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

Local dynamic stability versus kinematic variability of continuous overground and treadmill walking

This study quantified the relationships between local dynamic stabiliht and variabilitr during continuous overground and treadmill walking. Stride-to-stride standard deviations were computed from temporal and kinematic data. Marimum finite-time Lyapunov exponents were estimated to quantify local dynamic stability. Local stability of gait kinematics was shown to be achieved over multiple consecutive strides. Traditional measures of variability poorly predicted local stability. Treadmill walking was associated with significant changes in both variability and local stability. Thus, motorized treadmills may produce misleading or erroneous results in situations where changes in neuromuscular control are likely to affect the variability and/or stability of locomotion.

Modelling and prediction for chaotic fir laser attractor using rational function neural network

Many real-world systems such as irregular ECG signal, volatility of currency exchange rate and heated fluid reaction exhibit highly complex nonlinear characteristic known as chaos. These chaotic systems cannot be retreated satisfactorily using linear system theory due to its high dimensionality and irregularity. This research focuses on prediction and modelling of chaotic FIR (Far InfraRed) laser system for which the underlying equations are not given. This paper proposed a method for prediction and modelling a chaotic FIR laser time series using rational function neural network. Three network architectures, TDNN (Time Delayed Neural Network), RBF (radial basis function) network and the RF (rational function) network, are also presented. Comparisons between these networks performance show the improvements introduced by the RF network in terms of a decrement in network complexity and better ability of predictability.

Nonlinear dynamic analysis of the EEG in patients with Alzheimer's disease and vascular dementia

To assess nonlinear EEG activity in patients with Alzheimer's disease (AD) and vascular dementia (VaD), the authors estimated the correlation dimension (D2) and the first positive Lyapunov exponent (L1) of the EEGs in both patients and age-matched healthy control subjects. EEGs were recorded in 15 electrodes from 12 AD patients, 12 VaD patients, and 14 healthy subjects. The AD patients had significantly lower D2 values than the normal control subjects, (P < H > 0.05), except at the F7 and the O1 electrodes, and the VaD patients, except at the C3 and the C4 electrodes. The VaD patients had relatively increased values of D2 and L1 compared with the AD patients, and rather higher values of D2 than the normal control subjects at the F7, F4, F8, Fp2, O1, and O2 electrodes. The L1 values of the EEGs were also lower for the AD patients than for the normal control subjects, except in the O1 and the O2 channels, and for the VaD patients at all electrodes. The L1 values were higher for the VaD patients than for the normal control subjects (F3, F4, F8, O1, and O2). In addition, the authors detected that the VaD patients had an uneven distribution of D2 values over the regions than the AD patients and the normal control subjects, although the statistics do not confirm this. By contrast, AD patients had uniformly lower D2 values in most regions, indicating that AD patients have less complex temporal characteristics of the EEG in entire regions. These nonlinear analyses of the EEG may be helpful in understanding the nonlinear EEG activity in AD and VaD.

Experimental phase synchronization of a chaotic convective flow

We report experimental evidence of phase synchronization of high dimensional chaotic oscillators in a laboratory experiment. The experiment consists of a thermocapillary driven convective cell in a time dependent chaotic regime. The synchronized states emerge as a consequence of a localized temperature perturbation to the heater. The transition to phase synchronization is studied as a function of the external perturbations. The existence and stability conditions for this phenomenon are discussed.

Nonlinear time series analysis of normal and pathological human walking

Characterizing locomotor dynamics is essential for understanding the neuromuscular control of locomotion. In particular, quantifying dynamic stability during walking is important for assessing people who have a greater risk of falling. However, traditional biomechanical methods of defining stability have not quantified the resistance of the neuromuscular system to perturbations, suggesting that more precise definitions are required. For the present study, average maximum finite-time Lyapunov exponents were estimated to quantify the local dynamic stability of human walking kinematics. Local scaling exponents, defined as the local slopes of the correlation sum curves, were also calculated to quantify the local scaling structure of each embedded time series. Comparisons were made between overground and motorized treadmill walking in young healthy subjects and between diabetic neuropathic (NP) patients and healthy controls (CO) during overground walking. A modification of the method of surrogate data was developed to examine the stochastic nature of the fluctuations overlying the nominally periodic patterns in these data sets. Results demonstrated that having subjects walk on a motorized treadmill artificially stabilized their natural locomotor kinematics by small but statistically significant amounts. Furthermore, a paradox previously present in the biomechanical literature that resulted from mistakenly equating variability with dynamic stability was resolved. By slowing their self-selected walking speeds, NP patients adopted more locally stable gait patterns, even though they simultaneously exhibited greater kinematic variability than CO subjects. Additionally, the loss of peripheral sensation in NP patients was associated with statistically significant differences in the local scaling structure of their walking kinematics at those length scales where it was anticipated that sensory feedback would play the greatest role. Lastly, stride-to-stride fluctuations in the walking patterns of all three subject groups were clearly distinguishable from linearly autocorrelated Gaussian noise. As a collateral benefit of the methodological approach taken in this study, some of the first steps at characterizing the underlying structure of human locomotor dynamics have been taken. Implications for understanding the neuromuscular control of locomotion are discussed. (c) 2000 American Institute of Physics.

