest that nonlinear prediction can be a useful tool to characterize EEG dynamics during PS protocols, and may consequently constitute a complement of quantitative EEG analysis in clinical applications.

Modeling Multivariate Time Series on Manifolds with Skew Radial Basis Functions

We present an approach for constructing nonlinear empirical mappings from high-dimensional domains to multivariate ranges. We employ radial basis functions and skew radial basis functions for constructing a model using data that are potentially scattered or sparse. The algorithm progresses iteratively, adding a new function at each step to refine the model. The placement of the functions is driven by a statistical hypothesis test that accounts for correlation in the multivariate range variables. The test is applied on training and validation data and reveals nonstatistical or geometric structure when it fails. At each step, the added function is fit to data contained in a spatiotemporally defined local region to determine the parameters—in particular, the scale of the local model. The scale of the function is determined by the zero crossings of the autocorrelation function of the residuals. The model parameters and the number of basis functions are determined automatically from the given data, and there is no need to initialize any ad hoc parameters save for the selection of the skew radial basis functions. Compactly supported skew radial basis functions are employed to improve model accuracy, order, and convergence properties. The extension of the algorithm to higher-dimensional ranges produces reduced-order models by exploiting the existence of correlation in the range variable data. Structure is tested not just in a single time series but between all pairs of time series. We illustrate the new methodologies using several illustrative problems, including modeling data on manifolds and the prediction of chaotic time series.

Nonlinear dynamics of voices in esophageal phonation

The present study investigated the difference in voice perturbation measures and parameters obtained from nonlinear dynamic analysis between normal laryngeal phonation and standard esophageal (SE) phonation. Jitter, shimmer, correlation dimension and Kolmogorov entropy were measured from 10 SE and 10 normal male speakers of Cantonese. Jitter and shimmer values were significantly higher for SE than laryngeal voice. But jitter values were found to be significantly different when length of sound samples was altered. In addition, both correlation dimension and Kolmogorov entropy values were significantly higher for SE than laryngeal voice and sample length did not appear to affect the result. These results suggest that SE voices are more chaotic than laryngeal voice. It follows that the technique of nonlinear dynamic analysis may be more reliable and stable for evaluating the acoustic characteristics of SE voices.

Anterior-posterior and medial-lateral control of sway in infants during sitting acquisition does not become adult-like

We examined (1) how sitting postural control in infants develops in the anterior-posterior (A/P) and medial-lateral (M/L) directions of sway, and (2) whether this control is already adult-like during the late phase of infant's sitting acquisition. COP data were acquired from 14 healthy infants (from the onset of sitting until independent sitting) and 21 healthy adults while sitting on a force platform. Attractor dimensionality (CoD: correlation dimension), attractor predictability (LyE: largest Lyapunov exponent), and sway variability (RMS: root-mean square) were calculated from the COP data to evaluate postural control. In the A/P direction, sitting was mastered by the infants by decreasing the active degrees of freedom of the postural system (decreased CoD), using a more predictable and (locally) stable sway (decreased LyE), and increasing sway variability (increased RMS). Control of sitting became practically simple, stable and exploratory with infant development. This may support the hypothesis that the sitting posture serves as the foundation for the development of other motor skills, as reaching. In the M/L direction, only sway variability decreased with development, possibly due to changes in the infant's body dimensions. Taken together, these findings indicate that early in development the focus is more in the A/P than the M/L direction. Adults' postural control was found more adaptable than the infants in both directions, involving more active degrees of freedom and less predictable sway patterns. Identifying the factors that make the dynamics of the postural system adult-like requires further research.

Wireless gyroscope suit for gait stability estimation

Gait stability is primary in assessing individuals with high risk of falling, particularly the elderly. Custom made self-adjustable wireless gyroscope suit is used as a sensing device to quantify gait stability. A nonlinear time series analysis i.e. maximum Lyapunov exponent (λ*) was employed to estimate the short term and long term stability and it is closely related to the ability of human neuro-muscular control system in maintaining gait stability. Experimental analysis and tests validated the efficacy of this novel approach. The results achieved are comparable with the findings of multiple kinematic and dynamic parameters derived from optical motion capture system and force platform which are widely used as gold standard.

