EEG Inverse Problem Research Articles

A Sparse Dictionary Learning Framework to Discover Discriminative Source Activations in EEG Brain Mapping

Electroencephalography (EEG) source analysis is one of the most important noninvasive human brain imaging tools that provides millisecond temporal accuracy. However, discovering essential activated brain sources associated with different brain status is still a challenging problem. In this study, we propose for the first time that the ill-posed EEG inverse problem can be formulated and solved as a sparse over-complete dictionary learning problem. In particular, a novel supervised sparse dictionary learning framework was developed for EEG source reconstruction. A revised version of discriminative K-SVD (DK-SVD) algorithm is exploited to solve the formulated supervised dictionary learning problem. As the proposed learning framework incorporated the EEG label information of different brain status, it is capable of learning a sparse representation that reveal the most discriminative brain activity sources among different brain states. Compared to the state-of-the-art EEG source analysis methods, proposed sparse dictionary learning framework achieved significant superior performance in both computing speed and accuracy for the challenging EEG source reconstruction problem through extensive numerical experiments. More importantly, the experimental results also validated that the proposed sparse learning framework is effective to discover the discriminative task-related brain activation sources, which shows the potential to advance the high resolution EEG source analysis for real-time non-invasive brain imaging research.

Spatio-temporal reconstruction of brain dynamics from EEG with a Markov prior

Electroencephalography (EEG) can capture brain dynamics in high temporal resolution. By projecting the scalp EEG signal back to its origin in the brain also high spatial resolution can be achieved. Source localized EEG therefore has potential to be a very powerful tool for understanding the functional dynamics of the brain. Solving the inverse problem of EEG is however highly ill-posed as there are many more potential locations of the EEG generators than EEG measurement points. Several well-known properties of brain dynamics can be exploited to alleviate this problem. More short ranging connections exist in the brain than long ranging, arguing for spatially focal sources. Additionally, recent work (Delorme et al., 2012) argues that EEG can be decomposed into components having sparse source distributions. On the temporal side both short and long term stationarity of brain activation are seen. We summarize these insights in an inverse solver, the so-called “Variational Garrote” (Kappen and Gómez, 2013). Using a Markov prior we can incorporate flexible degrees of temporal stationarity. Through spatial basis functions spatially smooth distributions are obtained. Sparsity of these are inherent to the Variational Garrote solver. We name our method the MarkoVG and demonstrate its ability to adapt to the temporal smoothness and spatial sparsity in simulated EEG data. Finally a benchmark EEG dataset is used to demonstrate MarkoVG's ability to recover non-stationary brain dynamics.

S-SMOOTH: Sparsity and Smoothness Enhanced EEG Brain Tomography.

EEG source imaging enables us to reconstruct current density in the brain from the electrical measurements with excellent temporal resolution (~ ms). The corresponding EEG inverse problem is an ill-posed one that has infinitely many solutions. This is due to the fact that the number of EEG sensors is usually much smaller than that of the potential dipole locations, as well as noise contamination in the recorded signals. To obtain a unique solution, regularizations can be incorporated to impose additional constraints on the solution. An appropriate choice of regularization is critically important for the reconstruction accuracy of a brain image. In this paper, we propose a novel Sparsity and SMOOthness enhanced brain TomograpHy (s-SMOOTH) method to improve the reconstruction accuracy by integrating two recently proposed regularization techniques: Total Generalized Variation (TGV) regularization and ℓ1−2 regularization. TGV is able to preserve the source edge and recover the spatial distribution of the source intensity with high accuracy. Compared to the relevant total variation (TV) regularization, TGV enhances the smoothness of the image and reduces staircasing artifacts. The traditional TGV defined on a 2D image has been widely used in the image processing field. In order to handle 3D EEG source images, we propose a voxel-based Total Generalized Variation (vTGV) regularization that extends the definition of second-order TGV from 2D planar images to 3D irregular surfaces such as cortex surface. In addition, the ℓ1−2 regularization is utilized to promote sparsity on the current density itself. We demonstrate that ℓ1−2 regularization is able to enhance sparsity and accelerate computations than ℓ1 regularization. The proposed model is solved by an efficient and robust algorithm based on the difference of convex functions algorithm (DCA) and the alternating direction method of multipliers (ADMM). Numerical experiments using synthetic data demonstrate the advantages of the proposed method over other state-of-the-art methods in terms of total reconstruction accuracy, localization accuracy and focalization degree. The application to the source localization of event-related potential data further demonstrates the performance of the proposed method in real-world scenarios.

