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

Estimating good discrete partitions from observed data: symbolic false nearest neighbors

A symbolic analysis of observed time series requires a discrete partition of a continuous state space containing the dynamics. A particular kind of partition, called "generating," preserves all deterministic dynamical information in the symbolic representation, but such partitions are not obvious beyond one dimension. Existing methods to find them require significant knowledge of the dynamical evolution operator. We introduce a statistic and algorithm to refine empirical partitions for symbolic state reconstruction. This method optimizes an essential property of a generating partition, avoiding topological degeneracies, by minimizing the number of "symbolic false nearest neighbors." It requires only the observed time series and is sensible even in the presence of noise when no truly generating partition is possible.

Comparative study of embedding methods

Embedding experimental data is a common first step in many forms of dynamical analysis. The choice of appropriate embedding parameters (dimension and lag) is crucial to the success of the subsequent analysis. We argue here that the optimal embedding of a time series cannot be determined by criteria based solely on the time series itself. Therefore we base our analysis on an examination of systems that have explicit analytic representations. A comparison of analytically obtained results with those obtained by an examination of the corresponding time series provides a means of assessing the comparative success of different embedding criteria. The assessment also includes measures of robustness to noise. The limitations of this study are explicitly delineated. While bearing these limitations in mind, we conclude that for the examples considered here, the best identification of the embedding dimension was achieved with a global false nearest neighbors argument, and the best value of lag was identified by the mutual information function.

Low frequency oscillations in semi-insulating GaAs: a nonlinear analysis

We have observed low frequency current oscillations in a semi-insulating GaAs sample grown by low temperature molecular beam epitaxy. For this, an experimental setup proper to measure high impedance samples with small external noise was developed. Spontaneous oscillations in the current were observed for some bias conditions. Although measurements were carried out from room temperature down to liquid helium, the dynamical analysis was carried out around 200 K where the signal to noise ratio was fairly favorable. To increase the data quality we have also used a noise reduction algorithm suitably developed for nonlinear systems. We observed attractors having low embedding dimension, limit cycle bifurcations, and chaotic behavior characteristic of nonlinear dynamical processes in route to chaos. Attractor reconstruction, Poincare sections, Lyapunov exponents, and correlation dimension were also analyzed.

Signal nonlinearity in fMRI: a comparison between BOLD and MION

In this paper, we introduce a methodology for comparing the nonlinearities present in sets of time series using four different nonlinearity measures, one of which, the "delay vector variance" method, is a novel approach to the characterization of a time series. It is then applied to examine the difference in nonlinearity between functional magnetic resonance imaging (fMRI) signals that have been recorded using different contrast agents. Recently, an exogenous contrast agent, monocrystalline iron oxide particle (MION), has been introduced for fMRI, which has been shown to increase the functional sensitivity compared with the traditional blood oxygen level dependent (BOLD) technique. The resulting fMRI signals are influenced by cerebral blood volume, whereas the more traditionally recorded BOLD signals are influenced not only by cerebral blood volume, but also by the cerebral blood flow and the metabolic rate of oxygen. The proposed methodology is applied to address the question whether this difference in the number of physiological variables is reflected in a difference in the degree of nonlinearity. We therefore analyze two sets of fMRI signals, one from a BOLD and the other from a MION monkey study with similar experimental designs. In the neuroimaging context, the proposed nonlinearity analyses are different from those described in the literature, since no a priori model is assumed: rather than pinpointing the source(s) of nonlinearity, nonparametric analyses are performed on BOLD and MION fMRI signals. Furthermore, we introduce a strategy for analyzing a population of fMRI signals, rather than focusing the analysis on one signal, as is traditionally done in the domain of nonlinear signal processing. Our results show that, overall, the BOLD signals are more nonlinear in nature than the MION ones, which is in agreement with current hypotheses.

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.