Solar Cycle Prediction

A review of solar cycle prediction methods and their performance is given, including forecasts for cycle 24. The review focuses on those aspects of the solar cycle prediction problem that have a bearing on dynamo theory. The scope of the review is further restricted to the issue of predicting the amplitude (and optionally the epoch) of an upcoming solar maximum no later than right after the start of the given cycle. Prediction methods form three main groups. Precursor methods rely on the value of some measure of solar activity or magnetism at a specified time to predict the amplitude of the following solar maximum. Their implicit assumption is that each numbered solar cycle is a consistent unit in itself, while solar activity seems to consist of a series of much less tightly intercorrelated individual cycles. Extrapolation methods, in contrast, are based on the premise that the physical process giving rise to the sunspot number record is statistically homogeneous, i.e., the mathematical regularities underlying its variations are the same at any point of time and, therefore, it lends itself to analysis and forecasting by time series methods. Finally, instead of an analysis of observational data alone, model based predictions use physically (more or less) consistent dynamo models in their attempts to predict solar activity. In their overall performance during the course of the last few solar cycles, precursor methods have clearly been superior to extrapolation methods. Nevertheless, most precursor methods overpredicted cycle 23, while some extrapolation methods may still be worth further study. Model based forecasts have not yet had a chance to prove their skills. One method that has yielded predictions consistently in the right range during the past few solar cycles is that of K. Schatten et al., whose approach is mainly based on the polar field precursor. The incipient cycle 24 will probably mark the end of the Modern Maximum, with the Sun switching to a state of less strong activity. It will therefore be an important testbed for cycle prediction methods and, by inference, for our understanding of the solar dynamo.

Comparison of different threshold values r for approximate entropy: application to investigate the heart rate variability between heart failure and healthy control groups

Approximate entropy (ApEn) is widely accepted as a complexity measure of the heart rate variability (HRV) signal, but selecting the criteria for the threshold value r is controversial. This paper aims to verify whether Chon's method of forecasting the r(max) is an appropriate one for the HRV signal. The standard limb lead ECG signals of 120 subjects were recorded for 10 min in a supine position. The subjects were divided into two groups: the heart failure (22 females and 38 males, median age 62.4 ± 12.6) and healthy control group (33 females and 27 males, median age 51.5 ± 16.9). Three types of ApEn were calculated: the ApEn(0.2) using the recommended constant r = 0.2, the ApEn(chon) using Chon's method and the ApEn(max) using the true r(max). A Wilcoxon rank sum test showed that the ApEn(0.2) (p = 0.267) and the ApEn(max) (p = 0.813) had no statistical differences between the two groups, while the ApEn(chon) (p = 0.040) had. We generated a synthetic database to study the effect of two influential factors (the signal length N and the ratio of short- and long-term variability sd(1)/sd(2)) on the empirical formula in Chon's method (Chon et al 2009 IEEE Eng. Med. Biol. Mag. 28 18-23). The results showed that the empirical formula proposed by Chon et al is a good method for analyzing the random signal, but not an appropriate tool for analyzing nonlinear signals, such as the logistic or HRV signals.

Forecasting the evolution of nonlinear and nonstationary systems using recurrence-based local Gaussian process models

An approach based on combining nonparametric Gaussian process (GP) modeling with certain local topological considerations is presented for prediction (one-step look ahead) of complex physical systems that exhibit nonlinear and nonstationary dynamics. The key idea here is to partition system trajectories into multiple near-stationary segments by aligning the boundaries of the partitions with those of the piecewise affine projections of the underlying dynamic system, and deriving nonparametric prediction models within each segment. Such an alignment is achieved through the consideration of recurrence and other local topological properties of the underlying system. This approach was applied for state and performance forecasting in Lorenz system under different levels of induced noise and nonstationarity, synthetic heart-rate signals, and a real-world time-series from an industrial operation known to exhibit highly nonlinear and nonstationary dynamics. The results show that local Gaussian process can significantly outperform not just classical system identification, neural network and nonparametric models, but also the sequential Bayesian Monte Carlo methods in terms of prediction accuracy and computational speed.

