Estimation of neural dynamics from MEG/EEG cortical current density maps: application to the reconstruction of large-scale cortical synchrony

There is a growing interest in elucidating the role of specific patterns of neural dynamics--such as transient synchronization between distant cell assemblies--in brain functions. Magnetoencephalography (MEG)/electroencephalography (EEG) recordings consist in the spatial integration of the activity from large and multiple remotely located populations of neurons. Massive diffusive effects and poor signal-to-noise ratio (SNR) preclude the proper estimation of indices related to cortical dynamics from nonaveraged MEG/EEG surface recordings. Source localization from MEG/EEG surface recordings with its excellent time resolution could contribute to a better understanding of the working brain. We propose a robust and original approach to the MEG/EEG distributed inverse problem to better estimate neural dynamics of cortical sources. For this, the surrogate data method is introduced in the MEG/EEG inverse problem framework. We apply this approach on nonaveraged data with poor SNR using the minimum norm estimator and find source localization results weakly sensitive to noise. Surrogates allow the reduction of the source space in order to reconstruct MEG/EEG data with reduced biases in both source localization and time-series dynamics. Monte Carlo simulations and results obtained from real MEG data indicate it is possible to estimate non invasively an important part of cortical source locations and dynamic and, therefore, to reveal brain functional networks.     

Cross validation for selection of cortical interaction models from scalp EEG or MEG

A cross-validation (CV) method based on state-space framework is introduced for comparing the fidelity of different cortical interaction models to the measured scalp electroencephalogram (EEG) or magnetoencephalography (MEG) data being modeled. A state equation models the cortical interaction dynamics and an observation equation represents the scalp measurement of cortical activity and noise. The measured data are partitioned into training and test sets. The training set is used to estimate model parameters and the model quality is evaluated by computing test data innovations for the estimated model. Two CV metrics normalized mean square error and log-likelihood are estimated by averaging over different training/test partitions of the data. The effectiveness of this method of model selection is illustrated by comparing two linear modeling methods and two nonlinear modeling methods on simulated EEG data derived using both known dynamic systems and measured electrocorticography data from an epilepsy patient.     

Distinguishing between moving and stationary sources using EEG/MEG measurements with an application to epilepsy

Performances of electroencephalography (EEG) and magnetoencephalography (MEG) source estimation methods depend on the validity of the assumed model. In many cases, the model structure is related to physical information. We discuss a number of statistical selection methods to distinguish between two possible models using least-squares estimation and assuming a spherical head model. The first model has a single moving source whereas the second has two stationary sources; these may result in similar EEG/MEG measurements. The need to decide between such models occurs for example in Jacksonian seizures (e.g., epilepsy) or in intralobular activities, where a model with either two stationary dipole sources or a single moving dipole source may be possible. We also show that all of the selection methods discussed choose the correct model with probability one when the number of trials goes to infinity. Finally we present numerical examples and compare the performances of the methods by varying parameters such as the signal-to-noise ratio, source depth, and separation of sources, and also apply the methods to real MEG data for epilepsy.     

A multiresolution framework to MEG/EEG source imaging

A new method based on a multiresolution approach for solving the ill-posed problem of brain electrical activity reconstruction from electroencephaloram (EEG)/magnetoencephalogram (MEG) signals is proposed in a distributed source model. At each step of the algorithm, a regularized solution to the inverse problem is used to constrain the source space on the cortical surface to be scanned at higher spatial resolution. We present the iterative procedure together with an extension of the ST-maximum a posteriori method [1] that integrates spatial and temporal a priori information in an estimator of the brain electrical activity. Results from EEG in a phantom head experiment with a real human skull and from real MEG data on a healthy human subject are presented. The performances of the multiresolution method combined with a nonquadratic estimator are compared with commonly used dipolar methods, and to minimum-norm method with and without multiresolution. In all cases, the proposed approach proved to be more efficient both in terms of computational load and result quality, for the identification of sparse focal patterns of cortical current density, than the fixed scale imaging approach.     

