Practical Methods of Measuring the Generalized Dimension and the Largest Lyapunov Exponent in High Dimensional Chaotic Systems

Analyzing and identifying predictable time range for stress prediction based on chaos theory and deep learning

Propose

Stress is a common problem globally. Prediction of stress in advance could help people take effective measures to manage stress before bad consequences occur. Considering the chaotic features of human psychological states, in this study, we integrate deep learning and chaos theory to address the stress prediction problem.

Methods

Based on chaos theory, we embed one's seemingly disordered stress sequence into a high dimensional phase space so as to reveal the underlying dynamics and patterns of the stress system, and meanwhile are able to identify the stress predictable time range. We then conduct deep learning with a two-layer (dimension and temporal) attention mechanism to simulate the nonlinear state of the embedded stress sequence for stress prediction.

Results

We validate the effectiveness of the proposed method on the public available Tesserae dataset. The experimental results show that the proposed method outperforms the pure deep learning method and Chaos method in both 2-label and 3-label stress prediction.

Conclusion

Integrating deep learning and chaos theory for stress prediction is effective, and can improve the prediction accuracy over 2% and 8% more than those of the deep learning and the Chaos method respectively. Implications and further possible improvements are also discussed at the end of the paper.

Microscale Neuronal Activity Collectively Drives Chaotic and Inflexible Dynamics at the Macroscale in Seizures

Neuronal activity propagates through the network during seizures, engaging brain dynamics at multiple scales. Such propagating events can be described through the avalanches framework, which can relate spatiotemporal activity at the microscale with global network properties. Interestingly, propagating avalanches in healthy networks are indicative of critical dynamics, where the network is organized to a phase transition, which optimizes certain computational properties. Some have hypothesized that the pathologic brain dynamics of epileptic seizures are an emergent property of microscale neuronal networks collectively driving the brain away from criticality. Demonstrating this would provide a unifying mechanism linking microscale spatiotemporal activity with emergent brain dysfunction during seizures. Here, we investigated the effect of drug-induced seizures on critical avalanche dynamics, using in vivo whole-brain two-photon imaging of GCaMP6s larval zebrafish (males and females) at single neuron resolution. We demonstrate that single neuron activity across the whole brain exhibits a loss of critical statistics during seizures, suggesting that microscale activity collectively drives macroscale dynamics away from criticality. We also construct spiking network models at the scale of the larval zebrafish brain, to demonstrate that only densely connected networks can drive brain-wide seizure dynamics away from criticality. Importantly, such dense networks also disrupt the optimal computational capacities of critical networks, leading to chaotic dynamics, impaired network response properties and sticky states, thus helping to explain functional impairments during seizures. This study bridges the gap between microscale neuronal activity and emergent macroscale dynamics and cognitive dysfunction during seizures.SIGNIFICANCE STATEMENT Epileptic seizures are debilitating and impair normal brain function. It is unclear how the coordinated behavior of neurons collectively impairs brain function during seizures. To investigate this we perform fluorescence microscopy in larval zebrafish, which allows for the recording of whole-brain activity at single-neuron resolution. Using techniques from physics, we show that neuronal activity during seizures drives the brain away from criticality, a regime that enables both high and low activity states, into an inflexible regime that drives high activity states. Importantly, this change is caused by more connections in the network, which we show disrupts the ability of the brain to respond appropriately to its environment. Therefore, we identify key neuronal network mechanisms driving seizures and concurrent cognitive dysfunction.

Complexity of locomotion activities in an outside-of-the-lab wearable motion capture dataset

Gait complexity is widely used to understand risk factors for injury, rehabilitation, the performance of assistive devices, and other matters of clinical interest. We analyze the complexity of out-of-the-lab locomotion activities via measures that have previously been used in gait analysis literature, as well as measures from other domains of data analysis. We categorize these broadly as quantifying either the intrinsic dimensionality, the variability, or the regularity, periodicity, or self-similarity of the data from a nonlinear dynamical systems perspective. We perform this analysis on a novel full-body motion capture dataset collected in out-of-the-lab conditions for a variety of indoor environments. This is a unique dataset with a large amount (over 24 h total) of data from participants behaving without low-level instructions in out-of-the-lab indoor environments. We show that reasonable complexity measures can yield surprising, and even profoundly contradictory, results. We suggest that future complexity analysis can use these guidelines to be more specific and intentional about what aspect of complexity a quantitative measure expresses. This will become more important as wearable motion capture technology increasingly allows for comparison of ecologically relevant behavior with lab-based measurements.

