Non-linear and linear forecasting of the EEG time series

The method of non-linear forecasting of time series was applied to different simulated signals and EEG in order to check its ability of distinguishing chaotic from noisy time series. The goodness of prediction was estimated, in terms of the correlation coefficient between forecasted and real time series, for non-linear and autoregressive (AR) methods. For the EEG signal both methods gave similar results. It seems that the EEG signal, in spite of its chaotic character, is well described by the AR model.     

Non-linear analysis of the electroencephalogram in Creutzfeldt-Jakob disease

Creutzfeldt-Jakob disease is a rare, neurological, dementing disorder characterised by periodic sharp waves in the electroencephalogram (EEG). Non-linear analysis of these EEG changes may provide insight into the abnormal dynamics of cortical neural networks in this disorder. Babloyantz et al. have suggested that the periodic sharp waves reflect low-dimensional chaotic dynamics in the brain. In the present study this hypothesis was re-examined using newly developed techniques for non-linear time series analysis. We analysed the EEG of a patient with autopsy-proven Creutzfeldt-Jakob disease using the method of non-linear forecasting as introduced by Sugihara and May, and we tested for non-linearity with amplitude-adjusted, phase-randomised surrogate data. Two epochs with generalised periodic sharp waves showed clear evidence for non-linearity. These epochs could be predicted better and further ahead in time than most of the irregular background activity. Testing against cycle-randomised surrogate data and close inspection of the periodograms showed that the non-linearity of the periodic sharp waves may be better explained by quasi-periodicity than by low-dimensional chaos. The EEG further displayed at least one example of a sudden, large qualitative change in the dynamics, highly suggestive of a bifurcation. The presence of quasi-periodicity and bifurcations strongly argues for the use of a non-linear model to describe the EEG in Creutzfeldt-Jakob disease. Received: 28 October 1996 / Accepted in revised form: 8 July 1997     

Investigation of the dynamics underlying periodic complexes in the EEG

Periodic complexes (PC), occurring lateralised or diffuse, are relatively rare EEG phenomena which reflect acute severe brain disease. The pathophysiology is still incompletely understood. One hypothesis suggested by the alpha rhythm model of Lopes da Silva is that periodic complexes reflect limit cycle dynamics of cortical networks caused by excessive excitatory feedback. We examined this hypothesis by applying a recently developed technique to EEGs displaying periodic complexes and to periodic complexes generated by the model. The technique, non-linear cross prediction, characterises how well a time series can be predicted, and how much amplitude and time asymmetry is present. Amplitude and time asymmetry are indications of non-linearity. In accordance with the model, most EEG channels with PC showed clear evidence of amplitude and time asymmetry, pointing to non-linear dynamics. However, the non-linear predictability of true PC was substantially lower than that of PC generated by the model. Furthermore, no finite value for the correlation dimension could be obtained for the real EEG data, whereas the model time series had a dimension slighter higher than one, consistent with a limit cycle attractor. Thus we can conclude that PC reflect non-linear dynamics, but a limit cycle attractor is too simple an explanation. The possibility of more complex (high dimensional and spatio-temporal) non-linear dynamics should be investigated. Received: 26 February 1998 / Accepted in revised form: 24 August 1998     

A stochastic limit cycle oscillator model of the EEG

We present an empirical model of the electroencephalogram (EEG) signal based on the construction of a stochastic limit cycle oscillator using Itô calculus. This formulation, where the noise influences actually interact with the dynamics, is substantially different from the usual definition of measurement noise. Analysis of model data is compared with actual EEG data using both traditional methods and modern techniques from nonlinear time series analysis. The model demonstrates visually displayed patterns and statistics that are similar to actual EEG data. In addition, the nonlinear mechanisms underlying the dynamics of the model do not manifest themselves in nonlinear time series analysis, paralleling the situation with real, non-pathological EEG data. This modeling exercise suggests that the EEG is optimally described by stochastic limit cycle behavior.     

