Effect of Head Shape Variations Among Individuals on the EEG/MEG Forward and Inverse Problems

We study the effect of the head shape variations on the EEG/magnetoencephalography (MEG) forward and inverse problems. We build a random head model such that each sample represents the head shape of a different individual and solve the forward problem assuming this random head model, using a polynomial chaos expansion. The random solution of the forward problem is then used to quantify the effect of the geometry when the inverse problem is solved with a standard head model. The results derived with this approach are valid for a continuous family of head models, rather than just for a set of cases. The random model consists of three random surfaces that define layers of different electric conductivity, and we built an example based on a set of 30 deterministic models from adults. Our results show that for a dipolar source model, the effect of the head shape variations on the EEG/MEG inverse problem due to the random head model is slightly larger than the effect of the electronic noise present in the sensors. The variations in the EEG inverse problem solutions are due to the variations in the shape of the volume conductor, while the variations in the MEG inverse problem solutions, larger than the EEG ones, are caused mainly by the variations of the absolute position of the sources in a coordinate system based on anatomical landmarks, in which the magnetometers have a fixed position.     

Array response kernels for EEG and MEG in multilayer ellipsoidal geometry

We present forward modeling solutions in the form of array response kernels for electroencephalography (EEG) and magnetoencephalography (MEG), assuming that a multilayer ellipsoidal geometry approximates the anatomy of the head and a dipole current models the source. The use of an ellipsoidal geometry is useful in cases for which incorporating the anisotropy of the head is important but a better model cannot be defined. The structure of our forward solutions facilitates the analysis of the inverse problem by factoring the lead field into a product of the current dipole source and a kernel containing the information corresponding to the head geometry and location of the source and sensors. This factorization allows the inverse problem to be approached as an explicit function of just the location parameters, which reduces the complexity of the estimation solution search. Our forward solutions have the potential of facilitating the solution of the inverse problem, as they provide algebraic representations suitable for numerical implementation. The applicability of our models is illustrated with numerical examples on real EEG/MEG data of N20 responses. Our results show that the residual data after modeling the N20 response using a dipole for the source and an ellipsoidal geometry for the head is in average lower than the residual remaining when a spherical geometry is used for the same estimated dipole.     

A meshless method for solving the EEG forward problem

We present a numerical method to solve the quasistatic Maxwell equations and compute the electroencephalography (EEG) forward problem solution. More generally, we develop a computationally efficient method to obtain the electric potential distribution generated by a source of electric activity inside a three-dimensional body of arbitrary shape and layers of different electric conductivities. The method needs only a set of nodes on the surface and inside the head, but not a mesh connecting the nodes. This represents an advantage over traditional methods like boundary elements or finite elements since the generation of the mesh is typically computationally intensive. The performance of the proposed method is compared with the boundary element method (BEM) by numerically solving some EEG forward problems examples. For a large number of nodes and the same precision, our method has lower computational load than BEM due to a faster convergence rate and to the sparsity of the linear system to be solved.     

EEG and MEG: forward solutions for inverse methods

A solution of the forward problem is an important component of any method for computing the spatio-temporal activity of the neural sources of magnetoencephalography (MEG) and electroencephalography (EEG) data. The forward problem involves computing the scalp potentials or external magnetic field at a finite set of sensor locations for a putative source configuration. We present a unified treatment of analytical and numerical solutions of the forward problem in a form suitable for use in inverse methods. This formulation is achieved through factorization of the lead field into the product of the moment of the elemental current dipole source with a "kernel matrix" that depends on the head geometry and source and sensor locations, and a "sensor matrix" that models sensor orientation and gradiometer effects in MEG and differential measurements in EEG. Using this formulation and a recently developed approximation formula for EEG, based on the "Berg parameters," we present novel reformulations of the basic EEG and MEG kernels that dispel the myth that EEG is inherently more complicated to calculate than MEG. We also present novel investigations of different boundary element methods (BEM's) and present evidence that improvements over currently published BEM methods can be realized using alternative error-weighting methods. Explicit expressions for the matrix kernels for MEG and EEG for spherical and realistic head geometries are included.     