Identification of Interictal Epileptic Networks from Dense-EEG.

Epilepsy is a network disease. The epileptic network usually involves spatially distributed brain regions. In this context, noninvasive M/EEG source connectivity is an emerging technique to identify functional brain networks at cortical level from noninvasive recordings. In this paper, we analyze the effect of the two key factors involved in EEG source connectivity processing: (i) the algorithm used in the solution of the EEG inverse problem and (ii) the method used in the estimation of the functional connectivity. We evaluate four inverse solutions algorithms (dSPM, wMNE, sLORETA and cMEM) and four connectivity measures (r 2, h 2, PLV, and MI) on data simulated from a combined biophysical/physiological model to generate realistic interictal epileptic spikes reflected in scalp EEG. We use a new network-based similarity index to compare between the network identified by each of the inverse/connectivity combination and the original network generated in the model. The method will be also applied on real data recorded from one epileptic patient who underwent a full presurgical evaluation for drug-resistant focal epilepsy. In simulated data, results revealed that the selection of the inverse/connectivity combination has a significant impact on the identified networks. Results suggested that nonlinear methods (nonlinear correlation coefficient, phase synchronization and mutual information) for measuring the connectivity are more efficient than the linear one (the cross correlation coefficient). The wMNE inverse solution showed higher performance than dSPM, cMEM and sLORETA. In real data, the combination (wMNE/PLV) led to a very good matching between the interictal epileptic network identified from noninvasive EEG recordings and the network obtained from connectivity analysis of intracerebral EEG recordings. These results suggest that source connectivity method, when appropriately configured, is able to extract highly relevant diagnostic information about networks involved in interictal epileptic spikes from non-invasive dense-EEG data.

A simple method for EEG guided transcranial electrical stimulation without models

Objective. There is longstanding interest in using EEG measurements to inform transcranial Electrical Stimulation (tES) but adoption is lacking because users need a simple and adaptable recipe. The conventional approach is to use anatomical head-models for both source localization (the EEG inverse problem) and current flow modeling (the tES forward model), but this approach is computationally demanding, requires an anatomical MRI, and strict assumptions about the target brain regions. We evaluate techniques whereby tES dose is derived from EEG without the need for an anatomical head model, target assumptions, difficult case-by-case conjecture, or many stimulation electrodes. Approach. We developed a simple two-step approach to EEG-guided tES that based on the topography of the EEG: (1) selects locations to be used for stimulation; (2) determines current applied to each electrode. Each step is performed based solely on the EEG with no need for head models or source localization. Cortical dipoles represent idealized brain targets. EEG-guided tES strategies are verified using a finite element method simulation of the EEG generated by a dipole, oriented either tangential or radial to the scalp surface, and then simulating the tES-generated electric field produced by each model-free technique. These model-free approaches are compared to a ‘gold standard’ numerically optimized dose of tES that assumes perfect understanding of the dipole location and head anatomy. We vary the number of electrodes from a few to over three hundred, with focality or intensity as optimization criterion. Main results. Model-free approaches evaluated include (1) voltage-to-voltage, (2) voltage-to-current; (3) Laplacian; and two Ad-Hoc techniques (4) dipole sink-to-sink; and (5) sink to concentric. Our results demonstrate that simple ad hoc approaches can achieve reasonable targeting for the case of a cortical dipole, remarkably with only 2–8 electrodes and no need for a model of the head. Significance. Our approach is verified directly only for a theoretically localized source, but may be potentially applied to an arbitrary EEG topography. For its simplicity and linearity, our recipe for model-free EEG guided tES lends itself to broad adoption and can be applied to static (tDCS), time-variant (e.g., tACS, tRNS, tPCS), or closed-loop tES.