Analysis of Chaotic Dynamics by the Extended Entropic Chaos Degree

The Lyapunov exponent is the most-well-known measure for quantifying chaos in a dynamical system. However, its computation for any time series without information regarding a dynamical system is challenging because the Jacobian matrix of the map generating the dynamical system is required. The entropic chaos degree measures the chaos of a dynamical system as an information quantity in the framework of Information Dynamics and can be directly computed for any time series even if the dynamical system is unknown. A recent study introduced the extended entropic chaos degree, which attained the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. Moreover, an improved calculation formula for the extended entropic chaos degree was recently proposed to obtain appropriate numerical computation results for multidimensional chaotic maps. This study shows that all Lyapunov exponents of a chaotic map can be estimated to calculate the extended entropic chaos degree and proposes a computational algorithm for the extended entropic chaos degree; furthermore, this computational algorithm was applied to one and two-dimensional chaotic maps. The results indicate that the extended entropic chaos degree may be a viable alternative to the Lyapunov exponent for both one and two-dimensional chaotic dynamics.

Financial markets' deterministic aspects modeled by a low-dimensional equation

We ask whether empirical finance market data (Financial Stress Index, swap and equity, emerging and developed, corporate and government, short and long maturity), with their recently observed alternations between calm periods and financial turmoil, could be described by a low-dimensional deterministic model, or whether this requests a stochastic approach. We find that a deterministic model performs at least as well as one of the best stochastic models, but may offer additional insight into the essential mechanisms that drive financial markets.

An Improved Calculation Formula of the Extended Entropic Chaos Degree and Its Application to Two-Dimensional Chaotic Maps

The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.

Chaotic vibrations of size-dependent flexible rectangular plates

A mathematical model describing nonlinear vibrations of size-dependent rectangular plates is proposed. The plates are treated as the Cosserat continuum with bounded rotations of their particles (pseudo-continuum). The governing partial differential equations (PDEs) and boundary/initial conditions are obtained using the von Kármán geometric relations, and they are yielded by the energetic Hamilton principle. The derived mixed-form PDEs are reduced to ordinary differential equations and algebraic equations (AEs) using (i) the Galerkin-Krylov-Bogoliubov method (GKBM) in higher approximations, and then they are solved with the help of a combination of the Runge-Kutta methods of the second and fourth order, (ii) the finite difference method (FDM), and (iii) the Newmark method. The convergence of FDM vs the interval of the space coordinate grids and of GKBM vs the number of employed terms of the approximating function is investigated. The latter approach allows for achieving reliable results by taking account of almost infinite-degree-of-freedom approximation to the regular and chaotic dynamics of the studied plates. The problem of stability loss of the size-dependent plates under harmonic load is also tackled.

Detecting prediction limit of marked point processes using constrained random shuffle surrogate data

Marked point processes refer to time series of discrete events with additional information about the events. Seismic activities, neural activities, and price movements in financial markets are typical examples of marked point process data. In this paper, we propose a method for investigating the prediction limits of marked point process data, where random shuffle surrogate data with time window constraints are proposed and utilized to estimate the prediction limits. We applied the proposed method to the marked point process data obtained from several dynamical systems and investigated the relationship between the largest Lyapunov exponent and the prediction limit estimated by the proposed method. The results revealed a positive correlation between the reciprocal of the estimated prediction limit and the largest Lyapunov exponent of the underlying dynamical systems in marked point processes.

