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

Order patterns recurrence plots in the analysis of ERP data

Recurrence quantification analysis (RQA) is an established tool for data analysis in various behavioural sciences. In this article we present a refined notion of RQA based on order patterns. The use of order patterns is commonplace in time series analysis. Exploiting this concept in combination with recurrence plots (RP) and their quantification (RQA) allows for advances in contemporary EEG research, specifically in the analysis of event related potentials (ERP), as the method is known to be robust against non-stationary data. The use of order patterns recurrence plots (OPRPs) on EEG data recorded during a language processing experiment exemplifies the potentials of the method. We could show that the application of RQA to ERP data allows for a considerable reduction of the number of trials required in ERP research while still maintaining statistical validity.

Singing of Neoconocephalus robustus as an example of deterministic chaos in insects

We use nonlinear time series analysis methods to analyse the dynamics of the sound-producing apparatus of the katydid Neoconocephalus robustus. We capture the dynamics by analysing a recording of the singing activity. First, we reconstruct the phase space from the sound recording and test it against determinism and stationarity. After confirming determinism and stationarity, we show that the maximal Lyapunov exponent of the series is positive, which is a strong indicator for the chaotic behaviour of the system. We discuss that methods of nonlinear time series analysis can yield instructive insights and foster the understanding of acoustic communication among insects.

Observation of period multiplication and instability in a dc glow discharge

The temporal dynamics in the fluctuations of the plasma floating potentials from an undriven dc glow discharge argon plasma at an intermediate gas pressure of 250mTorr and at the range of discharge currents I=6-50mA are investigated. In this study, the discharge current I is used as the plasma system's bifurcation parameter in analogy with the parameter space of a numerical dynamical system. Over several regions of the discharge current, the floating potential fluctuation time series data has been indicative of random noise, periodic oscillations, and irregular fluctuations. As the bifurcation parameter (discharge current) is increased, the Fourier spectrum of the data shows increased signs of period multiplication, quasiperiodicity, and instabilities. In addition, the computations of the correlation dimension provide some insight into the complex nature of the instabilities in the glow discharge plasma.

Establishing the stochastic nature of intracellular calcium oscillations from experimental data

Calcium has been established as a key messenger in both intra- and intercellular signaling. Experimentally observed intracellular calcium responses to different agonists show a variety of behaviors from simple spiking to complex oscillatory regimes. Here we study typical experimental traces of calcium oscillations in hepatocytes obtained in response to phenylephrine and ATP. The traces were analyzed with methods of nonlinear time series analysis in order to determine the stochastic/deterministic nature of the intracellular calcium oscillations. Despite the fact that the oscillations appear, visually, to be deterministic yet perturbed by noise, our analyses provide strong evidence that the measured calcium traces in hepatocytes are prevalently of stochastic nature. In particular, bursting calcium oscillations are temporally correlated Gaussian series distorted by a monotonic, instantaneous, time-independent function, whilst the spiking behavior appears to have a dynamical nonlinear component whereby the overall determinism level is still low. The biological importance of this finding is discussed in relation to the mechanisms incorporated in mathematical models as well as the role of stochasticity and determinism at cellular and tissue levels which resemble typical statistical and thermodynamic effects in physics.

Analogical model for mechanical vibrations in flue organ pipes inferred by independent component analysis

Several experiments have been performed to investigate the mechanical vibrations associated with an organ pipe. The measurements have been made by using laser Doppler vibrometry, a well-known not-invasive optical measurement technique that is very widely used in structural dynamics. The recorded signals are analyzed by using a well-established decomposition method in time domain, i.e., independent component analysis. Asymptotic dynamics methods to recognize low-dimensional dynamic system associated with this wave field is then considered. The full-toned recorded signals appear decomposed into three independent components. The independent components are nonlinear due to the fractal dimension of the attractor. These results for the mechanic vibrational field are compared with those of the acoustic one. It is interesting to note that the two fields have many common characteristics. Finally, a low-dimensional dynamic system that reproduces the main characteristics of the mechanical wave field in the time and frequency domains is introduced.

Detecting determinism in short time series using a quantified averaged false nearest neighbors approach

We propose a criterion to detect determinism in short time series. This criterion is based on the estimation of the parameter E2 defined by the averaged false neighbors method for analyzing time series [Cao, Physica D 110, 43 (1997)]. Using surrogate data testing with several chaotic and stochastic simulated time series, we show that the variation coefficient of E2 over a few values of the embedding dimension d defines a suitable statistic to detect determinism in short data sequences. This result holds for a time series generated by a high-dimensional chaotic system such as the Mackey-Glass one. Different decreasing lengths of the time series are included in the numerical experiments for both synthetic and real-world data. We also investigate the robustness of the criterion in the case of deterministic time series corrupted by additive noise.

