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

Tissue specificity of nonlinear dynamics in baseline fMRI

In this work, recent advances in the field of nonlinear dynamics (NLD) were applied to fMRI data to examine the spatio-temporal properties of BOLD resting state fluctuations. Five human subjects were imaged during resting state (visual fixation) at 3T using single-shot gradient echo planar imaging (EPI). Respiration and cardiac signals were concurrently recorded for retrospectively removing fluctuations due to these physiologic activities. Patterns of singularity in the complex plane (PSC) and Lempel-Ziv complexity (LZ) were used to study the deterministic nonlinearity in resting state fMRI data. The results show that there is greater nonlinearity (higher PSC) and determinism (lower LZ) in gray matter compared to white matter and CSF. In addition, the removal of respiratory and cardiac pulsations decreases the nonlinearity and determinism but does not alter the relative difference between gray matter and white matter. Therefore, our results demonstrate that determinism and nonlinearity in the fMRI data are tissue-specific, suggesting that they reflect native physiologic and metabolic fluctuations and are not a result of physiologic artifacts due to respiration and cardiac pulsation.

Lyapunov exponents of laser Doppler flowmetry signals in healthy and type 1 diabetic subjects

The skin of diabetic subjects presents abnormalities in capillary blood flow and its regulation, often leading to the generation of plantar ulcers. In order to gain insight into this pathology for type 1 diabetic patients, Lyapunov exponents (LEs) of signals reflecting microvascular perfusion--laser Doppler flowmetry (LDF) signals--are calculated. The algorithm to compute LEs is first validated on simulated data and LDF surrogates. Then, LDF signals recorded at rest and during the application of local and progressive pressure of 11.1 Pa/s are processed. The exponents appear in pairs and are different for healthy and type 1 diabetic subjects at rest; P = 0.0556 for the 7th, 8th, and 9th LEs. Furthermore, progressive pressure has also a distinct effect on LEs. The difference is more pronounced for diabetic patients, for whom P = 0.0625 for the four LEs of highest absolute value. Because these differences arise from abnormalities in microvascular blood flow, they may help to explain the high prevalence of type 1 diabetic patients developing foot ulcers.

Nonlinear dynamical analysis of EEG and MEG: review of an emerging field

Many complex and interesting phenomena in nature are due to nonlinear phenomena. The theory of nonlinear dynamical systems, also called 'chaos theory', has now progressed to a stage, where it becomes possible to study self-organization and pattern formation in the complex neuronal networks of the brain. One approach to nonlinear time series analysis consists of reconstructing, from time series of EEG or MEG, an attractor of the underlying dynamical system, and characterizing it in terms of its dimension (an estimate of the degrees of freedom of the system), or its Lyapunov exponents and entropy (reflecting unpredictability of the dynamics due to the sensitive dependence on initial conditions). More recently developed nonlinear measures characterize other features of local brain dynamics (forecasting, time asymmetry, determinism) or the nonlinear synchronization between recordings from different brain regions. Nonlinear time series has been applied to EEG and MEG of healthy subjects during no-task resting states, perceptual processing, performance of cognitive tasks and different sleep stages. Many pathologic states have been examined as well, ranging from toxic states, seizures, and psychiatric disorders to Alzheimer's, Parkinson's and Cre1utzfeldt-Jakob's disease. Interpretation of these results in terms of 'functional sources' and 'functional networks' allows the identification of three basic patterns of brain dynamics: (i) normal, ongoing dynamics during a no-task, resting state in healthy subjects; this state is characterized by a high dimensional complexity and a relatively low and fluctuating level of synchronization of the neuronal networks; (ii) hypersynchronous, highly nonlinear dynamics of epileptic seizures; (iii) dynamics of degenerative encephalopathies with an abnormally low level of between area synchronization. Only intermediate levels of rapidly fluctuating synchronization, possibly due to critical dynamics near a phase transition, are associated with normal information processing, whereas both hyper-as well as hyposynchronous states result in impaired information processing and disturbed consciousness.

A Method for Segmentation of Switching Dynamic Modes in Time Series

A method to identify switching dynamics in time series, based on Annealed Competition of Experts algorithm (ACE), has been developed by Kohlmorgen et al. Incorrect selection of embedding dimension and time delay of the signal significantly affect the performance of the ACE method, however. In this paper, 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 deterministic annealing approach as well as phase space closeness measure. Using these improved implementations, we have enhanced the performance of the ACE algorithm in determining the location of the switching of dynamic modes in the time series. The application of the improved ACE method to heart rate 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 superior to the original ACE algorithm.