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.

Linear and nonlinear measures predict swimming in the leech

Stimulation of a trigger interneuron of an isolated nerve cord preparation of the medicinal leech, Hirudo medicinalis, sometimes leads to swimming; sometimes it does not. We investigate signals transmitted in the ventral cord of the leech after stimulation and seek quantitative measures that would make it possible to distinguish signals that predict swimming from those that do not. We find that a number of linear as well as nonlinear measures provide statistically significant distinctions between the two kinds of signals. The linear measures are the time dependence of (i) the standard deviation and (ii) the autocorrelation function at a small time delay. The nonlinear measures are (i) a measure of nonlinear predictability and (ii) the time dependence of a measure of the size of the embedded signal trajectory. Calculations using surrogate data suggest that the differences between the two classes of signals are dynamical as well as statistical.

Slower speeds in patients with diabetic neuropathy lead to improved local dynamic stability of continuous overground walking

Patients with diabetic peripheral neuropathy are significantly more likely to fall while walking than subjects with intact sensation. While it has been suggested that these patients walk slower to improve locomotor stability, slower speeds are also associated with increased locomotor variability, and increased variability has traditionally been equated with loss of stability. If the latter were true, this would suggest that slowing down, as a locomotor control strategy, should be completely antithetical to the goal of maintaining stability. The present study resolves these seemingly paradoxical findings by using methods from nonlinear time series analysis to directly quantify the sensitivity of the locomotor system to local perturbations that are manifested as natural kinematic variability. Fourteen patients with severe peripheral neuropathy and 12 gender-, age-, height-, and weight-matched non-diabetic controls participated. Sagittal plane angles of the right hip, knee, and ankle joints and tri-axial accelerations of the trunk were measured during 10 min of continuous overground walking at self-selected speeds. Maximum finite-time Lyapunov exponents were computed for each time series to quantify the local dynamic stability of these movements. Neuropathic patients exhibited slower walking speeds and better local dynamic stability of upper body movements in the horizontal plane than did control subjects. The differences in local dynamic stability were significantly predicted by differences in walking speed, but not by differences in sensory status. These results support the hypothesis that reductions in walking speed are a compensatory strategy used by neuropathic patients to maintain dynamic stability of the upper body during level walking.

Investigation of determinism in heart rate variability

The article searches for the possible presence of determinism in heart rate variability (HRV) signals by using a new approach based on NARMA (nonlinear autoregressive moving average) modeling and free-run prediction. Thirty-three 256-point HRV time series obtained from Wistar rats submitted to different autonomic blockade protocols are considered, and a collection of surrogate data sets are generated from each one of them. These surrogate sequences are assumed to be nondeterministic and therefore they may not be predictable. The original HRV time series and related surrogates are submitted to NARMA modeling and prediction. Special attention has been paid to the problem of stationarity. The results consistently show that the surrogate data sets cannot be predicted better than the trivial predictor-the mean-while most of the HRV control sequences are predictable to a certain degree. This suggests that the normal HRV signals have a deterministic signature. The HRV time series derived from the autonomic blockade segments of the experimental protocols do not show the same predictability performance, albeit the physiological interpretation is not obvious. These results have important implications to the methodology of HRV analysis, indicating that techniques from nonlinear dynamics and deterministic chaos may be applied to elicit more information about the autonomic modulation of the cardiovascular activity. (c) 2000 American Institute of Physics.

An estimation of the first positive Lyapunov exponent of the EEG in patients with schizophrenia

We studied the complexity of the electroencephalogram (EEG) in schizophrenic patients by estimating the first Lyapunov exponent (L1), which might serve as an indicator of the specific brain function in schizophrenia. We recorded the EEG from 25 schizophrenic patients (12 male, 13 female; age=25.1+/-7.0 years) fulfilling DSM-IV criteria and 15 healthy controls (9 male, 6 female; age=27. 8+/-4.2 years) at 16 electrodes, different from previous studies which recorded the EEGs at limited electrodes. We employed a method with an optimal embedding dimension to calculate the L1s. For limited noisy data, this algorithm was strikingly faster and more accurate than previous ones. Our results showed that the schizophrenic patients had lower values of the L1 at the left inferior frontal and anterior temporal regions compared with normal controls. These results for L1 in non-linear analysis have some differences from those for power ratios in linear analysis. These suggest that the non-linear analysis of the EEGs such as the estimation of the L1 might be a useful tool in analyzing EEG data to explore the neurodynamics of the brains of schizophrenic patients.