Movement patterns of peak-dose levodopa-induced dyskinesias in patients with Parkinson's disease

The present study characterized involuntary movements associated with levodopa-induced dyskinesias (LID) in patients with Parkinson's disease. We used amplitude, proportional energy, frequency dispersion and sample entropy to determine whether LID movement patterns are truly random, as clinical description seems to suggest, or possess some underlying pattern that is not visible to the naked eye. LID was captured using a magnetic tracker system, which provided 3D rendering of whole-body LID. Patients were instructed to maintain a standing position, with arms extended in front of them. We compared the measurements of the dyskinetic PD group (DPD) with 10 patients without dyskinesias (NDPD) and 10 control subjects. In comparison to the other two groups, movement patterns from the DPD group had significantly higher amplitude, confirming the presence of dyskinesias. In addition, higher frequency components in the power spectrum of velocity were detected, suggestive of higher velocity in LID movement. Furthermore, there was a concentration in narrow frequency bands, which suggested stable oscillatory activity. Finally, sample entropy revealed more regularity in the DPD group. Although not statistically significant, we found that the amplitude from the NDPD group had a tendency to be smaller than those of controls. As well, the spectra were often more dispersed for the NDPD group. In conclusion, the present results suggest that LID cannot be considered as purely random movement since they possess some deterministic pattern of motion. This may provide a way for patients to adapt to these involuntary movements while performing voluntary motor acts.

Analysis of the magnetoencephalogram background activity in Alzheimer's disease patients with auto-mutual information

The aim of the present study was to analyse the magnetoencephalogram (MEG) background activity in patients with Alzheimer's disease (AD), one of the most frequent disorders among elderly population. For this pilot study, we recorded the MEGs with a 148-channel whole-head magnetometer in 20 patients with probable AD and 21 age-matched control subjects. Artefact-free epochs of 3392 samples were analysed with auto-mutual information (AMI). Average AMI decline rates were lower for the AD patients' recordings than for control subjects' ones. Statistically significant differences were found using a Student's t-test (p<0.01) in 144 channels. Mean AMI values were analysed with a receiver operating characteristic curve. Sensitivity, specificity and accuracy values of 75%, 90.5% and 82.9% were obtained. Our results show that AMI estimations of the magnetic brain activity are different in both groups, hence indicating an abnormal type of dynamics associated with AD. This study suggests that AMI might help medical doctors in the diagnosis of the disease.

Reducing colored noise for chaotic time series in the local phase space

A two step method is proposed to reduce colored noise for chaotic data in the local phase space. With the observation that the energy of colored noise is mainly distributed in a low dimensional subspace, a noise dominated subspace is first estimated by the energy distribution of colored noise. At step 1, for the reference phase point, the components projected into the noise dominated subspace are deleted and the enhanced data are reconstructed with the remaining components. The residual error of the output of step 1 tends to distribute on each direction uniformly. So at step 2, the local projection method is further applied to the output of step 1, treating the residual error as white noise. Experiments show that our method performs well in eliminating colored noise for chaotic data.

Improved Estimation of Embedding Parameters of Nonlinear Time Series by Structural Learning of Neural Network with Fuzzy Regularizer

This work proposes an improved refinement scheme of estimation of optimal embedding parameters of a nonlinear time series by a feed-forward neural network trained by structural learning with a fuzzy regularizer (FR). The newly proposed fuzzy rules for tuning regularization parameter enables automatic selection of optimal model with lesser computational load than the basic refinement scheme with RNS proposed by authors earlier. From the simulation results, it has been found that the proposed scheme is very efficient in estimation of optimal embedding parameters in lesser computational time. The short term prediction results also show that the estimated embedding parameters produce better and stable one step prediction.

Nonlinear analysis of posturographic data

The aim of this work is to determine whether postural sway can be well described by nonlinear deterministic modelling. Since the results of nonlinear analysis depend on experimental data processing, emphasis was given to the assessment of a proper methodology to process posturographic data. Centre of Pressure (CoP) anterior-posterior (AP) displacements (stabilogram) were obtained by static posturography tests performed on control subjects. A nonlinear determinism test was applied to investigate the nature of data. A nonlinear filtering method allowed us to estimate properly the parameters of the nonlinear model without altering signal dynamics. The largest Lyapunov exponent (LLE) was estimated to quantify the chaotic behaviour of postural sway. LLE values were found to be positive although close to zero. This suggests that postural sway derives from a process exhibiting weakly chaotic dynamics.

From phase space to frequency domain: a time-frequency analysis for chaotic time series

Time-frequency analysis is performed for chaotic flow with a power spectrum estimator based on the phase-space neighborhood. The relation between the reference phase point and its nearest neighbors is demonstrated. The nearest neighbors, representing the state recurrences in the phase space reconstructed by time delay embedding, actually cover data segments with similar wave forms and thus possess redundant information, but recur with no obvious temporal regularity. To utilize this redundant recurrence information, a neighborhood-based spectrum estimator is devised. Then time-frequency analysis with this estimator is performed for the Lorenz time series, the Rössler time series, experimental laser data, and colored noise. Features revealed by the spectrogram can be used to distinguish noisy chaotic flow from colored noise.

