Re-examination of the evidence for low-dimensional, nonlinear structure in the human electroencephalogram

Cautionary Observations Concerning the Introduction of Psychophysiological Biomarkers into Neuropsychiatric Practice

The combination of statistical learning technologies with large databases of psychophysiological data has appropriately generated enthusiastic interest in future clinical applicability. It is argued here that this enthusiasm should be tempered with the understanding that significant obstacles must be overcome before the systematic introduction of psychophysiological measures into neuropsychiatric practice becomes possible. The objective of this study is to identify challenges to this effort. The nonspecificity of psychophysiological measures complicates their use in diagnosis. Low test-retest reliability complicates use in longitudinal assessment, and quantitative psychophysiological measures can normalize in response to placebo intervention. Ten cautionary observations are introduced and, in some instances, possible directions for remediation are suggested.

Random Neural Network Based Epileptic Seizure Episode Detection Exploiting Electroencephalogram Signals

Epileptic seizures are caused by abnormal electrical activity in the brain that manifests itself in a variety of ways, including confusion and loss of awareness. Correct identification of epileptic seizures is critical in the treatment and management of patients with epileptic disorders. One in four patients present resistance against seizures episodes and are in dire need of detecting these critical events through continuous treatment in order to manage the specific disease. Epileptic seizures can be identified by reliably and accurately monitoring the patients' neuro and muscle activities, cardiac activity, and oxygen saturation level using state-of-the-art sensing techniques including electroencephalograms (EEGs), electromyography (EMG), electrocardiograms (ECGs), and motion or audio/video recording that focuses on the human head and body. EEG analysis provides a prominent solution to distinguish between the signals associated with epileptic episodes and normal signals; therefore, this work aims to leverage on the latest EEG dataset using cutting-edge deep learning algorithms such as random neural network (RNN), convolutional neural network (CNN), extremely random tree (ERT), and residual neural network (ResNet) to classify multiple variants of epileptic seizures from non-seizures. The results obtained highlighted that RNN outperformed all other algorithms used and provided an overall accuracy of 97%, which was slightly improved after cross validation.

Network-based brain computer interfaces: principles and applications

Brain-computer interfaces (BCIs) make possible to interact with the external environment by decoding the mental intention of individuals. BCIs can therefore be used to address basic neuroscience questions but also to unlock a variety of applications from exoskeleton control to neurofeedback (NFB) rehabilitation. In general, BCI usability critically depends on the ability to comprehensively characterize brain functioning and correctly identify the user's mental state. To this end, much of the efforts have focused on improving the classification algorithms taking into account localized brain activities as input features. Despite considerable improvement BCI performance is still unstable and, as a matter of fact, current features represent oversimplified descriptors of brain functioning. In the last decade, growing evidence has shown that the brain works as a networked system composed of multiple specialized and spatially distributed areas that dynamically integrate information. While more complex, looking at how remote brain regions functionally interact represents a grounded alternative to better describe brain functioning. Thanks to recent advances in network science, i.e. a modern field that draws on graph theory, statistical mechanics, data mining and inferential modelling, scientists have now powerful means to characterize complex brain networks derived from neuroimaging data. Notably, summary features can be extracted from these networks to quantitatively measure specific organizational properties across a variety of topological scales. In this topical review, we aim to provide the state-of-the-art supporting the development of a network theoretic approach as a promising tool for understanding BCIs and improve usability.

Complex temporal patterns processing by a neural mass model of a cortical column

It is well known that neuronal networks are capable of transmitting complex spatiotemporal information in the form of precise sequences of neuronal discharges characterized by recurrent patterns. At the same time, the synchronized activity of large ensembles produces local field potentials that propagate through highly dynamic oscillatory waves, such that, at the whole brain scale, complex spatiotemporal dynamics of electroencephalographic (EEG) signals may be associated to sensorimotor decision making processes. Despite these experimental evidences, the link between highly temporally organized input patterns and EEG waves has not been studied in detail. Here, we use a neural mass model to investigate to what extent precise temporal information, carried by deterministic nonlinear attractor mappings, is filtered and transformed into fluctuations in phase, frequency and amplitude of oscillatory brain activity. The phase shift that we observe, when we drive the neural mass model with specific chaotic inputs, shows that the local field potential amplitude peak appears in less than one full cycle, thus allowing traveling waves to encode temporal information. After converting phase and amplitude changes obtained into point processes, we quantify input-output similarity following a threshold-filtering algorithm onto the amplitude wave peaks. Our analysis shows that the neural mass model has the capacity for gating the input signal and propagate selected temporal features of that signal. Finally, we discuss the effect of local excitatory/inhibitory balance on these results and how excitability in cortical columns, controlled by neuromodulatory innervation of the cerebral cortex, may contribute to set a fine tuning and gating of the information fed to the cortex.

Parameter estimation and identifiability in a neural population model for electro-cortical activity

Electroencephalography (EEG) provides a non-invasive measure of brain electrical activity. Neural population models, where large numbers of interacting neurons are considered collectively as a macroscopic system, have long been used to understand features in EEG signals. By tuning dozens of input parameters describing the excitatory and inhibitory neuron populations, these models can reproduce prominent features of the EEG such as the alpha-rhythm. However, the inverse problem, of directly estimating the parameters from fits to EEG data, remains unsolved. Solving this multi-parameter non-linear fitting problem will potentially provide a real-time method for characterizing average neuronal properties in human subjects. Here we perform unbiased fits of a 22-parameter neural population model to EEG data from 82 individuals, using both particle swarm optimization and Markov chain Monte Carlo sampling. We estimate how much is learned about individual parameters by computing Kullback-Leibler divergences between posterior and prior distributions for each parameter. Results indicate that only a single parameter, that determining the dynamics of inhibitory synaptic activity, is directly identifiable, while other parameters have large, though correlated, uncertainties. We show that the eigenvalues of the Fisher information matrix are roughly uniformly spaced over a log scale, indicating that the model is sloppy, like many of the regulatory network models in systems biology. These eigenvalues indicate that the system can be modeled with a low effective dimensionality, with inhibitory synaptic activity being prominent in driving system behavior.