Noise-induced chaos in an optically injected semiconductor laser model

The chaos induced by an intrinsic spontaneous-emission noise in an optically injected semiconductor laser is investigated through a single-mode injection model. A method is developed to quantitatively study the scale-dependent noise effect in general, and the noise-induced chaotic feature in particular. We find that noise at an experimentally measured level can induce chaos in the system. This suggests that noise-induced chaos may indeed exist in real systems. Certain required characteristics for noise to induce chaos are identified: the periodic state itself, when subject to weak noise, should undergo a process that is much more diffusive than the Brownian motion, and the adjacent chaotic states should still behave chaotically on certain finite scales when subject to noise. We believe they are generic features for noise to induce chaos. The correlation dimension of the clean and noisy attractors is also calculated to study noise-induced changes in the geometrical structure of the attractors.

Chaos and experimental resolution

We study systematically what levels of finite-resolution in measurements significantly affect the calculation of time delays, embedding dimensions, and Lyapunov exponent for two well-known chaotic systems. We find a tradeoff between the information contained in the measured time series and what is lost. Moreover, the "noise" series is low-dimensional and highly correlated.

Phase transitions in the common brainstem and related systems investigated by nonstationary time series analysis

Neuronal activities of the reticular formation (RF) of the lower brainstem and the nucleus tractus solitarii (NTS, first relay station of baroreceptor afferents) were recorded together in the anesthized dog with related parameters of EEG, respiration and cardiovascular system. The RF neurons are part of the common brainstem system (CBS) which participates in regulation and coordination of cardiovascular, respiratory, somatomotor systems, and vigilance. Multiple time series of these physiological subsystems yield useful information about internal dynamic coordination of the organism. Essential problems are nonlinearity and instationarity of the signals, due to the dynamic complexity of the systems. Several time-resolving methods are presented to describe nonlinear dynamic couplings in the time course, particularly during phase transitions. The methods are applied to the recorded signals representing the complex couplings of the physiological subsystems. Phase transitions in these systems are detected by recurrence plots of the instationary signals. The pointwise transinformation and the pointwise conditional coupling divergence are measures of the mutual interaction of the subsystems in the state space. If the signals show marked rhythms, instantaneous frequencies and their shiftings are demonstrated by time frequency distributions, and instantaneous phase differences show couplings of oscillating subsystems. Transient signal components are reconstructed by wavelet packet time selective transient reconstruction. These methods are useful means for analyzing coupling characteristics of the complex physiological system, and detailed analyses of internal dynamic coordination of subsystems become possible. During phase transitions of the functional organization (a) the rhythms of the central neuronal activities and the peripheral systems are altered, (b) changes in the coupling between CBS neurons and cardiovascular signals, respiration and the EEG, and (c) between NTS neurons (influenced by baroreceptor afferents) and CBS neurons occur, and (d) the processing of baroreceptor input at the NTS neurons changes. The results of this complex analysis, which could not be done formerly in this manner, confirm and complete former investigations on the dynamic organization of the CBS with its changing relations to peripheral and other central nervous subsystems.

On the frequency and amplitude spectrum and the fluctuations at the output of a communication receiver

A mixer cascaded with a low-pass filter is the basic piece of any communication receiver. The frequency and amplitude fluctuations of beat signals recorded at the output of the mixer plus filter set-up are investigated experimentally and explained on the basis of number theory in relation to the Riemann problem concerning the distribution of prime numbers. The frequency of the beat signal is obtained from a diophantine approximation of the frequency ratio of the input oscillators. The amplitude is defined globally from the position of resolved fractions with respect to an equally spaced graduation. Time series analysis methods show that frequency fluctuations present a transition from white frequency noise to 1/f frequency noise close to resonance; the latter is compatible with an underlying fractal attractor. The same transition is observed in the case of time series computed from continued fraction expansions.

Improvement of the local prediction of chaotic time series

In this paper we explore the effect of pseudofalse neighbor points, which are true neighbor points in the reconstructed attractor, but which are considered not suitable to be used when local methods are adopted to predict the chaotic time series. In our approach, the epsilon(p) neighbor points are used to reduce the influence of the pseudofalse neighbor points, thereby improving the performance of the local prediction of the chaotic time series.

Improved false nearest neighbor method to detect determinism in time series data

The false nearest neighbor method introduced by Kennel et al. [Phys. Rev. A 45, 3403 (1992)] is revisited and modified in order to ensure a correct distinction between low-dimensional chaotic data and noise. Still, correlated noise processes can yield vanishing percentages of false nearest neighbors for rather low embedding dimensions and can be mistaken for deterministic signals. Therefore, the false nearest neighbors method has always to be combined with a surrogate data test.

Dynamics of the human alpha rhythm: evidence for non-linearity?