Effect of chaotic vasomotion in skeletal muscle on tissue oxygenation

Vasomotion refers to spontaneous variations in the lumen size of small vessels, with a plausible role in regulation of various aspects of microcirculation. We propose a computational model of vasomotion in skeletal muscle in which the pattern of vasomotion is shown to critically determine the efficiency of oxygenation of a muscle fiber. In this model, precapillary sphincters are modeled as nonlinear oscillators. We hypothesize that these sphincters interact via exchange of vasoactive substances. As a consequence, vasomotion is described as a phenomenon associated with a network of nonlinear oscillators. As a specific instance, we model the vasomotion of precapillary sphincters surrounding an active fiber. The sphincters coordinate their rhythms so as to minimize oxygen deficit in the fiber. Our modeling studies indicate that efficient oxygenation of the fiber depends crucially on the mode of interaction among the vasomotions of individual sphincters. While chaotic forms of vasomotion enhanced oxygenation, regular patterns of vasomotion failed to meet the oxygenation demand accurately.

Sensitively recorded breathing signals of rats and their nonlinear dynamics

Nonlinear dynamical properties of sensitively recorded breathing signals (SRBS), which include cardiac induced air flow pulsations so-called pneumocardiogram (PNCG) signals, are investigated, in this methodological study. For this purpose, we assessed the SRBS of laboratory rat. The nonlinear behaviors of SRBS were investigated by the reconstructing phase space, using the autocorrelation function and the false nearest neighbor method. The chaotic SRBS attractors were discussed from the point of view of the cardiopulmonary system. This method can be used to assess the heart performance and respiratory mechanics, and might be useful to design for the physiological studies of cardiorespiratory system in small laboratory animals.

A dynamical systems analysis of spontaneous movements in newborn infants

The authors evaluated the characteristics of infants' spontaneous movements by using dynamical systems analysis. Participants were 6 healthy 1-month-old full-term newborn infants (3 males, 3 females). They used a triaxial accelerometer to measure limb acceleration in 3-dimensional space. Acceleration signals were recorded during 200 s from the right wrist when the infant was in an active alert state and lying supine (sampling rate 200 Hz). and was stored in the system's memory. Digitized data were transferred to a PC for subsequent processing with analysis software. The acceleration time series data were analyzed linearly and nonlinearly. Nonlinear time series analysis suggested that the infants' spontaneous movements are characterized by a nonlinear chaotic dynamics with 5 or 6 embedding dimensions. The production of infants'spontaneous movements involves chaotic dynamic systems that are capable of generating voluntary skill movements.

Quantitative and qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection

It has been suggested by experimentalists that a weakly nonlinear analysis of the recently introduced equations of motion for the nematic electroconvection by M. Treiber and L. Kramer [Phys. Rev. E 58, 1973 (1998)] has the potential to reproduce the dynamics of the zigzag-type extended spatiotemporal chaos and localized solutions observed near onset in experiments [M. Dennin, D. S. Cannell, and G. Ahlers, Phys. Rev. E 57, 638 (1998); J. T. Gleeson (private communication)]. In this paper, we study a complex spatiotemporal pattern, identified as spatiotemporal chaos, that bifurcates at the onset from a spatially uniform solution of a system of globally coupled complex Ginzburg-Landau equations governing the weakly nonlinear evolution of four traveling wave envelopes. The Ginzburg-Landau system can be derived directly from the weak electrolyte model for electroconvection in nematic liquid crystals when the primary instability is a Hopf bifurcation to oblique traveling rolls. The chaotic nature of the pattern and the resemblance to the observed experimental spatiotemporal chaos in the electroconvection of nematic liquid crystals are confirmed through a combination of techniques including the Karhunen-Loeve decomposition, time-series analysis of the amplitudes of the dominant modes, statistical descriptions, and normal form theory, showing good agreement between theory and experiments.

Cutting process dynamics by nonlinear time series and wavelet analysis

We have modeled the dynamics of a cutting process by a two-degree-of-freedom mass-spring system with dry friction. Using nonlinear time series and wavelet analysis, we have investigated the vibrational instabilities of the system for different values of the cutting force. By constructing the phase portraits and calculating the Lyapunov exponents we have delineated the conditions for which a periodic or chaotic motion can occur. The results are verified by means of a time-scale representation of the wavelet power spectrum.