Differential characterization of neural sources with the bimodaltruncated SVD pseudo-inverse for EEG and MEG measurements

A method for obtaining a practical inverse for the distribution of neural activity in the human cerebral cortex is developed for electric, magnetic, and bimodal data to exploit their complementary aspects. Intracellular current is represented by current dipoles uniformly distributed on two parallel sulci joined by a gyrus. Linear systems of equations relate electric, magnetic, and bimodal data to unknown dipole moments. The corresponding lead-field matrices are characterized by singular value decomposition (SVD). The optimal reference electrode location for electric data is chosen on the basis of the decay behavior of the singular values. The singular values of these matrices show better decay behavior with increasing number of measurements, however, that property is useful depending on the noise in the measurements. The truncated SVD pseudo-inverse is used to control noise artifacts in the reconstructed images. Simulations for single-dipole sources at different depths reveal the relative contributions of electric and magnetic measures. For realistic noise levels the performance of both unimodal and bimodal systems do not improve with an increase in the number of measurements beyond ~100. Bimodal image reconstructions are generally superior to unimodal ones in finding the center of activity     

MEG and EEG source localization in beamspace

Beamspace methods are applied to EEG/MEG source localization problems in this paper. Beamspace processing involves passing the data through a linear transformation that reduces the data dimension prior to applying a desired statistical signal processing algorithm. This process generally reduces the data requirements of the subsequent algorithm. We present one approach for designing beamspace transformations that are optimized to preserve source activity located within a given region of interest and show that substantial reductions in dimension are obtained with negligible signal loss. Beamspace versions of maximum likelihood dipole fitting, MUSIC, and minimum variance beamforming source localization algorithms are presented. The performance improvement offered by the beamspace approach with limited data is demonstrated by bootstrapping somatosensory data to evaluate the variability of the source location estimates obtained with each algorithm. The quantitative benefits of beamspace processing depend on the algorithm, signal to noise ratio, and amount of data. Dramatic performance improvements are obtained in scenarios with low signal to noise ratio and a small number of independent data samples.     

Maximum-likelihood estimation of low-rank signals for multiepoch MEG/EEG analysis

A maximum-likelihood-based algorithm is presented for reducing the effects of spatially colored noise in evoked response magneto- and electro-encephalography data. The repeated component of the data, or signal of interest, is modeled as the mean, while the noise is modeled as the Kronecker product of a spatial and a temporal covariance matrix. The temporal covariance matrix is assumed known or estimated prior to the application of the algorithm. The spatial covariance structure is estimated as part of the maximum-likelihood procedure. The mean matrix representing the signal of interest is assumed to be low-rank due to the temporal and spatial structure of the data. The maximum-likelihood estimates of the components of the low-rank signal structure are derived in order to estimate the signal component. The relationship between this approach and principal component analysis (PCA) is explored. In contrast to prestimulus-based whitening followed by PCA, the maximum-likelihood approach does not require signal-free data for noise whitening. Consequently, the maximum-likelihood approach is much more effective with nonstationary noise and produces better quality whitening for a given data record length. The efficacy of this approach is demonstrated using simulated and real MEG data.     

Sensitivity distributions of EEG and MEG measurements

It is generally believed that because the skull has low conductivity to electric current but is transparent to magnetic fields, the measurement sensitivity of the magnetoencephalography (MEG) in the brain region should be more concentrated than that of the electroencephalography (EEG). It is also believed that the information recorded by these techniques is very different. If this were indeed the case, it might be possible to justify the cost of MEG instrumentation which is at least 25 times higher than that of EEG instrumentation. The localization of measurement sensitivity using these techniques was evaluated quantitatively in an inhomogeneous spherical head model using a new concept called half-sensitivity volume (HSV). It is shown that the planar gradiometer has a far smaller HSV than the axial gradiometer. However, using the EEG it is possible to achieve even smaller HSVs than with whole-head planar gradiometer MEG devices. The micro-superconducting quantum interference device (SQUID) MEG device does have HSVs comparable to those of the EEG. The sensitivity distribution of planar gradiometers, however, closely resembles that of dipolar EEG leads and, therefore, the MEG and EEG record the electric activity of the brain in a very similar way     