Explicit-duration hidden Markov model inference of UP-DOWN states from continuous signals

Neocortical neurons show UP-DOWN state (UDS) oscillations under a variety of conditions. These UDS have been extensively studied because of the insight they can yield into the functioning of cortical networks, and their proposed role in putative memory formation. A key element in these studies is determining the precise duration and timing of the UDS. These states are typically determined from the membrane potential of one or a small number of cells, which is often not sufficient to reliably estimate the state of an ensemble of neocortical neurons. The local field potential (LFP) provides an attractive method for determining the state of a patch of cortex with high spatio-temporal resolution; however current methods for inferring UDS from LFP signals lack the robustness and flexibility to be applicable when UDS properties may vary substantially within and across experiments. Here we present an explicit-duration hidden Markov model (EDHMM) framework that is sufficiently general to allow statistically principled inference of UDS from different types of signals (membrane potential, LFP, EEG), combinations of signals (e.g., multichannel LFP recordings) and signal features over long recordings where substantial non-stationarities are present. Using cortical LFPs recorded from urethane-anesthetized mice, we demonstrate that the proposed method allows robust inference of UDS. To illustrate the flexibility of the algorithm we show that it performs well on EEG recordings as well. We then validate these results using simultaneous recordings of the LFP and membrane potential (MP) of nearby cortical neurons, showing that our method offers significant improvements over standard methods. These results could be useful for determining functional connectivity of different brain regions, as well as understanding network dynamics.     

Novel whole body plethysmography system for the continuous characterization of sleep and breathing in a mouse

Sleep is associated with marked alterations in ventilatory control that lead to perturbations in respiratory timing, breathing pattern, ventilation, pharyngeal collapsibility, and sleep-related breathing disorders (SRBD). Mouse models offer powerful insight into the pathogenesis of SRBD; however, methods for obtaining the full complement of continuous, high-fidelity respiratory, electroencephalographic (EEG), and electromyographic (EMG) signals in unrestrained mice during sleep and wake have not been developed. We adapted whole body plethysmography to record EEG, EMG, and respiratory signals continuously in unrestrained, unanesthetized mice. Whole body plethysmography tidal volume and airflow signals and a novel noninvasive surrogate for respiratory effort (respiratory movement signal) were validated against simultaneously measured gold standard signals. Compared with the gold standard, we validated 1) tidal volume (correlation, R(2) = 0.87, P < 0.001; and agreement within 1%, P < 0.001); 2) inspiratory airflow (correlation, R(2) = 0.92, P < 0.001; agreement within 4%, P < 0.001); 3) expiratory airflow (correlation, R(2) = 0.83, P < 0.001); and 4) respiratory movement signal (correlation, R(2) = 0.79-0.84, P < 0.001). The expiratory airflow signal, however, demonstrated a decrease in amplitude compared with the gold standard. Integrating respiratory and EEG/EMG signals, we fully characterized sleep and breathing patterns in conscious, unrestrained mice and demonstrated inspiratory flow limitation in a New Zealand Obese mouse. Our approach will facilitate studies of SRBD mechanisms in inbred mouse strains and offer a powerful platform to investigate the effects of environmental and pharmacological exposures on breathing disturbances during sleep and wakefulness.     

Topographic analysis of dimension estimates of EEG and filtered rhythms in epileptic patients with complex partial seizures

Nonlinear dynamic properties were analyzed on the EEG and filtered rhythms recorded from healthy subjects and epileptic patients with complex partial seizures. Estimates of correlation dimensions of control EEG, interictal EEG and ictal EEG were calculated. The values were demonstrated on topograms. The delta (0.5–4 Hz), theta (4–8 Hz), alpha (8–13 Hz), beta (13–30 Hz) and gamma (30–40 Hz) components were obtained and considered as signals from the cortex. Corresponding surrogate data was produced. Firstly, the influence of sampling parameters on the calculation was tested. The dimension estimates of the signals from the frontal, temporal, parietal and occipital regions were computed and compared with the results of surrogate data. In the control subjects, the estimates between the EEG and surrogate data did not differ (P > 0.05). The interictal EEG from the frontal region and occipital region, as well as its theta component from the frontal region, and temporal region, showed obviously low dimensions (P < 0.01). The ictal EEG exhibited significantly low-dimension estimates across the scalp. All filtered rhythms from the temporal region yielded lower results than those of the surrogate data (P < 0.01). The dimension estimates of the EEG and filtered components markedly changed when the neurological state varied. For each neurological state, the dimension estimates were not uniform among the EEG and frequency components. The signal with a different frequency range and in a different neurological state showed a different dimension estimate. Furthermore, the theta and alpha components demonstrated the same estimates not only within each neurological state, but also among the different states. These results indicate that the theta and alpha components may be caused by similar dynamic processes. We conclude that the brain function underlying the ictal EEG has a simple mechanism. Several heterogeneous dynamic systems play important roles in the generation of EEG. Received: 10 December 1999 / Accepted in revised form: 8 May 2000     