Global models from the Canadian lynx cycles as a direct evidence for chaos in real ecosystems

Real food chains are very rarely investigated since long data sequences are required. Typically, if we consider that an ecosystem evolves with a period corresponding to the time for maturation, possessing few dozen of cycles would require to count species over few centuries. One well known example of a long data set is the number of Canadian lynx furs caught by the Hudson Bay company between 1821 and 1935 as reported by Elton and Nicholson in 1942. In spite of the relative quality of the data set (10 undersampled cycles), two low-dimensional global models that settle to chaotic attractors were obtained. They are compared with an ad hoc 3D model which was proposed as a possible model for this data set. The two global models, which were estimated with no prior knowledge about the dynamics, can be considered as direct evidences of chaos in real ecosystems.

Recurrence plots and Shannon entropy for a dynamical analysis of asynchronisms in noninvasive mechanical ventilation

Recurrence plots were introduced to quantify the recurrence properties of chaotic dynamics. Hereafter, the recurrence quantification analysis was introduced to transform graphical interpretations into statistical analysis. In this spirit, a new definition for the Shannon entropy was recently introduced in order to have a measure correlated with the largest Lyapunov exponent. Recurrence plots and this Shannon entropy are thus used for the analysis of the dynamics underlying patient assisted with a mechanical noninvasive ventilation. The quality of the assistance strongly depends on the quality of the interactions between the patient and his ventilator which are crucial for tolerance and acceptability. Recurrence plots provide a global view of these interactions and the Shannon entropy is shown to be a measure of the rate of asynchronisms as well as the breathing rhythm.

Bifurcation analysis of bubble dynamics in fluidized beds

We use a low-dimensional, agent-based bubble model to study the changes in the global dynamics of fluidized beds in response to changes in the frequency of the rising bubbles. The computationally based bifurcation analysis shows that at low frequencies, the global dynamics is attracted towards a fixed point since the bubbles interact very little with one another. As the frequency of injection increases, however, the global dynamics undergoes a series of bifurcations to new behaviors that include highly periodic orbits, chaotic attractors, and intermittent behavior between periodic orbits and chaotic sets. Using methods from time-series analysis, we are able to approximate nonlinear models that allow for long-term predictions and the possibility of developing control algorithms.

A unified approach to attractor reconstruction

In the analysis of complex, nonlinear time series, scientists in a variety of disciplines have relied on a time delayed embedding of their data, i.e., attractor reconstruction. The process has focused primarily on intuitive, heuristic, and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has left several longstanding, but common problems unresolved in which the standard approaches produce inferior results or give no guidance at all. We view the current reconstruction process as unnecessarily broken into separate problems. We propose an alternative approach that views the problem of choosing all embedding parameters as being one and the same problem addressable using a single statistical test formulated directly from the reconstruction theorems. This allows for varying time delays appropriate to the data and simultaneously helps decide on embedding dimension. A second new statistic, undersampling, acts as a check against overly long time delays and overly large embedding dimension. Our approach is more flexible than those currently used, but is more directly connected with the mathematical requirements of embedding. In addition, the statistics developed guide the user by allowing optimization and warning when embedding parameters are chosen beyond what the data can support. We demonstrate our approach on uni- and multivariate data, data possessing multiple time scales, and chaotic data. This unified approach resolves all the main issues in attractor reconstruction.

State-space analysis of joint angle kinematics in normal treadmill walking

By restricting analysis to single averaged strides considered to be characteristic for the individual under investigation, current methods in gait analysis do not exploit the full dynamics of continuous locomotion. Therefore, a novel approach is presented that is based on long-term measurements of kinematic data during treadmill walking. The method consists of reconstructing the system attractor in the embedding space and then analyzing its geometric structure. Estimating the dimension of movement trajectories correlates well with the notion of controlling multiple degrees of freedom during performance of complex movement tasks such as walking. The influence of walking speed on the complexity of physiologic walking was investigated in 10 healthy subjects walking on a treadmill at seven fixed speeds. The results suggest that human walking becomes more complex at slower speeds. This may be associated with results from EMG studies demonstrating more irregular EMG patterns at very slow walking speeds. This study emphasizes that tools from non-linear dynamics are well suited for providing more insight into motor control in humans.

Test nonlinear determinacy of Electromyogram

To test whether electromyogram (EMG) is nonlinear deterministic signal or just random noise, we analyzed four EMGs data of an adult woman. EMG is assessed with the correlation dimension (D<inf>2</inf>), maximum Lyapunov exponent (&#955;<inf>1</inf>), approximate entropy(ApEn), recurrence plot analysis (RPA) and recurrence quantification analysis (RQA). From all the results of these assessments we can conclude that EMG obeys a certain nonlinear deterministic law and nonstationarity is significant within this signal.

