Tracking the time-varying cortical connectivity patterns by adaptive multivariate estimators

The directed transfer function (DTF) and the partial directed coherence (PDC) are frequency-domain estimators that are able to describe interactions between cortical areas in terms of the concept of Granger causality. However, the classical estimation of these methods is based on the multivariate autoregressive modelling (MVAR) of time series, which requires the stationarity of the signals. In this way, transient pathways of information transfer remains hidden. The objective of this study is to test a time-varying multivariate method for the estimation of rapidly changing connectivity relationships between cortical areas of the human brain, based on DTF/PDC and on the use of adaptive MVAR modelling (AMVAR) and to apply it to a set of real high resolution EEG data. This approach will allow the observation of rapidly changing influences between the cortical areas during the execution of a task. The simulation results indicated that time-varying DTF and PDC are able to estimate correctly the imposed connectivity patterns under reasonable operative conditions of signal-to-noise ratio (SNR) ad number of trials. An SNR of five and a number of trials of at least 20 provide a good accuracy in the estimation. After testing the method by the simulation study, we provide an application to the cortical estimations obtained from high resolution EEG data recorded from a group of healthy subject during a combined foot-lips movement and present the time-varying connectivity patterns resulting from the application of both DTF and PDC. Two different cortical networks were detected with the proposed methods, one constant across the task and the other evolving during the preparation of the joint movement.     

Estimation of time-varying connectivity patterns through the use of an adaptive directed transfer function

Frequency-derived identification of the propagation of information between brain regions has quickly become a popular area in the neurosciences. Of the various techniques used to study the propagation of activation within the central nervous system, the directed transfer function (DTF) has been well used to explore the functional connectivity during a variety of brain states and pathological conditions. However, the DTF method assumes the stationarity of the neural electrical signals and the time invariance of the connectivity among different channels over the investigated time window. Such assumptions may not be valid in the abnormal brain signals such as seizures and interictal spikes in epilepsy patients. In the present study, we have developed an adaptive DTF (ADTF) method through the use of a multivariate adaptive autoregressive model to study the time-variant propagation of seizures and interictal spikes in simulated electrocorticogram (ECoG) networks. The time-variant connectivity reconstruction is achieved by the Kalman filter algorithm, which can incorporate time-varying state equations. We study the performance of the proposed method through simulations with various propagation models using either sample seizures or interictal spikes as the source waveform. The present results suggest that the new ADTF method correctly captures the temporal dynamics of the propagation models, while the DTF method cannot, and even returns erroneous results in some cases. The present ADTF method was tested in real epileptiform ECoG data from an epilepsy patient, and the ADTF results are consistent with the clinical assessments performed by neurologists.      

A matrix-based algorithm for estimating multiple coherence of a periodic signal and its application to the multichannel EEG during sensory stimulation

The coherence between the stimulation signal and the electroencephalogram (EEG) has been used in the detection of evoked responses. The detector's performance, however, depends on both the signal-to-noise ratio (SNR) of the responses and the number of data segments (M) used in coherence estimation. In practical situations, when a given SNR occurs, detection can only be improved by increasing M and hence the total data length. This is particularly relevant when monitoring is the objective. In the present study, we propose a matrix-based algorithm for estimating the multiple coherence of the stimulation signal taking into account a set of N EEG channels as a way of increasing the detection rate for a fixed value of M. Monte Carlo simulations suggest that thresholds for such multivariate detector are the same as those for multiple coherence of Gaussian signals and that using more than six signals is not advisable for improving the detection rate with M = 10. The results with EEG from 12 normal subjects during photic stimulation at 10 Hz showed a maximum detection for N greater than 2 in 58% of the subjects with M = 10, and hence suggest that the proposed multivariate detector is valuable in evoked responses applications.     