Optimal phase-space projection for noise reduction

In this communication we will reexamine the widely studied technique of phase-space projection. By imposing a time-domain constraint on the residual noise, we deduce a more general version of the optimal projector, which includes those appearing in previous literature as subcases but does not assume the independence between the clean signal and the noise. As an application, we will apply this technique for noise reduction. Numerical results show that our algorithm has succeeded in augmenting the signal-to-noise ratio for simulated data from the Rössler system and experimental speech record.

Exploring Sitting Posture and Discomfort Using Nonlinear Analysis Methods

-The possibilities for describing sitting postural control using nonlinear methods was investigated during long-term driving. A total of 85 min of motorway driving intervals (n=12) were used for analysis. The results show that contrary to conventional analysis techniques, nonlinear measures were able to identify a threshold behavior describing the change in discomfort. Visual recurrence plots showed a clear change in the underlying dynamics after 1 hr of driving. The result was confirmed by the statistically significant differences in the stability and complexity of the COP time series, as explored using recurrence quantification analysis and spatio-temporal entropy. The findings of the experiment are consistent with the literature, and present a novel way to uncover transitions of discomfort stages in sitting behavioral research.

Computational analysis in vitro: dynamics and plasticity of a neuro-robotic system

When the brain interacts with the environment it constantly adapts by representing the environment in a form that is called an internal model. The neurobiological basis for internal models is provided by the connectivity and the dynamical properties of neurons. Thus, the interactions between neural tissues and external devices provide a fundamental means for investigating the connectivity and dynamical properties of neural populations. We developed this idea, suggested in the 1980s by Valentino Braitenberg, for investigating and representing the dynamical behavior of neuronal populations in the brainstem of the lamprey. The brainstem was maintained in vitro and connected in a closed loop with two types of artificial device: (a) a simulated dynamical system and (b) a small mobile robot. In both cases, the device was controlled by recorded extracellular signals and its output was translated into electrical stimuli delivered to the neural system. The goal of the first study was to estimate the dynamical dimension of neural preparation in a single-input/single-output configuration. The dynamical dimension is the number of state variables that together with the applied input determine the output of a system. The results indicate that while this neural system has significant dynamical properties, its effective complexity, as established by the dynamical dimension, is rather moderate. In the second study, we considered a more specific situation, in which the same portion of the nervous system controls a robotic device in a two-input/two-output configuration. We fitted the input-output data from the neuro-robotic preparation to neural network models having different internal dynamics and we observed the generalization error of each model. Consistent with the first study, this second experiment showed that a simple recurrent dynamical model was able to capture the behavior of the hybrid system. This experimental and computational framework provides the means for investigating neural plasticity and internal representations in the context of brain-machine interfaces.

Multivariate phase space reconstruction by nearest neighbor embedding with different time delays

A recently proposed nearest neighbor based selection of time delays for phase space reconstruction is extended to multivariate time series, with an iterative selection of variables and time delays. A case study of numerically generated solutions of the x- and z coordinates of the Lorenz system, and an application to heart rate and respiration data, are used for illustration.

Postprocessing methods for finding the embedding dimension of chaotic time series

One problem when using the global false nearest-neighbors (GFNN) method and Cao's method to estimate embedding dimension is that their effectiveness is affected by the ratio of signal power to noise power (SNR). Simple models are proposed to explain the curves commonly obtained when using the GFNN method and Cao's method. Methods are proposed for systematically estimating the embedding dimension. Prior information is incorporated to improve the estimates.

Statistically relaxing to generating partitions for observed time-series data

We introduce a relaxation algorithm to estimate approximations to generating partitions for observed dynamical time series. Generating partitions preserve dynamical information of a deterministic map in the symbolic representation. Our method optimizes an essential property of a generating partition: avoiding topological degeneracies. We construct an energy-like functional and use a nonequilibrium stochastic minimization algorithm to search through configuration space for the best assignment of symbols to observed data. As each observed point may be assigned a symbol, the partitions are not constrained to an arbitrary parametrization. We further show how to select particular generating partition solutions which also code low-order unstable periodic orbits in a given way, hence being able to enumerate through a number of potential generating partition solutions.