Fluctuations and determinism of respiratory impedance in asthma and chronic obstructive pulmonary disease

Asthma and COPD are chronic respiratory diseases that fluctuate widely with regard to clinical symptoms and airway obstruction, complicating treatment and prediction of exacerbations. Time series of respiratory impedance obtained by the forced oscillation technique are a convenient tool to study the respiratory system with high temporal resolution. In previous studies it was suggested that power-law-like fluctuations exist also in the healthy lung and that respiratory system impedance variability differs in asthma. In this study we elucidate such differences in a population of well-characterized subjects with asthma (n = 13, GINA 1+2), COPD (n = 12, GOLD I+II), and controls (n = 10) from time series at single frequency (12 min, f = 8 Hz). Maximum likelihood estimation did not rule out power-law behavior, accepting the null hypothesis in 17/35 cases (P > 0.05) and with significant differences in exponents for COPD (P < 0.03). Detrended fluctuation analysis exhibited scaling exponents close to 0.5, indicating few correlations, with no differences between groups (P > 0.14). In a second approach, we considered asthma and COPD as dynamic diseases, corresponding to changes of unknown parameters in a deterministic system. The similarity in shape between the combined probability distributions of normalized resistance and reactance was quantified by Wasserstein distances and reliably distinguished the two diseases (cross-validated predictive accuracy 0.80; sensitivity 0.83, specificity 0.77 for COPD). Wasserstein distances between 3+3 dimensional phase space reconstructions resulted in marginally better classification (accuracy 0.84, sensitivity 0.83, specificity 0.85). These latter findings suggest that the dynamics of respiratory impedance contain valuable information for the diagnosis and monitoring of patients with asthma and COPD, whereas the value of the stochastic approach is not clear presently.

Devaney's chaos on recurrence plots

Although the definition of deterministic chaos in the sense of Devaney is often employed for investigating dynamical systems, it has been rarely used for identifying deterministic chaos from time series. In this paper, we develop a set of methods that infers whether a given time series is consistent with deterministic chaos in the sense of Devaney or not. We define weaker notions of the three properties of Devaney's chaos which can be checked using a given time series of a finite length. We apply these methods for artificial and real data.

Characterizing the dynamics of postural sway in humans using smoothness and regularity measures

We investigate human postural sway velocity time series by computing two dynamical statistics quantifying the smoothness (the central tendency measure or CTM) and the regularity (the sample entropy or SampEn) of their underlying dynamics. The purpose of the study is to investigate the effect of aging and vision on the selected measures and to explore the nature of postural dynamics by performing surrogate data tests. A group of 14 young subjects was compared to a group of 11 older healthy subjects in two visual conditions: with eyes open (EO) and with eyes closed (EC). The results suggest that vision and age do not influence the two statistics of the velocity data in the same way. More specifically, the smoothness statistic is able to detect the aging effect. The regularity measure is sensitive to the visual feedback removal. In contrast with some findings in the literature, the results of the surrogate data tests indicate that the center of pressure velocity dynamics are stochastic and are not produced by a purely deterministic behavior. Finally, we discuss some potential implications of our results in terms of postural control mechanisms.

Motif distributions in phase-space networks for characterizing experimental two-phase flow patterns with chaotic features

The dynamics of two-phase flows have been a challenging problem in nonlinear dynamics and fluid mechanics. We propose a method to characterize and distinguish patterns from inclined water-oil flow experiments based on the concept of network motifs that have found great usage in network science and systems biology. In particular, we construct from measured time series phase-space complex networks and then calculate the distribution of a set of distinct network motifs. To gain insight, we first test the approach using time series from classical chaotic systems and find a universal feature: motif distributions from different chaotic systems are generally highly heterogeneous. Our main finding is that the distributions from experimental two-phase flows tend to be heterogeneous as well, suggesting the underlying chaotic nature of the flow patterns. Calculation of the maximal Lyapunov exponent provides further support for this. Motif distributions can thus be a feasible tool to understand the dynamics of realistic two-phase flow patterns.

Nonuniform state-space reconstruction and coupling detection

We investigate the state space reconstruction from multiple time series derived from continuous and discrete systems and propose a method for building embedding vectors progressively using information measure criteria regarding past, current, and future states. The embedding scheme can be adapted for different purposes, such as mixed modeling, cross-prediction and Granger causality. In particular, we apply this method in order to detect and evaluate information transfer in coupled systems. As a practical application, we investigate in records of scalp epileptic EEG the information flow across brain areas.