Estimating stationary dipoles from MEG/EEG data contaminated withspatially and temporally correlated background noise

The stationary dipole model for the inverse problem of magnetoencephalographic (MEG) and electroencephalographic (EEG) data is extended by including spatio-temporal correlations of the background noise. For that purpose, the spatio-temporal covariances are described as a Kronkecker product of a spatial and a temporal covariance matrix. The maximum likelihood method is used to estimate this Kronecker product from a series of trials of MEG/EEG data. A simulation study shows that the inclusion of the background noise generally improves the dipole estimate substantially. When the frequency of the source time functions, however, coincides with the frequency contents of the covariance function, the dipole estimate worsens when the temporal correlations are included. The inclusion of spatial correlations always improves the estimates     

Integrated MEG/EEG and fMRI model based on neural masses

We introduce a bottom-up model for integrating electroencephalography (EEG) or magnetoencephalography (MEG) with functional magnetic resonance imaging (fMRI). An extended neural mass model is proposed based on the physiological principles of cortical minicolumns and their connections. The fMRI signal is extracted from the proposed neural mass model by introducing a relationship between the stimulus and the neural activity and using the resultant neural activity as input of the extended Balloon model. The proposed model, validated using simulations, is instrumental in evaluating the upcoming combined methods for simultaneous analysis of MEG/EEG and fMRI.     

A Spatiotemporal Framework for Estimating Trial-to-Trial Amplitude Variation in Event-Related MEG/EEG

A spatiotemporal framework for estimating trial-to-trial variability in evoked response (ER) data is presented. Spatial and temporal bases capture the aspects of the response that are consistent across trials, while the basis expansion coefficients represent the variable components of the response. We focus on the simplest case of constant spatiotemporal response shape and varying amplitude across trials. Two different constraints on the amplitude evolution are employed to effectively integrate the individual responses and improve robustness at low SNR. The linear dynamical system response constraint estimates the current trial amplitude as an unknown constant scaling of the estimate in the previous trial plus zero-mean Gaussian noise with unknown variance. The independent response constraint estimates response amplitudes across trials as independent Gaussian random variables having unknown mean and variance. We develop a generalized expectation-maximization algorithm to obtain the maximum-likelihood (ML) estimates of the signal waveform, noise covariance matrix, and unknown constraint parameters. ML source localization is achieved by scanning the likelihood over different sets of spatial bases. We demonstrate the variability estimation and source localization effectiveness of the proposed algorithms using both real and simulated ER data.     

A maximum-likelihood estimator for trial-to-trial variations in noisy MEG/EEG data sets

The standard procedure to determine the brain response from a multitrial evoked magnetoencephalography (MEG) or electroencephalography (EEG) data set is to average the individual trials of these data, time locked to the stimulus onset. When the brain responses vary from trial-to-trial this approach is false. In this paper, a maximum-likelihood estimator is derived for the case that the recorded data contain amplitude variations. The estimator accounts for spatially and temporally correlated background noise that is superimposed on the brain response. The model is applied to a series of 17 MEG data sets of normal subjects, obtained during median nerve stimulation. It appears that the amplitude of late component (30-120 ms) shows a systematic negative trend indicating a weakening response during stimulation time. For the early components (20-35 ms) no such a systematic effect was found. The model is furthermore applied on a MEG data set consisting of epileptic spikes of constant spatial distribution but varying polarity. For these data, the advantage of applying the model is that positive and negative spikes can be processed with a single model, thereby reducing the number of degrees of freedom and increasing the signal-to-noise ratio.     

Conditional Correlation as a Measure of Mediated Interactivity in fMRI and MEG/EEG

Many measures have been proposed so far to extract brain functional interactivity from functional magnetic resonance imaging (fMRI) and magnetoencephalography/electroencephalography (MEG/EEG) data sets. Unfortunately, none has been able to provide a relevant, self-contained, and common definition of brain interaction. In this paper, we propose a first step in this direction. We first introduce a common terminology together with a cross-modal definition of interaction. In this setting, we investigate the commonalities shared by some measures of interaction proposed in the literature. We show that temporal correlation, nonlinear correlation, mutual information, generalized synchronization, phase synchronization, coherence, and phase locking value (PLV) actually measure the same quantity (namely correlation) when one is investigating linear interactions between independently and identically distributed Gaussian variables. We also demonstrate that these data-driven measures can only partly account for the interaction patterns that can be expressed by the effective connectivity of structural equation modeling (SEM) . To bridge this gap, we suggest the use of conditional correlation, which is shown to be related to mediated interaction.     