On the Usage of Linear Regression Models to Reconstruct Limb Kinematics from Low Frequency EEG Signals

Several works have reported on the reconstruction of 2D/3D limb kinematics from low-frequency EEG signals using linear regression models based on positive correlation values between the recorded and the reconstructed trajectories. This paper describes the mathematical properties of the linear model and the correlation evaluation metric that may lead to a misinterpretation of the results of this type of decoders. Firstly, the use of a linear regression model to adjust the two temporal signals (EEG and velocity profiles) implies that the relevant component of the signal used for decoding (EEG) has to be in the same frequency range as the signal to be decoded (velocity profiles). Secondly, the use of a correlation to evaluate the fitting of two trajectories could lead to overly-optimistic results as this metric is invariant to scale. Also, the correlation has a non-linear nature that leads to higher values for sinus/cosinus-like signals at low frequencies. Analysis of these properties on the reconstruction results was carried out through an experiment performed in line with previous studies, where healthy participants executed predefined reaching movements of the hand in 3D space. While the correlations of limb velocity profiles reconstructed from low-frequency EEG were comparable to studies in this domain, a systematic statistical analysis revealed that these results were not above the chance level. The empirical chance level was estimated using random assignments of recorded velocity profiles and EEG signals, as well as combinations of randomly generated synthetic EEG with recorded velocity profiles and recorded EEG with randomly generated synthetic velocity profiles. The analysis shows that the positive correlation results in this experiment cannot be used as an indicator of successful trajectory reconstruction based on a neural correlate. Several directions are herein discussed to address the misinterpretation of results as well as the implications on previous invasive and non-invasive works.     

Comparison of different methods of time shift measurement in EEG

Digital signal processing techniques are often used for measurement of small time shifts between EEG signals. In our work we tested properties of linear cross-correlation and phase/coherence method. The last mentioned method was used in two versions. The first version used fast Fourier transform (FFT) algorithm and the second was based on autoregressive modeling with fixed or adaptive model order. Methods were compared on several testing signals mimicking real EEG signals. The accuracy index for each method was computed. Results showed that for long signal segments all methods bring comparably good results. Accuracy of FFT phase/coherence method significantly decreased when very short segments were used and also decreased with an increasing level of the additive noise. The best results were obtained with autoregressive version of phase/coherence. This method is more reliable and may be used with high accuracy even in very short signals segments and it is also resistant to additive noise.     

Exploring Age-Related Changes in Dynamical Non-Stationarity in Electroencephalographic Signals during Early Adolescence

Dynamics of brain signals such as electroencephalogram (EEG) can be characterized as a sequence of quasi-stable patterns. Such patterns in the brain signals can be associated with coordinated neural oscillations, which can be modeled by non-linear systems. Further, these patterns can be quantified through dynamical non-stationarity based on detection of qualitative changes in the state of the systems underlying the observed brain signals. This study explored age-related changes in dynamical non-stationarity of the brain signals recorded at rest, longitudinally with 128-channel EEG during early adolescence (10 to 13 years of age, 56 participants). Dynamical non-stationarity was analyzed based on segmentation of the time series with subsequent grouping of the segments into clusters with similar dynamics. Age-related changes in dynamical non-stationarity were described in terms of the number of stationary states and the duration of the stationary segments. We found that the EEG signal became more non-stationary with age. Specifically, the number of states increased whereas the mean duration of the stationary segment decreased with age. These two effects had global and parieto-occipital distribution, respectively, with the later effect being most dominant in the alpha (around 10 Hz) frequency band.     