Major depression with ischemic heart disease: effects of paroxetine and nortriptyline on measures of nonlinearity and chaos of heart rate

Depression is associated with increased cardiovascular mortality in patients with preexisting cardiac illness. A decrease in cardiac vagal function as suggested by a decrease in heart rate variability (HRV) or heart period variability has been linked to sudden death in patients with cardiac disease as well as in normal controls. Recent studies have shown decreased vagal function in cardiac patients with depression as well as in depressed patients without cardiac illness. In this study, we compared 20 h awake and sleep heart period nonlinear measures using quantification of nonlinearity and chaos in two groups of patients with major depression and ischemic heart disease (mean age 59-60 years) before and after 6 weeks of treatment with paroxetine or nortriptyline. Patients received paroxetine, 20-30 mg/day or nortriptyline targeted to 190-570 nmol/l for 6 weeks. For HRV analysis, 24 patients were included in the paroxetine treatment study and 20 patients in the nortriptyline study who had at least 20000 s of awake data. The ages of these groups were 60.4 +/- 10.5 years for paroxetine and 60.8 +/- 13.4 years for nortriptyline. There was a significant decrease in the largest Lyapunov exponent (LLE) after treatment with nortriptyline but not paroxetine. There were also significant decreases in nonlinearity scores on S(netPR) and S(netGS) after nortriptyline, which may be due to a decrease in cardiac vagal modulation of HRV. S(netGS) and awake LLE were the most significant variables that contributed to the discrimination of postparoxetine and postnortriptyline groups even with the inclusion of time and frequency domain measures. These findings suggest that nortriptyline decreases the measures of chaos probably through its stronger vagolytic effects on cardiac autonomic function compared with paroxetine, which is in agreement with previous clinical and preclinical reports. Nortriptyline was also associated with a significant decrease in nonlinearity scores, which may be due to anticholinergic and/or sympatholytic effects. As depression is associated with a strong risk factor for cardiovascular mortality, one should be careful about using any drug that adversely affects cardiac vagal function.

Nonlinear measures of respiration: respiratory irregularity and increased chaos of respiration in patients with panic disorder

BACKGROUND

Respiratory irregularity has been previously reported in patients with panic disorder using time domain measures. However, the respiratory signal is not entirely linear and a few previous studies used approximate entropy (APEN), a measure of regularity of time series. We have been studying APEN and other nonlinear measures including a measure of chaos, the largest Lyapunov exponent (LLE) of heart rate time series, in some detail. In this study, we used these measures of respiration to compare normal controls (n = 18) and patients with panic disorder (n = 22) in addition to the traditional time domain measures of respiratory rate and tidal volume.

METHODS

Respiratory signal was obtained by the Respitrace system using a thoracic and an abdominal belt, which was digitized at 500 Hz. Later, the time series were constructed at 4 Hz, as the highest frequency in this signal is limited to 0.5 Hz. We used 256 s of data (1024 points) during supine and standing postures under normal breathing and controlled breathing at 12 breaths/min.

RESULTS

APEN was significantly higher in patients in standing posture during normal as well as controlled breathing (p = 0.002 and 0.02, respectively). LLE was also significantly higher in standing posture during normal breathing (p = 0.009). Similarly, the time domain measures of standard deviations and the coefficient of variation (COV) of tidal volume (TV) were significantly higher in the patient group (p = 0.02 and 0.004, respectively). The frequency of sighs was also higher in the patient group in standing posture (p = 0.02). In standing posture, LLE (p < 0.05) as well as APEN (p < 0.01) contributed significantly toward the separation of the two groups over and beyond the linear measure, i.e. the COV of TV.

CONCLUSION

These findings support the previously described respiratory irregularity in patients with panic disorder and also illustrate the utility of nonlinear measures such as APEN and LLE as additional measures toward a better understanding of the abnormalities of respiratory physiology in similar patient populations as the correlation between LLE, APEN and some of the time domain measures only explained up to 50-60% of the variation.

The algorithmic complexity of multichannel EEGs is sensitive to changes in behavior

Symbolic measures of complexity provide a quantitative characterization of the sequential structure of symbol sequences. Promising results from the application of these methods to the analysis of electroencephalographic (EEG) and event-related brain potential (ERP) activity have been reported. Symbolic measures used thus far have two limitations, however. First, because the value of complexity increases with the length of the message, it is difficult to compare signals of different epoch lengths. Second, these symbolic measures do not generalize easily to the multichannel case. We address these issues in studies in which both single and multichannel EEGs were analyzed using measures of signal complexity and algorithmic redundancy, the latter being defined as a sequence-sensitive generalization of Shannon's redundancy. Using a binary partition of EEG activity about the median, redundancy was shown to be insensitive to the size of the data set while being sensitive to changes in the subject's behavioral state (eyes open vs. eyes closed). The covariance complexity, calculated from the singular value spectrum of a multichannel signal, was also found to be sensitive to changes in behavioral state. Statistical separations between the eyes open and eyes closed conditions were found to decrease following removal of the 8- to 12-Hz content in the EEG, but still remained statistically significant. Use of symbolic measures in multivariate signal classification is described.