Whole cell stochastic model reproduces the irregularities found in the membrane potential of bursting neurons

Irregular intrinsic behavior of neurons seems ubiquitous in the nervous system. Even in circuits specialized to provide periodic and reliable patterns to control the repetitive activity of muscles, such as the pyloric central pattern generator (CPG) of the crustacean stomatogastric ganglion (STG), many bursting motor neurons present irregular activity when deprived from synaptic inputs. Moreover, many authors attribute to these irregularities the role of providing flexibility and adaptation capabilities to oscillatory neural networks such as CPGs. These irregular behaviors, related to nonlinear and chaotic properties of the cells, pose serious challenges to developing deterministic Hodgkin-Huxley-type (HH-type) conductance models. Only a few deterministic HH-type models based on experimental conductance values were able to show such nonlinear properties, but most of these models are based on slow oscillatory dynamics of the cytosolic calcium concentration that were never found experimentally in STG neurons. Based on an up-to-date single-compartment deterministic HH-type model of a STG neuron, we developed a stochastic HH-type model based on the microscopic Markovian states that an ion channel can achieve. We used tools from nonlinear analysis to show that the stochastic model is able to express the same kind of irregularities, sensitivity to initial conditions, and low dimensional dynamics found in the neurons isolated from the STG. Without including any nonrealistic dynamics in our whole cell stochastic model, we show that the nontrivial dynamics of the membrane potential naturally emerge from the interplay between the microscopic probabilistic character of the ion channels and the nonlinear interactions among these elements. Moreover, the experimental irregular behavior is reproduced by the stochastic model for the same parameters for which the membrane potential of the original deterministic model exhibits periodic oscillations.

Nearest neighbor embedding with different time delays

A nearest neighbor based selection of time delays for phase space reconstruction is proposed and compared to the standard use of time delayed mutual information. The possibility of using different time delays for consecutive dimensions is considered. A case study of numerically generated solutions of the Lorenz system is used for illustration. The effect of contamination with various levels of additive Gaussian white noise is discussed.

Detecting synchronizations in an asymmetric vocal fold model from time series data

A nonlinear modeling approach is presented for the reconstruction of the synchronization structure in an asymmetric two-mass model from time series data. The asymmetric two-mass model describes a variety of normal and pathological human voices associated with synchronous and desynchronous oscillations of the two asymmetric vocal folds. Our technique recovers the synchronization diagram, which yields the regimes of synchronization as well as desynchronization, which are dependent upon the asymmetry parameter and the subglottal pressure. This allows the prediction of the regime of pathological phonation associated with desynchronization of the vocal folds from a few sets of recorded time series. It is shown that the modeling is quite effective when the time series data are chaotic and if they are taken from a regime of desynchronization. We discuss the applicability of the present approach as a diagnostic tool for voice pathologies.

Effect of prolonged free-walking fatigue on gait and physiological rhythm

This study examined the ways in which gait patterns and physiological rhythms such as those of muscle activity (tibialis anterior (TA) and biceps femoris (BF)) and cardiac activity are affected by the fatigue induced by prolonged free walking. Twelve normal subjects who walked for 3 h at their preferred pace were divided into two groups according to whether their mean gait cycle time (reciprocal of stride rate) during the second 90 min was higher (Group A: n=8) or lower (Group B: n=4) than that during the first 90 min. For Group A, the level of subjective fatigue during the walking task was significantly higher and the heart rate at rest was significantly lower than Group B. In Group A, prolonged walking significantly decreased the mean power frequency of the electromyography from TA, increased the variability of gait rhythm, decreased the largest Lyapunov exponent of the vertical component of back-waist acceleration, and decreased the amplitude of the vertical component of back-waist acceleration. Taking the onset timings of these changes into account, we propose that subjects who tire easily during prolonged walking first show local muscle fatigue at TA followed by instability of gait rhythm and then they slow their gait rhythm to enhance local dynamic stability. For both groups we constructed a physical fatigue index described by linear regression of gait and physiological variables. When we compared the subjective fatigue level with the fatigue level predicted using the index, we obtained a relatively high correlation coefficient for both groups (r=0.77).