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

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

EEG Analysis in Structural Focal Epilepsy Using the Methods of Nonlinear Dynamics (Lyapunov Exponents, Lempel-Ziv Complexity, and Multiscale Entropy)

This paper analyzes a case with the patient having focal structural epilepsy by processing electroencephalogram (EEG) fragments containing the "sharp wave" pattern of brain activity. EEG signals were recorded using 21 channels. Based on the fact that EEG signals are time series, an approach has been developed for their analysis using nonlinear dynamics tools: calculating the Lyapunov exponent's spectrum, multiscale entropy, and Lempel-Ziv complexity. The calculation of the first Lyapunov exponent is carried out by three methods: Wolf, Rosenstein, and Sano-Sawada, to obtain reliable results. The seven Lyapunov exponent spectra are calculated by the Sano-Sawada method. For the observed patient, studies showed that with medical treatment, his condition did not improve, and as a result, it was recommended to switch from conservative treatment to surgical. The obtained results of the patient's EEG study using the indicated nonlinear dynamics methods are in good agreement with the medical report and MRI data. The approach developed for the analysis of EEG signals by nonlinear dynamics methods can be applied for early detection of structural changes.

Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems

The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the classical problems governed by difference and differential equations (Hénon map, hyperchaotic Hénon map, logistic map, Rössler attractor, Lorenz attractor) and with the use of both Fourier spectra and Gauss wavelets. It has been shown that a modification of the neural network method makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyperchaos, and others. The aim of the comparison was to evaluate the considered algorithms, study their convergence, and also identify the most suitable algorithms for specific system types and objectives. Moreover, an algorithm of calculation of the spectrum of Lyapunov exponents based on a trained neural network has been proposed. It has been proven that the developed method yields good results for different types of systems and does not require a priori knowledge of the system equations.

Dynamical complexity and computation in recurrent neural networks beyond their fixed point

Spontaneous activity found in neural networks usually results in a reduction of computational performance. As a consequence, artificial neural networks are often operated at the edge of chaos, where the network is stable yet highly susceptible to input information. Surprisingly, regular spontaneous dynamics in Neural Networks beyond their resting state possess a high degree of spatio-temporal synchronization, a situation that can also be found in biological neural networks. Characterizing information preservation via complexity indices, we show how spatial synchronization allows rRNNs to reduce the negative impact of regular spontaneous dynamics on their computational performance.

Randomness evaluation for an optically injected chaotic semiconductor laser by attractor reconstruction

State-space reconstruction is investigated for evaluating the randomness generated by an optically injected semiconductor laser in chaos. The reconstruction of the attractor requires only the emission intensity time series, allowing both experimental and numerical evaluations with good qualitative agreement. The randomness generation is evaluated by the divergence of neighboring states, which is quantified by the time-dependent exponents (TDEs) as well as the associated entropies. Averaged over the entire attractor, the mean TDE is observed to be positive as it increases with the evolution time through chaotic mixing. At a constant laser noise strength, the mean TDE for chaos is observed to be greater than that for periodic dynamics, as attributed to the effect of noise amplification by chaos. After discretization, the Shannon entropies continually generated by the laser for the output bits are estimated in providing a fundamental basis for random bit generation, where a combined output bit rate reaching 200 Gb/s is illustrated using practical tests. Overall, based on the reconstructed states, the TDEs and entropies offer a direct experimental verification of the randomness generated in the chaotic laser.

Computational study of the piccolo: Evidence for chaotic tones

Direct numerical solutions of the Navier-Stokes equations have been used to compute tones produced by a model of the piccolo. The behavior depends on the angle at which the air jet is directed at the embouchure hole and the displacement of the jet center from the embouchure edge. Values of the jet angle and displacement are found that produce pure tones with spectra similar to those of real piccolo tones. The behavior of calculated tones as a function of the blowing speed u is studied, and it is found that unsteady behavior can occur depending on the value of u, the jet angle, and the jet displacement. Detailed analysis of these unsteady tones suggests that the model piccolo studied here exhibits chaotic behavior with a positive Lyapunov exponent. The implications for piccolo tones produced with real piccolos by real piccolo players are discussed.