Stochastic resonance mechanism in aerosol index dynamics

We consider satellite time series concerning the atmospheric aerosol content. We prove that these time series are well described by a stochastic dynamical model. The principal peak in the power spectrum of these signals can be explained by stochastic resonance, linking variable external factors, such as Sun-Earth radiation budget and local insolation, to fluctuations on smaller spatial and temporal scale due to internal weather and antrophic components.

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.

Onset of chaotic dynamics in a ball mill: Attractors merging and crisis induced intermittency

In mechanical treatment carried out by ball milling, powder particles are subjected to repeated high-energy mechanical loads which induce heavy plastic deformations together with fracturing and cold-welding events. Owing to the continuous defect accumulation and interface renewal, both structural and chemical transformations occur. The nature and the rate of such transformations have been shown to depend on variables, such as impact velocity and collision frequency that depend, in turn, on the whole dynamics of the system. The characterization of the ball dynamics under different impact conditions is then to be considered a necessary step in order to gain a satisfactory control of the experimental set up. In this paper we investigate the motion of a ball in a milling device. Since the ball motion is governed by impulsive forces acting during each collision, no analytical expression for the complete ball trajectory can be obtained. In addition, mechanical systems exhibiting impacts are strongly nonlinear due to sudden changes of velocities at the instant of impact. Many different types of periodic and chaotic impact motions exist indeed even for simple systems with external periodic excitation forces. We present results of the analysis on the ball trajectory, obtained from a suitable numerical model, under growing degree of impact elasticity. A route to high dimensional chaos is obtained. Crisis and attractors merging are also found. (c) 2002 American Institute of Physics.

Non-linear prediction for oesophageal voice analysis

Herein, non-linear prediction methods are applied to oesophageal voice analysis. The research aims to investigate normal and pathological subjects, in order to improve knowledge of the oesophageal voice behaviour. Analysis is performed in the reconstructed phase space, using both non-linear prediction with local linear approximation and the S-Map method. Preliminary results seem to confirm that in normal subjects a non-linear stable deterministic behaviour takes place, while in pathological subjects the non-linear contribution reduces while the time series becomes unstable.

Stress fluctuations in sheared Stokesian suspensions

We report an analysis, using the tools of nonlinear dynamics and chaos theory, of the fluctuations in the stress determined from simulations of shear flow of Stokesian suspensions. The simulations are for shear between plane parallel walls of a suspension of rigid identical spheres in a Newtonian fluid, over a range of particle concentration. By analyzing the time series of the stress, we find that the dynamics underlying these fluctuations is deterministic, low-dimensional, and chaotic. We use the dynamic and metric invariants of the underlying dynamics as a means of characterizing suspension behavior. The dimension of the chaotic attractor increases with particle concentration, indicating the increasing influence of multiple-body interactions on the rheology of the suspension with rise in particle concentration. We use our analysis to make accurate predictions of the short-term evolution of a stress component from its preceding time series, and predict the evolution of one component of the stress using the time series of another. We comment on the physical origin of the chaotic stress fluctuations, and on the implications of our results on the relation between the microstructure and the stress.

False neighbors and false strands: a reliable minimum embedding dimension algorithm

The time-delay reconstruction of the state space of a system from observed scalar data requires a time lag and an integer embedding dimension. We demonstrate a reliable method to estimate the minimum necessary embedding dimension that improves upon previous methods by correcting for systematic effects due to temporal oversampling, autocorrelation, and changing time lag. The method gives a sharp and reliable indication of the proper dimension. With little computational cost, the method also distinguish easily between infinite-dimensional colored noise-including noisy periodicity-and low-dimensional dynamics.