OBJECT

For a better understanding of the physiological mechanisms responsible for alpha rhythms it is important to know whether non-linear processes play a role in their generation. We used non-linear forecasting in combination with surrogate data testing to investigate the prevalence and nature of alpha rhythm non-linearity, based on EEG recordings from humans. We interpreted these findings using computer simulations of the alpha rhythm model of Lopes da Silva et al. (1974).

METHODS

EEGs were recorded at 02 and O1 in 60 healthy subjects (30 males; 30 females; age: 49.28 years; range 11-84) during a resting eyes-closed state. Four artefact-free epochs (2.5 s; sample frequency 200 Hz) from each subject were tested for non-linearity using a non-linear prediction statistic and phase-randomized surrogate data. A similar type of analysis was done on the output of the alpha model for different values of input.

RESULTS

In the 480 (60 subjects, 2 derivations, 4 blocks) epochs studied, the null hypothesis that the alpha rhythms can result from linearly filtered noise, could be rejected in 6 cases (1.25%). The alpha model showed a bifurcation from a point attractor to a limit cycle at an input pulse density of 615 pps. Non-linearity could only be detected in the model output close to and beyond this bifurcation point. The sources of the non-linearity are the sigmoidal relationships between average membrane potential and output pulse density of the various cells of the neuronal populations.

CONCLUSION

The alpha rhythm is a heterogeneous entity dynamically: 98.75% of the epochs (type I alpha) cannot be distinguished from filtered noise. Apparently, during these epochs the activity of the brain has such a high complexity that it cannot be distinguished from a random process. In 1.25% of the epochs (type II alpha) non-linearity was found which may be explained by dynamics in the vicinity of a bifurcation to a limit cycle. There is thus experimental evidence from the point of view of dynamics for the existence of the two types of alpha rhythm and the bifurcation predicted by the model.

Phase space reconstruction using input-output time series data

In this paper we suggest that an extension of a procedure recently proposed by Wayland et al. [Phys. Rev. Lett. 70, 580 (1993)] for recognizing determinism in an autonomous time series can also be used as a diagnostic for determining an appropriate embedding dimension for driven ("input-output") systems. We compare the results of this extension to the results produced by the extensions to the method of false nearest neighbors put forward by Rhodes and Morari [Proceedings of the American Control Conference, Seattle, edited by The American Automatic Control Council (IEEE, Piscataway, 1995)] and the method of averaged false nearest neighbors by Cao et al. [Int. J. Bifurcation Chaos 8, 1491 (1998)].

Dynamics underlying rhythmic and non-rhythmic variants of abnormal, waking delta activity

OBJECT

We applied a new test, nonlinear cross prediction (NLCP), to investigate whether or polymorphic delta activity (PDA) and frontal intermittent rhythmic delta activity (FIRDA) reflect linear or nonlinear brain dynamics. Furthermore realistic models were constructed to explain the dynamical properties of PDA and FIRDA.

METHODS

Forty-nine EEG time series with FIRDA and 40 time series with PDA were studied with the NLCP algorithm. This characterizes a time series in terms of its predictability, amplitude asymmetry, and time asymmetry, with the latter two measures reflecting nonlinearity. Parameters of an EEG model proposed by Lopes da Silva were adjusted to obtain time series resembling PDA and FIRDA.

RESULTS

FIRDA was more predictable than PDA. Most PDA segments could not be distinguished from linearly filtered noise. In contrast, FIRDA activity showed strong evidence of nonlinear dynamics. These dynamical properties of PDA and FIRDA could be reproduced by the Lopes da Silva model. PDA and FIRDA reflect a point attractor and a limit cycle attractor, respectively, perturbed by dynamical noise.

CONCLUSION

Experimental analysis and modeling of the data suggest that PDA and FIRDA reflect fundamentally different types of brain dynamics. While PDA is filtered noise, reflecting low-level, random input to cortical networks, FIRDA may reflect limit-cycle oscillations due to increased excitation.

Nonlinear dynamics of giant resonances in atomic nuclei

The dynamics of monopole giant resonances in nuclei is analyzed in the time-dependent relativistic mean-field model. The phase spaces of isoscalar and isovector collective oscillations are reconstructed from the time series of dynamical variables that characterize the proton and neutron density distributions. The analysis of the resulting recurrence plots and correlation dimensions indicates regular motion for the isoscalar mode, and chaotic dynamics for the isovector oscillations. Information-theoretic functionals identify and quantify the nonlinear dynamics of giant resonances in quantum systems that have spatial as well as temporal structure.

Extracting messages masked by chaotic signals of time-delay systems

We show how to extract messages masked by a chaotic signal of a time-delay system with very high dimensions and many positive Lyapunov exponents. Using a special embedding coordinate, the infinite-dimensional phase space of the time-delay system is projected onto a special three-dimensional space, which enables us to identify the time delay of the system from the transmitted signal and reconstruct the chaotic dynamics to unmask the hidden message successfully. The message extraction procedure is illustrated by simulations with the Mackey-Glass time-delay system for two types of masking schemes and different kinds of messages.