Detecting epileptic seizure from scalp EEG using Lyapunov spectrum

One of the inherent weaknesses of the EEG signal processing is noises and artifacts. To overcome it, some methods for prediction of epilepsy recently reported in the literature are based on the evaluation of chaotic behavior of intracranial electroencephalographic (EEG) recordings. These methods reduced noises, but they were hazardous to patients. In this study, we propose using Lyapunov spectrum to filter noise and detect epilepsy on scalp EEG signals only. We determined that the Lyapunov spectrum can be considered as the most expected method to evaluate chaotic behavior of scalp EEG recordings and to be robust within noises. Obtained results are compared to the independent component analysis (ICA) and largest Lyapunov exponent. The results of detecting epilepsy are compared to diagnosis from medical doctors in case of typical general epilepsy.

[Analysis of electrogastrography in gastrectomized subjects]

OBJECTIVES

To establish a method for the development of a mathematical model of autonomic activity in gastrointestinal movements and to basically evaluate of the application of the nonlinear analysis method to electrogastrography, we performed feature extraction of electrogastrographic changes in healthy elderly and gastrectomized subjects.

METHODS

The subjects consisted of 9 healthy elderly males and 3 elderly males without constipation who had undergone resection of 2/3 of the stomach. Electrogastrograms were obtained in a sitting position for 30 minutes and in a supine position for 90 minutes. Spectrum analyses of electrographic time series, the maximum Lyapunov exponent for the evaluation of the chaos of dynamic systems forming time series, and translation error for the evaluation of the smoothness of the attractor orbit were performed.

RESULTS

The maximum Lyapunov exponent was a positive number in all analysis intervals in all subjects. This suggests the irregularity of electrogastrograms in gastrectomized subjects. The translation error in the gastrectomized subjects was higher than that in the healthy elderly subjects, showing irregularity. However, as a result of spectrum analysis, gastric electric activity was predominant on electrogastrograms of the healthy elderly subjects, but intestinal electric activity was predominant in the gastrectomized subjects.

CONCLUSIONS

Differentiation between healthy and gastrectomized elderly people is difficult using only one of the spectrum analysis methods, the maximum Lyapunov exponent, or the translation error. However, since the extracted features differed among the 3 analysis methods, differentiation and diagnosis may be possible using a combination of these methods.

[Application of electrogastrography to public health]

In general, gastrointestinal motility tests cause pain; therefore, the establishment of noninvasive methods is desired. Noninvasive methods facilitate the measurement of motility close to the normal physiological state, can provide new findings, and may contribute to the development of associated fields. Electrogastrography (EGG) is a gastrointestinal motility test in which gastrointestinal electric activity is measured. Compared with other gastrointestinal motility measurement methods such as the gastric emptying and internal pressure measurement methods, EGG is noninvasive and allows measurement under minimum restriction; therefore, measurement for a long time is also possible. In addition, since gastrointestinal electric activity, which cannot be quantified using other methods, can be measured, EGG is applicable to the evaluation of the state of the body and pathological conditions, and may provide new findings such as those useful for the prevention of gastrointestinal dysfunction associated with various disorders. EGG is also useful for preventing disorders associated with abnormal gastrointestinal activity such as functional dyspepsia, which has been more frequently observed in recent years, and constipation, which is an extremely frequent complaint in the elderly. Thus, EGG is of marked importance in public health. However, the range of EGG utilizations and applications is still limited at present. Therefore, we outlined the measurement/analysis methods, the advantages and problems of EGG and electrogastroenterography (EGEG), described their clinical importance, and also commented on forefront studies on EGG and evaluated its prospects.

Regularity analysis of an individual-based ecosystem simulation

We analyze the results of a large simulation of an evolving ecosystem to evaluate its complexity. In particular, we are interested to know how close to a stochastic or a deterministic behavior our simulation is. Four methods have been used for this analysis: Higuchi fractal dimension, correlation dimension, largest Lyapunov exponent, and P&H method. Besides, we use a surrogate data test to reach a final decision about analysis. As we expect, our results show that there is a deterministic and chaotic behavior in ecosystem simulation.