Topographic organization of nonlinear interdependence in multichannel human EEG

This paper investigates the spatial organization of nonlinear interactions between different brain regions in healthy human subjects. This is achieved by studying the topography of nonlinear interdependence in multichannel EEG data, acquired from 40 healthy human subjects at rest. An algorithm for the detection and quantification of nonlinear interdependence is applied to four pairs of bipolar electrode derivations to detect posterior and anterior interhemispheric and left and right intrahemispheric interdependences. Multivariate surrogate data sets are constructed to control for linear coherence and finite sample size. Nonlinear interdependence is shown to occur in a small but statistically robust number of epochs. The occurrence of nonlinear interdependence in any region is correlated with the concurrent presence of nonlinear interdependence in other regions at high levels of significance. The strength, direction and topography of the interdependences are also correlated. For example, posterior interhemispheric interdependence from right-to-left is strongly correlated with right intrahemispheric interdependence from back-to-front. There is a subtle change in these correlations when subjects open their eyes. These results suggest that nonlinear interdependence in the human brain has a specific topographic organization which reflects simple cognitive changes. It sometimes occurs as an isolated phenomenon between two brain regions, but often involves concurrent interdependences between multiple brain regions.

Evidence of state-dependent interhemispheric relationships in lizard EEG during the awake state

The electroencephalogram (EEG) from both hemispheres of eight Canary lizards Gallotia galloti was registered at 25 degrees C and 35 degrees C during awake state with open eyes (OE) and closed eyes. The possible interdependence between the recorded EEGs was assessed by means of newly developed methods of multivariate nonlinear time-series analysis. The interdependence turned out to be significant and greater for the OE state at both temperatures, although it was of linear type in most of the cases. We conclude that the methods can be successfully applied to study the interdependence between noisy, low-amplitude EEGs.

Detection and description of non-linear interdependence in normal multichannel human EEG data

OBJECTIVES

This study examines human scalp electroencephalographic (EEG) data for evidence of non-linear interdependence between posterior channels. The spectral and phase properties of those epochs of EEG exhibiting non-linear interdependence are studied.

METHODS

Scalp EEG data was collected from 40 healthy subjects. A technique for the detection of non-linear interdependence was applied to 2.048 s segments of posterior bipolar electrode data. Amplitude-adjusted phase-randomized surrogate data was used to statistically determine which EEG epochs exhibited non-linear interdependence.

RESULTS

Statistically significant evidence of non-linear interactions were evident in 2.9% (eyes open) to 4.8% (eyes closed) of the epochs. In the eyes-open recordings, these epochs exhibited a peak in the spectral and cross-spectral density functions at about 10 Hz. Two types of EEG epochs are evident in the eyes-closed recordings; one type exhibits a peak in the spectral density and cross-spectrum at 8 Hz. The other type has increased spectral and cross-spectral power across faster frequencies. Epochs identified as exhibiting non-linear interdependence display a tendency towards phase interdependencies across and between a broad range of frequencies.

CONCLUSIONS

Non-linear interdependence is detectable in a small number of multichannel EEG epochs, and makes a contribution to the alpha rhythm. Non-linear interdependence produces spatially distributed activity that exhibits phase synchronization between oscillations present at different frequencies. The possible physiological significance of these findings are discussed with reference to the dynamical properties of neural systems and the role of synchronous activity in the neocortex.

Markov models from data by simple nonlinear time series predictors in delay embedding spaces

We analyze prediction schemes for stochastic time series data. We propose that under certain conditions, a scalar time series, obtained from a vector-valued Markov process can be modeled as a finite memory Markov process in the observable. The transition rules of the process are easily computed using simple nonlinear time series predictors originally proposed for deterministic chaotic signals. The optimal time lag entering the embedding procedure is shown to be significantly smaller than the deterministic case. The concept is illustrated for simulated data and for surface wind velocity data, for which the deterministic part of the dynamics is shown to be nonlinear.

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.

Reconstructing embedding spaces of coupled dynamical systems from multivariate data

A method for reconstructing dimensions of subspaces for weakly coupled dynamical systems is offered. The tool is able to extrapolate the subspace dimensions from the zero coupling limit, where the division of dimensions as per the algorithm is exact. Implementation of the proposed technique to multivariate data demonstrates its effectiveness in disentangling subspace dimensionalities also in the case of emergent synchronized motions, for both numerical and experimental systems.