Psychosomatic symptoms as biomarkers: transcending the psyche-soma dichotomy

Following the advancement in understanding dynamical systems, the author presents a novel metaphor of psychosomatic symptoms as low-dimensional biomarkers. This metaphor, which transcends the old binary of psyche-soma, resonates with classical psychoanalytic concepts and with Matte-Blanco's idea of repetition as indicative of dimensionality reduction. The relevance of this metaphor for explanation, diagnosis, and treatment is illustrated through a case study of a male patient suffering from hyperprolactinemia.

The dynamics of laser droplet generation

We propose an experimental setup allowing for the characterization of laser droplet generation in terms of the underlying dynamics, primarily showing that the latter is deterministically chaotic by means of nonlinear time series analysis methods. In particular, we use a laser pulse to melt the end of a properly fed vertically placed metal wire. Due to the interplay of surface tension, gravity force, and light-metal interaction, undulating pendant droplets are formed at the molten end, which eventually completely detach from the wire as a consequence of their increasing mass. We capture the dynamics of this process by employing a high-speed infrared camera, thereby indirectly measuring the temperature of the wire end and the pendant droplets. The time series is subsequently generated as the mean value over the pixel intensity of every infrared snapshot. Finally, we employ methods of nonlinear time series analysis to reconstruct the phase space from the observed variable and test it against determinism and stationarity. After establishing that the observed laser droplet generation is a deterministic and dynamically stationary process, we calculate the spectra of Lyapunov exponents. We obtain a positive largest Lyapunov exponent and a negative divergence, i.e., sum of all the exponents, thus indicating that the observed dynamics is deterministically chaotic with an attractor as solution in the phase space. In addition to characterizing the dynamics of laser droplet generation, we outline industrial applications of the process and point out the significance of our findings for future attempts at mathematical modeling.

Chaotic operation by a single transistor circuit in the reverse active region

In this paper, we present an externally triggered experimental chaotic circuit with a bipolar junction transistor operating in its reverse active region in order to investigate for possible control features in its output phase portraits. Nonlinear time series modeling techniques are applied to analyze the circuit's output voltage oscillations and reveal the presence of chaos, while the chaos itself is achieved by controlling the amplitude of the applied input signal. The phase space, which describes the behavior evolution of a nonlinear system, is reconstructed using the delay embedding theorem suggested by Takens. The time delay used for this reconstruction is chosen after examining the first minimum of the collected data average mutual information, while the sufficient embedding dimension is estimated using the false nearest-neighbor algorithm which has a value of 5. Also the largest Lyapunov exponent is estimated and found equal to 0.020 48. Finally, the phase space embedding based weight predictor algorithm is employed to make a short-term prediction of the chaotic time series for which the system's governing equations may be unknown.

Non-linear Analysis of Single Electroencephalography (EEG) for Sleep-Related Healthcare Applications

OBJECTIVES

Soft-computing techniques are commonly used to detect medical phenomena and to help with clinical diagnoses and treatment. The purpose of this paper is to analyze the single electroencephalography (EEG) signal with the chaotic methods in order to identify the sleep stages.

METHODS

Data acquisition (polysomnography) was performed on four healthy young adults (all males with a mean age of 27.5 years). The evaluated algorithm was designed with a correlation dimension and Lyapunov's exponent using a single EEG signal that detects differences in chaotic characteristics.

RESULTS

The change of the correlation dimension and the largest Lyapunov exponent over the whole night sleep EEG was performed. The results show that the correlation dimension and largest Lyapunov exponent decreased from light sleep to deep sleep and they increased during the rapid eye movement stage.

CONCLUSIONS

These results suggest that chaotic analysis may be a useful adjunct to linear (spectral) analysis for identifying sleep stages. The single EEG based nonlinear analysis is suitable for u-healthcare applications for monitoring sleep.

Recurrence Quantification Analysis of Spontaneous Electrophysiological Activity during Development: Characterization of In Vitro Neuronal Networks Cultured on Multi Electrode Array Chips