Entropy-Based Quantitative Electroencephalogram Analysis for Diagnosing Attention-Deficit Hyperactivity Disorder in Girls

Diagnosis of attention-deficit hyperactivity disorder (ADHD) is currently based on core symptoms or checklists; however, the inevitability of practitioner subjectivity leads to over- and underdiagnosis. Although the Federal Drug Administration has approved an elevated theta/beta ratio (TBR) of the electroencephalogram (EEG) band as a tool for assisting ADHD diagnosis, several studies have reported no significant differences of the TBR between ADHD and control subjects. This study detailed the development of a method based on approximate entropy (ApEn) analysis of EEG to compare ADHD and control groups. Differences between ADHD presentation in boys and girls indicate the necessity of separate investigations. This study enrolled 30 girls with ADHD and 30 age-matched controls. The results revealed significantly higher ApEn values in most brain areas in the control group than in the ADHD group. Compared with TBR-related feature descriptors, ApEn-related feature descriptors can produce the higher average true positive rate (0.846), average true negative rate (0.814), average accuracy (0.817), and average area under the receiver operating characteristic curve value (0.862). Therefore, compared with TBR, ApEn possessed the better potential for differentiating between girls with ADHD and controls.

Presaccadic EEG activity predicts visual saliency in free-viewing contour integration

While viewing a scene, the eyes are attracted to salient stimuli. We set out to identify the brain signals controlling this process. In a contour integration task, in which participants searched for a collinear contour in a field of randomly oriented Gabor elements, a previously established model was applied to calculate a visual saliency value for each fixation location. We studied brain activity related to the modeled saliency values, using coregistered eye tracking and EEG. To disentangle EEG signals reflecting salience in free viewing from overlapping EEG responses to sequential eye movements, we adopted generalized additive mixed modeling (GAMM) to single epochs of saccade-related EEG. We found that, when saliency at the next fixation location was high, amplitude of the presaccadic EEG activity was low. Since presaccadic activity reflects covert attention to the saccade target, our results indicate that larger attentional effort is needed for selecting less salient saccade targets than more salient ones. This effect was prominent in contour-present conditions (half of the trials), but ambiguous in the contour-absent condition. Presaccadic EEG activity may thus be indicative of bottom-up factors in saccade guidance. The results underscore the utility of GAMM for EEG-eye movement coregistration research.

Information-theoretic model selection for optimal prediction of stochastic dynamical systems from data

In the absence of mechanistic or phenomenological models of real-world systems, data-driven models become necessary. The discovery of various embedding theorems in the 1980s and 1990s motivated a powerful set of tools for analyzing deterministic dynamical systems via delay-coordinate embeddings of observations of their component states. However, in many branches of science, the condition of operational determinism is not satisfied, and stochastic models must be brought to bear. For such stochastic models, the tool set developed for delay-coordinate embedding is no longer appropriate, and a new toolkit must be developed. We present an information-theoretic criterion, the negative log-predictive likelihood, for selecting the embedding dimension for a predictively optimal data-driven model of a stochastic dynamical system. We develop a nonparametric estimator for the negative log-predictive likelihood and compare its performance to a recently proposed criterion based on active information storage. Finally, we show how the output of the model selection procedure can be used to compare candidate predictors for a stochastic system to an information-theoretic lower bound.

Economy, Movement Dynamics, and Muscle Activity of Human Walking at Different Speeds

The complex behaviour of human walking with respect to movement variability, economy and muscle activity is speed dependent. It is well known that a U-shaped relationship between walking speed and economy exists. However, it is an open question if the movement dynamics of joint angles and centre of mass and muscle activation strategy also exhibit a U-shaped relationship with walking speed. We investigated the dynamics of joint angle trajectories and the centre of mass accelerations at five different speeds ranging from 20 to 180% of the predicted preferred speed (based on Froude speed) in twelve healthy males. The muscle activation strategy and walking economy were also assessed. The movement dynamics was investigated using a combination of the largest Lyapunov exponent and correlation dimension. We observed an intermediate stage of the movement dynamics of the knee joint angle and the anterior-posterior and mediolateral centre of mass accelerations which coincided with the most energy-efficient walking speed. Furthermore, the dynamics of the joint angle trajectories and the muscle activation strategy was closely linked to the functional role and biomechanical constraints of the joints.

Multifractal surrogate-data generation algorithm that preserves pointwise Hölder regularity structure, with initial applications to turbulence

An algorithm is described that can generate random variants of a time series while preserving the probability distribution of original values and the pointwise Hölder regularity. Thus, it preserves the multifractal properties of the data. Our algorithm is similar in principle to well-known algorithms based on the preservation of the Fourier amplitude spectrum and original values of a time series. However, it is underpinned by a dual-tree complex wavelet transform rather than a Fourier transform. Our method, which we term the iterated amplitude adjusted wavelet transform can be used to generate bootstrapped versions of multifractal data, and because it preserves the pointwise Hölder regularity but not the local Hölder regularity, it can be used to test hypotheses concerning the presence of oscillating singularities in a time series, an important feature of turbulence and econophysics data. Because the locations of the data values are randomized with respect to the multifractal structure, hypotheses about their mutual coupling can be tested, which is important for the velocity-intermittency structure of turbulence and self-regulating processes.

On the estimation of phase synchronization, spurious synchronization and filtering

Phase synchronization, viz., the adjustment of instantaneous frequencies of two interacting self-sustained nonlinear oscillators, is frequently used for the detection of a possible interrelationship between empirical data recordings. In this context, the proper estimation of the instantaneous phase from a time series is a crucial aspect. The probability that numerical estimates provide a physically relevant meaning depends sensitively on the shape of its power spectral density. For this purpose, the power spectrum should be narrow banded possessing only one prominent peak [M. Chavez et al., J. Neurosci. Methods 154, 149 (2006)]. If this condition is not fulfilled, band-pass filtering seems to be the adequate technique in order to pre-process data for a posterior synchronization analysis. However, it was reported that band-pass filtering might induce spurious synchronization [L. Xu et al., Phys. Rev. E 73, 065201(R), (2006); J. Sun et al., Phys. Rev. E 77, 046213 (2008); and J. Wang and Z. Liu, EPL 102, 10003 (2013)], a statement that without further specification causes uncertainty over all measures that aim to quantify phase synchronization of broadband field data. We show by using signals derived from different test frameworks that appropriate filtering does not induce spurious synchronization. Instead, filtering in the time domain tends to wash out existent phase interrelations between signals. Furthermore, we show that measures derived for the estimation of phase synchronization like the mean phase coherence are also useful for the detection of interrelations between time series, which are not necessarily derived from coupled self-sustained nonlinear oscillators.