Lyapunov spectrum from time series using moving boxes

We present a very fast algorithm (few seconds) for estimating full Lyapunov spectrum from time series. The method requires a smaller number of parameters than other time-average algorithms, and is tested for data with different numerical precision, sampling frequencies, total sampling times, and presence of noise. We report conclusive results for the electron density broadband fluctuations of a plasma at the edge of tokamak TBR-1.

Non-linear EEG measures during sleep: effects of the different sleep stages and cyclic alternating pattern

The objective of this work was to study the non-linear aspects of sleep EEG, taking into account the different sleep stages and the peculiar organization of its phasic events in ordered sequences (CAP) by applying a series of new non-linear measures (non-linear cross prediction or NLCP), which appear more reliable for the detection and characterization of non-linear structures in experimental data than the commonly used correlation dimension. Eight healthy subjects aged 18-20 years participated in this study. Polysomnography was performed in all subjects; signals were sampled at 128 Hz and stored on hard disk. The C3 or C4 derivation was used for all the subsequent computational steps, which were performed on EEG epochs (4096 data points) selected from sleep stage 2 (S2) and slow-wave sleep (SWS), in both CAP and non-CAP (NCAP) conditions. Also, epochs from sleep stage 1 (S1), REM and wakefulness preceding sleep were recorded. The dynamic properties of the EEG were assessed by means of the non-linear cross-prediction test, which uses three different 'model' time series in order to predict non-linearly the original data set (Pred, Ama, and Tir). Pred is a measure of the predictability of the time series, and Ama and Tir are measures of asymmetry, indicating non-linear structure. The non-linear measures applied in this study indicate that sleep EEG tends to show non-linear structure only during CAP periods, both during S2 and SWS. Moreover, during CAP periods, non-linearity can only be detected during the phase A1 subtypes (and partially A2) of CAP. The A3 phases show characteristics of non-stationarity and bear some resemblance to wakefulness. Based on the results of this study, sleep might be considered as a dynamically evolving sequence of different states of the EEG, which we could track by detecting non-linearity, mostly in association with CAP. Our results clearly show that detectable non-linearity in the EEG is closely related to the occurrence of the phase A of CAP.

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.

Changing complexity in human behavior and physiology through aging and disease

Lipsitz and Goldberger proposed that there is a loss in the complexity of physiological and behavioral systems with aging and disease. Here, we show that this unidirectional view of the change in system complexity is too narrow in its consideration of the actual changes that occur with aging and disease. An increase or decrease in the complexity of a behavioral or physiological system output can occur and the direction of change is dependent on the confluence of constraints that channel the system dynamics. It is postulated that the observed increase or decrease in complexity with aging and disease is dependent on the nature of both the intrinsic dynamics of the system and the short-term change required to realize a local task demand.

Detection of hemodynamic turbulence in experimental stenosis: an in vivo study in the rat carotid artery

In a previous study, we have demonstrated that turbulences in arteries less than 1.5 mm in diameter perfused ex vivo by an oscillated flow can be determined using nonlinear mathematical models. The hypothesis tested here was that nonlinear analyses enable the in vivo detection of turbulences in blood flow in vessels with dimensions affected by microsurgery. Twenty Wistar male rats were studied. After an intraperitoneal anesthesia (50 mg/kg sodium pentobarbital), each right carotid artery was dissected and a transit time flowmeter was used to measure blood flow. Arteries were studied in control conditions and graded stenoses, which were performed by a collar system and progressively increased from 0 to 95%. For each flow signal, time delay, false-nearest neighbors, correlation dimension and the largest Lyapunov exponent were computed to characterize the level of turbulence. The level of turbulence was highly correlated with the degree of stenosis induced (p < 0.001). The correlation dimensions of all the flow signals varied between 3 and 5, thus implying that more than three independent noninteger control variables were necessary to account for the complexity of rat carotid artery dynamics. Turbulence flow significantly increased in arteries with 40-95% stenosis. Nonlinear analysis may be useful to determine the turbulence level of an in vivo flow in arteries less than 1.5 mm in diameter and might be clinical useful for turbulence detection after microsurgery.