Comparison of tests for embeddings

It is possible to compare results for the classical tests for embeddings of chaotic data with the results of a recently proposed test. The classical tests, which depend on real numbers (fractal dimensions, Lyapunov exponents) averaged over an attractor, are compared with a topological test that depends on integers. The comparison can only be done for mappings into three dimensions. We find that the classical tests fail to predict when a mapping is an embedding and when it is not. We point out the reasons for this failure, which are not restricted to three dimensions.

A new algorithm developed based on a mixture of spectral and nonlinear techniques for the analysis of heart rate variability

In this paper, an algorithm based on a joint use of spectral and nonlinear techniques for heart rate variability (HRV) analysis is proposed. First, the measured RR data are passed into a trimmed moving average (TMA)-based filtering system to generate a lower frequency (LF) time series and a higher frequency (HF) one that approximately reflect the sympathetic and vagal activities, respectively. Since the Lyapunov exponent can be used to characterize the level of chaos in complex physiological systems, the largest Lyapunov exponents corresponding to the complex sympathetic and vagal systems are then estimated from the LF and HF time series, respectively, using an existing algorithm. Numerical results of a postural maneuver experiment indicate that both characteristic exponents or their combinations might serve as a set of innovative and robust indicators for HRV analysis, even under the contamination of sparse impulses due to aberrant beats in the RR data.

Message encoding and decoding using an asynchronous chaotic laser diode transmitter-receiver array

We have numerically investigated a chaotic laser diode transmitter-receiver array scheme (CLDTRAS), which is a secure digital communication scheme using a difference between two types of transmitter-receiver array consisting of two self-pulsating laser diodes (LDs), i.e., a receiver LD and a transmitter LD. By analyzing the bit error rate, particularly its dependence on the parameter mismatches of the hardware and channel noise and on the correlation coefficient between a transmitter LD and receiver LD, we examined the problems of sensitivity to parameter mismatches and channel noise and a dependence on chaos synchronization between a transmitter LD and a receiver LD. The former makes communication difficult, and the latter makes it possible for an eavesdropper to estimate the receiver LD using chaos synchronization and to forge the hardware. Then we studied the effects of the bit error rate for various values of the threshold, which determines a binary message, and for various numbers of transmitters-receivers making up a LD transmitter-receiver array. It has been shown that a highly noise-tolerant and hardware-dependent communication scheme can be achieved with the LD transmitter-receiver array, whose transmitter and receiver LDs are asynchronous with respect to each other, by choosing the proper threshold and increasing the number of LD transmitters-receivers. Since it is possible to communicate without chaos synchronization, it becomes difficult to forge hardware and to eavesdrop with the forged hardware even if the key is stolen.

Generalized synchronization in fractional order systems

The generalized synchronization of fractional order systems is investigated, including the synchronization between two different fractional order systems with the same order, the synchronization between two different fractional order systems with no orders the same, and the synchronization between a classical system and its corresponding fractional order system with mismatched parameters. The mechanism for the occurrence of generalized synchronization of fractional order systems is clarified, the necessary and sufficient conditions are given, and several methods to detect generalized synchronization are discussed. The relationship between generalized synchronization and the equivalence of fractional order systems is also considered.

On the bound of the Lyapunov exponents for continuous systems

In this paper, both upper bounds and lower bounds for all the Lyapunov exponents of continuous differential systems are determined. Several examples are given to show the application of the estimates derived here.

Linear and nonlinear measures of blood pressure variability: increased chaos of blood pressure time series in patients with panic disorder