Assessing spatial coupling in complex population dynamics using mutual prediction and continuity statistics

A number of important questions in ecology involve the possibility of interactions or "coupling" among potential components of ecological systems. The basic question of whether two components are coupled (exhibit dynamical interdependence) is relevant to investigations of movement of animals over space, population regulation, food webs and trophic interactions, and is also useful in the design of monitoring programs. For example, in spatially extended systems, coupling among populations in different locations implies the existence of redundant information in the system and the possibility of exploiting this redundancy in the development of spatial sampling designs. One approach to the identification of coupling involves study of the purported mechanisms linking system components. Another approach is based on time series of two potential components of the same system and, in previous ecological work, has relied on linear cross-correlation analysis. Here we present two different attractor-based approaches, continuity and mutual prediction, for determining the degree to which two population time series (e.g., at different spatial locations) are coupled. Both approaches are demonstrated on a one-dimensional predator-prey model system exhibiting complex dynamics. Of particular interest is the spatial asymmetry introduced into the model as linearly declining resource for the prey over the domain of the spatial coordinate. Results from these approaches are then compared to the more standard cross-correlation analysis. In contrast to cross-correlation, both continuity and mutual prediction are clearly able to discern the asymmetry in the flow of information through this system.

Routes to spatiotemporal chaos in the rheology of nematogenic fluids

With a view to understanding the "rheochaos" observed in recent experiments in a variety of orientable fluids, we study numerically the equations of motion of the spatiotemporal evolution of the traceless symmetric order parameter of a sheared nematogenic fluid. In particular we establish, by decisive numerical tests, that the irregular oscillatory behavior seen in a region of parameter space where the nematic is not stably flow-aligning is in fact spatiotemporal chaos. We outline the dynamical phase diagram of the model and study the route to the chaotic state. We find that spatiotemporal chaos in this system sets in via a regime of spatiotemporal intermittency, with a power-law distribution of the widths of laminar regions, as in H. Chaté and P. Manneville, Phys. Rev. Lett. 58, 112 (1987). Further, the evolution of the histogram of band sizes shows a growing length scale as one moves from the chaotic towards the flow-aligned phase. Finally we suggest possible experiments in which one can observe the intriguing behaviors discussed here.

Surrogate test to distinguish between chaotic and pseudoperiodic time series

In this paper a different algorithm is proposed to produce surrogates for pseudoperiodic time series. By imposing a few constraints on the noise components of pseudoperiodic data sets, we devise an effective method to generate surrogates. Unlike other algorithms, this method properly copes with pseudoperiodic orbits contaminated with linear colored observational noise. We will demonstrate the ability of this algorithm to distinguish chaotic orbits from pseudoperiodic orbits through simulation data sets from the Rössler system. As an example of application of this algorithm, we will also employ it to investigate a human electrocardiogram record.

Decreased nonlinear complexity and chaos during sleep in first episode schizophrenia: a preliminary report

Schizophrenia is characterized by disturbed sleep architecture. It has been thought that sleep abnormalities may underlie information processing deficits associated with this disorder. Nonlinear analyses of sleep data can provide valuable information on sleep characteristics that may be relevant to the functions of sleep. This study examined the predictability and nonlinear complexity of sleep EEG time series in two EEG channels (C4 and F4) using measures of nonlinearity, such as symbolic dynamics and the largest Lyapunov exponent (LLE) in schizophrenia. A series of antipsychotic naive patients with first episode of schizophrenia or schizoaffective disorder and age-matched healthy controls were studied during awake period, stage 1/2, slow wave sleep (stage 3/4) and rapid eye movement (REM) sleep. Nonlinearity scores were significantly lower during awake stage in patients compared to controls suggesting that there may be a diminished interplay between various parameters for the genesis of waking EEG. Symbolic dynamics and LLE were significantly lower in patients during REM compared to healthy controls, suggesting decreased nonlinear complexity of the EEG time series and diminished chaos in schizophrenia. Decreased nonlinear complexity was also correlated with neurocognitive deficits as assessed by the Wisconsin card sorting test. Diminished complexity of EEG time series during awake and REM sleep in patients with schizophrenia may underlie the impaired ability to process information in psychotic disorders such as schizophrenia.

"Dynamic" connectivity in neural systems: theoretical and empirical considerations

The study of functional interdependences between brain regions is a rapidly growing focus of neuroscience research. This endeavor has been greatly facilitated by the appearance of a number of innovative methodologies for the examination of neurophysiological and neuroimaging data. The aim of this article is to present an overview of dynamical measures of interdependence and contrast these with statistical measures that have been more widely employed. We first review the motivation, conceptual basis, and experimental approach of dynamical measures of interdependence and their application to the study of neural systems. A consideration of boot-strap "surrogate data" techniques, which facilitate hypothesis testing of dynamical measures, is then used to clarify the difference between dynamical and statistical measures of interdependence. An overview of some of the most active research areas such as the study of the "synchronization manifold," dynamical interdependence in neurophysiology data and the putative role of nonlinear desynchronization is then given. We conclude by suggesting that techniques based on dynamical interdependence--or "dynamical connectivity"--show significant potential for extracting meaningful information from functional neuroimaging data.