Evaluating the Performance of Kalman-Filter-Based EEG Source Localization

Electroencephalographic (EEG) source localization is an important tool for noninvasive study of brain dynamics, due to its ability to probe neural activity more directly, with better temporal resolution than other imaging modalities. One promising technique for solving the EEG inverse problem is Kalman filtering, because it provides a natural framework for incorporating dynamic EEG generation models in source localization. Here, a recently developed inverse solution is introduced, which uses spatiotemporal Kalman filtering tuned through likelihood maximization. Standard diagnostic tests for objectively evaluating Kalman filter performance are then described and applied to inverse solutions for simulated and clinical EEG data. These tests, employed for the first time in Kalman-filter-based source localization, check the statistical properties of the innovation and validate the use of likelihood maximization for filter tuning. However, this analysis also reveals that the filter's existing space- and time-invariant process model, which contains a single fixed-frequency resonance, is unable to completely model the complex spatiotemporal dynamics of EEG data. This finding indicates that the algorithm could be improved by allowing the process model parameters to vary in space.      

A method for localizing EEG sources in realistic head models

A computationally practical method for performing moving dipole calculations to localize EEG sources in realistic, boundary element (integral equation) type of head models is presented. This method makes use of a rapid method of solving the forward problem of calculating the EEG's produced by a dipole in a realistic head model. This rapid forward calculation method allows the use of standard Simplex search techniques to solve the inverse problem of localizing electrical sources in the brain from EEG's measured on the scalp     

A recursive algorithm for the three-dimensional imaging of brain electric activity: Shrinking LORETA-FOCUSS

Estimation of intracranial electric activity from the scalp electroencephalogram (EEG) requires a solution to the EEG inverse problem, which is known as an ill-conditioned problem. In order to yield a unique solution, weighted minimum norm least square (MNLS) inverse methods are generally used. This paper proposes a recursive algorithm, termed Shrinking LORETA-FOCUSS, which combines and expands upon the central features of two well-known weighted MNLS methods: LORETA and FOCUSS. This recursive algorithm makes iterative adjustments to the solution space as well as the weighting matrix, thereby dramatically reducing the computation load, and increasing local source resolution. Simulations are conducted on a 3-shell spherical head model registered to the Talairach human brain atlas. A comparative study of four different inverse methods, standard Weighted Minimum Norm, L1-norm, LORETA-FOCUSS and Shrinking LORETA-FOCUSS are presented. The results demonstrate that Shrinking LORETA-FOCUSS is able to reconstruct a three-dimensional source distribution with smaller localization and energy errors compared to the other methods.     

Spatio-temporal cortical source imaging of brain electrical activity by means of time-varying parametric projection filter

In the present study, we explore suitable spatio-temporal filters for inverse estimation of an equivalent dipole-layer distribution from the scalp electroencephalogram (EEG) for imaging of brain electric sources. We propose a time-varying parametric projection filter (tPPF) for the spatio-temporal EEG analysis. The performance of this tPPF algorithm was evaluated by computer simulation studies. An inhomogeneous three-concentric-spheres model was used in the present simulation study to represent the head volume conductor. An equivalent dipole layer was used to represent equivalently brain electric sources and estimated from the scalp potentials. The tPPF filter was tested to remove time-varying noise such as instantaneous artifacts caused by eyes-blink. The present simulation results indicate that the proposed time-variant tPPF method provides enhanced performance in rejecting time-varying noise, as compared with the time-invariant parametric projection filter.     

脑电逆问题的延时相关阵子空间分解算法

 根据头表观测电位反演脑电源的空间信息是脑电研究中的一个重要问题.本文提出了脑电逆问题的延时相关阵子空间分解算法.通过在三层同心球头模型上,与现行延时为零的相关阵子空间分解算法的对比研究表明,该方法能更好的压制空间相干噪音,显示了一定的应用前景.     

Application of quasi-static magnetic reciprocity to finite element models of the MEG lead-field

The reconstruction of neuronal current sources from magneto- and/or electroencephalography (MEG/EEG) measurements is referred to as an inverse problem. A precursor to most inverse algorithms is a forward transfer, or lead-field, matrix, in which the rows correspond to MEG and/or EEG measurement sites, and each column captures the linear response to a particular unit source. Simple models of the head, such as concentric spheres, result in analytic expressions for the lead-field. More realistic head models, such as those based on medical imagery, require numeric simulations. A straightforward, though inefficient, way to obtain the lead-field is to perform one forward simulation for each source, resulting in one column of the lead-field. For MEG/EEG inverse problems, however, the potential sources (rows) far outnumber the measurement sites (columns). Two approaches have been described for computing the EEG lead-field with a number of forward simulations equal to the number of measurement rows, rather than the number of source columns. One of these approaches is based on the principle of electric reciprocity, and the other approach is based on linear-algebraic manipulations of the forward problem. For the MEG lead-field, only a linear-algebraic approach has been described for numeric approaches such as the finite element method. This paper describes a reciprocal approach for the MEG lead-field and discusses implementation details for both approaches.     