On the tracking of rapid dynamic changes in seizure EEG

Estimation of autospectra and coherence and phase spectra of the seizure electroencephalograph (EEG), using the fast Fourier transform (FFT) technique, will cause smearing of the rapid dynamic changes which occur during the seizure. This is inherent to FFT spectral estimation, due to the averaging process which is necessary in order to get consistent spectral estimates. A different approach suggested in the present study is to carry out multivariate autoregressive modeling of the multichannel seizure EEG, combined with adaptive segmentation. In order to obtain good estimates in cases of short record length, the vectorial autoregressive (AR) modeling was based on residual energy ratios. The method has been tested on multichannel seizure EEG recordings from rats with focal epilepsy, caused by intracerebral administration of Kainic acid, and in-depth EEG recordings in patients with temporal lobe epilepsy     

Propagation pattern analysis during atrial fibrillation based on sparse modeling

In this study, sparse modeling is introduced for the estimation of propagation patterns in intracardiac atrial fibrillation (AF) signals. The estimation is based on the partial directed coherence function, derived from fitting a multivariate autoregressive model to the observed signal using least-squares (LS) estimation. The propagation pattern analysis incorporates prior information on sparse coupling as well as the distance between the recording sites. Two optimization methods are employed for estimation of the model parameters, namely, the adaptive group least absolute selection and shrinkage operator (aLASSO), and a novel method named the distance-adaptive group LASSO (dLASSO). Using simulated data, both optimization methods were superior to LS estimation with respect to detection and estimation performance. The normalized error between the true and estimated model parameters dropped from 0.20 ± 0.04 for LS estimation to 0.03 ± 0.01 for both aLASSO and dLASSO when the number of available data samples exceeded the number of model parameters by a factor of 5. For shorter data segments, the error reduction was more pronounced and information on the distance gained in importance. Propagation pattern analysis was also studied on intracardiac AF data, the results showing that the identification of propagation patterns is substantially simplified by the sparsity assumption.     

Spectral analysis of a thalamus-to-cortex seizure pathway

Physiological evidence has shown that the anterior thalamus (AN) and its associated efferents/afferents constitute an important propagation pathway for one animal model of generalized tonic clonic epileptic seizures. In this study the authors extend and confirm the support for AN's role by examining neuro-electric signal indicators during seizure episodes. They show that the electroencephalogram (EEG) recorded from AN is highly coherent with the EEG derived from the cortex (CTX). By removing the effects of another thalamic nucleus, posterior thalamus (PT)-unaffiliated with the tract linking AN to cortex-partial coherence analysis leaves the CTX/AN coherence undiminished. The most robust band of strong CTX-AN coherence is centered around the spike wave pacing frequency of 1-3 Hz. Partial-multiple coherence analysis techniques are used to remove the possible signal contributions from hippocampus in addition to PT. The CTX-AN coherence still remains undiminished in the low-frequency bands. Conclusive evidence from coherence studies and other spectral measures reaffirm the special role of the AN in the propagation of seizure activity from subcortex to cortex     

Conventional and wavelet coherence applied to sensory-evoked electrical brain activity

The use of coherence is a well-established standard approach for the analysis of biomedical signals. Being entirely based on frequency analysis, i.e., on spectral properties of the signal, it is not possible to obtain any information about the temporal structure of coherence which is useful in the study of brain dynamics, for example. Extending the concept of coherence as a measure of linear dependence between realizations of a random process to the wavelet transform, this paper introduces a new approach to coherence analysis which allows to monitor time-dependent changes in the coherence between electroenecphalographic (EEG) channels. Specifically, we analyzed multichannel EEG data of 26 subjects obtained in an experiment on associative learning, and compare the results of Fourier coherence and wavelet coherence, showing that wavelet coherence detects features that were inaccessible by application of Fourier coherence.     