Nonlinear analysis of EEG during NREM sleep reveals changes in functional connectivity due to natural aging

The spatial organization of nonlinear interactions between different brain regions during the first NREM sleep stage is investigated. This is achieved via consideration of four bipolar electrode derivations, Fp1F3, Fp2F4, O1P3, O2P4, which are used to compare anterior and posterior interhemispheric interactions and left and right intrahemispheric interactions. Nonlinear interdependence is detected via application of a previously written algorithm, along with appropriately generated surrogate data sets. It is now well understood that the output of neural systems does not scale linearly with inputs received and, thus, the study of nonlinear interactions in EEG is crucial. This approach also offers significant advantages over standard linear techniques, in that the strength, direction, and topography of the interdependencies can all be calculated and considered. Previous research has linked delta activity during the first NREM sleep stage to performance on frontally activating tasks during waking hours. We demonstrate that nonlinear mechanisms are the driving force behind this delta activity. Furthermore, evidence is presented to suggest that the aging brain calls upon the right parietal region to assist the pre-frontal cortex. This is highlighted by statistically significant differences in the rates of interdependencies between the left pre-frontal cortex and the right parietal region when comparing younger subjects (<23 years) with older subjects (>60 years). This assistance has been observed in brain-imaging studies of sleep-deprived young adults, suggesting that similar mechanisms may play a role in the event of healthy aging. Additionally, the contribution to the delta rhythm via nonlinear mechanisms is observed to be greater in older subjects.

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

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

Measure of predictability

Many techniques have been developed to measure the difficulty of forecasting data from an observed time series. This paper introduces a measure which we call the "forecast entropy" designed to measure the predictability of a time series. We use attractors reconstructed from the time series and the distributions in the regular and tangent spaces of the data which comprise the attractor. We then consider these distributions on different scales. We present a formula for calculating the forecast entropy. To provide a standard of predictability, we define an idealized random system whose forecast entropy will be maximal; we then use this measure to rescale the forecast entropy to lie in the range [0,1]. The time series obtained from several chaotic systems as well as from a pseudorandom system are studied using this measure. We present evidence that the forecast entropy can be used as a tool for determining optimal delays and embedding dimensions used for reconstructing better attractors. We also show that the forecast entropy of a random system has completely different characteristics from that of a deterministic one.

PHASE SPACE RECONSTRUCTION AND NONLINEAR PREDICTIONS FOR STATIONARY AND NONSTATIONARY MARKOVIAN PROCESSES

We analyze prediction schemes for stochastic time series data. We propose that under certain conditions a scalar time series, obtained from a vector-valued Markov process, can be modeled as a finite memory Markov process in the observable. The transition rules of the process are easily computed using simple nonlinear time series predictors originally proposed for deterministic chaotic signals. The optimal time lag entering the embedding procedure is shown to be significantly smaller than in the deterministic case. The same concept can be extended to nonstationary stochastic processes, where an increase of the embedding dimension can help to identify instantaneous dynamical properties, and where redundant information in the past is exploited in an optimal way.

Using chaotic interrogation and attractor nonlinear cross-prediction error to detect fastener preload loss in an aluminum frame

Structural health monitoring is an important field concerned with assessing the current state (or "health") of a structural system or component with regard to its ability to perform its intended function appropriately. One approach to this problem is identifying appropriate features obtained from time series vibration responses of the structure that change as structural degradation occurs. In this work, we present a novel technique adapted from the nonlinear time series prediction community whereby the structure is excited by an applied chaotic waveform, and predictive maps built between structural response attractors are used as the feature space. The structural response is measured at several points on the structure, and pairs of attractors are used to predict each other. As the dynamics of the structure change due to damage, the prediction error rises. This approach is applied to detecting the preload loss in a bolted joint in an aluminum frame structure.