Improving the false nearest neighbors method with graphical analysis

We introduce a graphical presentation for the false nearest neighbors (FNN) method. In the original method only the percentage of false neighbors is computed without regard to the distribution of neighboring points in the time-delay coordinates. With this presentation it is much easier to distinguish deterministic chaos from noise. The graphical approach also serves as a tool to determine better conditions for detecting low-dimensional chaos, and to get a better understanding on the applicability of the FNN method.

Test for low-dimensional determinism in electroencephalograms

We tested low-dimensional determinism in an electroencephalogram (EEG), based on the fact that smoothness (continuity) on an embedded phase space is enough to imply determinism within time series. A modified version of the method developed by Salvino and Cawley [Phys. Rev. Lett. 73, 1091 (1994)] was used. In our method, we chose a box randomly and then estimated the mean directional element in the box containing the d+1 data points, where d is the embedding dimension. The global average for the mean local directional elements over the boxes, W, is a measure for smoothness. The nonlinear noise reduction method developed by Sauer [Physica D 58, 193 (1992)] is then applied to the EEG. We also compared the results for the EEG with those for its surrogate data. We found that the W values for the noise-reduced EEG had stable values around 0.35, which means that the EEG is not a low-dimensional deterministic signal. However, this method may not be applicable to the time series generated from high-dimensional deterministic systems. We cannot exclude the possibility that the determinism in the EEG may be too high-dimensional to be detected with current methods.

Practical implementation of nonlinear time series methods: The TISEAN package

We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature. (c) 1999 American Institute of Physics.

Hyperchaotic qualities of the ball motion in a ball milling device

Ball collisions in milling devices are governed by complex dynamics ruled by impredictable impulsive forces. In this paper, nonlinear dynamics techniques are employed to analyze the time series describing the trajectory of a milling ball in an empty container obtained from a numerical model. The attractor underlying the system dynamics was reconstructed by the time delay method. In order to characterize the system dynamics the calculation of the spectrum of Lyapunov exponents was performed. Six Lyapunov exponents, divided into two terns with opposite sign, were obtained. The detection of the positive tern demonstrates the occurrence of the hyperchaotic qualities of the ball motion. A fractal Lyapunov dimension, equal to 5.62, was also obtained confirming the strange features of the attractor. (c) 1999 American Institute of Physics.

Global and local dimensions of vocal dynamics

The global embedding dimension (dE) and the local dynamical dimension (dL) are calculated from the microphone and electroglottographic (EGG) signals elicited from five healthy subjects and seven dysphonic subjects with laryngeal pathology during phonation of sustained/a/. The data from each pathologic subject contain at least one bifurcation and are divided into periodic and irregular segments for analysis. The dE values from both the microphone and EGG signals elicited from the healthy subjects indicate that a relatively small coordinate space can be used to reconstruct the attractor, with little residual noise. Consistent across all healthy subjects, three dominant degrees of freedom (dL) are found to govern local dynamics of the trajectories on the attractor. From the pathologic subjects, many of the dE values suggest the presence of a high-dimensional component in the signals. However, the noise does not completely obscure the deterministic dynamics of the source signal or prevent extraction of an optimal global embedding dimension. The data do not reveal consistent differences in degrees of freedom between healthy and pathologic phonation, or between different modes of pathologic phonation. However, the dL values suggest that the pathologic vocal fold vibration of these subjects, even highly irregular vibration, is governed locally by a low number of dominant degrees of freedom, sometimes no greater than those calculated from the signals of healthy subjects. Only in the cases of severe breathiness are the microphone and EGG signals sufficiently contaminated by noise to obscure any deterministic component.

Recurrence plots of experimental data: To embed or not to embed?

A recurrence plot is a visualization tool for analyzing experimental data. These plots often reveal correlations in the data that are not easily detected in the original time series. Existing recurrence plot analysis techniques, which are primarily application oriented and completely quantitative, require that the time-series data first be embedded in a high-dimensional space, where the embedding dimension d(E) is dictated by the dimension d of the data set, with d(E)>/=2d+1. One such set of recurrence plot analysis tools, recurrence quantification analysis, is particularly useful in finding locations in the data where the underlying dynamics change. We have found that for certain low-dimensional systems the same results can be obtained with no embedding. (c) 1998 American Institute of Physics.