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.     

Discretization Error Analysis and Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography: Part I

For imaging problems in which numerical solutions need to be computed for both the inverse and the underlying forward problems, discretization can be a major factor that determines the accuracy of imaging. In this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography. We model the forward problem by a pair of diffusion equations at the excitation and emission wavelengths and consider a finite element discretization method for the numerical solution of the forward problem. For the inverse problem, we use an optimization framework which allows incorporation of a priori information in the form of zeroth- and first-order Tikhonov regularization terms. Next, we convert the inverse problem into a variational problem and use Galerkin projection to discretize the inverse problem. Following the discretization, we analyze the error in reconstructed images due to the discretization of the forward and inverse problems and present two theorems which point out the factors that may lead to high error such as the mutual dependence of the forward and inverse problems, the number of sources and detectors, their configuration and their positions with respect to fluorophore concentration, and the formulation of the inverse problem. Finally, we demonstrate the results and implications of our error analysis by numerical experiments. In the second part of the paper, we apply our results to design novel adaptive discretization algorithms.      

The forward EEG solutions can be computed using artificial neural networks

Study of electroencenphalography (EEG) is the one of the most utilized methods in both basic brain research and clinical diagnosis of neurological disorders. Recent technological advances in computer and electronic systems have allowed the EEG to be recorded from large electrode arrays. Modeling the brain waves using a head volume conductor model provides an effective method to localize functional generators within the brain. However, the forward solutions to this model, which represent theoretical potentials in response to current sources within the volume conductor, are difficult to compute because of time-consuming numerical procedures utilized in either the boundary element method (BEM) or the finite element method (FEM). This paper presents a novel computational approach using an artificial neural network (ANN) to map two vectors of forward solutions. These two vectors correspond to different head models but with respect to the same current source. The input vector to the ANN is based on the spherical head model, which can be computed efficiently but involves large errors. The output vector from the ANN is based on the spheroidal model, which is more precise, but difficult to compute directly using the traditional means. Our experiments indicate that this ANN approach provides a remarkable improvement over the BEM and FEM methods: 1) the mean-square error of computation was only approximately 0.3% compared to the exact solution; 2) the online computation was extremely efficient, requiring only 168 floating point operations per channel to compute the forward solution, and 10.2 K-bytes of storage to represent the entire ANN. Using this approach it is possible to perform real-time EEG modeling accurately on personal computers.     

A Global Sensitivity Analysis of Three- and Four-Layer EEG Conductivity Models

The accuracy of forward models for EEG partly depends on the conductivity values of the head tissues. Yet, the influence of the conductivities on the model output is still not well understood. In this paper, we apply a variance-based sensitivity analysis method to the most common EEG forward models (three or four layers). This method is global because it quantifies the influence of each parameter with all the parameters varying at the same time. With nonlinear models, it helps to understand the interaction between parameters, which is not possible with simple sensitivity analyses (one-at-a-time variations, derivatives, and perturbations). By analyzing the potential topographies at the electrodes, we obtained several results. For a shallow dipole, the EEG topographies are mainly sensitive to the interaction between skull and scalp conductivities. It means that the variability of the EEG topographies is driven mostly by a function of skull and scalp conductivities. Similar results are presented for skull anisotropy and a current injection as performed in electrical impedance tomography. This global sensitivity analysis gives new information about EEG forward models—it identifies the main input parameters that need model refinement—and directions on how to calibrate these models.      