Cognitive neuroscience of creativity: EEG based approaches

Cognitive neuroscience of creativity has been extensively studied using non-invasive electrical recordings from the scalp called electroencephalograms (EEGs) and event related potentials (ERPs). The paper discusses major aspects of performing research using EEG/ERP based experiments including the recording of the signals, removing noise, estimating ERP signals, and signal analysis for better understanding of the neural correlates of processes involved in creativity. Important factors to be kept in mind to record clean EEG signal in creativity research are discussed. The recorded EEG signal can be corrupted by various sources of noise and methodologies to handle the presence of unwanted artifacts and filtering noise are presented followed by methods to estimate ERPs from the EEG signals from multiple trials. The EEG and ERP signals are further analyzed using various techniques including spectral analysis, coherence analysis, and non-linear signal analysis. These analysis techniques provide a way to understand the spatial activations and temporal development of large scale electrical activity in the brain during creative tasks. The use of this methodology will further enhance our understanding the processes neural and cognitive processes involved in creativity.     

Nonlinearity in normal human EEG: cycles,temporal asymmetry,nonstationarity and randomness,not chaos

Two-hour vigilance and sleep electroencephalogram (EEG) recordings from five healthy volunteers were analyzed using a method for identifying nonlinearity and chaos which combines the redundancy–linear redundancy approach with the surrogate data technique. A nonlinear component in the EEG was detected, however, inconsistent with the hypothesis of low-dimensional chaos. A possibility that a temporally asymmetric process may underlie or influence the EEG dynamics was indicated. A process that merges nonstationary nonlinear deterministic oscillations with randomness is proposed for an explanation of observed properties of the analyzed EEG signals. Taking these results into consideration, the use of dimensional and related chaos-based algorithms in quantitative EEG analysis is critically discussed. Received: 25 September 1994 / Accepted in revised form: 10 July 1996     

The short-time Fourier transform and muscle fatigue assessment in dynamic contractions

The mean frequency of the power spectrum of an electromyographic signal is an accepted index for monitoring fatigue in static contractions. There is however, indication that it may be a useful index even in dynamic contractions in which muscle length and/or force may vary. The objective of this investigation was to explore this possibility. An examination of the effects of amplitude modulation on modeled electromyographic signals revealed that changes in variance created in this way do not sufficiently affect characteristic frequency data to obscure a trend with fatigue. This validated the contention that not all non-stationarities in signals necessarily manifest in power spectral parameters. While an investigation of the nature and effects of non-stationarities in real electromyographic signals produced from dynamic contractions indicated that a more complex model is warranted, the results also indicated that averaging associated with estimating spectral parameters with the short-time Fourier transform can control the effects of the more complex non-stationarities. Finally, a fatigue test involving dynamic contractions at a force level under 30% of peak voluntary dynamic range, validated that it was possible to track fatigue in dynamic contractions using a traditional short-time Fourier transform methodology.     

Time-Dependent Statistical and Correlation Properties of Neural Signals during Handwriting

To elucidate the cortical control of handwriting, we examined time-dependent statistical and correlational properties of simultaneously recorded 64-channel electroencephalograms (EEGs) and electromyograms (EMGs) of intrinsic hand muscles. We introduced a statistical method, which offered advantages compared to conventional coherence methods. In contrast to coherence methods, which operate in the frequency domain, our method enabled us to study the functional association between different neural regions in the time domain. In our experiments, subjects performed about 400 stereotypical trials during which they wrote a single character. These trials provided time-dependent EMG and EEG data capturing different handwriting epochs. The set of trials was treated as a statistical ensemble, and time-dependent correlation functions between neural signals were computed by averaging over that ensemble. We found that trial-to-trial variability of both the EMGs and EEGs was well described by a log-normal distribution with time-dependent parameters, which was clearly distinguished from the normal (Gaussian) distribution. We found strong and long-lasting EMG/EMG correlations, whereas EEG/EEG correlations, which were also quite strong, were short-lived with a characteristic correlation durations on the order of 100 ms or less. Our computations of correlation functions were restricted to the spectral range (13–30 Hz) of EEG signals where we found the strongest effects related to handwriting. Although, all subjects involved in our experiments were right-hand writers, we observed a clear symmetry between left and right motor areas: inter-channel correlations were strong if both channels were located over the left or right hemispheres, and 2–3 times weaker if the EEG channels were located over different hemispheres. Although we observed synchronized changes in the mean energies of EEG and EMG signals, we found that EEG/EMG correlations were much weaker than EEG/EEG and EMG/EMG correlations. The absence of strong correlations between EMG and EEG signals indicates that (i) a large fraction of the EEG signal includes electrical activity unrelated to low-level motor variability; (ii) neural processing of cortically-derived signals by spinal circuitry may reduce the correlation between EEG and EMG signals.     