Arterial blood pressure (BP) variability increases progressively with the development of hypertension and an increase in BP variability is associated with end organ damage and cardiovascular morbidity. On the other hand, a decrease in heart rate (HR) variability is associated with significant cardiovascular mortality. There is a strong association between cardiovascular mortality and anxiety. Several previous studies have shown decreased HR variability in patients with anxiety. In this study, we investigated beat-to-beat variability of systolic and diastolic BP (SBP and DBP) in normal controls and patients with panic disorder during normal breathing and controlled breathing at 12, and 20 breaths per minute using linear as well as nonlinear techniques. Finger BP signal was obtained noninvasively using Finapres. Standing SBPvi and DBP BPvi (log value of BP variance corrected for mean BP divided by HR variance corrected for mean HR) were significantly higher in patients compared to controls. Largest Lyapunov exponent (LLE) of SBP and DBP, a measure of chaos, was significantly higher in patients in supine as well as standing postures. The ratios of LLE (SBP/HR) and LLE (DBP/HR) were also significantly higher (P<.001) in patients compared to controls. These findings further suggest dissociation between HR and BP variability and a possible relative increase in sympathetic function in anxiety. This increase in BP variability may partly explain the increase in cardiovascular mortality in this group of patients.

Effect of nortriptyline and paroxetine on measures of chaos of heart rate time series in patients with panic disorder

Tricyclic antidepressants have notable cardiac side effects, and this issue has become important due to the recent reports of increased cardiovascular mortality in patients with depression and anxiety. Several previous studies indicate that serotonin reuptake inhibitors (SRIs) do not appear to have such adverse effects. Apart from the effects of these drugs on routine 12-lead ECG, the effects on beat-to-beat heart rate (HR) and QT interval time series provide more information on the side effects related to cardiac autonomic function. In this study, we evaluated the effects of two antidepressants, nortriptyline (n=13), a tricyclic, and paroxetine (n=16), an SRI inhibitor, on HR variability in patients with panic disorder, using a measure of chaos, the largest Lyapunov exponent (LLE) using pre- and posttreatment HR time series. Our results show that nortriptyline is associated with a decrease in LLE of high frequency (HF: 0.15-0.5 Hz) filtered series, which is most likely due to its anticholinergic effect, while paroxetine had no such effect. Paroxetine significantly decreased sympathovagal ratios as measured by a decrease in LLE of LF/HF. These results suggest that paroxetine appears to be safer in regards to cardiovascular effects compared to nortriptyline in this group of patients.

Nonlinear measures of QT interval series: novel indices of cardiac repolarization lability: MEDqthr and LLEqthr

In this study, we investigated nonlinear measures of chaos of QT interval time series in 28 normal control subjects, 36 patients with panic disorder and 18 patients with major depression in supine and standing postures. We obtained the minimum embedding dimension (MED) and the largest Lyapunov exponent (LLE) of instantaneous heart rate (HR) and QT interval series. MED quantifies the system's complexity and LLE predictability. There was a significantly lower MED and a significantly increased LLE of QT interval time series in patients. Most importantly, nonlinear indices of QT/HR time series, MEDqthr (MED of QT/HR) and LLEqthr (LLE of QT/HR), were highly significantly different between controls and both patient groups in either posture. Results remained the same even after adjusting for age. The increased LLE of QT interval time series in patients with anxiety and depression is in line with our previous findings of higher QTvi (QT variability index, a log ratio of QT variability corrected for mean QT squared divided by heart rate variability corrected for mean heart rate squared) in these patients, using linear techniques. Increased LLEqthr (LLE of QT/HR) may be a more sensitive tool to study cardiac repolarization and a valuable addition to the time domain measures such as QTvi. This is especially important in light of the finding that LLEqthr correlated poorly and nonsignificantly with QTvi. These findings suggest an increase in relative cardiac sympathetic activity and a decrease in certain aspects of cardiac vagal function in patients with anxiety as well as depression. The lack of correlation between QTvi and LLEqthr suggests that this nonlinear index is a valuable addition to the linear measures. These findings may also help to explain the higher incidence of cardiovascular mortality in patients with anxiety and depressive disorders.