Nonlinear analysis of the EEG of schizophrenics with optimal embedding dimension

We estimated the correlation dimensions of EEGs in patients with schizophrenia to investigate the dynamical properties underlying the EEG. We employed a new method, proposed by Kennel et al. (Kennel MB, Brown R, Abarbanel HDI. Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys Rev A 1992;45:3403-11), to calculate the correlation dimension D2. That method determined the proper minimum embedding dimension by looking at the behaviour of nearest neighbours under a change in the embedding dimension d from d to d + 1. We demonstrated that for limited noisy data, our algorithm was strikingly faster and more accurate than previous ones. We estimated the D2 of EEGs from 16 channels in patients with schizophrenia according to DSM-IV whereas previous studies, which estimated chaoticity of EEG in schizophrenia, recorded EEG only in a limited number of channels. Schizophrenic patients had a lower correlation dimension in the left inferior frontal and anterior temporal regions compared with controls. Our finding of decreased left frontal and temporal chaotic activity in schizophrenics is in line with the findings of a hypofrontality and hypotemporality reported in previous clinical studies such as EEG, blood flow, brain MRI and positron emission tomography studies in schizophrenia. This result suggests that chaos analysis may be a useful tool in analysing EEG data to explore the brain mechanism of schizophrenia.

Quantification of sympathetic and parasympathetic tones by nonlinear indexes in normotensive rats

Because the use of spectral powers of blood pressure (BP) and R-R interval (RR) in the low (LF) and high frequencies (HF) to quantify sympathetic and parasympathetic activities is still under debate, we questioned whether nonlinear methods may give better results. The BP signal was recorded for 30 min before and after intravenous injection of hexamethonium (20 mg/kg), atropine (0.5 mg/kg), atenolol (1 mg/kg), and prazosin (1 mg/kg) in conscious, normotensive Wistar-Kyoto rats. Three nonlinear indexes [percentage of recurrence, percentage of determinism, and length index (Lmax)] extracted from the recurrence plot method were used to analyze the BP signal. Sympathetic but not parasympathetic blockade reduced BP level and its LF component. RR increased and decreased after beta- and alpha-blockades, respectively. Hexamethonium increased HF, and atropine reduced LF, of RR. Sympathetic blockade and, in particular, alpha-sympathetic blockade increased nonlinear indexes of BP. In contrast, parasympathetic blockade by atropine increased nonlinear indexes of RR. These results suggest that, compared with spectral indexes, nonlinear indexes may be more specific markers of sympathetic and parasympathetic tones.

Preliminary report of detecting microembolic signals in transcranial Doppler time series with nonlinear forecasting

BACKGROUND AND PURPOSE

Most algorithms used for automatic detection of microembolic signals (MES) are based on power spectral analysis of the Doppler shift. However, controversies exist as to whether these algorithms can replace the human expert. Therefore, a different algorithm was applied that takes advantage of the periodicity of the MES. This so-called nonlinear forecasting (NLF) is able to detect periodicity in a time series, and it is hypothesized that this technique has the potential to detect MES. Moreover, because of the lack of prominent periodicity in both the normal Doppler signals (DS) and movement artifacts (MA), the NLF has a potential to differentiate MES from normal blood flow variations and MA.

METHODS

Twenty single MES and 100 MA were selected by 2 human experts. NLF was applied to MES and MA and compared with 200 randomly chosen DS. NLF resulted in a so-called prediction value that ranges from + 1 in signals with prominent periodicity to 0 in signals that lack periodicity.

RESULTS

NLF revealed that MES are more predictable than the normal Doppler signals (prediction [MES]=0.829+/-0.084 versus prediction [DS]= -0.060+/-0.228; P<0.0001). Moreover, MES are more predictable than the MA (prediction [MA]=-0.034+/-0.223; P<0.0001). No difference in prediction could be found between DS and MA.

CONCLUSIONS

This preliminary report shows that MES can be separated from DS and MA by NLF. Research is needed as to whether this technology can be further developed for automatic detection of MES.

Chaos suppression in gas-solid fluidization

Fluidization in granular materials occurs primarily as a result of a dynamic balance between gravitational forces and forces resulting from the flow of a fluid through a bed of discrete particles. For systems where the fluidizing medium and the particles have significantly different densities, density wave instabilities create local pockets of very high void fraction termed bubbles. The fluidization regime is termed the bubbling regime. Such a system is appropriately termed a self-excited nonlinear system. The present study examines chaos suppression resulting from an opposing oscillatory flow in gas-solid fluidization. Time series data representing local, instantaneous pressure were acquired at the surface of a horizontal cylinder submerged in a bubbling fluidized bed. The particles had a weight mean diameter of 345 &mgr;m and a narrow size distribution. The state of fluidization corresponded to the bubbling regime and total air flow rates employed in the present study ranged from 10% to 40% greater than that required for minimum fluidization. The behavior of time-varying local pressure in fluidized beds in the absence of a secondary flow is consistent with deterministic chaos. Kolmogorov entropy estimates from local, instantaneous pressure suggest that the degree of chaotic behavior can be substantially suppressed by the presence of an opposing, oscillatory secondary flow. Pressure signals clearly show a "phase-locking" phenomenon coincident with the imposed frequency. In the present study, the greatest degree of suppression occurred for operating conditions with low primary and secondary flow rates, and a secondary flow oscillation frequency of 15 Hz. (c) 1998 American Institute of Physics.