Analysis and reliability of posturographic parameters in Parkinson patients at an early stage

Aim of this work was to evaluate Parkinsonian (PARK) patients at their initial stage of the disease by static posturography in order to study: the stability of posture system; the role of visual input; the influence of an acute administration of levodopa. This would allow the creation of a database that can be useful to follow the progression of the disease. Attention was also focused on the assessment of the reliability of quantitative posturographic parameters (PP); a large number of PP that can be derived using classical, structural, and nonlinear approaches have been considered. Particular attention has been given to this latter approach. The class of PD subjects examined in this work resulted to be very different from classical Parkinsonian subjects studied in literature.

Detection and segmentation of changes of dynamics in biological time series

A method to identify switching dynamics in time series, based on annealed competition of experts algorithm (ACE), has been developed by J. Kohlmorgen, et al (2000). Incorrect selection of embedding dimension and time delay of the signal significantly affect the performance of the ACE method, however. We utilize systematic approaches based on mutual information and false nearest neighbor to determine appropriate embedding dimension and time delay. Moreover, we obtained further improvements to the original ACE method by incorporating a phase space closeness measure during the training procedure as well as deterministic annealing approach. Using these ameliorated implementations, we have enhanced the performance of the ACE algorithm in determining the location of the switching of dynamic modes in time series. The application of the improved ACE method to RR interval data obtained from rats during control and administration of double autonomic blockade conditions indicate that the improved ACE algorithm is able to segment dynamic mode changes with pinpoint accuracy and that its performance is far superior to the original ACE algorithm.

A nonlinear approach to tracking slow-time-scale changes in movement kinematics

Degenerative processes like repetitive strain injuries (RSIs) cause normal movement patterns to change slowly over time. Accurately tracking how these disease/injury processes evolve over time and predicting their future progression could allow early intervention and prevent further deterioration. However, these processes often cannot be measured directly and first-principles models of these processes and how they affect movement control are highly complex and difficult to derive analytically. This study was conducted to determine if algorithms developed to track damage accumulation in mechanical systems without requiring first-principles models or direct measurements of the damage itself could also track a similar "hidden" process in a biomechanical context. Five healthy adults walked on a motorized treadmill at their preferred speed, while the treadmill inclination angle was slowly increased from 0 degrees (level) to approximately +8 degrees . Sagittal plane kinematics for the left hip, knee, and ankle joints were computed. The treadmill inclination angle was independently recorded and defined the "damage" to be tracked. Scalar tracking metrics were computed from the lower extremity walking kinematics. These metrics exhibited strong cubic relationships with treadmill inclination (88.9%<or=r(2)<or=98.2%; p<0.001). These results demonstrate that the proposed approach may also be well suited to tracking and predicting slow-time-scale degenerative biological processes like muscle fatigue or RSIs. This possibility is potentially quite powerful because it suggests that easily obtainable biomechanical data can provide unique and valuable insights into the dynamics of "hidden" biological processes that cannot be easily measured themselves.

Differences Between Local and Orbital Dynamic Stability During Human Walking

Currently there is no commonly accepted way to define, much less quantify, locomotor stability. In engineering, “orbital stability” is defined using Floquet multipliers that quantify how purely periodic systems respond to perturbations discretely from one cycle to the next. For aperiodic systems, “local stability” is defined by local divergence exponents that quantify how the system responds to very small perturbations continuously in real time. Triaxial trunk accelerations and lower extremity sagittal plane joint angles were recorded from ten young healthy subjects as they walked for 10min over level ground and on a motorized treadmill at the same speed. Maximum Floquet multipliers (Max FM) were computed at each percent of the gait cycle (from 0% to 100%) for each time series to quantify the orbital stability of these movements. Analyses of variance comparing Max FM values between walking conditions and correlations between Max FM values and previously published local divergence exponent results were computed. All subjects exhibited orbitally stable walking kinematics (i.e., magnitudes of Max FM<1.0), even though these same kinematics were previously found to be locally unstable. Variations in orbital stability across the gait cycle were generally small and exhibited no systematic patterns. Walking on the treadmill led to small, but statistically significant improvements in the orbital stability of mediolateral (p=0.040) and vertical (p=0.038) trunk accelerations and ankle joint kinematics (p=0.002). However, these improvements were not exhibited by all subjects (p⩽0.012 for subject × condition interaction effects). Correlations between Max FM values and previously published local divergence exponents were inconsistent and 11 of the 12 comparisons made were not statistically significant (r2⩽19.8%; p⩾0.049). Thus, the variability inherent in human walking, which manifests itself as local instability, does not substantially adversely affect the orbital stability of walking. The results of this study will allow future efforts to gain a better understanding of where the boundaries lie between locally unstable movements that remain orbitally stable and those that lead to global instability (i.e., falling).