Decreased chaos of heart rate time series in children of patients with panic disorder

This study examined the differences of heart rate variability measures between children of parents with panic disorder and children of healthy controls using linear as well as nonlinear techniques. Supine and standing heart rate variability indices were measured in all children using power spectral analysis and a measure of chaos, the largest Lyapunov exponent (LLE) of heart rate time series. No significant differences emerged between the children of panic disorder parents and children of normal controls on any of the spectral heart rate variability measures. However, children of patients with panic disorder had significantly lower LLE of heart rate time series in supine posture, suggesting a relative decrease of cardiac vagal function in this group of children. This suggests a possible heritable effect of certain measures of heart rate variability, as previous studies showed decreased heart rate variability in patients with panic disorder using spectral as well as nonlinear techniques. Recent evidence also suggests that some of these nonlinear measures are superior or of additional value to the traditional time and frequency domain measures of heart rate variability to predict serious ventricular arrhythmias and sudden death.

Diminished chaos of heart rate time series in patients with major depression

BACKGROUND

Depression and anxiety have been linked to serious cardiovascular events in patients with preexisting cardiac illness. A decrease in cardiac vagal function as suggested by a decrease in heart rate (HR) variability has been linked to sudden death.

METHODS

We compared LLE and nonlinearity scores of the unfiltered (UF) and filtered time series (very low, low, and high frequency; VLF, LF and HF) of HR between patients with depression (n = 14) and healthy control subjects (n = 18).

RESULTS

We found significantly lower LLE of the unfiltered series in either posture, and HF series in patients with major depression in supine posture (p <.002). LLE (LF/UF), which may indicate relative sympathetic activity was also significantly higher in supine and standing postures in patients (p <.05); LF/HF (LLE) was also higher in patients (p <.05) in either posture.

CONCLUSIONS

These findings suggest that major depression is associated with decreased cardiac vagal function and a relative increase in sympathetic function, which may be related to the higher risk of cardiovascular mortality in this group and illustrates the usefulness of nonlinear measures of chaos such as LLE in addition to the commonly used spectral measures.

Heart rate time series: decreased chaos after intravenous lactate and increased non-linearity after isoproterenol in normal subjects

In this study, we reanalyzed our previous heart rate time series data on the effects of intravenous sodium lactate (n=9) and intravenous isoproterenol (n=11) using non-linear techniques. Our prior findings of significantly higher baseline non-linear scores (NL: S(netGS)) and significantly lower largest Lyapunov exponents in supine posture in patients with panic disorder compared to control subjects prompted this study. We obtained the largest Lyapunov exponent (LLE), and a measure of non-linearity (NL: S(netGS)) of heart rate time series. LLE quantifies predictability and NL quantifies the deviation from linear processes. There was a significant increase in NL score, (S(netGS)) after isoproterenol infusions and a significant decrease in LLE (an increase in predictability indicating decreased chaos), after intravenous lactate in supine posture in normal control subjects. Increased NL scores in supine posture after intravenous isoproterenol may be due to a relative increase in cardiac sympathetic activity or a decrease in vagal activity at least in certain circumstances, and an overall decrease in LLE may indicate an impaired cardiac autonomic flexibility after intravenous sodium lactate, as LLE is diminished by autonomic blockade by atropine. Band analysis of LLE (LF/HF) (LF: 0.04-0.15 Hz and HF: 0.15-0.5 Hz) showed an increase of these ratios during either condition with a higher sympathovagal interaction after the drug administration. These findings may throw new light on the association of anxiety and significant cardiovascular events.

Decreased chaos and increased nonlinearity of heart rate time series in patients with panic disorder

In this study, we investigated measures of nonlinear dynamics and chaos of heart rate time series in 30 normal control subjects and 36 age-matched patients with panic disorder in supine and standing postures. We obtained minimum embedding dimension (MED), largest Lyapunov exponent (LLE) and measures of nonlinearity (NL) of heart rate time series. MED quantifies system's complexity, LLE predictability and NL, deviation from linear processes. There was a significant increase in complexity (p < 0.00001), an increase in predictability (decreased chaos) (p < 0.00001) and an increase in nonlinearity (Snet GS) (p = 0.00001), especially in supine posture in patients with panic disorder. Increased NL score in supine posture may be due to a relative increase in cardiac sympathetic activity and an overall decrease in LLE may indicate an impaired cardiac autonomic flexibility in these patients due possibly to a decrease in cardiac vagal activity. These findings may further explain the reported higher incidence of cardiovascular mortality in patients with anxiety disorders.