Estimating evoked dipole responses in unknown spatially correlatednoise with EEG/MEG arrays

We present maximum likelihood (ML) methods for estimating evoked dipole responses using electroencephalography (EEG) and magnetoencephalography (MEG) arrays, which allow for spatially correlated noise between sensors with unknown covariance. The electric source is modeled as a collection of current dipoles at fixed locations and the head as a spherical conductor. We permit the dipoles' moments to vary with time by modeling them as linear combinations of parametric or nonparametric basis functions. We estimate the dipoles' locations and moments and derive the Cramer-Rao bound for the unknown parameters. We also propose an ML based method for scanning the brain response data, which can be used to initialize the multidimensional search required to obtain the true dipole location estimates. Numerical simulations demonstrate the performance of the proposed methods     

Spatiotemporal EEG/MEG source analysis based on a parametric noise covariance model

A method is described to incorporate the spatiotemporal noise covariance matrix into a spatiotemporal source analysis. The essential feature is that the estimation problem is split into two parts. First, a model is fitted to the observed noise covariance matrix. This model is a Kronecker product of a spatial and a temporal matrix. The spatial matrix models the spatial covariances by a function dependent on sensor distance. The temporal matrix models the temporal covariances as lag dependent. In the second part, sources are estimated given this noise model, which can be done very efficiently due to the Kronecker formulation. An application to real electroencephalogram (EEG) data shows that the noise model fits the data very well. Simulation results show that the resulting source estimates are more precise than those obtained from a standard analysis neglecting the noise covariance. In addition, the estimated standard errors of the source parameter estimates are far more precise than those obtained from a standard analysis. Finally, the source parameter standard errors are used to investigate the effects of temporal sampling. It is shown that increasing the sampling by a factor x, decreases the standard errors of all source parameters with the square root of x.     

A state-space modeling approach for localization of focal current sources from MEG

State-space modeling is a promising approach for current source reconstruction from magnetoencephalography (MEG) because it constrains the spatiotemporal behavior of inverse solutions in a flexible manner. However, state-space model-based source localization research remains underdeveloped; extraction of spatially focal current sources and handling of the high dimensionality of the distributed source model remain problematic. In this study, we propose a novel state-space model-based method that resolves these problems, extending our previous source localization method to include a temporal constraint by state-space modeling. To enable focal current reconstruction, we account for spatially inhomogeneous temporal dynamics by introducing dynamics model parameters that differ for each cortical position. The model parameters and the intensity of the current sources are jointly estimated according to a bayesian framework. We circumvent the high dimensionality of the problem by assuming prior distributions of the model parameters to reduce the sensitivity to unmodeled components, and by adopting variational bayesian inference to reduce the computational cost. Through simulation experiments and application to real MEG data, we have confirmed that our proposed method successfully reconstructs focal current activities, which evolve with their temporal dynamics.     

Performance analysis of reduced-rank beamformers for estimating dipole source signals using EEG/MEG

We study the performance of various beamformers for estimating a current dipole source at a known location using electroencephalography (EEG) and magnetoencephalography (MEG). We present our beamformers in the form of the generalized sidelobe canceler (GSC). Under this structure, the beamformer can be solved by finding a filter that achieves the minimum mean-squared error (MMSE) between the mainbeam response and filtered observed signal. We express the MMSE as a function of the filter's rank and use it as a criterion to evaluate the performance of the beamformers. We do not make any assumptions on the rank of the interference-plus-noise covariance matrix. Instead, we treat it as low-rank and derive a general expression for the MMSE. We present numerical examples to compare the MSE performance of beamformers commonly studied in the literature: principal components (PCs), cross-spectral metrics (CSMs), and eigencanceler (EIG) beamformers. Our results show that good estimates of the dipole source signals can be achieved using reduced-rank beamformers even for low signal-to-noise ratio (SNR) values.     