Synchronization likelihood with explicit time-frequency priors

Cognitive processing requires integration of information processed simultaneously in spatially distinct areas of the brain. The influence that two brain areas exert on each others activity is usually governed by an unknown function, which is likely to have nonlinear terms. If the functional relationship between activities in different areas is dominated by the nonlinear terms, linear measures of correlation may not detect the statistical interdependency satisfactorily. Therefore, algorithms for detecting nonlinear dependencies may prove invaluable for characterizing the functional coupling in certain neuronal systems, conditions or pathologies. Synchronization likelihood (SL) is a method based on the concept of generalized synchronization and detects nonlinear and linear dependencies between two signals (Stam, C.J., van Dijk, B.W., 2002. Synchronization likelihood: An unbiased measure of generalized synchronization in multivariate data sets. Physica D, 163: 236-241.). SL relies on the detection of simultaneously occurring patterns, which can be complex and widely different in the two signals. Clinical studies applying SL to electro- or magnetoencephalography (EEG/MEG) signals have shown promising results. In previous implementations of the algorithm, however, a number of parameters have lacked a rigorous definition with respect to the time-frequency characteristics of the underlying physiological processes. Here we introduce a rationale for choosing these parameters as a function of the time-frequency content of the patterns of interest. The number of parameters that can be arbitrarily chosen by the user of the SL algorithm is thereby decreased from six to two. Empirical evidence for the advantages of our proposal is given by an application to EEG data of an epileptic seizure and simulations of two unidirectionally coupled Hénon systems.

The effects of sensory loss and walking speed on the orbital dynamic stability of human walking

Peripheral sensory feedback is believed to contribute significantly to maintaining walking stability. Patients with diabetic peripheral neuropathy have a greatly increased risk of falling. Previously, we demonstrated that slower walking speeds in neuropathic patients lead to improved local dynamic stability. However, all subjects exhibited significant local instability during walking, even though no subject fell or stumbled during testing. The present study was conducted to determine if and how significant changes in peripheral sensation and walking speed affect orbital stability during walking. Trunk and lower extremity kinematics were examined from two prior experiments that compared patients with significant neuropathy to healthy controls and walking at multiple different speeds in young healthy subjects. Maximum Floquet multipliers were computed for each time series to quantify the orbital stability of these movements. All subjects exhibited orbitally stable walking kinematics, even though these same kinematics were previously shown to be locally unstable. Differences in orbital stability between neuropathic and control subjects were small and, with the exception of knee joint movements (p=0.001), not statistically significant (0.380p0.946). Differences in knee orbital stability were not mediated by differences in walking speed. This was supported by our finding that although orbital stability improved slightly with slower walking speeds, the correlations between walking speed and orbital stability were generally weak (r(2)16.7%). Thus, neuropathic patients do not gain improved orbital stability as a result of slowing down and do not experience any loss of orbital stability because of their sensory deficits.

Improving the global fitting method on nonlinear time series analysis

We are concerned with improving the forecast capabilities of the global approach to time series. We assume that the normal techniques of global mapping are applied, the noise reduction is performed, etc. Then, using the mathematical foundations behind such approaches, we propose a method that, without a great computational cost, greatly increases the accuracy of the corresponding forecasting.

Phase space warping: nonlinear time-series analysis for slowly drifting systems

A new general dynamical systems approach to data analysis is presented that allows one to track slowly evolving variables responsible for non-stationarity in a fast subsystem. The method is based on the idea of phase space warping, which refers to the small distortions in the fast subsystem's phase space that results from the slow drift, and uses short-time reference model prediction error as its primary measurement of this phenomenon. The basic theory is presented and the issues associated with its implementation in a practical algorithm are discussed. A vector-tracking version of the procedure, based on smooth orthogonal decomposition analysis, is applied to the study of a nonlinear vibrating beam experiment in which a crack propagates to complete fracture. Our method shows that the damage evolution is governed by a scalar process, and we are able to give real-time estimates of the current damage state and identify the governing damage evolution model. Using a final recursive estimation step based on this model, the time to failure is continuously and accurately predicted well in advance of actual failure.

Chaotic dynamics of resting ventilatory flow in humans assessed through noise titration

The mammalian ventilatory behaviour exhibits nonlinear dynamics as reflected by certain nonlinearity or complexity indicators (e.g. correlation dimension, approximate entropy, Lyapunov exponents, etc.) but this is not sufficient to determine its possible chaotic nature. To address this, we applied the noise titration technique, previously shown to discern and quantify chaos in short and noisy time series, to ventilatory flow recordings obtained in quietly breathing normal humans. Nine subjects (8 men and 1 woman, 24-42 years) were studied during 15-min epochs of ventilatory steady-state (10.1+/-3.0 breaths/min, tidal volume 0.63+/-0.2 L). Noise titration applied to the unfiltered signals subsampled at 5 Hz detected nonlinearity in all cases (noise limit 20.2+/-12.5%). Noise limit values were weakly correlated to the correlation dimension and the largest Lyapunov exponent of the signals. This study shows that the noise titration approach evidences a chaotic dimension to the behavior of ventilatory flow over time in normal humans during tidal breathing.