Surrogate analysis for detecting nonlinear dynamics in normal vowels

Normal vowels are known to have irregularities in the pitch-to-pitch variation which is quite important for speech signals to be perceived as natural human sound. Such pitch-to-pitch variation of vowels is studied in the light of nonlinear dynamics. For the analysis, five normal vowels recorded from three male and two female subjects are exploited, where the vowel signals are shown to have normal levels of the pitch-to-pitch variation. First, by the false nearest-neighbor analysis, nonlinear dynamics of the vowels are shown to be well analyzed by using a relatively low-dimensional reconstructing dimension of 4 < or = d < or = 7. Then, we further studied nonlinear dynamics of the vowels by spike-and-wave surrogate analysis. The results imply that there exists nonlinear dynamical correlation between one pitch-waveform pattern to another in the vowel signals. On the basis of the analysis results, applicability of the nonlinear prediction technique to vowel synthesis is discussed.

Reconstruction of chaotic dynamics by on-line EM algorithm

In this paper, we discuss the reconstruction of chaotic dynamics by using a normalized Gaussian network (NGnet), which is a network of local linear regression units. The NGnet is trained by an on-line EM algorithm in order to learn the vector field of the chaotic dynamics. We investigate the robustness of our approach to two kinds of noise processes: system noise and observation noise. System noise disturbs the system dynamics itself. Observation noise is added when the state variables are observed. It is shown that the trained NGnet is able to reproduce a chaotic attractor, which approximates the complexity and instability of the original chaotic attractor well, even under various noise conditions. The trained NGnet also shows good prediction performance. When only part of the dynamical variables are observed, the delay embedding method is used. The NGnet is then trained to learn the vector field in the delay coordinate space. Experiments show that the chaotic dynamics embedded in the delay coordinate space is able to be learned under the two types of noise.

Interdependencies in the spontaneous EEG while listening to music

We studied the patterns of interdependency between different brain regions during the performance of higher cognitive functions. Our goal was to check the existence in these patterns of both task-related differences (e.g. listening to music vs. rest) and training-related differences (musicians vs. non-musicians). For this purpose, a non-linear measure, called similarity index (S.I.), was used to detect asymmetric interdependencies between different brain regions by means of EEG signals. Relatively active and passive regions of the brain were found where the degree of activity was represented by excited degrees of freedom. The S.I. obtained during listening to different kinds of music was compared statistically with the S.I. with eyes closed, and significant changes (P< or = 0.05) were entered into schematic brain maps. A topographical representation of the S.I. yielded differences in the interdependency while performing different cognitive tasks. The results demonstrate the occurrence of task-related differences in both groups of subjects. Furthermore, subjects with musical training possessed significantly higher degrees of interdependencies than such without musical training while listening to music but not to text. We conclude that the new measure can be successfully applied for studying the dynamical co-operation between cortical areas during higher cognitive functioning.

Embedding of multidimensional time-dependent observations

A method is proposed to reconstruct dynamic attractors by embedding of multivariate observations of dynamic nonlinear processes. The Takens embedding theory is combined with independent component analysis to transform the embedding into a vector space of linearly independent vectors (phase variables). The method is successfully tested against prediction of the unembedded state vector in two case studies of simulated chaotic processes.

Apamin-induced irregular firing in vitro and irregular single-spike firing observed in vivo in dopamine neurons is chaotic