Multivariate autoregressive models for classification ofspontaneous electroencephalographic signals during mental tasks

This article explores the use of scalar and multivariate autoregressive (AR) models to extract features from the human electroencephalogram (EEG) with which mental tasks can be discriminated. This is part of a larger project to investigate the feasibility of using EEG to allow paralyzed persons to control a device such as a wheelchair. EEG signals from four subjects were recorded while they performed two mental tasks. Quarter-second windows of six-channel EEG were transformed into four different representations: scalar AR model coefficients, multivariate AR coefficients, eigenvalues of a correlation matrix, and the Karhunen-Loeve transform of the multivariate AR coefficients. Feature vectors defined by these representations were classified with a standard, feedforward neural network trained via the error backpropagation algorithm. The four representations produced similar results, with the multivariate AR coefficients performing slightly better and more consistently with an average classification accuracy of 91.4% on novel, untrained, EEG signals     

On time delay estimation of epileptic EEG

The present study deals with comparative evaluation of various methods for time delay estimations applied to multichannel seizure EEG. The different methods included block algorithms, both in time and frequency domains (such as general crosscorrelation, FFT, AR), and a new method for time delay estimation based on adaptive least-squares filtering. The various time delay estimators were tested on simulated signals and on real multichannel EEG recorded from rats having generalized seizures with focal onset. The adaptive least-squares filtering method (the lattice-ladder type) has been found to be the most efficient for time delay estimation     

Spatial filtering and neocortical dynamics: estimates of EEGcoherence

The spatial statistics of scalp electroencephalogram (EEG) are usually presented as coherence in individual frequency bands. These coherences result both from correlations among neocortical sources and volume conduction through the tissues of the head. The scalp EEG is spatially low-pass filtered by the poorly conducting skull, introducing artificial correlation between the electrodes. A four concentric spheres (brain, CSF, skull, and scalp) model of the head and stochastic field theory are used here to derive an analytic estimate of the coherence at scalp electrodes due to volume conduction of uncorrelated source activity, predicting that electrodes within 10-12 cm can appear correlated. The surface Laplacian estimate of cortical surface potentials spatially bandpass filters the scalp potentials reducing this artificial coherence due to volume conduction. Examination of EEG data confirms that the coherence estimates from raw scalp potentials and Laplacians are sensitive to different spatial bandwidths and should be used in parallel in studies of neocortical dynamic function     

The Canonical Bicoherence—Part I: Definition, Multitaper Estimation, and Statistics

The central theme of this pair of papers (Parts I and II) [IEEE Transactions on Signal Processing, vol. 57, no. 4, April 2009] is a new definition: the canonical bicoherence, a combination of the canonical coherence and the bicoherence. The canonical bicoherence is an effective tool for analyzing quadratic nonlinearity in multivariate signals. In this first part, the definition and properties of the canonical bicoherence are presented. The feasibility of the canonical bicoherence in detecting quadratic phase coupling (QPC) of multivariate signals is explained theoretically, illustrated by an example, and verified by numerical simulations. Multitaper methods and a sequence of three singular value decompositions (SVD's) are used to estimate canonical bicoherences, to achieve reliable estimates with a reasonable amount of memory and computation time. Finally, we show that the canonical bicoherence estimate has an approximate asymptotic $kappachi_{nu}^{2}$-distribution, and the weighted jackknife method, used over tapers and segments, is applied to estimate variances of multitaper canonical bicoherence estimates.      

Fourier-Based Feature Extraction for Classification of EEG Signals Using EEG Rhythms

In this paper, we propose a method for the analysis and classification of electroencephalogram (EEG) signals using EEG rhythms. The EEG rhythms capture the nonlinear complex dynamic behavior of the brain system and the nonstationary nature of the EEG signals. This method analyzes common frequency components in multichannel EEG recordings, using the filter bank signal processing. The mean frequency (MF) and RMS bandwidth of the signal are estimated by applying Fourier-transform-based filter bank processing on the EEG rhythms, which we refer intrinsic band functions, inherently present in the EEG signals. The MF and RMS bandwidth estimates, for the different classes (e.g., ictal and seizure-free, open eyes and closed eyes, inter-ictal and ictal, healthy volunteers and epileptic patients, inter-ictal epileptogenic and opposite to epileptogenic zone) of EEG recordings, are statistically different and hence used to distinguish and classify the two classes of signals using a least-squares support vector machine classifier. Experimental results, with 100 % classification accuracy, on a real-world EEG signals database analysis illustrate the effectiveness of the proposed method for EEG signal classification.     