On the dynamics of ocean ambient noise: Two decades later

Two decades ago, it was shown that ambient noise exhibits low dimensional chaotic behavior. Recent new techniques in nonlinear science can effectively detect the underlying dynamics in noisy time series. In this paper, the presence of low dimensional deterministic dynamics in ambient noise is investigated using diverse nonlinear techniques, including correlation dimension, Lyapunov exponent, nonlinear prediction, and entropy based methods. The consistent interpretation of different methods demonstrates that ambient noise can be best modeled as nonlinear stochastic dynamics, thus rejecting the hypothesis of low dimensional chaotic behavior. The ambient noise data utilized in this study are of duration 60 s measured at South China Sea.

Procedures for the Comparative Testing of Noninvasive Neuroassessment Devices

A sequential process for comparison testing of noninvasive neuroassessment devices is presented. Comparison testing of devices in a clinical population should be preceded by computational research and reliability testing with healthy populations, as opposed to proceeding immediately to testing with clinical participants. A five-step process is outlined as follows: 1. Complete a preliminary literature review identifying candidate measures. 2. Conduct systematic simulation studies to determine the computational properties and data requirements of candidate measures. 3. Establish the test-retest reliability of each measure in a healthy comparison population and the clinical population of interest. 4. Investigate the clinical validity of reliable measures in appropriately defined clinical populations. 5. Complete device usability assessment (weight, simplicity of use, cost effectiveness, ruggedness) only for devices and measures that are promising after steps 1 through 4 are completed. Usability may be considered throughout the device evaluation process but such considerations are subordinate to the higher priorities addressed in steps 1 through 4.

How to estimate the correlation dimension of high-dimensional signals?

The paper presents improvements to the Takens-Ellner (TE) algorithm estimating the correlation dimension (d) of high-dimensional signals. The signal being the sum of 4 Lorenz signals and possessing the correlation dimension d approximately equal to 8 was analyzed. The conversion of TE to the classic Grassberger-Proccacia (GP) algorithm is presented that shows the advantage of TE over the GP algorithm. The maximal d estimated for the given number of points in phase space is significantly higher for the TE algorithm than for the GP algorithm. The formula for the precision of individual d estimation is presented. The paper shows, how to estimate the distance corresponding to the end of the Linear Scaling Region in the correlation integral function, even before starting the procedure of d estimation. It makes it possible to reject the majority of longer distances from the analysis reducing the computation time considerably.

A complex systems analysis of stick-slip dynamics of a laboratory fault

We study the stick-slip behavior of a granular bed of photoelastic disks sheared by a rough slider pulled along the surface. Time series of a proxy for granular friction are examined using complex systems methods to characterize the observed stick-slip dynamics of this laboratory fault. Nonlinear surrogate time series methods show that the stick-slip behavior appears more complex than a periodic dynamics description. Phase space embedding methods show that the dynamics can be locally captured within a four to six dimensional subspace. These slider time series also provide an experimental test for recent complex network methods. Phase space networks, constructed by connecting nearby phase space points, proved useful in capturing the key features of the dynamics. In particular, network communities could be associated to slip events and the ranking of small network subgraphs exhibited a heretofore unreported ordering.

Detection of weak chaos in infant respiration

This paper concerns the application of newly developed methods for decomposition of an infant respiratory signal into locally stable nonsinusoidal periodic components. Each estimated component has dynamical variation in its three periodicity attributes, i.e., periodicity, scaling factors, and the waveform or pattern associated with the successive segments. Earlier, it has been reported with the application of conventional surrogate analysis and with the cylindrical basis function modeling that the underlying system is distinctly different from linearly filtered Gaussian process, and most probably the human respiratory system behaves as a nonlinear periodic oscillator with two or three degrees of freedom being driven by a high-dimensional noise source. Here, the surrogate analysis is extended and four new types of nonlinear surrogates have been proposed, which are produced by randomizing one or multiple periodicity attributes while preserving certain individual relationships. In this way, a new type of dissection of dynamics is possible, which can lead to a proper understanding of couplings between different controlling parameters.

IDENTIFICATION OF NONLINEAR OSCILLATORY ACTIVITY EMBEDDED IN BROADBAND NEURAL SIGNALS

Oscillatory phenomena in the brain activity and their synchronization are frequently studied using mathematical models and analytic tools derived from nonlinear dynamics. In many experimental situations, however, neural signals have a broadband character and if oscillatory activity is present, its dynamical origin is unknown. To cope with these problems, a framework for detecting nonlinear oscillatory activity in broadband time series is presented. First, a narrow-band oscillatory mode is extracted from a broadband background. Second, it is tested whether the extracted mode is significantly different from linearly filtered noise, modelled as a linear stochastic process possibly passed through a static nonlinear transformation. If a nonlinear oscillatory mode is positively detected, further analysis using nonlinear approaches such as the phase synchronization analysis can potentially bring new information. For linear processes, however, standard approaches such as the coherence analysis are more appropriate and provide sufficient description of underlying interactions with smaller computational effort. The method is illustrated in a numerical example and applied to analyze experimentally obtained human EEG time series from a sleeping subject.

Evolution of genuine cross-correlation strength of focal onset seizures

To quantify the evolution of genuine zero-lag cross-correlations of focal onset seizures, we apply a recently introduced multivariate measure to broad band and to narrow-band EEG data. For frequency components below 12.5 Hz, the strength of genuine cross-correlations decreases significantly during the seizure and the immediate postseizure period, while higher frequency bands show a tendency of elevated cross-correlations during the same period. We conclude that in terms of genuine zero-lag cross-correlations, the electrical brain activity as assessed by scalp electrodes shows a significant spatial fragmentation, which might promote seizure offset.