Optically injected laser system: characterization of chaos, bifurcation, and control

A single mode semiconductor laser subjected to optical injection, described by a set of three coupled nonlinear ordinary differential equations, exhibiting chaos is considered. By means of a recurrence analysis, quantification of the strange attractor is made. Analytical studies of the system using asymptotic averaging technique, derive certain conditions describing the prediction of 1-->2 bifurcation, which have subsequently been verified on numerical simulation. Furthermore, the locus of points on the parameter phase space representing Hopf bifurcation has been derived. The problem of control of chaos by a new procedure based on adaptive stabilization is also addressed. The results of such control are shown explicitly. Though this analysis deals with a very specific set of equations, the overall features that come out of the study remains valid for almost all laser systems.

Estimation of dynamical invariants without embedding by recurrence plots

In this paper we show that two dynamical invariants, the second order Renyi entropy and the correlation dimension, can be estimated from recurrence plots (RPs) with arbitrary embedding dimension and delay. This fact is interesting as these quantities are even invariant if no embedding is used. This is an important advantage of RPs compared to other techniques of nonlinear data analysis. These estimates for the correlation dimension and entropy are robust and, moreover, can be obtained at a low numerical cost. We exemplify our results for the Rossler system, the funnel attractor and the Mackey-Glass system. In the last part of the paper we estimate dynamical invariants for data from some fluid dynamical experiments and confirm previous evidence for low dimensional chaos in this experimental system.

Dimensional complexity of the EEG in patients with posttraumatic stress disorder

Recent electrophysiological studies have reported evidence of information processing abnormalities in patients with posttraumatic stress disorder (PTSD). The aim of this study is to examine dynamical complexity of the EEG in PTSD patients, which is thought to reflect information processing of the brain. Resting EEG recordings (32,800 data points acquired continuously from 82 s of an EEG record) were obtained in 16 channels of 27 patients with PTSD from a mixed civilian trauma population and 14 healthy subjects. The correlation dimension (D2) of the EEG was used to quantify the complexity of the cortical dynamics underlying the EEG signal. The PTSD patients were found to have lower D2 values than those of the healthy subjects in most channels (Fp1, F8, C4, P4, T3, T4, T5, T6, and O1), indicating that PTSD patients have globally reduced complexity in their EEG waveforms. This study supports the hypotheses that PTSD patients exhibit disturbed cortical information processing, and that non-linear dynamical analysis of the EEG can be a tool for detecting changes in neurodynamics of the brain in PTSD.

Aperiodic flow-induced oscillations of collapsible tubes: a critical reappraisal

The evidence for the aperiodic self-excited oscillations of flow-conveying collapsible tubes being mathematically chaotic is re-examined. Many cases which powerfully suggest nonlinear deterministic behaviour have not been recorded over time-spans which allow their exhaustive examination. The present investigation centred on a previously recorded robust and generic oscillation, but more recent and more discerning tests were applied. Despite hints that a low embedding dimension might suffice, the data appeared on most indices high-dimensional. A U-shaped return map was found and modelled using both radial basis functions and polynomials, but lack of detailed structure in the map prevented effective parameter estimation. On the basis of power-law rather than exponential divergence of nearby trajectories, and of inability to discriminate against behaviour which would also be manifested by a surrogate consisting of a noise-perturbed nonlinear periodic oscillator, it is concluded that the data do not support the idea that the aperiodicity in the particular oscillation examined is caused by deterministic chaos. There was evidence that the distributed nature of the physical system might underlie aspects of the high dimensionality. We advocate equally searching testing of any future candidate chaotic oscillations in the investigation of collapsed-tube flows.

Detecting local synchronization in coupled chaotic systems

We introduce a technique to detect and quantify local functional dependencies between coupled chaotic systems. The method estimates the fraction of locally synchronized configurations, in a pair of signals with an arbitrary state of global synchronization. Application to a pair of interacting Rössler oscillators shows that our method is able to quantify the number of dynamical configurations where a local prediction task is possible, as well as in the absence of global synchronization features.

Spatiotemporal rheochaos in nematic hydrodynamics

Motivated by recent observations of rheochaos in sheared wormlike micelles, we study the coupled nonlinear partial differential equations for the hydrodynamic velocity and order-parameter fields in a sheared nematogenic fluid. In a suitable parameter range, we find irregular, dynamic shear banding and establish by decisive numerical tests that the chaos we observe in the model is spatiotemporal in nature.