Reconstructing state spaces from multivariate data using variable delays

We study two methods for constructing a nonuniform embedding for multivariate data. A nonuniform embedding is a state space reconstruction which is more flexible than the common delay coordinates with fixed delays since it contains variable delays. Using these methods, we can extract causal relationships among many variables in a more suitable way. We demonstrate that the proposed methods can give more precise predictions and simpler models than some previous methods.

Chaos without nonlinear dynamics

A linear, second-order filter driven by randomly polarized pulses is shown to generate a waveform that is chaotic under time reversal. That is, the filter output exhibits determinism and a positive Lyapunov exponent when viewed backward in time. The filter is demonstrated experimentally using a passive electronic circuit, and the resulting waveform exhibits a Lorenz-like butterfly structure. This phenomenon suggests that chaos may be connected to physical theories whose underlying framework is not that of a traditional deterministic nonlinear dynamical system.

Estimating the Shannon entropy: recurrence plots versus symbolic dynamics

Recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics. A few years later, the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis. Among the different measures introduced, a Shannon entropy was found to be correlated with the inverse of the largest Lyapunov exponent. The discrepancy between this and the usual interpretation of a Shannon entropy is solved here by using a new definition--still based on the recurrence plots--and it is verified that this new definition is correlated with the largest Lyapunov exponent, as expected from the Pesin conjecture. A comparison with a Shannon entropy computed from symbolic dynamics is also provided.

Method for measuring unstable dimension variability from time series

Many of the results in the theory of dynamical systems rely on the assumption of hyperbolicity. One of the possible violations of this condition is the presence of unstable dimension variability (UDV), i.e., the existence in a chaotic attractor of sets of unstable periodic orbits, each with a different number of expanding directions. It has been shown that the presence of UDV poses severe limitations to the length of time for which a numerically generated orbit can be assumed to lie close to a true trajectory of such systems (the shadowing time). In this work we propose a method to detect the presence of UDV in real systems from time series measurements. Variations in the number of expanding directions are detected by determining the local topological dimension of the unstable space for points along a trajectory on the attractor. We show for a physical system of coupled electronic oscillators that with this method it is possible to decompose attractors into subsets with different unstable dimension and from this gain insight into the times a typical trajectory spends in each region.

Chaos in computer performance

Modern computer microprocessors are composed of hundreds of millions of transistors that interact through intricate protocols. Their performance during program execution may be highly variable and present aperiodic oscillations. In this paper, we apply current nonlinear time series analysis techniques to the performances of modern microprocessors during the execution of prototypical programs. Our results present pieces of evidence strongly supporting that the high variability of the performance dynamics during the execution of several programs display low-dimensional deterministic chaos, with sensitivity to initial conditions comparable to textbook models. Taken together, these results show that the instantaneous performances of modern microprocessors constitute a complex (or at least complicated) system and would benefit from analysis with modern tools of nonlinear and complexity science.

Complexity analysis of the magnetoencephalogram background activity in Alzheimer's disease patients

The aim of the present study was to analyse the magnetoencephalogram (MEG) background activity in patients with Alzheimer's disease (AD) using the Lempel-Ziv (LZ) complexity. This non-linear method measures the complexity of finite sequences and is related to the number of distinct substrings and the rate of their occurrence along the sequence. The MEGs were recorded with a 148-channel whole-head magnetometer (MAGNES 2500 WH, 4D Neuroimaging) in 21 patients with AD and in 21 age-matched control subjects. Artefact-free epochs were selected for complexity analysis. Results showed that MEG signals from AD patients had lower complexity than control subjects' MEGs and the differences were statistically significant (p<0.01). In order to reduce the dimension of the LZ complexity results, a principal components analysis (PCA) was applied, and only the first principal component was retained. The first component score from PCA was graphically analysed using a box plot and a receiver-operating characteristic (ROC) curve. A specificity of 85.71%, a sensitivity of 80.95% and an area under the ROC curve of 0.9002 were obtained. These preliminary results suggest that cognitive dysfunction in AD is associated with a decreased LZ complexity in the MEG signals.