An attempt to localize brain electrical activity sources using EEG with limited number of electrodes

A very interesting research goal is to find underlying sources generating the EEG signal – referred to as the “EEG inverse problem”. Its aim is to determine spatial distribution of brain activity, described by local brain currents density, on the basis of potentials measured on the scalp as EEG signal. The purpose of the research presented in the article was to check whether the results of the inverse problem solution, obtained by the LORETA algorithm for the reduced set of 8 electrodes selected by the authors will be close to the results for the initial set of 32 electrodes. EEG signals were registered during the BCI operation based on ERD/ERS potentials. Obtained results showed no significant differences in the location of the most important sources in both cases. It is worth emphasizing that reducing the number of electrodes would have a significant impact on an BCI ergonomics.

High-resolution imaging-guided electroencephalography source localization: temporal effect regularization incorporation in LORETA inverse solution

Electroencephalography (EEG) and magnetic resonance imaging (MRI) are noninvasive neuroimag- ing modalities. They are widely used and could be complementary. The fusion of these modalities may enhance some emerging research fields targeting the exploration better brain activities. Such research attracted various scientific investigators especially to provide a convivial and helpful advanced clinical-aid tool enabling better neurological explorations. Our present research was, in fact, in the context of EEG inverse problem resolution and investigated an advanced estimation methodology for the localization of the cerebral activity. Our focus was, therefore, on the integration of temporal priors to low-resolution brain electromagnetic tomography (LORETA) formalism and to solve the inverse problem in the EEG. The main idea behind our proposed method was in the integration of a temporal projection matrix within the LORETA weighting matrix. A hyperparameter is the principal fact for such a temporal integration, and its importance would be obvious when obtaining a regularized smooth- ness solution. Our experimental results clearly confirmed the impact of such an optimization procedure adopted for the temporal regularization parameter comparatively to the LORETA method. © 2015 SPIE and IS&T (DOI: 10

The performance of the spatiotemporal Kalman filter and LORETA in seizure onset localization.

The assumption of spatial-smoothness is often used to solve the bioelectric inverse problem during electroencephalographic (EEG) source imaging, e.g., in low resolution electromagnetic tomography (LORETA). Since the EEG data show a temporal structure, the combination of the temporal-smoothness and the spatial-smoothness constraints may improve the solution of the EEG inverse problem. This study investigates the performance of the spatiotemporal Kalman filter (STKF) method, which is based on spatial and temporal smoothness, in the localization of a focal seizure's onset and compares its results to those of LORETA. The main finding of the study was that the STKF with an autoregressive model of order two significantly outperformed LORETA in the accuracy and consistency of the localization, provided that the source space consists of a whole-brain volumetric grid. In the future, these promising results will be confirmed using data from more patients and performing statistical analyses on the results. Furthermore, the effects of the temporal smoothness constraint will be studied using different types of focal seizures.

Identification of epileptogenic networks from dense EEG: A model-based study.

Epilepsy is a network disease. Identifying the epileptogenic networks from noninvasive recordings is a challenging issue. In this context, M/EEG source connectivity is a promising tool to identify brain networks with high temporal and spatial resolution. In this paper, we analyze the impact of the two main factors that intervene in EEG source connectivity processing: i) the algorithm used to solve the EEG inverse problem and ii) the method used to measure the functional connectivity. We evaluate three inverse solutions algorithms (dSPM, wMNE and cMEM) and three connectivity measures (r(2), h(2) and MI) on data simulated from a combined biophysical/physiological model able to generate realistic interictal epileptic spikes reflected in scalp EEG. The performance criterion used here is the similarity between the network identified by each of the inverse/connectivity combination and the original network used in the source model. Results show that the choice of the combination has a high impact on the identified network. Results suggest also that nonlinear methods (nonlinear correlation coefficient and mutual information) for measuring the connectivity are more efficient than the linear one (the cross correlation coefficient). The dSPM as inverse solution shows the lowest performance compared to cMEM and wMNE.