Use of nonlinear methods to assess effects of clonidine on blood pressure in spontaneously hypertensive rats

In spontaneously hypertensive rats (SHR), chronic infusion of clonidine failed to decrease blood pressure and blood pressure variability. We used nonlinear methods to get a deeper insight on the effects of clonidine on blood pressure dynamics. For 24 h and 4 wk, clonidine (0.1 mg . kg-1 . day-1 sc) was infused by minipumps in the conscious SHRs, and, for comparison, a vehicle was infused in SHRs and in Wistar-Kyoto rats. Blood pressure was recorded for 30 min before and after treatments. We used the Lyapunov exponent, approximated by the inverse of the lmax index derived from the recurrence plot method, to characterize nonlinear dynamics. Before treatment, lmax index of blood pressure was lower (P < 0.01) in the SHRs than in the Wistar-Kyoto rats. Clonidine significantly increased lmax (P < 0.01) to the level observed in normotensive rats, at 24 h and up to 4 wk after infusion. We conclude that clonidine has a significant chronic effect on blood pressure dynamics, as evidenced by nonlinear methods. Our study also suggests that the mechanisms governing blood pressure variations are nonlinear.

Nonlinear dynamical analysis of periodic lateralized epileptiform discharges

Nonlinear time series analysis can be used to investigate the dynamics underlying the generation of EEG signal. In the present study we used this approach to study the pathophysiology of PLEDs. We calculated the correlation dimension D2 of an EEG with typical PLEDs, and compared the results with those obtained for surrogate data. These surrogate data have the same power spectrum and amplitude distribution as the original EEG data, but are otherwise random. By construction, such surrogate data can be described by a linear model. Our results showed that D2 estimations for PLEDs were low, on the order of one, and that the results for EEG and the surrogate data were clearly different, indicating that the EEG with PLEDs reflects nonlinear dynamics of the underlying neural networks.

Non-linear dynamical analysis of the EEG in Alzheimer's disease with optimal embedding dimension

We used non-linear analysis to investigate the dynamical properties underlying the EEG in patients with Alzheimer's disease. We calculated the correlation dimension D2 and the first positive Lyapunov exponent L1. We employed a new method, which was proposed by Kennel et al., to calculate the non-linear invariant measures. That method determined the proper minimum embedding dimension by looking at the behavior of nearest neighbors under a change in the embedding dimension d from d to d + 1. We demonstrated that for limited noisy data, our algorithm was strikingly faster and more accurate than previous ones. Also, we found that, in almost all channels, patients with Alzheimer's disease had significantly lower D2 and L1 values than those for age-approximated healthy controls. These results suggest that brains afflicted by Alzheimer's disease show behaviors which are less chaotic than those of normal healthy brains. In this paper, we show that non-linear analysis can provide a fruitful tool for detecting relative changes, which cannot be detected by conventional linear analysis, in the complexity of brain dynamics. We propose that non-linear dynamical analyses of the EEGs from patients with Alzheimer's disease will be a diagnostic modality in the appropriate clinical setting.

Chaotic dynamics of sea clutter

The notion that a deterministic nonlinear dynamical system (with relatively few degrees of freedom) can display aperiodic behavior has a strong bearing on sea clutter characterization: random-looking sea clutter may be the outcome of a chaotic process. This new approach envisages deterministic rules for the underlying sea clutter dynamics, in contrast to the stochastic approach where sea clutter is viewed as a random process with a large number of degrees of freedom. In this paper, we demonstrate, convincingly for the first time, the chaotic dynamics of sea clutter. We say so on the basis of results obtained using radar data collected from a series of extensive and thorough experiments, which have been carried out with ground-truthed sea clutter data sets at three different sites. The study includes correlation dimension analysis (based on the maximum likelihood principle) and Lyapunov spectrum analysis. The Lyapunov (Kaplan-Yorke) dimension, which is a byproduct of Lyapunov spectrum analysis, shows that it is indeed a good estimator of the correlation dimension. The Lyapunov spectrum also reveals that sea clutter is produced by a coupled system of nonlinear differential equations of order five or six. (c) 1997 American Institute of Physics.

