Test for low-dimensional determinism in electroencephalograms

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.

A method for detecting nonlinear determinism in normal and epileptic brain EEG signals

A robust method of detecting determinism for short time series is proposed and applied to both healthy and epileptic EEG signals. The method provides a robust measure of determinism through characterizing the trajectories of the signal components which are obtained through singular value decomposition. Robustness of the method is shown by calculating proposed index of determinism at different levels of white and colored noise added to a simulated chaotic signal. The method is shown to be able to detect determinism at considerably high levels of additive noise. The method is then applied to both intracranial and scalp EEG recordings collected in different data sets for healthy and epileptic brain signals. The results show that for all of the studied EEG data sets there is enough evidence of determinism. The determinism is more significant for intracranial EEG recordings particularly during seizure activity.

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

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

A Nonstationary Model of Newborn EEG

The detection of seizure in the newborn is a critical aspect of neurological research. Current automatic detection techniques are difficult to assess due to the problems associated with acquiring and labelling newborn electroencephalogram (EEG) data. A realistic model for newborn EEG would allow confident development, assessment and comparison of these detection techniques. This paper presents a model for newborn EEG that accounts for its self-similar and nonstationary nature. The model consists of background and seizure submodels. The newborn EEG background model is based on the short-time power spectrum with a time-varying power law. The relationship between the fractal dimension and the power law of a power spectrum is utilized for accurate estimation of the short-time power law exponent. The newborn EEG seizure model is based on a well-known time-frequency signal model. This model addresses all significant time-frequency characteristics of newborn EEG seizure which include; multiple components or harmonics, piecewise linear instantaneous frequency laws and harmonic amplitude modulation. Estimates of the parameters of both models are shown to be random and are modelled using the data from a total of 500 background epochs and 204 seizure epochs. The newborn EEG background and seizure models are validated against real newborn EEG data using the correlation coefficient. The results show that the output of the proposed models have a higher correlation with real newborn EEG than currently accepted models (a 10% and 38% improvement for background and seizure models, respectively).

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.

Reliability of the 0-1 test for chaos

In time series analysis, it has been considered of key importance to determine whether a complex time series measured from the system is regular, deterministically chaotic, or random. Recently, Gottwald and Melbourne have proposed an interesting test for chaos in deterministic systems. Their analyses suggest that the test may be universally applicable to any deterministic dynamical system. In order to fruitfully apply their test to complex experimental data, it is important to understand the mechanism for the test to work, and how it behaves when it is employed to analyze various types of data, including those not from clean deterministic systems. We find that the essence of their test can be described as to first constructing a random walklike process from the data, then examining how the variance of the random walk scales with time. By applying the test to three sets of data, corresponding to (i) 1/falpha noise with long-range correlations, (ii) edge of chaos, and (iii) weak chaos, we show that the test mis-classifies (i) both deterministic and weakly stochastic edge of chaos and weak chaos as regular motions, and (ii) strongly stochastic edge of chaos and weak chaos, as well as 1/falpha noise as deterministic chaos. Our results suggest that, while the test may be effective to discriminate regular motion from fully developed deterministic chaos, it is not useful for exploratory purposes, especially for the analysis of experimental data with little a priori knowledge. A few speculative comments on the future of multiscale nonlinear time series analysis are made.

EEG dynamics in patients with Alzheimer's disease

Alzheimer's disease (AD) is the most common neurodegenerative disorder characterized by cognitive and intellectual deficits and behavior disturbance. The electroencephalogram (EEG) has been used as a tool for diagnosing AD for several decades. The hallmark of EEG abnormalities in AD patients is a shift of the power spectrum to lower frequencies and a decrease in coherence of fast rhythms. These abnormalities are thought to be associated with functional disconnections among cortical areas resulting from death of cortical neurons, axonal pathology, cholinergic deficits, etc. This article reviews main findings of EEG abnormalities in AD patients obtained from conventional spectral analysis and nonlinear dynamical methods. In particular, nonlinear alterations in the EEG of AD patients, i.e. a decreased complexity of EEG patterns and reduced information transmission among cortical areas, and their clinical implications are discussed. For future studies, improvement of the accuracy of differential diagnosis and early detection of AD based on multimodal approaches, longitudinal studies on nonlinear dynamics of the EEG, drug effects on the EEG dynamics, and linear and nonlinear functional connectivity among cortical regions in AD are proposed to be investigated. EEG abnormalities of AD patients are characterized by slowed mean frequency, less complex activity, and reduced coherences among cortical regions. These abnormalities suggest that the EEG has utility as a valuable tool for differential and early diagnosis of AD.