Solving the EEG inverse problem based on space–time–frequency structured sparsity constraints

We introduce STOUT (spatio-temporal unifying tomography), a novel method for the source analysis of electroencephalograpic (EEG) recordings, which is based on a physiologically-motivated source representation. Our method assumes that only a small number of brain sources are active throughout a measurement, where each of the sources exhibits focal (smooth but localized) characteristics in space, time and frequency. This structure is enforced through an expansion of the source current density into appropriate spatio-temporal basis functions in combination with sparsity constraints. This approach combines the main strengths of two existing methods, namely Sparse Basis Field Expansions (Haufe et al., 2011) and Time–Frequency Mixed-Norm Estimates (Gramfort et al., 2013). By adjusting the ratio between two regularization terms, STOUT is capable of trading temporal for spatial reconstruction accuracy and vice versa, depending on the requirements of specific analyses and the provided data. Due to allowing for non-stationary source activations, STOUT is particularly suited for the localization of event-related potentials (ERP) and other evoked brain activity. We demonstrate its performance on simulated ERP data for varying signal-to-noise ratios and numbers of active sources. Our analysis of the generators of visual and auditory evoked N200 potentials reveals that the most active sources originate in the temporal and occipital lobes, in line with the literature on sensory processing.

EEG source connectivity analysis: from dense array recordings to brain networks.

The recent past years have seen a noticeable increase of interest for electroencephalography (EEG) to analyze functional connectivity through brain sources reconstructed from scalp signals. Although considerable advances have been done both on the recording and analysis of EEG signals, a number of methodological questions are still open regarding the optimal way to process the data in order to identify brain networks. In this paper, we analyze the impact of three factors that intervene in this processing: i) the number of scalp electrodes, ii) the combination between the algorithm used to solve the EEG inverse problem and the algorithm used to measure the functional connectivity and iii) the frequency bands retained to estimate the functional connectivity among neocortical sources. Using High-Resolution (hr) EEG recordings in healthy volunteers, we evaluated these factors on evoked responses during picture recognition and naming task. The main reason for selection this task is that a solid literature background is available about involved brain networks (ground truth). From this a priori information, we propose a performance criterion based on the number of connections identified in the regions of interest (ROI) that belong to potentially activated networks. Our results show that the three studied factors have a dramatic impact on the final result (the identified network in the source space) as strong discrepancies were evidenced depending on the methods used. They also suggest that the combination of weighted Minimum Norm Estimator (wMNE) and the Phase Synchronization (PS) methods applied on High-Resolution EEG in beta/gamma bands provides the best performance in term of topological distance between the identified network and the expected network in the above-mentioned cognitive task.

A regularized matrix factorization approach to induce structured sparse-low-rank solutions in the EEG inverse problem

We consider the estimation of the Brain Electrical Sources (BES) matrix from noisy electroencephalographic (EEG) measurements, commonly named as the EEG inverse problem. We propose a new method to induce neurophysiological meaningful solutions, which takes into account the smoothness, structured sparsity, and low rank of the BES matrix. The method is based on the factorization of the BES matrix as a product of a sparse coding matrix and a dense latent source matrix. The structured sparse-low-rank structure is enforced by minimizing a regularized functional that includes the ℓ21-norm of the coding matrix and the squared Frobenius norm of the latent source matrix. We develop an alternating optimization algorithm to solve the resulting nonsmooth-nonconvex minimization problem. We analyze the convergence of the optimization procedure, and we compare, under different synthetic scenarios, the performance of our method with respect to the Group Lasso and Trace Norm regularizers when they are applied directly to the target matrix.