Dopamine neurons of the substantia nigra often fire action potentials irregularly in vivo, but in vitro fire in a regular, pacemaker-like firing pattern. Bath application of apamin, a blocker of calcium-activated potassium channels, can shift a dopamine neuron from pacemaker-like to irregular firing. To determine whether the irregular firing was caused by intrinsic cellular mechanisms rather than random synaptic input or some other form of noise, spike density functions of interspike interval records were analyzed using non-linear forecasting methods to quantify any non-linear (non-periodic) structure. Intrinsic cellular mechanisms are capable of producing chaotic firing, which is deterministic, non-linear, and loses predictability exponentially with increasing forecast time.To determine whether forecasting spike density functions could reliably measure predictability, forecasting was first applied to spike density functions produced by computer simulations of pacemaker-like, chaotic, and random firing, as well as pacemaker-like and chaotic firing that were randomly synaptically driven. Exponential loss of predictability was successfully detected in both chaotic and randomly driven chaotic firing. Predictability scaled faster than exponentially for random spiking, and linearly (slower than exponentially) for randomly driven pacemaker firing. The method was then applied to experimental records of apamin-induced irregular firing of rat dopaminergic neurons of the substantia nigra in vitro and in vivo. Exponential loss of predictability was detected in both cases, consistent with chaotic firing. Experimental records of pacemaker-like firing in vitro showed linear scaling, consistent with a randomly driven pacemaker. Several schemes for neural encoding of synaptic inputs have been suggested, such as rate codes or temporal codes. However, our results suggest that under some conditions, the irregular firing of dopamine neurons does not reflect the random temporal dynamics of its inputs, but rather the intrinsic, deterministic dynamics of dopamine cells themselves, under the tonic neuromodulatory influence of apamin in vitro and possibly that of an unidentified endogenous modulatory substance in vivo.

Non-linear dynamic analysis of clozapine-induced electroencephalographic changes in schizophrenic patients--a preliminary study

1. In order to find the electroencephalographic (EEG) parameters that reflect the effect of clozapine in schizophrenic patients, the authors applied various non-linear analyses on multi-channel EEG data drawn from patients before and after a therapeutic trial of clozapine. 2. The correlation dimension was difficult to extract from our limited time series EEG data and the authors did not find a meaningful association with clozapine use. The primary Lyapunov exponent could be reliably calculated but also did not reflect the effect of clozapine. 3. However, the mutual cross-prediction (MCP) algorithm showed potentially meaningful results. The driving system was shifted to the frontal channels after a 4-week trial with clozapine. Moreover, MCP might have a value as a predictor of treatment response. 4. Although preliminary in nature, the MCP might have greater power for interpreting complex changes from channel to channel in EEG induced by clozapine.

Positive Lyapunov exponents calculated from time series of strange nonchaotic attractors

Time-series methods for estimating Lyapunov exponents may give a positive exponent when they are applied to the time series of strange nonchaotic systems. Strange nonchaotic systems are characterized by expanding and contracting regions in phase space that result in repeatedly expanding or contracting trajectories. Using time-series methods, the maximum time-series Lyapunov exponent is calculated as an average of the locally most expanding exponents that characterize the divergence of nearby trajectories following a reconstructed attractor over time. A positive exponent is reported by time-series methods for trajectories in an expanding region. While in a converging region, the most expanding dynamics are related to the quasiperiodic driving force. Statistically, a zero exponent related to the quasiperiodic force is obtained through time-series methods within converging regions. As a result, the calculated maximum Lyapunov exponent is positive.

Spatiotemporal nonlinear dynamics of a magnetoelastic ribbon

Magnetoelastic materials have a strong coupling between strain and magnetization, so applying a magnetic field to a magnetoelastic material can change its shape. This coupling leads to interesting dynamics. We have studied the dynamics of a wide ribbon of Metglass 2605sc which was driven by a magnetic field. The ribbon was suspended as a pendulum in a set of Helmholtz coils, which provided both dc and ac magnetic fields. Laser light was reflected off the ribbon to measure its angular displacement. Two points on the ribbon could be simultaneously illuminated, and one of the laser beams could be scanned over the ribbon. We observed quasiperiodic bifurcations in the motion of the ribbon, and characterized the spatial aspect of the motion with some recently developed statistics.

Attractor reconstruction for non-linear systems: a methodological note

Attractor reconstruction is an important step in the process of making predictions for non-linear time-series and in the computation of certain invariant quantities used to characterize the dynamics of such series. The utility of computed predictions and invariant quantities is dependent on the accuracy of attractor reconstruction, which in turn is determined by the methods used in the reconstruction process. This paper suggests methods by which the delay and embedding dimension may be selected for a typical delay coordinate reconstruction. A comparison is drawn between the use of the autocorrelation function and mutual information in quantifying the delay. In addition, a false nearest neighbor (FNN) approach is used in minimizing the number of delay vectors needed. Results highlight the need for an accurate reconstruction in the computation of the Lyapunov spectrum and in prediction algorithms.

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.