New insights into anterior cruciate ligament deficiency and reconstruction through the assessment of knee kinematic variability in terms of nonlinear dynamics

PURPOSE

Injuries to the anterior cruciate ligament (ACL) occur frequently, particularly in young adult athletes, and represent the majority of the lesions of knee ligaments. Recent investigations suggest that the assessment of kinematic variability using measures of nonlinear dynamics can provide with important insights with respect to physiological and pathological states. The purpose of the present article was to critically review and synthesize the literature addressing ACL deficiency and reconstruction from a nonlinear dynamics standpoint.

METHODS

A literature search was carried out in the main medical databases for studies published between 1990 and 2010.

RESULTS

Seven studies investigated knee kinematic variability in ACL patients. Results provided support for the theory of "optimal movement variability". Practically, loss below optimal variability is associated with a more rigid and very repeatable movement pattern, as observed in the ACL-deficient knee. This is a state of low complexity and high predictability. On the other hand, increase beyond optimal variability is associated with a noisy and irregular movement pattern, as found in the ACL-reconstructed knee, regardless of which type of graft is used. This is a state of low complexity and low predictability. In both cases, the loss of optimal variability and the associated high complexity lead to an incapacity to respond appropriately to the environmental demands, thus providing an explanation for vulnerability to pathological changes following injury.

CONCLUSION

Subtle fluctuations that appear in knee kinematic patterns provide invaluable insight into the health of the neuromuscular function after ACL rupture and reconstruction. It is thus critical to explore them in longitudinal studies and utilize nonlinear measures as an important component of post-reconstruction medical assessment.

LEVEL OF EVIDENCE

II.

Uniform approach to linear and nonlinear interrelation patterns in multivariate time series

Currently, a variety of linear and nonlinear measures is in use to investigate spatiotemporal interrelation patterns of multivariate time series. Whereas the former are by definition insensitive to nonlinear effects, the latter detect both nonlinear and linear interrelation. In the present contribution we employ a uniform surrogate-based approach, which is capable of disentangling interrelations that significantly exceed random effects and interrelations that significantly exceed linear correlation. The bivariate version of the proposed framework is explored using a simple model allowing for separate tuning of coupling and nonlinearity of interrelation. To demonstrate applicability of the approach to multivariate real-world time series we investigate resting state functional magnetic resonance imaging (rsfMRI) data of two healthy subjects as well as intracranial electroencephalograms (iEEG) of two epilepsy patients with focal onset seizures. The main findings are that for our rsfMRI data interrelations can be described by linear cross-correlation. Rejection of the null hypothesis of linear iEEG interrelation occurs predominantly for epileptogenic tissue as well as during epileptic seizures.

Complexity measures of brain wave dynamics

To understand the nature of brain dynamics as well as to develop novel methods for the diagnosis of brain pathologies, recently, a number of complexity measures from information theory, chaos theory, and random fractal theory have been applied to analyze the EEG data. These measures are crucial in quantifying the key notions of neurodynamics, including determinism, stochasticity, causation, and correlations. Finding and understanding the relations among these complexity measures is thus an important issue. However, this is a difficult task, since the foundations of information theory, chaos theory, and random fractal theory are very different. To gain significant insights into this issue, we carry out a comprehensive comparison study of major complexity measures for EEG signals. We find that the variations of commonly used complexity measures with time are either similar or reciprocal. While many of these relations are difficult to explain intuitively, all of them can be readily understood by relating these measures to the values of a multiscale complexity measure, the scale-dependent Lyapunov exponent, at specific scales. We further discuss how better indicators for epileptic seizures can be constructed.

Partial seizures are associated with early increases in signal complexity

OBJECTIVES

Partial seizures are often believed to be associated with EEG signals of low complexity because seizures are associated with increased neural network synchrony. The investigations reported here provide an assessment of the signal complexity of epileptic seizure onsets using newly developed quantitative measures.

METHODS

Using the Gabor atom density (GAD) measure of signal complexity, 339 partial seizures in 45 patients with intracranial electrode arrays were analyzed. Segmentation procedures were applied to determine the timing and amplitude of GAD changes relative to the electrographic onset of the seizure.

RESULTS

Three hundred and thirty out of 339 seizures have significant complexity level changes, with 319 (97%) having an increase in complexity. GAD increases occur within seconds of the onset of the partial seizure but are not observed in channels remote from the focus. The complexity increase is similar for seizures from mesial temporal origin, neocortical temporal and extra-temporal origin.

CONCLUSIONS

Partial onset seizures are associated with early increases in signal complexity as measured by GAD. This increase is independent of the location of the seizure focus.

SIGNIFICANCE

Despite the often predominant rhythmic activity that characterizes onset and early evolution of epileptic seizures, partial seizure onset is associated with an early increase in complexity. These changes are common to partial seizures originating from different brain regions, indicating a similar seizure dynamic.

Quantification of brain macrostates using dynamical nonstationarity of physiological time series

The brain shows complex, nonstationarity temporal dynamics, with abrupt micro- and macrostate transitions during its information processing. Detecting and characterizing these transitions in dynamical states of the brain is a critical issue in the field of neuroscience and psychiatry. In the current study, a novel method is proposed to quantify brain macrostates (e.g., sleep stages or cognitive states) from shifts of dynamical microstates or dynamical nonstationarity. A ``dynamical microstate'' is a temporal unit of the information processing in the brain with fixed dynamical parameters and specific spatial distribution. In this proposed approach, a phase-space-based dynamical dissimilarity map (DDM) is used to detect transitions between dynamically stationary microstates in the time series, and Tsallis time-dependent entropy is applied to quantify dynamical patterns of transitions in the DDM. We demonstrate that the DDM successfully detects transitions between microstates of different temporal dynamics in the simulated physiological time series against high levels of noise. Based on the assumption of nonlinear, deterministic brain dynamics, we also demonstrate that dynamical nonstationarity analysis is useful to quantify brain macrostates (sleep stages I, II, III, IV, and rapid eye movement (REM) sleep) from sleep EEGs with an overall accuracy of 77%. We suggest that dynamical nonstationarity is a useful tool to quantify macroscopic mental states (statistical integration) of the brain using dynamical transitions at the microscopic scale in physiological data.