Efficient estimation of a time-varying dimension parameter and its application to EEG analysis

This paper considers the problem of estimating the dimension of nonstationary electroencephalogram (EEG) signals and describes the implementation of an efficient algorithm to calculate a time-varying dimension estimate. The algorithm allows the practical calculation of a dimension estimate and its statistical significance over large data sets with a high temporal resolution. The method is applied to EEG recordings from patients with temporal lobe epilepsy and in one case the results of the analysis are compared with those obtained from an existing method of computing the correlation density.

Discerning nonstationarity from nonlinearity in seizure-free and preseizure EEG recordings from epilepsy patients

A number of recent studies indicate that nonlinear electroencephalogram (EEG) analyses allow to define a state predictive of an impending epileptic seizure. In this paper, we combine a method for detecting nonlinear determinism with a novel test for stationarity to characterize EEG recordings from both the seizure-free interval and the preseizure phase. We discuss differences between these periods, particularly an increased occurrence of stationary, nonlinear segments prior to seizures. These differences seem most prominent for recording sites within the seizure-generating area and for EEG segments less than one minute's length.

A method for determinism in short time series, and its application to stationary EEG

A novel method for detecting determinism in short time series is developed and applied to investigate determinism in stationary electroencephalogram (EEG) recordings. This method is based on the observation that the trajectory of a time series generated from a differentiable dynamical system behaves smoothly in an embedded state space. The angles between two successive tangent vectors in the trajectory reconstructed from the time series is calculated as a function of time. The irregularity of the angle variations obtained from the time series is estimated using second-order difference plots, and compared with that of the corresponding surrogate data. Using this method, we demonstrate that scalp EEG recordings from normal subjects do not exhibit a low-dimensional deterministic structure. This method can be useful for analyzing determinism in short time series, such as those from physiological recordings.

Correlation-dimension and autocorrelation fluctuations in epileptic seizure dynamics

We focus on an anomalous scaling region in correlation integral [C(epsilon)] analysis of electrocorticogram in epilepsy patients. We find that epileptic seizures typically are accompanied by wide fluctuations in the slope of this scaling region. An explanation, based on analyzing the interplay between the autocorrelation and C(epsilon), is provided for these fluctuations. This anomalous slope appears to be a sensitive measure for tracking (but not predicting) seizures.

Detecting determinism in high-dimensional chaotic systems

A method based upon the statistical evaluation of the differentiability of the measure along the trajectory is used to identify determinism in high-dimensional systems. The results show that the method is suitable for discriminating stochastic from deterministic systems even if the dimension of the latter is as high as 13. The method is shown to succeed in identifying determinism in electroencephalogram signals simulated by means of a high-dimensional system.

Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: dependence on recording region and brain state

We compare dynamical properties of brain electrical activity from different recording regions and from different physiological and pathological brain states. Using the nonlinear prediction error and an estimate of an effective correlation dimension in combination with the method of iterative amplitude adjusted surrogate data, we analyze sets of electroencephalographic (EEG) time series: surface EEG recordings from healthy volunteers with eyes closed and eyes open, and intracranial EEG recordings from epilepsy patients during the seizure free interval from within and from outside the seizure generating area as well as intracranial EEG recordings of epileptic seizures. As a preanalysis step an inclusion criterion of weak stationarity was applied. Surface EEG recordings with eyes open were compatible with the surrogates' null hypothesis of a Gaussian linear stochastic process. Strongest indications of nonlinear deterministic dynamics were found for seizure activity. Results of the other sets were found to be inbetween these two extremes.

Chaos in the brain: a short review alluding to epilepsy, depression, exercise and lateralization

Electroencephalograms (EEGs) reflect the electrical activity of the brain. Even when they are analyzed from healthy individuals, they manifest chaos in the nervous system. EEGs are likely to be produced by a nonlinear system, since a nonlinear system with at least 3 degrees of freedom (or state variables) may exhibit chaotic behavior. Furthermore, such systems can have multiple stable states governed by "chaotic" ("strange") attractors. A key feature of chaotic systems is the presence of an infinite number of unstable periodic fixed points, which are found in spontaneously active neuronal networks (e.g., epilepsy). The brain has chemicals called neurotransmitters that convey the information through the 10(16) synapses residing there. However, each of these neurotransmitters acts through various receptors and their numerous subtypes, thereby exhibiting complex interactions. Albeit in epilepsy the role of chaos and EEG findings are well proven, in another condition, i.e., depression, the role of chaos is slowly gaining ground. The multifarious roles of exercise, neurotransmitters and (cerebral) hemispheric lateralization, in the case of depression, are also being established. The common point of reference could be nonlinear dynamics. The purpose of this review is to study those nonlinear/chaotic interactions and point towards new theoretical models incorporating the oscillation caused by the same neurotransmitter acting on its different receptor subtypes. This may lead to a better understanding of brain neurodynamics in health and disease.