Evaluation of different measures of functional connectivity using a neural mass model

We use a neural mass model to address some important issues in characterising functional integration among remote cortical areas using magnetoencephalography or electroencephalography (MEG or EEG). In a previous paper [Neuroimage (in press)], we showed how the coupling among cortical areas can modulate the MEG or EEG spectrum and synchronise oscillatory dynamics. In this work, we exploit the model further by evaluating different measures of statistical dependencies (i.e., functional connectivity) among MEG or EEG signals that are mediated by neuronal coupling. We have examined linear and nonlinear methods, including phase synchronisation. Our results show that each method can detect coupling but with different sensitivity profiles that depended on (i) the frequency specificity of the interaction (broad vs. narrow band) and (ii) the nature of the coupling (linear vs. nonlinear). Our analyses suggest that methods based on the concept of generalised synchronisation are the most sensitive when interactions encompass different frequencies (broadband analyses). In the context of narrow-band analyses, mutual information was found to be the most sensitive way to disclose frequency-specific couplings. Measures based on generalised synchronisation and phase synchronisation are the most sensitive to nonlinear coupling. These different sensitivity profiles mean that the choice of coupling measures can have dramatic effects on the cortical networks identified. We illustrate this using a single-subject MEG study of binocular rivalry and highlight the greater recovery of statistical dependencies among cortical areas in the beta band when mutual information is used.

Controlling system dimension: a class of real systems that obey the Kaplan-Yorke conjecture

The Kaplan-Yorke conjecture suggests a simple relationship between the fractal dimension of a system and its Lyapunov spectrum. This relationship has important consequences in the broad field of nonlinear dynamics where dimension and Lyapunov exponents are frequently used descriptors of system dynamics. We develop an experimental system with controllable dimension by making use of the Kaplan-Yorke conjecture. A rectangular steel plate is driven with the output of a chaotic oscillator. We controlled the Lyapunov exponents of the driving and then computed the fractal dimension of the plate's response. The Kaplan-Yorke relationship predicted the system's dimension extremely well. This finding strongly suggests that other driven linear systems will behave similarly. The ability to control the dimension of a structure's vibrational response is important in the field of vibration-based structural health monitoring for the robust extraction of damage-sensitive features.

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

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

A disturbance of nonlinear interdependence in scalp EEG of subjects with first episode schizophrenia

It has been proposed that schizophrenia arises through a disturbance of coupling between large-scale cortical systems. This "disconnection hypothesis" is tested by applying a measure of dynamical interdependence to scalp EEG data. EEG data were collected from 40 subjects with a first episode of schizophrenia and 40 matched healthy controls. An algorithm for the detection of dynamical interdependence was applied to six pairs of bipolar electrodes in each subject. The topographic organization of the interdependence was calculated and served as the principle measure of cortical integration. The rate of occurrence of dynamical interdependence did not statistically differ between subject groups at any of the sites. However, the topography across the scalp was significantly different between the two groups. Specifically, nonlinear interdependence tended to occur in larger concurrent "clusters" across the scalp in schizophrenia than in the healthy subjects. This disturbance was reflected most strongly in left intrahemispheric coupling and did not differ significantly according to symptomatology. Medication dose and subject arousal were not observed to be confounding factors. The study of dynamical interdependence in scalp EEG data does not support a straightforward interpretation of the disconnection hypothesis-that there is a decrease in the strength of functional coupling between adjacent cortical regions. Rather, it suggests a dysregulation in the organization of dynamical interactions across supraregional brain systems.

Quasicycles revisited: apparent sensitivity to initial conditions

Environmental noise is known to sustain cycles by perturbing a deterministic approach to equilibrium that is itself oscillatory. Quasicycles produced in this way display a regular period but varied amplitude. They were proposed by Nisbet and Gurney (Nature 263 (1976) 319) as one possible explanation for population fluctuations in nature. Here, we revisit quasicyclic dynamics from the perspective of nonlinear time series analysis. Time series are generated with a predator-prey model whose prey's growth rate is driven by environmental noise. A method for the analysis of short and noisy data provides evidence for sensitivity to initial conditions, with a global Lyapunov exponent often close to zero characteristic of populations 'at the edge of chaos'. Results with methods restricted to long time series are consistent with a finite-dimensional attractor on which dynamics are sensitive to initial conditions. These results are compared with those previously obtained for quasicycles in an individual-based model with heterogeneous spatial distributions. Patterns of sensitivity to initial conditions are shown to differentiate phase-forgetting from phase-remembering quasicycles involving a periodic driver. The previously reported mode at zero of Lyapunov exponents in field and laboratory populations may reflect, in part, quasicyclic dynamics.