Evaluation of boundary element methods for the EEG forward problem:effect of linear interpolation

The authors implement the approach for solving the boundary integral equation for the electroencephalography (EEG) forward problem proposed by de Munck (1992), in which the electric potential varies linearly across each plane triangle of the mesh. Previous solutions have assumed the potential is constant across an element. The authors calculate the electric potential and systematically investigate the effect of different mesh choices and dipole locations by using a three concentric sphere head model for which there is an analytic solution. Implementing the linear interpolation approximation results in errors that are approximately half those of the same mesh when the potential is assumed to be constant, and provides a reliable method for solving the problem     

Conventional and reciprocal approaches to the inverse dipole localization problem of electroencephalography

Forward transfer matrices relating dipole source to surface potentials can be determined via conventional or reciprocal approaches. In numerical simulations with a triangulated boundary-element three-concentric-spheres head model, we compare four inverse electroencephalogram (EEG) solutions: those obtained utilizing conventional or reciprocal forward transfer matrices, relating in each case source dipole components to potentials at either triangle centroids or triangle vertices. Single-dipole inverse solutions were obtained using simplex optimization with an additional position constraint limiting solution dipoles to within the brain region. Dipole localization errors are presented in all four cases, for varying dipole eccentricity and two different values of skull conductivity. Both conventional and reciprocal forward transfer matrices yielded inverse dipole solutions of comparable accuracy. Localization errors were low even for highly eccentric source dipoles on account of the nonlinear nature of the single-dipole solution and the position constraint. In the presence of Gaussian noise, both conventional and reciprocal approaches were also found to be equally robust to skull conductivity errors.     

Krylov subspace iterative techniques: on the detection of brain activity with electrical impedance tomography

In this paper, we review some numerical techniques based on the linear Krylov subspace iteration that can be used for the efficient calculation of the forward and the inverse electrical impedance tomography problems. Exploring their computational advantages in solving large-scale systems of equations, we specifically address their implementation in reconstructing localized impedance changes occurring within the human brain. If the conductivity of the head tissues is assumed to be real, the pre-conditioned conjugate gradients (PCGs) algorithm can be used to calculate efficiently the approximate forward solution to a given error tolerance. The performance and the regularizing properties of the PCG iteration for solving ill-conditioned systems of equations (PCGNs) is then explored, and a suitable preconditioning matrix is suggested in order to enhance its convergence rate. For image reconstruction, the nonlinear inverse problem is considered. Based on the Gauss-Newton method for solving nonlinear problems we have developed two algorithms that implement the PCGN iteration to calculate the linear step solution. Using an anatomically detailed model of the human head and a specific scalp electrode arrangement, images of a simulated impedance change inside brain's white matter have been reconstructed.     

Effects of head shape on EEGs and MEGs

This paper presents results of computer modeling studies of the effects of head shape on electroencephalograms (EEG's) and magnetoencephalograms (MEG's) and on the localization of electrical sources in the brain using these measurements. The effects of general, nonspherical head shape on EEG's and MEG's are determined by comparisons of EEG and MEG maps from nonspherical head models with corresponding maps from a spherical head model. The effects on source localization accuracy are determined by calculating moving dipole inverse solutions in a spherical head model using EEG's and MEG's from the nonspherical models and comparing the solutions with the known sources. It was found that nonspherical head shape can produce significant changes in the maps produced by some sources in the cortical region of the brain. However, it was also found that such deviations of the head from sphericity produce localization errors of less than approximately 1 cm. No significant differences in the effects of such deviations on EEG's and MEG's were found. Finally, it was found that most such deviations do not cause a dipolar source which is perpendicular to the surface of the head model to produce a significant magnetic field; such a source produces zero magnetic field in a sphere.     

Moving dipole inverse solutions using realistic torso models

A noniterative numerical solution for the potentials on the surfaces of a piecewise homogeneous volume conductor due to a current dipole is described. This forward solution has been used in electric and magnetic single moving dipole (SMD) inverse solutions that employ a torso volume conductor model whose boundaries are specified numerically. Thus, the volume conductor model used by the inverse solutions need not be limited to simple geometric shapes; torso models of realistic shape can be used.     

Estimating parametric line-source models with electroencephalography

We develop three parametric models for electroencephalography (EEG) to estimate current sources that are spatially distributed on a line. We assume a realistic head model and solve the EEG forward problem using the boundary element method (BEM). We present the models with increasing degrees of freedom, provide the forward solutions, and derive the maximum-likelihood estimates as well as Cramér-Rao bounds of the unknown source parameters. A series of experiments are conducted to evaluate the applicability of the proposed models. We use numerical examples to demonstrate the usefulness of our line-source models in estimating extended sources. We also apply our models to the real EEG data of N20 response that is known to have an extended source. We observe that the line-source models explain the N20 measurements better than the dipole model.