The epileptic process as nonlinear deterministic dynamics in a stochastic environment: an evaluation on mesial temporal lobe epilepsy

The theory of deterministic chaos addresses simple deterministic dynamics in which nonlinearity gives rise to complex temporal behavior. Although biological neuronal networks such as the brain are highly complicated, a number of studies provide growing evidence that nonlinear time series analysis of brain electrical activity in patients with epilepsy is capable of providing potentially useful diagnostic information. In the present study, this analysis framework was extended by introducing a new measure xi, designed to discriminate between nonlinear deterministic and linear stochastic dynamics. For the evaluation of its discriminative power, xi was extracted from intracranial multi-channel EEGs recorded during the interictal state in 25 patients with unilateral mesial temporal lobe epilepsy. Strong indications of nonlinear determinism were found in recordings from within the epileptogenic zone, while EEG signals from other sites mainly resembled linear stochastic dynamics. In all investigated cases, this differentiation allowed to retrospectively determine the side of the epileptogenic zone in full agreement with results of the presurgical workup.

Epileptic event forewarning from scalp EEG

The authors present a model-independent approach to quantify changes in the dynamics underlying nonlinear time-serial data. From time-windowed datasets, the authors construct discrete distribution functions on the phase space. Condition change between base case and test case distribution functions is assessed by dissimilarity measures via L1 distance and chi2 statistic. The discriminating power of these measures is first tested on noiseless data from the Lorenz and Bondarenko models, and is then applied to detecting dynamic change in multichannel clinical scalp EEG data. The authors compare the dissimilarity measures with the traditional nonlinear measures used in the analysis of chaotic systems. They also assess the potential usefulness of the new measures for robust, accurate, and timely forewarning of epileptic events.

Nonlinear EEG analysis in epilepsy: its possible use for interictal focus localization, seizure anticipation, and prevention

Several recent studies emphasize the high value of nonlinear EEG analysis particularly for improved characterization of epileptic brain states. In this review the authors report their work to increase insight into the spatial and temporal dynamics of the epileptogenic process. Specifically, they discuss possibilities for seizure anticipation, which is one of the most challenging aspects of epileptology. Although there are numerous studies exploring basic neuronal mechanisms that are likely to be associated with seizures, to date no definite information is available regarding how, when, or why a seizure occurs. Nonlinear EEG analysis now provides strong evidence that the interictal-ictal state transition is not an abrupt phenomenon. Rather, findings indicate that it is indeed possible to detect a preseizure phase. The unequivocal definition of such a state with a sufficient length would enable investigations of basic mechanisms leading to seizure initiation in humans, and development of adequate seizure prevention strategies.

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

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

Timely detection of dynamical change in scalp EEG signals

We present a robust, model-independent technique for quantifying changes in the dynamics underlying nonlinear time-serial data. After constructing discrete density distributions of phase-space points on the attractor for time-windowed data sets, we measure the dissimilarity between density distributions via L(1)-distance and chi(2) statistics. The discriminating power of the new measures is first tested on data generated by the Bondarenko "synthetic brain" model. We also compare traditional nonlinear measures and the new dissimilarity measures to detect dynamical change in scalp EEG data. The results demonstrate a clear superiority of the new measures in comparison to traditional nonlinear measures as robust and timely discriminators of changing dynamics. (c) 2000 American Institute of Physics.

Spatio-temporal characterizations of non-linear changes in intracranial activities prior to human temporal lobe seizures

Recent studies have shown that non-linear analysis of intracranial activities can detect a 'pre-ictal phase' preceding the epileptic seizure. Nevertheless, the dynamical nature of the underlying neuronal process and the spatial extension of this pre-ictal phase still remain unknown. In this paper, we address these aspects using a new non-linear measure of dynamic similarity between different parts of intracranial recordings of nine patients with medial temporal lobe epilepsy recorded during transitions to seizure. Our results confirm that non-linear changes in neuronal dynamics allow, in most cases (16 out of 17), a seizure anticipation several minutes in advance. Furthermore, we show that the spatial distribution of pre-ictal changes often involves an extended network projecting beyond the limits of the epileptogenic region. Finally, the pre-ictal phase could frequently (13 out of 17) be characterized with a marked shift toward slower frequencies in upper delta or theta frequency range.