Transitions of macroscopic structures and self-induced chaos observed in plasmas by a dc hollow cathode discharge having features of nonlinear open systems

A novel experimental investigation is presented on the connection between discontinuous transitions of macroscopic structures of plasma and self-induced chaotic oscillations characterized by the positive Lyapunov exponents lambda(L) through the period-doubling route in a dc hollow cathode discharge, which has features of nonlinear open systems. We have clarified experimentally that there appear different discharge modes accompanying the discontinuous transitions, and detailed qualitative explanations are presented about the mechanism of those transitions. It is shown that fundamental frequencies of the self-induced periodic oscillations with nonpositive lambda(L) change with the changes of discharge current, and the amplitude of chaotic oscillations with the positive lambda(L) jumps up almost one order higher than that of nonchaotic ones with the nonpositive lambda(L). The self-induced chaotic oscillations with the positive lambda(L) have been observed near two edges of discontinuous transitions of plasma structures, suggesting that the chaotic mode is associated with the discontinuous transition of macroscopic structures in some nonlinear open systems.

Chaotic behavior in the locomotion of Amoeba proteus

The locomotion of Amoeba proteus has been investigated by algorithms evaluating correlation dimension and Lyapunov spectrum developed in the field of nonlinear science. It is presumed by these parameters whether the random behavior of the system is stochastic or deterministic. For the analysis of the nonlinear parameters, n-dimensional time-delayed vectors have been reconstructed from a time series of periphery and area of A. proteus images captured with a charge-coupled-device camera, which characterize its random motion. The correlation dimension analyzed has shown the random motion of A. proteus is subjected only to 3-4 macrovariables, though the system is a complex system composed of many degrees of freedom. Furthermore, the analysis of the Lyapunov spectrum has shown its largest exponent takes positive values. These results indicate the random behavior of A. proteus is chaotic and deterministic motion on an attractor with low dimension. It may be important for the elucidation of the cell locomotion to take account of nonlinear interactions among a small number of dynamics such as the sol-gel transformation, the cytoplasmic streaming, and the relating chemical reaction occurring in the cell.

Nonlinear measures of heart rate time series: influence of posture and controlled breathing

In this study, we investigated measures of nonlinear dynamics and chaos theory in regards to heart rate variability in 27 normal control subjects in supine and standing postures, and 14 subjects in spontaneous and controlled breathing conditions. We examined minimum embedding dimension (MED), largest Lyapunov exponent (LLE) and measures of nonlinearity (NL) of heart rate time series. MED quantifies the system's complexity, LLE predictability and NL, a measure of deviation from linear processes. There was a significant decrease in complexity (P < 0.00001), a decrease in predictability (P < 0.00001) and an increase in nonlinearity (P = 0.00001) during the change from supine to standing posture. Decrease in MED, and increases in NL score and LLE in standing posture appear to be partly due to an increase in sympathetic activity of the autonomous nervous system in standing posture. An improvement in predictability during controlled breathing appears to be due to the introduction of a periodic component.

On the evidence of deterministic chaos in ECG: Surrogate and predictability analysis

The question whether the human cardiac system is chaotic or not has been an open one. Recent results in chaos theory have shown that the usual methods, such as saturation of correlation dimension D(2) or the existence of positive Lyapunov exponent, alone do not provide sufficient evidence to confirm the presence of deterministic chaos in an experimental system. The results of surrogate data analysis together with the short-term prediction analysis can be used to check whether a given time series is consistent with the hypothesis of deterministic chaos. In this work nonlinear dynamical tools such as surrogate data analysis, short-term prediction, saturation of D(2) and positive Lyapunov exponent have been applied to measured ECG data for several normal and pathological cases. The pathology presently studied are PVC (Premature Ventricular Contraction), VTA (Ventricular Tachy Arrhythmia), AV (Atrio-Ventricular) block and VF (Ventricular Fibrillation). While these results do not prove that ECG time series is definitely chaotic, they are found to be consistent with the hypothesis of chaotic dynamics. (c) 1998 American Institute of Physics.