Reticular activation and the dynamics of neuronal networks

It is postulated that during arousal the cortical system is driven by a spatially and temporally noisy signal arising from non-specific reticulo-cortical pathways. An elementary unit of cortical neuroanatomy is assumed, which permits non-linear dynamics to be represented by stochastic linear equations. Under these assumptions the resonant modes of the system of cortical dendrites approach thermodynamic equilibrium. Specific sensory signals perturb the dendritic system about equilibrium, generate low frequency, linear, non-dispersive waves corresponding to the EEG, which in turn regulate action potential sequences, and instantiate internal inputs to the dendritic field. A large and distributed memory capacity in axo-synaptic couplings, resistance to interference between functionally separate logical operations, and a very large next-state function set emerge as properties of the network. The model is able to explain the close association of the EEG with cognition, the channel of low capacity corresponding to the field of immediate attention, the low overall correlation of action potentials with EEG, and specificity of action potentials in some neurons during particular cognitive activity. Predictions made from hypothesis include features of thermal equilibrium in EEG (determinable by autoregression) and expectation that the cortical evoked response can be accounted for as the response to a sensory impulse of specific time characteristics.     

Nonlinear analysis of EEG signals: Surrogate data analysis

ObjectivesThe electroencephalogram (EEG) signal contains information about the state and condition of the brain. The aim of the study is to conduct a nonlinear analysis of the EEG signals and to compare the differences in the nonlinear characteristics of the EEG during normal state and the epileptic state.DataThe EEG data used for this study – which consisted of epileptic EEG and normal EEG – were obtained from the EEG database available with the Bonn University, Germany.ResultsThe attractors seen in normal and epileptic human brain dynamics were studied and compared. Surrogate data analyses were conducted on two nonlinear measures, namely the largest Lyapunov exponent and the correlation dimension, to test the hypothesis whether EEG signals were in accordance with linear stochastic models.DiscussionsThe existence of deterministic chaos in brain activity is confirmed by the existence of a chaotic attractor; also, saturation of the correlation dimension towards a definite value is the manifestation of a deterministic dynamics. Also a reduction is observed between the dimensionalities of the brain attractors from normal state to the epileptic state. The evaluation of the largest Lyapunov exponent also confirms the lowering of complexity during an episode of seizure.ConclusionIn case of Lyapunov exponent of EEG data, the change due to surrogating is small suggesting that it is not representing the system complexity properly but there is a marked change in the case of correlation dimension value due to surrogating.     

Correlation dimension based lossless compression of EEG signals

Transmission of long duration EEG signals without loss of information is essential for telemedicine based applications. In this work, a lossless compression scheme for EEG signals based on neural network predictors using the concept of correlation dimension (CD) is proposed. EEG signals which are considered as irregular time series of chaotic processes can be characterized by the non-linear dynamic parameter CD which is a measure of the correlation among the EEG samples. The EEG samples are first divided into segments of 1 s duration and for each segment, the value of CD is calculated. Blocks of EEG samples are then constructed such that each block contains segments with closer CD values. By arranging the EEG samples in this fashion, the accuracy of the predictor is improved as it makes use of highly correlated samples. As a result, the magnitude of the prediction error decreases leading to less number of bits for transmission. Experiments are conducted using EEG signals recorded under different physiological conditions. Different neural network predictors as well as classical predictors are considered. Experimental results show that the proposed CD based preprocessing scheme improves the compression performance of the predictors significantly.     

