Effects of geometric head model perturbations on the EEG forward and inverse problems

We study the effect of geometric head model perturbations on the electroencephalography (EEG) forward and inverse problems. Small magnitude perturbations of the shape of the head could represent uncertainties in the head model due to errors on images or techniques used to construct the model. They could also represent small scale details of the shape of the surfaces not described in a deterministic model, such as the sulci and fissures of the cortical layer. We perform a first-order perturbation analysis, using a meshless method for computing the sensitivity of the solution of the forward problem to the geometry of the head model. The effect on the forward problem solution is treated as noise in the EEG measurements and the Cramér-Rao bound is computed to quantify the effect on the inverse problem performance. Our results show that, for a dipolar source, the effect of the perturbations on the inverse problem performance is under the level of the uncertainties due to the spontaneous brain activity. Thus, the results suggest that an extremely detailed model of the head may be unnecessary when solving the EEG inverse problem.     

Comparison between Equivalent Charge Model and Equivalent Dipole Model on Realistic Head Model

Equivalent source layer(ESL) imaging is an important kind of high-resolution electroencephalogram(EEG) imaging.It consists of two categories:equivalent dipole layer(EDL) and equivalent charge layer(ECL).Both of them are assumed to be located on or near the cortical surface and have been proposed as high-resolution imaging modalities or as intermediate steps to estimate the epicortical potential.Here,EDL and ECL based on a realistic head model are presented,both simulations and real data experiment are done to compare these two models.The results show that ECL can provide higher spatial resolution about source location than EDL does.     

Improved surface laplacian estimates of cortical potential using realistic models of head geometry

Surface Laplacian of scalp EEG can be used to estimate the potential distribution on the cortical surface as an alternative to invasive approaches. However, the accuracy of surface Laplacian estimation depends critically on the geometric shape of the head model. This paper presents a new method for computing the surface Laplacian of scalp potential directly on realistic scalp surfaces in the form of a triangular mesh reconstructed from MRI scans. Unlike previous methods, this algorithm does not resort to any surface fitting proxy and can improve the surface Laplacian estimation of cortical potential patterns by as much as 34% on realistically shaped head models. Simulations and experimental data are presented to demonstrate the advantage of the proposed method over the conventional spherical approximation and the utility of a more accurate surface Laplacian method for estimating cortical potentials from scalp electrodes.     

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.      

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     

Estimating brain conductivities and dipole source signals with EEG arrays

Techniques based on electroencephalography (EEG) measure the electric potentials on the scalp and process them to infer the location, distribution, and intensity of underlying neural activity. Accuracy in estimating these parameters is highly sensitive to uncertainty in the conductivities of the head tissues. Furthermore, dissimilarities among individuals are ignored when standarized values are used. In this paper, we apply the maximum-likelihood and maximum a posteriori (MAP) techniques to simultaneously estimate the layer conductivity ratios and source signal using EEG data. We use the classical 4-sphere model to approximate the head geometry, and assume a known dipole source position. The accuracy of our estimates is evaluated by comparing their standard deviations with the Cramér-Rao bound (CRB). The applicability of these techniques is illustrated with numerical examples on simulated EEG data. Our results show that the estimates have low bias and attain the CRB for sufficiently large number of experiments. We also present numerical examples evaluating the sensitivity to imprecise assumptions on the source position and skull thickness. Finally, we propose extensions to the case of unknown source position and present examples for real data.     

Effect of skull resistivity on the spatial resolutions of EEG and MEG

The resistivity values of the different tissues of the head affect the lead fields of electroencephalography (EEG). When the head is modeled with a concentric spherical model, the different resistivity values have no effect on the lead fields of the magnetoencephalography (MEG). Recent publications indicate that the resistivity of the skull is much lower than what was estimated by Rush and Driscoll. At the moment, this information on skull resistivity is, however, slightly controversial. We have compared the spatial resolution of EEG and MEG for cortical sources by calculating the half-sensitivity volumes (HSVs) of EEG and MEG as a function of electrode and magnetometer distance, respectively, with the relative skull resistivity as a parameter. Because the spatial resolution is related to the HSV, these data give an overview of the effect of these parameters on the spatial resolution of both techniques. Our calculations show that, with the new information on the resistivity of the skull, in the spherical model for cortical sources the spatial resolution of the EEG is better than that of the MEG.     

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.     

EEG/MEC error bounds for a static dipole source with a realistichead model

We derive Cramer-Rao bounds (CRBs) on the errors of estimating the parameters (location and moment) of a static current dipole source using data from electro-encephalography (EEG), magneto-encephalography (MEG), or the combined EEG/MEG modality. We use a realistic head model based on knowledge of surfaces separating tissues of different conductivities obtained from magnetic resonance (MR) or computer tomography (CT) imaging systems. The electric potentials and magnetic field components at the respective sensors are functions of the source parameters through integral equations. These potentials and field are formulated for solving them by the boundary or the finite element method (BEM or FEM) with a weighted residuals technique. We present a unified framework for the measurements computed by these methods that enables the derivation of the bounds. The resulting bounds may be used, for instance, to choose the best configuration of the sensors for a given patient and region of expected source location. Numerical results are used to demonstrate an application for showing expected accuracies in estimating the source parameters as a function of its position in the brain, based on real EEG/MEG system and MR or CT images     

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.     