A resampling method for estimating the signal subspace of spatio-temporal EEG/MEG data

Source localization using spatio-temporal electroencephalography (EEG) and magnetoencephalography (MEG) data is usually performed by means of signal subspace methods. The first step of these methods is the estimation of a set of vectors that spans a subspace containing as well as possible the signal of interest. This estimation is usually performed by means of a singular value decomposition (SVD) of the data matrix: The rank of the signal subspace (denoted by r) is estimated from a plot in which the singular values are plotted against their rank order, and the signal subspace itself is estimated by the first r singular vectors. The main problem with this method is that it is strongly affected by spatial covariance in the noise. Therefore, two methods are proposed that are much less affected by this spatial covariance, and old and a new method. The old method involves prewhitening of the data matrix, making use of an estimate of the spatial noise covariance matrix. The new method is based on the matrix product of two average data matrices, resulting from a random partition of a set of stochastically independent replications of the spatio-temporal data matrix. The estimated signal subspace is obtained by first filtering out the asymmetric and negative definite components of this matrix product and then retaining the eigenvectors that correspond to the r largest eigenvalues of this filtered data matrix. The main advantages of the partition-based eigen decomposition over prewhited SVD is that 1) it does not require an estimate of the spatial noise covariance matrix and 2b) that it allows one to make use of a resampling distribution (the so-called partitioning distribution) as a natural quantification of the uncertainty in the estimated rank. The performance of three methods (SVD with and without prewhitening, and the partition-based method) is compared in a simulation study. From this study, it could be concluded that prewhited SVD and the partition-based eigen decomposition perform equally well when the amplitude time series are constant, but that the partition-based method performs better when the amplitude time series are variable.     

Hierarchical clustering and filtering in half-inverse space for MEG and/or EEG hypothesis-free analysis

We propose hierarchical clustering and filtering methods for the analysis of spatio-temporal multidimensional time series, where both methods are based on a new pseudo distance. The pseudo distance is determined between orthogonal matrices, which are derived by eigenvalue decomposition of the variance-covariance matrix of the time series. Because the grouping algorithm is also important in clustering, a modified Ward method grouping criterion is used here. The filtering derives temporal similarity information between two time series, providing information that cannot be evaluated by the clustering. If the time series to be clustered and filtered cannot be obtained directly, different time series reflecting the original time series are used instead. There exists a transform between the time series, and hence, scaling distortion occurs. We also propose a scaling normalization method. As an application example, we present an analysis of a multichannel magnetoencephalography (MEG) and/or electroencephalography (EEG) time series. Each of the MEG and EEG generations is a transform from the same electrophysiological brain activity. We applied these methods to sound localization-related MEG time series and evaluated their effectiveness. These methods may be useful for discovering similarity among many multidimensional time series without a priori information and/or hypotheses.     

Effort/gains dynamics in heterogeneous networks

We investigate the behaviour of TCP(α, β) protocols in the presence of wireless networks. We seek an answer to strategic issues of maximizing energy and bandwidth exploitation, without damaging the dynamics of multiple‐flow equilibrium. We take a fresh perspective on protocol design: What is the return of the effort that a protocol expends? Can we achieve more gains with less effort? We study first the design assumptions of TCP(α, β) protocols and discuss the impact of equation‐based modulation of α and β on protocol efficiency. We introduce two new measures to capture protocol behaviour: the ‘Extra Energy Expenditure’ and the ‘Unexploited Available Resource Index’. We confirm experimentally that, in general, smoothness and responsiveness constitute a tradeoff; however, we show that this tradeoff does not graft its dynamics into a conservative/aggressive behaviour, as it is traditionally believed. We uncover patterns of unjustified tactics; our results suggest that an adaptive congestion control algorithm is needed to integrate the dynamics of heterogeneous networks into protocol behaviour. Copyright © 2007 John Wiley & Sons, Ltd.