Time-frequency characterization of interdependencies in nonstationary signals: application to epileptic EEG

For the past decades, numerous works have been dedicated to the development of signal processing methods aimed at measuring the degree of association between electroencephalographic (EEG) signals. This interdependency parameter, which may be defined in various ways, is often used to characterize a functional coupling between different brain structures or regions during either normal or pathological processes. In this paper, we focus on the time-frequency characterization of the interdependency between signals. Particularly, we propose a novel estimator of the linear relationship between nonstationary signals based on the cross correlation of narrow band filtered signals. This estimator is compared to a more classical estimator based on the coherence function. In a simulation framework, results show that it may exhibit better statistical performances (bias and variance or mean square error) when a priori knowledge about time delay between signals is available. On real data (intracerebral EEG signals), results show that this estimator may also enhance the readability of the time-frequency representation of relationship and, thus, can improve the interpretation of nonstationary interdependencies in EEG signals. Finally, we illustrate the importance of characterizing the relationship in both time and frequency domains by comparing with frequency-independent methods (linear and nonlinear).     

The Influence of Model Parameters on BEG/MEG Single Dipole Source Estimation

This paper deals with source localization and strength estimation based on EEG and MEG data. It describes an estimation method (inverse procedure) which uses a four-spheres model of the head and a single current dipole. The dependency of the inverse solution on model parameters is investigated. It is found that sphere radii and conductivities influence especially the strength of the EEG equivalent dipole and not its location or direction. The influence on the equivalent dipole of the gradiometer is investigated. In general the MEG produces better location estimates than the EEG whereas the reverse is found for the component estimates. An inverse solution simultaneously based on EEG and MEG data appears slightly better than the average of separate EEG and MEG solutions. Variances of parameter estimators which can be calculated on the basis of a linear approximation of the model, were tested by Monte Carlo simulations.     

On the errors of two estimators of sub-pixel fractional cover whenmixing is linear

The authors consider the problem of estimating ground cover at sub-pixel scales from remotely sensed imagery. In particular, they examine two strategies that make use of a set of reference pixels, or training pixels, for which fractional ground cover is already known. These strategies are the so-called classical and inverse methods. The former proceeds by assuming that signals received at a sensor are area-weighted averages of characteristic signals for each ground cover component, estimates those characteristic signals from the training pixels, and predicts fractions for a general pixel to be those that give the best match of the modeled and observed image signals. The latter approach proceeds by direct multivariate regression of ground cover proportions on pixel spectral values. The authors show that when ground cover types are spectrally well separated, and mixing is indeed linear, the difference between the two estimators is much smaller than the prediction error associated with either. This means that it is perfectly acceptable to use standard methods of multivariate regression to perform spectral unmixing. They also show that the inverse estimator can be regarded as a regularized form of the classical estimator and that the supposed optimality of the inverse method may be compromised if the training dataset is not a random subset of the complete set of image pixels     

Bounds on estimating the acoustic attenuation of small tissue regions from reflected ultrasound

The acoustic attenuation coefficient of most soft biological tissues has an approximately linear-with-frequency attenuation characteristic. The slope of the attenuation coefficient with frequency, denoted by β, has been observed to vary with the severity of liver disease. Two approaches for estimating the β value from reflected ultrasound signals are examined: the spectral-difference approach, which estimates β from the slope of the difference between log-spectra from two locations within the liver, and the spectral-shift approach, which estimates β from the downward shift experienced by the propagating pulse spectrum with penetration into the liver. This paper considers signals reflected from a small tissue region, defined by a cell measuring D by D centimeters in the plane of the sonogram, thus determining the feasibility of producing attenuation images. Lower bounds on the variance of the two β estimators are calculated by deriving maximum-likelihood estimators and by locating the tissue cell in the focal plane of the transducer. If W denotes the usable bandwidth in the reflected signals, the bounds are shown to be proportional to (WD)^-3for β estimates calculated from individual reflected signals, and (WD)^-4for estimates from all the signals reflected from the tissue region. With currently available technology, clinically useful results can be obtained for cell sizes measuring approximately 2.0 cm on a side.     