The combination of a nonlinear time series analysis technique, Recurrence Quantification Analysis (RQA) based on Recurrence Plots (RPs), and traditional statistical analysis for neuronal electrophysiology is proposed in this paper as an innovative paradigm for studying the variation of spontaneous electrophysiological activity of in vitro Neuronal Networks (NNs) coupled to Multielectrode Array (MEA) chips. Recurrence, determinism, entropy, distance of activity patterns, and correlation in correspondence to spike and burst parameters (e.g., mean spiking rate, mean bursting rate, burst duration, spike in burst, etc.) have been computed to characterize and assess the daily changes of the neuronal electrophysiology during neuronal network development and maturation. The results show the similarities/differences between several channels and time periods as well as the evolution of the spontaneous activity in the MEA chip. RPs could be used for graphically exploring possible neuronal dynamic breaking/changing points, whereas RQA parameters are suited for locating them. The combination of RQA with traditional approaches improves the identification, description, and prediction of electrophysiological changes and it will be used to allow intercomparison between results obtained from different MEA chips. Results suggest the proposed processing paradigm as a valuable tool to analyze neuronal activity for screening purposes (e.g., toxicology, neurodevelopmental toxicology).

Feature analysis of epileptic EEG using nonlinear prediction method

We propose a feature extraction method based on the Volterra autoregressive model's prediction power and the data's predictability for the EEG signals to automatically detect the epileptic EEG signals from the EEG recordings. The method of determining the embedding dimension based on nonlinear prediction is applied to choose the embedding dimension of the EEG data. The proposed feature extraction method is used to extract the feature for three groups of EEG time series composing epileptic seizure. We divide the EEG data into segments, and respectively compute the feature values of each segment, where the length of data segment respectively takes the value of 250, 500, 1000 points. To investigate the robustness of our method under noises, we also analyze the three EEG time series with additive white Gaussian noise. The experiment results show that the feature values extracted with the proposed method could obviously distinguish the epileptic EEG signals from the normal EEG signals. The proposed method is effective for short time series, insensitive to the length of data segment, and robust to the additive white noise, and it could differentiate the epileptic EEG from the normal EEG when the signal-to-noise ratio is low.

Modeling the intercellular exchange of signaling molecules depending on intra- and inter-cellular environmental parameters

The exchange of biochemical substances is essential in establishing communication between bacterial cells. It is noticeable that all phases of the process are heavily influenced by perturbations of either internal or external parameters. Therefore, instead of developing an accurate quantitative model of substance exchange between bacterial cells, we are interested in the formalization of the basic shape of the process, and creating the appropriate strategy that allows further investigation of synchronization. Using a form of coupled difference logistic equations we investigated the synchronization of substance exchange between abstract cells and its sensitivity to fluctuations of environmental parameters using methods of nonlinear dynamics.

Extracting the causality of correlated motions from molecular dynamics simulations

The information theory measure of transfer entropy is used to extract the causality of correlated motions from molecular dynamics simulations. For each pair of correlated residues, the method quantifies which residue drives the correlated motions, and which residue responds. The measure reveals how correlated motions are used to transmit information through the system, and helps to clarify the link between correlated motions and biological function in biomolecular systems. The method is illustrated by its application to the Ets-1 transcription factor, which partially unfolds upon binding DNA. The calculations show dramatic changes in the direction of information flow upon DNA binding, and elucidate how the presence of DNA is communicated from the DNA binding H1 and H3 helices to inhibitory helix HI-1. Helix H4 is shown to act as a relay, which is attenuated in the apo state.

Approximate entropy for all signals

Calculation of approximate entropy (ApEn) requires a priori determination of two unknown parameters, m and r. While the recommended values of r, in the range of 0.1¿0.2 times the standard deviation of the signal, have been shown to be applicable for a wide variety of signals, in certain cases, r values within this prescribed range can lead to an incorrect assessment of the complexity of a given signal. To circumvent this limitation, we recently advocated finding the maximum ApEn value by assessing all values of r from 0 to 1 and found that the maximum ApEn does not always occur within the prescribed range of r values. Our results indicate that finding the maximum ApEn leads to the correct interpretation of a signal's complexity. One major limitation, however, is that the calculation of all choices of r values is often impractical because of the computational burden. Our new method, based on a heuristic stochastic model, overcomes this computational burden and leads to the automatic selection of the maximum ApEn value for any given signal. On the basis of Monte Carlo simulations, we derive general equations that can be used to estimate the maximum ApEn with high accuracy for a given value of m. Application to both synthetic and experimental data confirmed the advantages claimed with the proposed approach.