Localization of brain electrical activity sources and hemodynamic activity foci during motor imagery

Studied are sources of brain activity contributing to EEG patterns which correspond to motor imagery. The accuracy of their classification determines the efficiency of brain-computer interface (BCI) allowing for controlling external technical devices directly by brain signals without involving muscle activity. Sources of brain activity are identified by Independent Component Analysis. Those independent components for which the BCI classification accuracy are at maximum are treated as relevant for motor imagery task. Two of the most relevant sources demonstrate strictly exposed event related desynchronization and synchronization of mu--rhythm during imagery of contra--and ipsilateral hands. These sources are localized by solving inverse EEG problem taking into account individual geometry of brain and its covers provided by anatomical MRI images. The sources are shown to be localized in BA 3A relating to proprioceptive sensitivity of the contralateral hand. Their positions are closed to foci of BOLD activity obtained by fMRI.

L1 Norm based common spatial patterns decomposition for scalp EEG BCI

BackgroundBrain computer interfaces (BCI) is one of the most popular branches in biomedical engineering. It aims at constructing a communication between the disabled persons and the auxiliary equipments in order to improve the patients’ life. In motor imagery (MI) based BCI, one of the popular feature extraction strategies is Common Spatial Patterns (CSP). In practical BCI situation, scalp EEG inevitably has the outlier and artifacts introduced by ocular, head motion or the loose contact of electrodes in scalp EEG recordings. Because outlier and artifacts are usually observed with large amplitude, when CSP is solved in view of L2 norm, the effect of outlier and artifacts will be exaggerated due to the imposing of square to outliers, which will finally influence the MI based BCI performance. While, L1 norm will lower the outlier effects as proved in other application fields like EEG inverse problem, face recognition, etc.MethodsIn this paper, we present a new CSP implementation using the L1 norm technique, instead of the L2 norm, to solve the eigen problem for spatial filter estimation with aim to improve the robustness of CSP to outliers. To evaluate the performance of our method, we applied our method as well as the standard CSP and the regularized CSP with Tikhonov regularization (TR-CSP), on both the peer BCI dataset with simulated outliers and the dataset from the MI BCI system developed in our group. The McNemar test is used to investigate whether the difference among the three CSPs is of statistical significance.ResultsThe results of both the simulation and real BCI datasets consistently reveal that the proposed method has much higher classification accuracies than the conventional CSP and the TR-CSP.ConclusionsBy combining L1 norm based Eigen decomposition into Common Spatial Patterns, the proposed approach can effectively improve the robustness of BCI system to EEG outliers and thus be potential for the actual MI BCI application, where outliers are inevitably introduced into EEG recordings.

Detection of correlated sources in EEG using combination of beamforming and surface Laplacian methods

Beamforming offers a way to estimate the solution to the inverse problem in EEG and MEG but is also known to perform poorly in the presence of highly correlated sources, e.g. during binaural auditory stimulation, when both left and right primary auditory cortices are activated simultaneously. Surface Laplacian, or the second spatial derivative calculated from the electric potential, allows for deblurring of EEG potential recordings reducing the effects of low skull conductivity and is independent of the reference electrode location. We show that anatomically constrained beamforming in conjunction with the surface Laplacian allows for detection of both locations and dynamics of temporally correlated sources in EEG. Whole-head 122 channel binaural stimulus EEG data were simulated using a boundary element method (BEM) and realistic geometry forward model. We demonstrate that in contrast to conventional potential-based EEG beamforming, Laplacian beamforming allows to determine locations of correlated source dipoles without any a priori assumption about the number of sources. We also show (by providing simulations of auditory evoked potentials) that the dynamics at the detected source locations can be derived from subsets of electrodes. Deblurring auditory evoked potential maps subdivides EEG signals from each hemisphere and allows for the beamformer to be applied separately for left and right hemispheres.

EEG-fMRI fusion on the cortical surface using Coupled Tensor-Matrix Factorization: A simulation study

EEG-fMRI fusion on the cortical surface using Coupled Tensor-Matrix Factorization: A simulation study

Model and Simulation of the Brain Scalp Potential Analysis

Based on spherical head models, this paper, by employing the finite element method (FEM), analyzes the potential distribution of the brain scalp surface and attempts to work out the electroencephalography (EEG) forward problem, in hope of finding out the impact the dipole parameters has on it. According to the electromagnetism theory, this paper discusses the general resolution of EEG, it requires electric potentials of the globe's surface, and graphically displays results of computation through finite element post-processing, which tests their effectiveness. Furthermore, it analyzes the influences of dipole parameters on the potential distribution of scalp surface, such as position, direction and strength, which attempts to provide an effective method to solve EEG forward problem, based on diversified head model, and also proposes a prior information to the solution to EEG inverse problem.