Role of the autonomic nervous system in generating non-linear dynamics in short-term heart period variability.

We evaluated the role played by the autonomic nervous system in producing non-linear dynamics in short heart period variability (HPV) series recorded in healthy young humans. Non-linear dynamics are detected using an index of predictability based on a local non-linear predictor and a surrogate data approach. Different types of surrogates are utilized: (i) phase-randomized Fourier-transform based (FT) data; (ii) amplitude-adjusted FT (AAFT) data; and (iii) iteratively refined AAFT (IAAFT) data of two types (IAAFT-1 and IAAFT-2). The approach was applied to experimental protocols activating or blocking the sympathetic or parasympathetic branches of the autonomic nervous system or periodically perturbing cardiovascular control via paced respiration at different breathing rates. We found that short-term HPV was mostly linear at rest. Experimental protocols activating the sympathetic or parasympathetic nervous system did not produce non-linear dynamics. In contrast, paced respiration, especially at slow breathing rates, elicited significantly non-linear dynamics. Therefore, in short-term HPV ( approximately 300 beats) the use of non-linear models is not supported by the data, except under conditions whereby the subject is constrained to a slow respiratory rate.     

Adaptive Filtering Methods for Identifying Cross-Frequency Couplings in Human EEG

Oscillations have been increasingly recognized as a core property of neural responses that contribute to spontaneous, induced, and evoked activities within and between individual neurons and neural ensembles. They are considered as a prominent mechanism for information processing within and communication between brain areas. More recently, it has been proposed that interactions between periodic components at different frequencies, known as cross-frequency couplings, may support the integration of neuronal oscillations at different temporal and spatial scales. The present study details methods based on an adaptive frequency tracking approach that improve the quantification and statistical analysis of oscillatory components and cross-frequency couplings. This approach allows for time-varying instantaneous frequency, which is particularly important when measuring phase interactions between components. We compared this adaptive approach to traditional band-pass filters in their measurement of phase-amplitude and phase-phase cross-frequency couplings. Evaluations were performed with synthetic signals and EEG data recorded from healthy humans performing an illusory contour discrimination task. First, the synthetic signals in conjunction with Monte Carlo simulations highlighted two desirable features of the proposed algorithm vs. classical filter-bank approaches: resilience to broad-band noise and oscillatory interference. Second, the analyses with real EEG signals revealed statistically more robust effects (i.e. improved sensitivity) when using an adaptive frequency tracking framework, particularly when identifying phase-amplitude couplings. This was further confirmed after generating surrogate signals from the real EEG data. Adaptive frequency tracking appears to improve the measurements of cross-frequency couplings through precise extraction of neuronal oscillations.     

Relevance of nonlinear lumped-parameter models in the analysis of depth-EEG epileptic signals

In the field of epilepsy, the analysis of stereoelectroencephalographic (SEEG, intra-cerebral recording) signals with signal processing methods can help to better identify the epileptogenic zone, the area of the brain responsible for triggering seizures, and to better understand its organization. In order to evaluate these methods and to physiologically interpret the results they provide, we developed a model able to produce EEG signals from “organized” networks of neural populations. Starting from a neurophysiologically relevant model initially proposed by Lopes Da Silva et al. [Lopes da Silva FH, Hoek A, Smith H, Zetterberg LH (1974) Kybernetic 15: 27–37] and recently re-designed by Jansen et al. [Jansen BH, Zouridakis G, Brandt ME (1993) Biol Cybern 68: 275–283] the present study demonstrates that this model can be extended to generate spontaneous EEG signals from multiple coupled neural populations. Model parameters related to excitation, inhibition and coupling are then altered to produce epileptiform EEG signals. Results show that the qualitative behavior of the model is realistic; simulated signals resemble those recorded from different brain structures for both interictal and ictal activities. Possible exploitation of simulations in signal processing is illustrated through one example; statistical couplings between both simulated signals and real SEEG signals are estimated using nonlinear regression. Results are compared and show that, through the model, real SEEG signals can be interpreted with the aid of signal processing methods. Received: 3 January 2000 / Accepted: 24 March 2000