Dynamics and stability of muscle activations during walking in healthy young and older adults

To facilitate stable walking, humans must generate appropriate motor patterns and effective corrective responses to perturbations. Yet most EMG analyses do not address the continuous nature of muscle activation dynamics over multiple strides. We compared muscle activation dynamics in young and older adults by defining a multivariate state space for muscle activity. Eighteen healthy older and 17 younger adults walked on a treadmill for 2 trials of 5 min each at each of 5 controlled speeds (80-120% of preferred). EMG linear envelopes of v. lateralis, b. femoris, gastrocnemius, and t. anterior of the left leg were obtained. Interstride variability, local dynamic stability (divergence exponents), and orbital stability (maximum Floquet multipliers; FM) were calculated. Both age groups exhibited similar preferred walking speeds (p=0.86). Amplitudes and variability of individual EMG linear envelopes increased with speed (p<0.01) in all muscles but gastrocnemius. Older adults also exhibited greater variability in b. femoris and t. anterior (p<0.004). When comparing continuous multivariate EMG dynamics, older adults demonstrated greater local and orbital instability of their EMG patterns (p<0.01). We also compared how muscle activation dynamics were manifested in kinematics. Local divergence exponents were strongly correlated between kinematics and EMG, independent of age and walking speed, while variability and max FM were not. These changes in EMG dynamics may be related to increased neuromotor noise associated with aging and may indicate subtle deterioration of gait function that could lead to future functional declines.

Using the minimum description length principle for global reconstruction of dynamic systems from noisy time series

An alternative approach to determining embedding dimension when reconstructing dynamic systems from a noisy time series is proposed. The available techniques of determining embedding dimension (the false nearest-neighbor method, calculation of the correlation integral, and others) are known [H. D. I. Abarbanel, (Springer-Verlag, New York, 1997)] to be inefficient, even at a low noise level. The proposed approach is based on constructing a global model in the form of an artificial neural network. The required amount of neurons and the embedding dimension are chosen so that the description length should be minimal. The considered approach is shown to be appreciably less sensitive to the level and origin of noise, which makes it also a useful tool for determining embedding dimension when constructing stochastic models.

Computer systems are dynamical systems

In this paper, we propose a nonlinear dynamics-based framework for modeling and analyzing computer systems. Working with this framework, we use a custom measurement infrastructure and delay-coordinate embedding to study the dynamics of these complex nonlinear systems. We find strong indications, from multiple corroborating methods, of low-dimensional dynamics in the performance of a simple program running on a popular Intel computer-including the first experimental evidence of chaotic dynamics in real computer hardware. We also find that the dynamics change completely when we run the same program on a different type of Intel computer, or when that program is changed slightly. This not only validates our framework; it also raises important issues about computer analysis and design. These engineered systems have grown so complex as to defy the analysis tools that are typically used by their designers: tools that assume linearity and stochasticity and essentially ignore dynamics. The ideas and methods developed by the nonlinear dynamics community, applied and interpreted in the context of the framework proposed here, are a much better way to study, understand, and design modern computer systems.

Complex network from time series based on phase space reconstruction

We propose in this paper a reliable method for constructing complex networks from a time series with each vector point of the reconstructed phase space represented by a single node and edge determined by the phase space distance. Through investigating an extensive range of network topology statistics, we find that the constructed network inherits the main properties of the time series in its structure. Specifically, periodic series and noisy series convert into regular networks and random networks, respectively, and networks generated from chaotic series typically exhibit small-world and scale-free features. Furthermore, we associate different aspects of the dynamics of the time series with the topological indices of the network and demonstrate how such statistics can be used to distinguish different dynamical regimes. Through analyzing the chaotic time series corrupted by measurement noise, we also indicate the good antinoise ability of our method.

Detection of the type of intermittency using characteristic patterns in recurrence plots

One of the common routes to chaos is intermittency. Identification of the intermittency type is usually made using the properties of the probability distribution of laminar phases and of the average length of the laminar phases. Both have a statistical character and to obtain them a long time series has to be examined. Here, we present a recurrence plot method applicable to the analysis of short time series and through which the type of intermittency may be identified. The three types of intermittency introduced by Pomeau and Manneville and a chaos-chaos intermittency induced by interior crisis were examined. The identification of the type of intermittency is equivalent to the identification of the bifurcation associated with it. Our result seems particularly interesting as our method allows the analysis of short time series. The effect of the measurement noise on the effectiveness of the method is also discussed. An application of the method to the detection of type I intermittency in measured heart rate variability data is discussed.