Sample entropy tracks changes in electroencephalogram power spectrum with sleep state and aging

The regularity of electroencephalogram signals was compared between middle-aged (47.2 +/- 2.0 years) and elderly (78.4 +/- 3.8 years) female subjects in wake, nonrapid eye movement stages 2 and 3 (S-2, S-3), and rapid eye movement sleep. Signals from C3A2 leads of healthy subjects, acquired from polysomnograms obtained from the Sleep Heart Health Study, were analyzed using both sample entropy (SaEn) and power spectral analysis (delta, theta, alpha, and beta frequency band powers). SaEn changed systematically and significantly (P < 0.001) with sleep state in both age groups, following the relationships wake > rapid eye movement > S-2 > S-3. SaEn was found to be negatively correlated with delta power and positively correlated with beta power. Small changes in SaEn seem to reflect changes in spectral content rather than changes in regularity of the signal. A better predictor of SaEn than the frequency band powers was the logarithm of the power ratio (alpha + beta)/(delta + theta). Thus, SaEn seems to reflect the balance between sleep-promoting and alertness-promoting mechanisms. SaEn of the elderly was larger than that of middle-aged subjects in S-2 (P = 0.029) and rapid eye movement (P = 0.001), suggesting that cortical state is shifted toward alertness in elderly subjects in these sleep states compared with the middle-aged subjects.

Deterministic dynamics of neural activity during absence seizures in rats

The study of brain electrical activities in terms of deterministic nonlinear dynamics has recently received much attention. Forbidden ordinal patterns (FOP) is a recently proposed method to investigate the determinism of a dynamical system through the analysis of intrinsic ordinal properties of a nonstationary time series. The advantages of this method in comparison to others include simplicity and low complexity in computation without further model assumptions. In this paper, the FOP of the EEG series of genetic absence epilepsy rats from Strasbourg was examined to demonstrate evidence of deterministic dynamics during epileptic states. Experiments showed that the number of FOP of the EEG series grew significantly from an interictal to an ictal state via a preictal state. These findings indicated that the deterministic dynamics of neural networks increased significantly in the transition from the interictal to the ictal states and also suggested that the FOP measures of the EEG series could be considered as a predictor of absence seizures.

Brain-machine interactions for assessing the dynamics of neural systems

A critical advance for brain–machine interfaces is the establishment of bi-directional communications between the nervous system and external devices. However, the signals generated by a population of neurons are expected to depend in a complex way upon poorly understood neural dynamics. We report a new technique for the identification of the dynamics of a neural population engaged in a bi-directional interaction with an external device. We placed in vitro preparations from the lamprey brainstem in a closed-loop interaction with simulated dynamical devices having different numbers of degrees of freedom. We used the observed behaviors of this composite system to assess how many independent parameters − or state variables − determine at each instant the output of the neural system. This information, known as the dynamical dimension of a system, allows predicting future behaviors based on the present state and the future inputs. A relevant novelty in this approach is the possibility to assess a computational property – the dynamical dimension of a neuronal population – through a simple experimental technique based on the bi-directional interaction with simulated dynamical devices. We present a set of results that demonstrate the possibility of obtaining stable and reliable measures of the dynamical dimension of a neural preparation.

Mental states as macrostates emerging from brain electrical dynamics

Psychophysiological correlations form the basis for different medical and scientific disciplines, but the nature of this relation has not yet been fully understood. One conceptual option is to understand the mental as "emerging" from neural processes in the specific sense that psychology and physiology provide two different descriptions of the same system. Stating these descriptions in terms of coarser- and finer-grained system states (macro- and microstates), the two descriptions may be equally adequate if the coarse-graining preserves the possibility to obtain a dynamical rule for the system. To test the empirical viability of our approach, we describe an algorithm to obtain a specific form of such a coarse-graining from data, and illustrate its operation using a simulated dynamical system. We then apply the method to an electroencephalographic recording, where we are able to identify macrostates from the physiological data that correspond to mental states of the subject.

Nonlinear analysis and modeling of cortical activation and deactivation patterns in the immature fetal electrocorticogram

An approach combining time-continuous nonlinear stability analysis and a parametric bispectral method was introduced to better describe cortical activation and deactivation patterns in the immature fetal electroencephalogram (EEG). Signal models and data-driven investigations were performed to find optimal parameters of the nonlinear methods and to confirm the occurrence of nonlinear sections in the fetal EEG. The resulting measures were applied to the in utero electrocorticogram (ECoG) of fetal sheep at 0.7 gestation when organized sleep states were not developed and compared to previous results at 0.9 gestation. Cycling of the nonlinear stability of the fetal ECoG occurred already at this early gestational age, suggesting the presence of premature sleep states. This was accompanied by cycling of the time-variant biamplitude which reflected ECoG synchronization effects during premature sleep states associated with nonrapid eye movement sleep later in gestation. Thus, the combined nonlinear and time-variant approach was able to provide important insights into the properties of the immature fetal ECoG.

Nonlinear analysis of electroencephalogram and magnetoencephalogram recordings in patients with Alzheimer's disease

The aim of the present study is to show the usefulness of nonlinear methods to analyse the electroencephalogram (EEG) and magnetoencephalogram (MEG) in patients with Alzheimer's disease (AD). The following nonlinear methods have been applied to study the EEG and MEG background activity in AD patients and control subjects: approximate entropy, sample entropy, multiscale entropy, auto-mutual information and Lempel-Ziv complexity. We discuss why these nonlinear methods are appropriate to analyse the EEG and MEG. Furthermore, the performance of all these methods has been compared when applied to the same databases of EEG and MEG recordings. Our results show that EEG and MEG background activities in AD patients are less complex and more regular than in healthy control subjects. In line with previous studies, our work suggests that nonlinear analysis techniques could be useful in AD diagnosis.