In vivo measurement of the brain and skull resistivities using an EIT-based method and the combined analysis of SEF/SEP data

Results of "in vivo" measurements of the skull and brain resistivities are presented for six subjects. Results are obtained using two different methods, based on spherical head models. The first method uses the principles of electrical impedance tomography (EIT) to estimate the equivalent electrical resistivities of brain (rhobrain), skull (rhoskull) and skin (rhoskin) according to. The second one estimates the same parameters through a combined analysis of the evoked somatosensory cortical response, recorded simultaneously using magnetoencephalography (MEG) and electroencephalography (EEG). The EIT results, obtained with the same relative skull thickness (0.05) for all subjects, show a wide variation of the ratio rhoskull/rhobrain among subjects (average = 72, SD = 48%). However, the rhoskull/rhobrain ratios of the individual subjects are well reproduced by combined analysis of somatosensory evoked fields (SEF) and somatosensory evoked potentials (SEP). These preliminary results suggest that the rhoskull/rhobrain variations over subjects cannot be disregarded in the EEG inverse problem (IP) when a spherical model is used. The agreement between EIT and SEF/SEP points to the fact that whatever the source of variability, the proposed EIT-based method 相似文献   

Effect of conductivity uncertainties and modeling errors on EEGsource localization using a 2-D model

This paper presents a sensitivity study electroencephalography-based source localization due errors in the head-tissue conductivities and to errors in modeling the conductivity variation inside the brain and scalp. The study is conducted using a two-dimensional (2-D) finite element model obtained from a magnetic resonance imaging (MRI) scan of a head cross section. The effect of uncertainty in the following tissues is studied: white matter, gray matter, cerebrospinal fluid (CSF), skull, and fat. The distribution of source location errors, assuming a single-dipole source model, is examined in detail for different dipole locations over the entire brain region. We also present a detailed analysis of the effect of conductivity on source localization for a four-layer cylinder model and a four-layer sphere model. These two simple models provide insight into how the effect of conductivity on boundary potential translates into source location errors; and also how errors in a 2-D model compare to errors in a three-dimensional model. Results presented in this paper clearly point to the following conclusion: unless the conductivities of the head tissues and the distribution of these tissues throughout the head are modeled accurately, the goal of achieving localization accuracy to within a few millimeters is unattainable     

EEG Electrode Sensitivity?An Application of Reciprocity

In this paper, the reciprocity theorem is used to determine the sensitivity of EEG leads to the location and orientation of sources in the brain. Quantitative information used in determining the sensitivity is derived from constant potential plots of a three-concentric-sphere mathematical model of the head with current applied through surface leads (the reciprocal problem), and from an electrolytic tank employing a human skull. Advantages of the reciprocal or lead field approach are outlined and the following conclusions are drawn. 1) Leads placed at the end of a diameter through the center of the brain have a range of sensitivity due to source location of only 3 to 1. 2) For the same electrode placement, sensitivity is maximum to sources oriented parallel to the line of the electrodes regardless of source location. 3) Electrodes spaced 5 cm apart are about ten times more sensitive to proximal cortical sources (by virtue of position) than to sources near the center of the brain.     

Spatiotemporal forward solution of the EEG and MEG using network modeling

Dynamic systems have proven to be well suited to describe a broad spectrum of human coordination behavior such synchronization with auditory stimuli. Simultaneous measurements of the spatiotemporal dynamics of electroencephalographic (EEG) and magnetoencephalographic (MEG) data reveals that the dynamics of the brain signals is highly ordered and also accessible by dynamic systems theory. However, models of EEG and MEG dynamics have typically been formulated only in terms of phenomenological modeling such as fixed-current dipoles or spatial EEG and MEG patterns. In this paper, it is our goal to connect three levels of organization, that is the level of coordination behavior, the level of patterns observed in the EEG and MEG and the level of neuronal network dynamics. To do so, we develop a methodological framework, which defines the spatiotemporal dynamics of neural ensembles, the neural field, on a sphere in three dimensions. Using magnetic resonance imaging we map the neural field dynamics from the sphere onto the folded cortical surface of a hemisphere. The neural field represents the current flow perpendicular to the cortex and, thus, allows for the calculation of the electric potentials on the surface of the skull and the magnetic fields outside the skull to be measured by EEG and MEG, respectively. For demonstration of the dynamics, we present the propagation of activation at a single cortical site resulting from a transient input. Finally, a mapping between finger movement profile and EEG/MEG patterns is obtained using Volterra integrals.     

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.     

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     

Effect of electrode density and measurement noise on the spatial resolution of cortical potential distribution

The purpose of the present study was to examine the spatial resolution of electroencephalography (EEG) by means of inverse cortical EEG solution. The main interest was to study how the number of measurement electrodes and the amount of measurement noise affects the spatial resolution. A three-layer spherical head model was used to obtain the source-field relationship of cortical potentials and scalp EEG field. Singular value decomposition was used to evaluate the spatial resolution with various measurement noise estimates. The results suggest that as the measurement noise increases the advantage of dense electrode systems is decreased. With low realistic measurement noise, a more accurate inverse cortical potential distribution can be obtained with an electrode system where the distance between two electrodes is as small as 16 mm, corresponding to as many as 256 measurement electrodes. In clinical measurement environments, it is always beneficial to have at least 64 measurement electrodes.     

Effect of source location on the scalp potential asymmetry in anumerical model of the head

The correlation between source asymmetry in the brain and the potential amplitude asymmetry on the scalp was studied by a two-dimensional (2-D) numerical model of the head. The model employed computerized tomography (CT) images to define the different compartments of the head. The source was modeled by a dipole layer in the occiput for an occipital source (visual evoked potential generators) or a dipole layer around the cortex representing spontaneous activity generators. The volume conductor equation for the potential distribution was solved numerically using a finite volume method for two CT images; one had relatively symmetric left-right anatomy while the other had a fair deviation of 6° between the occiput and the nasion-inion line. By examining several arrangements of sources, it has been demonstrated that source asymmetry can cause nonnegligible asymmetry in the potential amplitude at the homotopic points on the scalp. This asymmetry, that is not related to real physiologic or psychological origin, should be taken into consideration in any EEG potential distribution analysis     

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.