Spatiotemporal complexity of ventricular fibrillation revealed by tissue mass reduction in isolated swine right ventricle. Further evidence for the quasiperiodic route to chaos hypothesis

We have presented evidence that ventricular fibrillation is deterministic chaos arising from quasiperiodicity. The purpose of this study was to determine whether the transition from chaos (ventricular fibrillation, VF) to periodicity (ventricular tachycardia) through quasiperiodicity could be produced by the progressive reduction of tissue mass. In isolated and perfused swine right ventricular free wall, recording of single cell transmembrane potentials and simultaneous mapping (477 bipolar electrodes, 1.6 mm resolution) were performed. The tissue mass was then progressively reduced by sequential cutting. All isolated tissues fibrillated spontaneously. The critical mass to sustain VF was 19.9 +/- 4.2 g. As tissue mass was decreased, the number of wave fronts decreased, the life-span of reentrant wave fronts increased, and the cycle length, the diastolic interval, and the duration of action potential lengthened. There was a parallel decrease in the dynamical complexity of VF as measured by Kolmogorov entropy and Poincaré plots. A period of quasiperiodicity became more evident before the conversion from VF (chaos) to a more regular arrhythmia (periodicity). In conclusion, a decrease in the number of wave fronts in ventricular fibrillation by tissue mass reduction causes a transition from chaotic to periodic dynamics via the quasiperiodic route.

Detecting noise in a time series

A numerical algorithm is presented for estimating whether, and roughly to what extent, a time series is noise corrupted. Using phase-randomized surrogates constructed from the original signal, metrics are defined which can be used to quantify the noise level. A saturation occurs in these metrics at signal to noise ratios (SNRs) of around 0 dB and below, and also at around 20 dB and above. In between these two regions there is a monotonic transition in the value of the metrics from one region to the other corresponding to changes in the SNR. (c) 1997 American Institute of Physics.

Procedure for estimating the correlation dimension of optokinetic nystagmus signals

In this study, optokinetic nystagmus (OKN) is hypothesized to be controlled by a low-dimensional deterministic and possibly chaotic generator. A procedure for quantifying the presumably low-dimensional structure of the OKN signal, based on the Singular Spectrum Approach and the Grassberger--Procaccia algorithm for estimating the correlation dimension, v, is described. The procedure developed showed robustness against noise. Applying this method to OKN signals from 10 healthy subjects and 10 patients suffering from vertigo showed a statistically significant lower mean v value for the patients.

Quasiperiodicity and chaos in cardiac fibrillation

In cardiac fibrillation, disorganized waves of electrical activity meander through the heart, and coherent contractile function is lost. We studied fibrillation in three stationary forms: in human chronic atrial fibrillation, in a stabilized form of canine ventricular fibrillation, and in fibrillation-like activity in thin sheets of canine and human ventricular tissue in vitro. We also created a computer model of fibrillation. In all four studies, evidence indicated that fibrillation arose through a quasiperiodic stage of period and amplitude modulation, thus exemplifying the "quasiperiodic transition to chaos" first suggested by Ruelle and Takens. This suggests that fibrillation is a form of spatio-temporal chaos, a finding that implies new therapeutic approaches.

Experimental measurements of dimensionality and spatial coherence in the dynamics of a flexible-beam impact oscillator

Experiments on a flexible-beam impact oscillator are described in which the spatial structure of typical motions is explored. The beam is held in a fixed mount with clamped-free boundary conditions, and the beam is driven by impacts between its free end and a sinusoidally driven impactor. Bifurcation diagrams using impactor frequency and offset as the bifurcation parameter are obtained using a computer-driven data acquisition system. The dimensionality of the system is studied by analysis of delay-reconstructed time series of experimental data. Valid delay reconstructions are obtained using mutual information and false nearest neighbour algorithms, and the correlation dimensions is estimated for the resulting experimental attractors. The relation of these topological characterizations of the system to the spatial structure of the vibrations is studied using two-point spatial correlation measurements and the proper orthogonal decomposition. It is shown that over 90% of the mean square response amplitude is captured by the first proper orthogonal mode for the cases examined and that the spatial coherence measurements can be used to distinguish between responses with similar dimensionality.

Synchronized action of synaptically coupled chaotic model neurons

Experimental observations of the intracellular recorded electrical activity in individual neurons show that the temporal behavior is often chaotic. We discuss both our own observations on a cell from the stomatogastric central pattern generator of lobster and earlier observations in other cells. In this paper we work with models with chaotic neurons, building on models by Hindmarsh and Rose for bursting, spiking activity in neurons. The key feature of these simplified models of neurons is the presence of coupled slow and fast subsystems. We analyze the model neurons using the same tools employed in the analysis of our experimental data. We couple two model neurons both electrotonically and electrochemically in inhibitory and excitatory fashions. In each of these cases, we demonstrate that the model neurons can synchronize in phase and out of phase depending on the strength of the coupling. For normal synaptic coupling, we have a time delay between the action of one neuron and the response of the other. We also analyze how the synchronization depends on this delay. A rich spectrum of synchronized behaviors is possible for electrically coupled neurons and for inhibitory coupling between neurons. In synchronous neurons one typically sees chaotic motion of the coupled neurons. Excitatory coupling produces essentially periodic voltage trajectories, which are also synchronized. We display and discuss these synchronized behaviors using two "distance" measures of the synchronization.