An investigation of EEG dynamics in an animal model of temporal lobe epilepsy using the maximum Lyapunov exponent

Analysis of intracranial electroencephalographic (iEEG) recordings in patients with temporal lobe epilepsy (TLE) has revealed characteristic dynamical features that distinguish the interictal, ictal, and postictal states and inter-state transitions. Experimental investigations into the mechanisms underlying these observations require the use of an animal model. A rat TLE model was used to test for differences in iEEG dynamics between well-defined states and to test specific hypotheses: 1) the short-term maximum Lyapunov exponent (STL(max)), a measure of signal order, is lowest and closest in value among cortical sites during the ictal state, and highest and most divergent during the postictal state; 2) STL(max) values estimated from the stimulated hippocampus are the lowest among all cortical sites; and 3) the transition from the interictal to ictal state is associated with a convergence in STL(max) values among cortical sites. iEEGs were recorded from bilateral frontal cortices and hippocampi. STL(max) and T-index (a measure of convergence/divergence of STL(max) between recorded brain areas) were compared among the four different periods. Statistical tests (ANOVA and multiple comparisons) revealed that ictal STL(max) was lower (p<0.05) than other periods, STL(max) values corresponding to the stimulated hippocampus were lower than those estimated from other cortical regions, and T-index values were highest during the postictal period and lowest during the ictal period. Also, the T-index values corresponding to the preictal period were lower than those during the interictal period (p<0.05). These results indicate that a rat TLE model demonstrates several important dynamical signal characteristics similar to those found in human TLE and support future use of the model to study epileptic state transitions.

Using recurrence plot for determinism analysis of EEG recordings in genetic absence epilepsy rats

OBJECTIVE

Understanding the transition of brain activity towards an absence seizure is a challenging task. In this paper, we use recurrence quantification analysis to indicate the deterministic dynamics of EEG series at the seizure-free, pre-seizure and seizure states in genetic absence epilepsy rats.

METHODS

The determinism measure, DET, based on recurrence plot, was applied to analyse these three EEG datasets, each dataset containing 300 single-channel EEG epochs of 5-s duration. Then, statistical analysis of the DET values in each dataset was carried out to determine whether their distributions over the three groups were significantly different. Furthermore, a surrogate technique was applied to calculate the significance level of determinism measures in EEG recordings.

RESULTS

The mean (+/-SD) DET of EEG was 0.177+/-0.045 in pre-seizure intervals. The DET values of pre-seizure EEG data are significantly higher than those of seizure-free intervals, 0.123+/-0.023, (P<0.01), but lower than those of seizure intervals, 0.392+/-0.110, (P<0.01). Using surrogate data methods, the significance of determinism in EEG epochs was present in 25 of 300 (8.3%), 181 of 300 (60.3%) and 289 of 300 (96.3%) in seizure-free, pre-seizure and seizure intervals, respectively.

CONCLUSIONS

Results provide some first indications that EEG epochs during pre-seizure intervals exhibit a higher degree of determinism than seizure-free EEG epochs, but lower than those in seizure EEG epochs in absence epilepsy.

SIGNIFICANCE

The proposed methods have the potential of detecting the transition between normal brain activity and the absence seizure state, thus opening up the possibility of intervention, whether electrical or pharmacological, to prevent the oncoming seizure.

Quantitative complexity analysis in multi-channel intracranial EEG recordings form epilepsy brains

Epilepsy is a brain disorder characterized clinically by temporary but recurrent disturbances of brain function that may or may not be associated with destruction or loss of consciousness and abnormal behavior. Human brain is composed of more than 10 to the power 10 neurons, each of which receives electrical impulses known as action potentials from others neurons via synapses and sends electrical impulses via a sing output line to a similar (the axon) number of neurons. When neuronal networks are active, they produced a change in voltage potential, which can be captured by an electroencephalogram (EEG). The EEG recordings represent the time series that match up to neurological activity as a function of time. By analyzing the EEG recordings, we sought to evaluate the degree of underlining dynamical complexity prior to progression of seizure onset. Through the utilization of the dynamical measurements, it is possible to classify the state of the brain according to the underlying dynamical properties of EEG recordings. The results from two patients with temporal lobe epilepsy (TLE), the degree of complexity start converging to lower value prior to the epileptic seizures was observed from epileptic regions as well as non-epileptic regions. The dynamical measurements appear to reflect the changes of EEG's dynamical structure. We suggest that the nonlinear dynamical analysis can provide a useful information for detecting relative changes in brain dynamics, which cannot be detected by conventional linear analysis.

Nonlinear dynamical analysis of the neonatal EEG time series: the relationship between neurodevelopment and complexity

OBJECTIVE

To investigate the relationship between the complexity of sleep EEG time series and neurodevelopment for premature or full-term neonates.

METHODS

Nonlinear dynamical analysis of neonatal sleep EEG time series is used to compute the correlation dimension D2 which is an index of the complexity of the dynamics of the developing brain. The dimensional complexity is estimated using Theiler's modification of the Grassberger-Procaccia algorithm for two different values of Theiler's w parameter. The hypothesis that neonatal EEG data during sleep contains nonlinear features is verified by means of surrogate data testing.

RESULTS

The dimensional complexity of the neonatal EEG increases with neurodevelopment and brain maturation. There is furthermore a statistically significant difference between the dimensional complexity of the EEG for neonates born prematurely when compared to full-term neonates at the same postmenstrual age (PMA). The neonatal EEG time series data used in this study proved to contain nonlinear features where the 'null hypothesis' of surrogate data testing is rejected with p<<0.0001.

CONCLUSIONS

A relationship between neurodevelopment and brain maturation and the complexity of the dynamics of the brain as measured by the dimensional complexity of the sleep EEG time series has been established. In particular, the dimensional complexity tends to increase with neurodevelopment and maturation as indicated by their PMA and birth status (premature or full-term). In particular, the brain dynamics of neonates born prematurely is less complex than the brain dynamics of neonates born full-term even at the same PMA. We attribute this to differences in the neurodevelopment between these two cohorts. We propose that the dimensional complexity can be used as an index for quantifying neurodevelopment.