Prediction of chaotic dynamics in sheared liquid crystalline polymers

A rheological model for rodlike polymers in the nematic liquid-crystalline phase is analyzed to characterize irregular dynamical response under pure shear flows. The model is studied with a continuation approach, and a period doubling scenario is detected. Time series generated via simulation are studied with nonlinear analysis tools to prove the existence of chaotic regimes.

Exercise induced muscle damage and recovery assessed by means of linear and non-linear sEMG analysis and ultrasonography

This study was aimed at investigating the time-course and recovery from eccentric (EC) exercise induced muscle damage by means of surface electromyography (sEMG), ultrasonography (US), and blood enzymes. Five subjects (EC Group) performed two bouts of 35 EC maximum contractions with the biceps brachii of their non dominant arm, five subjects were tested without performing EC (Control Group: CNT). The maximal isometric force (MVC) was measured. Force and sEMG signals were recorded during 80% MVC isometric contractions. In EC and CNT subjects US assessment on non-dominant biceps brachii was performed; creatin kinase (CK) and lactic dehydrogenasis (LDH) plasma levels were also assessed. Force, sEMG and CK-LDH measurements were performed before EC and after it periodically for 4 weeks. The sEMG was analysed in time and frequency domains; a non-linear analysis (Lyapunov 1st exponent, L1) of sEMG was also performed. After EC, the MVC was reduced by 40% on average with respect to the pre-EC values. A significant decrease in the initial frequency content, and in the MDF and L1 decay (13-42% less than the pre-EC values, respectively) was also observed. The sEMG amplitude (Root Mean Square, RMS) was unchanged after EC. The US revealed an increase in muscle belly thickness and in local muscle blood flow after EC. A complete recovery of all the considered parameters was achieved in two weeks. In conclusion sEMG analysis was confirmed as an early indicator of muscle damage. Muscle recovery from damage is followed by both sEMG and US and this may have useful clinical implications. Non linear analysis (L1) was revealed to be sensitive to early sEMG modifications induced by EC as well as able to follow the post EC changes in the sEMG.

The nonlinear dynamical analysis of the EEG in schizophrenia with temporal and spatial embedding dimension

We applied nonlinear dynamics theory to EEG analysis of schizophrenic patients and estimated the correlation dimension with both temporal embedding and spatial embedding. A higher D2 was found when using a time-delay embedding method. Especially at F7 and Fp1, a significant increase showed. We concluded that more complex activity occurred in certain lobes of schizophrenic patients. Using the spatial embedding method, a relative lower global correlation dimension was obtained. This shows that there might be a diffuse slow wave activity through a schizophrene's global cerebrum. Finally, we discuss the study from three angles of clinical semiology, spectrum analysis and neuropsychology and draw some conclusions about the relationship between the nonlinear analysis of schizophrenia EEG and clinical research. It seems that the theory of a nonlinear dynamics system is a powerful tool for EEG research and may prove useful in complementing visual analysis of EEG accompanied with other study means for brain electrical activity.

Nonlinear dynamic properties of low calcium-induced epileptiform activity

The analysis of the dynamic properties of epileptiform activity in vitro has led to a better understanding of the time course of neural synchronization and seizure states. Nonlinear analysis is thus potentially useful for the prediction of seizure onset. We have used nonlinear analysis methods to investigate the development of activity in the low calcium model of epilepsy in brain slices. This model is particularly interesting since neurons synchronize in the absence of synaptic transmission. The dynamic properties calculated from extracellular recordings of activity were used to analyze the transition to synchronous firing and their relation to neuronal excitability. The global embedding dimension, local dimension and the Lyapunov exponent were calculated from time segments corresponding to the onset, transition and fully developed stages of activity. The analysis was repeated for recordings made in the presence of various levels of DC electric fields to modulate neuronal excitability. The global and local dimensions did not change once activity was first initiated, even in the presence of the electric field. The maximum Lyapunov exponents increased during the onset of activity but decreased when the applied hyperpolarizing electric field was large enough to partially suppress the activity. These findings establish a relationship between neuronal excitability and the maximum Lyapunov exponent, and suggest that the Lyapunov exponent may be used to distinguish between various states of the neural network and might be important in seizure prediction and control.