Conventional and Reciprocal Approaches to the Inverse Dipole Localization Problem for N20–P20 Somatosensory Evoked Potentials

The non-invasive localization of the primary sensory hand area can be achieved by solving the inverse problem of electroencephalography (EEG) for N(20)-P(20) somatosensory evoked potentials (SEPs). This study compares two different mathematical approaches for the computation of transfer matrices used to solve the EEG inverse problem. Forward transfer matrices relating dipole sources to scalp potentials are determined via conventional and reciprocal approaches using individual, realistically shaped head models. The reciprocal approach entails calculating the electric field at the dipole position when scalp electrodes are reciprocally energized with unit current-scalp potentials are obtained from the scalar product of this electric field and the dipole moment. Median nerve stimulation is performed on three healthy subjects and single-dipole inverse solutions for the N(20)-P(20) SEPs are then obtained by simplex minimization and validated against the primary sensory hand area identified on magnetic resonance images. Solutions are presented for different time points, filtering strategies, boundary-element method discretizations, and skull conductivity values. Both approaches produce similarly small position errors for the N(20)-P(20) SEP. Position error for single-dipole inverse solutions is inherently robust to inaccuracies in forward transfer matrices but dependent on the overlapping activity of other neural sources. Significantly smaller time and storage requirements are the principal advantages of the reciprocal approach. Reduced computational requirements and similar dipole position accuracy support the use of reciprocal approaches over conventional approaches for N(20)-P(20) SEP source localization.

Hierarchical Bayesian inference for the EEG inverse problem using realistic FE head models: Depth localization and source separation for focal primary currents

The estimation of the activity-related ion currents by measuring the induced electromagnetic fields at the head surface is a challenging and severely ill-posed inverse problem. This is especially true in the recovery of brain networks involving deep-lying sources by means of EEG/MEG recordings which is still a challenging task for any inverse method. Recently, hierarchical Bayesian modeling (HBM) emerged as a unifying framework for current density reconstruction (CDR) approaches comprising most established methods as well as offering promising new methods. Our work examines the performance of fully-Bayesian inference methods for HBM for source configurations consisting of few, focal sources when used with realistic, high-resolution finite element (FE) head models. The main foci of interest are the correct depth localization, a well-known source of systematic error of many CDR methods, and the separation of single sources in multiple-source scenarios. Both aspects are very important in the analysis of neurophysiological data and in clinical applications. For these tasks, HBM provides a promising framework and is able to improve upon established CDR methods such as minimum norm estimation (MNE) or sLORETA in many aspects. For challenging multiple-source scenarios where the established methods show crucial errors, promising results are attained. Additionally, we introduce Wasserstein distances as performance measures for the validation of inverse methods in complex source scenarios.

Subspace electrode selection methodology for the reduction of the effect of uncertain conductivity values in the EEG dipole localization: a simulation study using a patient-specific head model

The simulation of the electroencephalogram (EEG) using a realistic head model needs the correct conductivity values of several tissues. However, these values are not precisely known and have an influence on the accuracy of the EEG source analysis problem. This paper presents a novel numerical methodology for the increase of accuracy of the EEG dipole source localization problem. The presented subspace electrode selection (SES) methodology is able to limit the effect of uncertain conductivity values on the solution of the EEG inverse problem, yielding improved source localization accuracy. We redefine the traditional cost function associated with the EEG inverse problem and introduce a selection procedure of EEG potentials. In each iteration of the presented EEG cost function minimization procedure, potentials are selected that are affected as little as possible by the uncertain conductivity value. Using simulation data, the proposed SES methodology is able to enhance the neural source localization accuracy dependent on the dipole location, the assumed versus actual conductivity and the possible noise in measurements.