SIGNIFICANCE

The dimensional complexity as measured by the correlation dimension of the sleep EEG time series may potentially be a useful measure for quantifying neurodevelopment in neonates. Future work is directed at the analysis of other EEG channels to understand the relationship between complexity in different regions of the brain and maturation and neurodevelopment, along with the utility of complexity to relate to neurodevelopment at older ages as measured by the Bayley score.

Detrended fluctuation analysis of resting EEG in depressed outpatients and healthy controls

OBJECTIVE

Recent findings have demonstrated that the EEG possesses long-range temporal (auto-) correlations (LRTC) in the dynamics of broad band oscillations. The analysis of LRTC provides a quantitative index of statistical dependencies in oscillations on different time scales. We analyzed LRTC in resting EEG signals in depressed outpatients and healthy controls.

METHODS

The participants in this study were 11 non-depressed, age-matched controls, and 11 unmedicated unipolar depressed patients. EEG data were obtained from each participant during 5-min resting baseline periods with eyes closed and then analyzed with detrended fluctuation analysis (DFA), a scaling analysis method that quantifies a simple parameter to represent the correlation properties of a time series. The scaling exponent, the result of DFA, provides a quantitative measure of LRTC from the EEG.

RESULTS

The present study demonstrates that all the scaling exponents in depressed patients and healthy controls were greater than 0.5 and less than 1.0, regardless of condition. Furthermore, the scaling exponents of depressed patients have relatively higher values in whole brain regions compared to healthy controls, with significant differences at F3, C3, T3, T4 and O1 channels (p<0.05). Finally, a significant linear correlation was observed between the severity of depression and the scaling exponent over most of the channels, except O2.

CONCLUSIONS

These results suggest that the brain affected by a major depressive disorder shows slower decay of the LRTC, and that the persistence of the LRTC of EEG in depressed patients was associated with the severity of depression over most of the cortical areas.

SIGNIFICANCE

The DFA method may broaden our understanding of the psychophysiological basis of depression.

Approximate entropy detects the effect of a secondary cognitive task on postural control in healthy young adults: a methodological report

BACKGROUND

Biomechanical measures of postural stability, while generally useful in neuroscience and physical rehabilitation research, may be limited in their ability to detect more subtle influences of attention on postural control. Approximate entropy (ApEn), a regularity statistic from nonlinear dynamics, recently has demonstrated relatively good measurement precision and shown promise for detecting subtle change in postural control after cerebral concussion. Our purpose was to further explore the responsiveness of ApEn by using it to evaluate the immediate, short-term effect of secondary cognitive task performance on postural control in healthy, young adults.

METHODS

Thirty healthy, young adults performed a modified version of the Sensory Organization Test featuring single (posture only) and dual (posture plus cognitive) task trials. ApEn values, root mean square (RMS) displacement, and equilibrium scores (ES) were calculated from anterior-posterior (AP) and medial-lateral (ML) center of pressure (COP) component time series. For each sensory condition, we compared the ability of the postural control parameters to detect an effect of cognitive task performance.

RESULTS

COP AP time series generally became more random (higher ApEn value) during dual task performance, resulting in a main effect of cognitive task (p = 0.004). In contrast, there was no significant effect of cognitive task for ApEn values of COP ML time series, RMS displacement (AP or ML) or ES.

CONCLUSION

During dual task performance, ApEn revealed a change in the randomness of COP oscillations that occurred in a variety of sensory conditions, independent of changes in the amplitude of COP oscillations. The finding expands current support for the potential of ApEn to detect subtle changes in postural control. Implications for future studies of attention in neuroscience and physical rehabilitation are discussed.

Differentiation of MEG activity in multiple sclerosis patients with the use of nonlinear analysis

The aim of this study is to investigate if there is any nonlinearity in the magnetoencephalographic recordings of patients with multiple sclerosis in comparison with controls in order to find out the differences in the mechanisms underlying their brain waves. Five multiple sclerosis patients and five controls were included in this study. Chaotic activity of multiple sclerosis patients is lower than in the normal brain. Nonlinear analysis may offer fertile perspectives for understanding the features of patients with multiple sclerosis.

Relative performance of mutual information estimation methods for quantifying the dependence among short and noisy data

Commonly used dependence measures, such as linear correlation, cross-correlogram, or Kendall's tau , cannot capture the complete dependence structure in data unless the structure is restricted to linear, periodic, or monotonic. Mutual information (MI) has been frequently utilized for capturing the complete dependence structure including nonlinear dependence. Recently, several methods have been proposed for the MI estimation, such as kernel density estimators (KDEs), k -nearest neighbors (KNNs), Edgeworth approximation of differential entropy, and adaptive partitioning of the XY plane. However, outstanding gaps in the current literature have precluded the ability to effectively automate these methods, which, in turn, have caused limited adoptions by the application communities. This study attempts to address a key gap in the literature-specifically, the evaluation of the above methods to choose the best method, particularly in terms of their robustness for short and noisy data, based on comparisons with the theoretical MI estimates, which can be computed analytically, as well with linear correlation and Kendall's tau . Here we consider smaller data sizes, such as 50, 100, and 1000, and within this study we characterize 50 and 100 data points as very short and 1000 as short. We consider a broader class of functions, specifically linear, quadratic, periodic, and chaotic, contaminated with artificial noise with varying noise-to-signal ratios. Our results indicate KDEs as the best choice for very short data at relatively high noise-to-signal levels whereas the performance of KNNs is the best for very short data at relatively low noise levels as well as for short data consistently across noise levels. In addition, the optimal smoothing parameter of a Gaussian kernel appears to be the best choice for KDEs while three nearest neighbors appear optimal for KNNs. Thus, in situations where the approximate data sizes are known in advance and exploratory data analysis and/or domain knowledge can be used to provide a priori insights into the noise-to-signal ratios, the results in the paper point to a way forward for automating the process of MI estimation.