Kinematic variability and local dynamic stability of upper body motions when walking at different speeds

A ubiquitous characteristic of elderly and patients with gait disabilities is that they walk slower than healthy controls. Many clinicians assume these patients walk slower to improve their stability, just as healthy people slow down when walking across ice. However, walking slower also leads to greater variability, which is often assumed to imply deteriorated stability. If this were true, then slowing down would be completely antithetical to the goal of maintaining stability. This study sought to resolve this paradox by directly quantifying the sensitivity of the locomotor system to local perturbations that are manifested as natural kinematic variability. Eleven young healthy subjects walked on a motorized treadmill at five different speeds. Three-dimensional movements of a single marker placed over the first thoracic vertebra were recorded during continuous walking. Mean stride-to-stride standard deviations and maximum finite-time Lyapunov exponents were computed for each time series to quantify the variability and local dynamic stability, respectively, of these movements. Quadratic regression analyses of the dependent measures vs. walking speed revealed highly significant U shaped trends for all three mean standard deviations, but highly significant linear trends, with significant or nearly significant quadratic terms, for five of the six finite-time Lyapunov exponents. Subjects exhibited consistently better local dynamic stability at slower speeds for these five measures. These results support the clinically based intuition that people who are at increased risk of falling walk slower to improve their stability, even at the cost of increased variability.

A Robust Method for Detection of Linear and Nonlinear Interactions: Application to Renal Blood Flow Dynamics

We have developed a method that can identify switching dynamics in time series, termed the improved annealed competition of experts (IACE) algorithm. In this paper, we extend the approach and use it for detection of linear and nonlinear interactions, by employing histograms showing the frequency of switching modes obtained from the IACE, then examining time-frequency spectra. This extended approach is termed Histogram of improved annealed competition of experts-time frequency (HIACE-TF). The hypothesis is that frequent switching dynamics in HIACE-TF results are due to interactions between different dynamic components. To validate this assertion, we used both simulation examples as well as application to renal blood flow data. We compared simulation results to a time-phase bispectrum (TPB) approach, which can also be used to detect time-varying quadratic phase coupling between various components. We found that the HIACE-TF approach is more accurate than the TPB in detecting interactions, and remains accurate for signal-to-noise ratios as low as 15 dB. With all 10 data sets, comprised of volumetric renal blood flow data, we also validated the feasibility of the HIACE-TF approach in detecting nonlinear interactions between the two mechanisms responsible for renal autoregulation. Further validation of the HIACE-TF approach was achieved by comparing it to a realistic mathematical model that has the capability to generate either the presence or the absence of nonlinear interactions between two renal autoregulatory mechanisms.

Dynamical heterogeneity of suprachiasmatic nucleus neurons based on regularity and determinism

The suprachiasmatic nucleus (SCN) is known to be the master biological clock in mammals. Despite the periodic mean firing rate, interspike interval (ISI) patterns of SCN neurons are quite complex and irregular. The aim of the present study was to investigate the existence of nonlinear determinism in the complex ISI patterns of SCN neurons. ISI sequences were recorded from 173 neurons in rat hypothalamic slice preparations using a cell-attached patch recording technique. Their correlation dimensions (D2) were estimated, and were then compared with those of the randomly-shuffled surrogate data. We found that only 16 neurons (16/173) exhibited deterministic ISI patterns of spikes. In addition, clustering analysis revealed that SCN neurons could be divided into two subgroups of neurons each having distinct values of coefficient of variation (CV) and skewness (SK). Interestingly, most deterministic SCN neurons (14/16) belonged to the group of irregularly spiking neurons having large CV and SK values. To see if the neuronal coupling mediated by the gamma-aminobutyric acid (GABA), the major neurotransmitter in the SCN, contributed to the deterministic nature, we examined the effect of the GABAA receptor antagonist bicuculline on D2 values of 56 SCN neurons. 8 SCN neurons which were originally stochastic became to exhibit deterministic characteristics after the bicuculline application. This result suggests that the deterministic nature of the SCN neurons arises not from GABAergic synaptic interactions, but likely from properties inherent to neurons themselves.

The nonlinear theory of schizophrenia

OBJECTIVE

Nonlinear properties exist within the brain across a hierarchy of scales and within a variety of critical neural processes. Only a few studies of brain activity in schizophrenia, however, have used nonlinear methods. This review paper evaluates the contribution of the nonlinear sciences towards understanding schizophrenia.

METHOD

Applications of nonlinear methods to the study of schizophrenia symptoms and to healthy and schizophrenia functional neuroscience data are reviewed. The main flaws of nonlinear algorithms and recent methods to correct these are also appraised.

RESULTS

Initial research methods utilized in the study of nonlinearity in schizophrenia have fundamental methodological limitations. In the last decade, many of these problems have been addressed, facilitating future progress. Research incorporating these improvements has been applied to normal electroencephalogram (EEG) data and to the symptoms of schizophrenia, but not systematically to brain imaging data collected from patients with schizophrenia.

CONCLUSION

There is strong statistical evidence for weak nonlinearity in normal EEG and in the fluctuations of the symptoms of schizophrenia. However, the contribution of nonlinear processes to brain dysfunction in schizophrenia is yet to be properly established or accurately quantified. Despite this, recent methodological advances suggest that a 'nonlinear theory' of schizophrenia may be helpful in understanding this disorder.