Monitoring neuronal oscillations and signal transmission betweencortical regions using time-frequency analysis ofelectroencephalographic activity

Oscillatory states in the electroencephalogram (EEG) reflect the rhythmic synchronous activity in large networks of neurons. Time-frequency (TF) methods, which quantify the spectral content of the EEG as a function of time, are well suited as tools for the study of spontaneous and induced changes in oscillatory states. The use of these methods provides insights into the temporal dynamics of EEG activity in both humans and experimental animals, and aids the study of the neuronal mechanisms that generate rhythmic EEG activity. Further the use of TF coherence analysts, which quantifies the consistency of phase relationships in multichannel EEG recordings, may contribute to the understanding of signal transmission between neuronal populations in different parts of the brain. We have used TF techniques to analyze the flow of activity patterns between two strongly connected brain structures: the entorhinal cortex and the hippocampus. Both of these structures are believed to be involved in information storage. By applying various frequencies of stimulation, we have found a peak in the spectral power in both sites at around 18 Hz, but the coherence between the EEG signals recorded from these sites was found to increase monotonically up to about 35 Hz. We have also found that long-term potentiation, a strong increase in the efficacy of excitatory synapses between these sites, either had no effect or decreased coherence     

On polynomial phase signals with time-varying amplitudes

We address the parameter estimation problem for a class of nonstationary signals modeled as polynomial phase signals with time-varying amplitudes. Exponentially damped polynomial phase signals are treated as a special case and are analyzed in detail. High-order instantaneous moments provide the basic analytical tool, but links are shown to exist with either the usually employed FFT-based technique or the high-resolution Kumaresan-Tufts (1982), MUSIC, and matrix pencil methods. Asymptotic properties of the relevant estimators are established, Cramer-Rao lower bounds on the amplitude and phase parameter estimates are derived, and computer simulations are carried out to evaluate the performance of various schemes. We focus on parametric modeling of AM-FM signals, mainly because parametric techniques offer parsimony and allow for theoretically unlimited resolution     

Approximate Time-Variable Coherence Analysis of Multichannel Signals

We present a new method for signal extraction from noisy multichannel epileptic seizure onset EEG signals. These signals are non-stationary which makes time-invariant filtering unsuitable. The new method assumes a signal model and performs denoising by filtering the signal of each channel using a time-variable filter which is an estimate of the Wiener filter. The approximate Wiener filters are obtained using the time-frequency coherence functions between all channel pairs, and a fix-point algorithm. We estimate the coherence functions using the multiple window method, after which the fix-point algorithm is applied. Simulations indicate that this method improves upon its restriction to assumed stationary signals for realistically non-stationary data, in terms of mean square error, and we show that it can also be used for time-frequency representation of noisy multichannel signals. The method was applied to two epileptic seizure onset signals, and it turned out that the most informative output of the method are the filters themselves studied in the time-frequency domain. They seem to reveal hidden features of the epileptic signal which are otherwise invisible. This algorithm can be used as preprocessing for seizure onset EEG signals prior to time-frequency representation and manual or algorithmic pattern classification.     

Performance Bounds for MIMO-OFDM Channel Estimation

The performance of a mobile multiple-input multiple-output orthogonal-frequency-division multiplexing (MIMO-OFDM) system depends on the ability of the system to accurately account for the effects of the frequency-selective time-varying channel at every symbol time and at every frequency subcarrier. Typically, pilot symbols are strategically placed at various times over various subcarriers in order to calculate sample channel estimates, and then these estimates are interpolated or extrapolated forward to provide channel estimates where no pilot data was transmitted. Performance is highly dependent on the distribution of the pilots with respect to the coherence time and coherence bandwidth of the channel, and on the chosen channel parameterization. In this paper, a vector formulation of the Cramer-Rao bound (CRB) for biased estimators and for functions of parameters is used to derive a lower bound on the channel estimation and prediction error of such a system. Numerical calculations using the bound demonstrate the benefits of multiple antennas for channel estimation and prediction and illustrate the impact of modeling errors on estimation performance when using channel models based on calibrated arrays.