Spatio-Temporal Clustering of Epileptic ECOG

The spatio-temporal mechanisms underlying the generation of epileptic seizures is not yet clearly understood. In this study, we attempt to quantify the spatio-temporal interactions of an epileptic brain by using a previously proposed SOM-based Similarity Index (SI) measure. We further show that spectral clustering approach can be appropriately used to determine the average spatial mappings in the brain at different stages of a seizure, by interpreting the SOM-SI values as affinity matrices. Results involving two pairs of seizures of an epileptic patient suggest that there may not be a regular pattern associated with channels's spatio-temporal dynamics during the inter-ictal to prepost ictal transition.

Stimuli of varying spatial scale induce gamma activity with distinct temporal characteristics in human visual cortex

Gamma activity to stationary grating stimuli was studied non-invasively using MEG recordings in humans. Using a spatial filtering technique, we localized gamma activity to primary visual cortex. We tested the hypothesis that spatial frequency properties of visual stimuli may be related to the temporal frequency characteristics of the associated cortical responses. We devised a method to assess temporal frequency differences between stimulus-related responses that typically exhibit complex spectral shapes. We applied this methodology to either single-trial (induced) or time-averaged (evoked) responses in four frequency ranges (0-40, 20-60, 40-80 and 60-100 Hz) and two time windows (either the entire duration of stimulus presentation or the first second following stimulus onset). Our results suggest that stimuli of varying spatial frequency induce responses that exhibit significantly different temporal frequency characteristics. These effects were particularly accentuated for induced responses in the classical gamma frequency band (20-60 Hz) analyzed over the entire duration of stimulus presentation. Strikingly, examining the first second of the responses following stimulus onset resulted in significant loss in stimulus specificity, suggesting that late signal components contain functionally relevant information. These findings advocate a functional role of gamma activity in sensory representation. We suggest that stimulus specific frequency characteristics of MEG signals can be mapped to processes of neuronal synchronization within the framework of coupled dynamical systems.

Detecting temporal and spatial correlations in pseudoperiodic time series

Recently there has been much attention devoted to exploring the complicated possibly chaotic dynamics in pseudoperiodic time series. Two methods [Zhang, Phys. Rev. E 73, 016216 (2006); Zhang and Small, Phys. Rev. Lett. 96, 238701 (2006)] have been forwarded to reveal the chaotic temporal and spatial correlations, respectively, among the cycles in the time series. Both these methods treat the cycle as the basic unit and design specific statistics that indicate the presence of chaotic dynamics. In this paper, we verify the validity of these statistics to capture the chaotic correlation among cycles by using the surrogate data method. In particular, the statistics computed for the original time series are compared with those from its surrogates. The surrogate data we generate is pseudoperiodic type (PPS), which preserves the inherent periodic components while destroying the subtle nonlinear (chaotic) structure. Since the inherent chaotic correlations among cycles, either spatial or temporal (which are suitably characterized by the proposed statistics), are eliminated through the surrogate generation process, we expect the statistics from the surrogate to take significantly different values than those from the original time series. Hence the ability of the statistics to capture the chaotic correlation in the time series can be validated. Application of this procedure to both chaotic time series and real world data clearly demonstrates the effectiveness of the statistics. We have found clear evidence of chaotic correlations among cycles in human electrocardiogram and vowel time series. Furthermore, we show that this framework is more sensitive to examine the subtle changes in the dynamics of the time series due to the match between PPS surrogate and the statistics adopted. It offers a more reliable tool to reveal the possible correlations among cycles intrinsic to the chaotic nature of the pseudoperiodic time series.

Linear and nonlinear causality between signals: methods, examples and neurophysiological applications

In this paper, we will present and review the most usual methods to detect linear and nonlinear causality between signals: linear Granger causality test (Geweke in J Am Stat Assoc 77:304-313, 1982) extended to direct causality in multivariate case (LGC), directed coherence (DCOH, Saito and Harashima in Recent advances in EEG and EMG data processing, Elsevier, Amsterdam, 1981), partial directed coherence (PDC, Sameshima and Baccala 1999) and nonlinear Granger causality test of Baek and Brock (in Working Paper University of Iowa, 1992) extended to direct causality in multivariate case (partial nonlinear Granger causality, PNGC). All these methods are tested and compared on several ARX, Poisson and nonlinear models, and on neurophysiological data (depth EEG). The results show that LGC, DCOH and PDC are not very robust in relation to nonlinear linkages but they seem to correctly find linear linkages if only the autoregressive parts are nonlinear. PNGC is extremely dependent on the choice of parameters. Moreover, LGC and PNGC may give misleading results in the case of causality on a spectral band, which is illustrated by our neurophysiological database.

Dimensional complexity of neuromagnetic activity reduced during finger movement of greater difficulty

OBJECTIVE

We investigated the variation in dimensionality (D2) of neuromagnetic activity over the primary sensorimotor cortex (SM1) in healthy adults performing motor tasks of different difficulty.

METHODS

Magnetoencephalography (MEG) was used to record neuromagnetic activity during self-paced, brisk unimanual finger extension at a rate of 1 and 2 Hz using the index finger of the dominant and non-dominant hands in 16 healthy subjects. Motor task difficulty was rated by the relative difference in time measurement between 1 and 2 Hz finger movements of both hands. The relative difference in dimensionality of SM1 activity was calculated by subtracting the D2 value in 2 Hz movement from that in 1 Hz one within subjects.

RESULTS

Simple regression analyses show a significantly negative relationship between the relative dimensional complexity and the relative motor task difficulty in the contralateral SM1 for the left- (p<0.05), but not the right- (p=0.447) hand movement.

CONCLUSIONS

The present data suggest that a motor task of greater difficulty may engender a reduction of simultaneously active quasi-independent neuronal generators in the contralateral SM1 underpinned by stronger neuronal connectivity of a relatively low dimensionality.

SIGNIFICANCE

The decrease in dimensional complexity of MEG activity associated with a motor task of greater difficulty gives new insights to motor control strategy.

Testing for nonlinearity in irregular fluctuations with long-term trends

We describe a method for investigating nonlinearity in irregular fluctuations (short-term variability) of time series even if the data exhibit long-term trends (periodicities). Such situations are theoretically incompatible with the assumption of previously proposed methods. The null hypothesis addressed by our algorithm is that irregular fluctuations are generated by a stationary linear system. The method is demonstrated for numerical data generated by known systems and applied to several actual time series.