Extent of cortical generators visible on the scalp: effect of a subdural grid.

The effect of the non-conducting substrate of a subdural grid on the scalp electric potential distribution is studied through simulations. Using a detailed head model and the finite element method we show that the governing physics equations predict an important attenuation in the scalp potential for generators located under the grid, and an amplification for generators located under holes in the skull filled with conductive media. These effects are spatially localized and do not cancel each other. A 4 × 8 cm grid can produce attenuations of 2 to 3 times, and an 8 × 8 cm grid attenuation of up to 8 times. As a consequence, when there is no subdural grid, generators of 4 to 8 cm(2) produce scalp potentials of the same maximum amplitude as generators of 10 to 20 cm(2) under the center of a subdural grid. This means that the minimum cortical extents necessary to produce visible scalp activity determined from simultaneous scalp and subdural recordings can be overestimations.

Improvements on spatial coverage and focality of deep brain stimulation in pre-surgical epilepsy mapping.

Objective.Electrical stimulation mapping (ESM) of the brain using stereo-electroencephalography (SEEG) intracranial electrodes, also known as depth-ESM (DESM), is being used as part of the pre-surgical planning for brain surgery in drug-resistant epilepsy patients. Typically, DESM consists in applying the electrical stimulation using adjacent contacts of the SEEG electrodes and in recording the EEG responses to those stimuli, giving valuable information of critical brain regions to better delimit the region to resect. However, the spatial extension or coverage of the stimulated area is not well defined even though the precise electrode locations can be determined from computed tomography images.Approach.We first conduct electrical simulations of DESM for different shapes of commercial SEEG electrodes showing the stimulation extensions for different intensities of injected current. We then evaluate the performance of DESM in terms of spatial coverage and focality on two realistic head models of real patients undergoing pre-surgical evaluation. We propose a novel strategy for DESM that consist in applying the current using contacts of different SEEG electrodes (x-DESM), increasing the versatility of DESM without implanting more electrodes. We also present a clinical case where x-DESM replicated the full semiology of an epilepsy seizure using a very low-intensity current injection, when typical adjacent DESM only reproduced partial symptoms with much larger intensities. Finally, we show one example of DESM optimal stimulation to achieve maximum intensity, maximum focality or intermediate solution at a pre-defined target, and one example of temporal interference in DESM capable of increasing focality in brain regions not immediately touching the electrode contacts.Main results.It is possible to define novel current injection patterns using contacts of different electrodes (x-DESM) that might improve coverage and/or focality, depending on the characteristics of the candidate brain. If individual simulations are not possible, we provide the estimated radius of stimulation as a function of the injected current and SEEG electrode brand as a reference for the community.Significance.Our results show that subject-specific electrical stimulations are a valuable tool to use in the pre-surgical planning to visualize the extension of the stimulated regions. The methods we present here are also applicable to pre-surgical planning of tumor resections and deep brain stimulation treatments.

A novel bounded EIT protocol to generate inhomogeneous skull conductivity maps non-invasively.

Electrical Impedance Tomography (EIT) can be used to estimate the electrical properties of the head tissues in a parametric approach. This modality is called parametric EIT or bounded EIT (bEIT). Typical bEIT protocols alternate between several current injection patterns with two current injection electrodes each: one source and one sink ("1-to-1"), while the rest of the electrodes measure the resulting electric potential. Then, one value of conductivity per tissue (e.g. scalp and/or skull) is estimated independently for each current injection pair. With these protocols, it is difficult to obtain local estimates of the skull tissue. Thus, the grand average of the estimates obtained from each pair is assigned to each tissue modeling them as homogeneous. However, it is known that these tissues are inhomogeneous within the same subject. We propose the use of current injection patterns with one source and many sinks ("1to-N") located at the opposite side of the head to build individual and inhomogeneous skull conductivity maps. We validate the method with simulations and compare its performance with equivalent maps generated by using the classical "1-to-1" patterns. The map generated by the novel method shows better spatial correlation with the more conductive spongy bone presence.Clinical Relevance- The novel bEIT protocol allows to map individual head models with spatially resolved skull conductivities in vivo and non-invasively for use in electroencephalography (EEG) source localization, transcranial electrical stimulation (TES) dose calculations and TES pattern optimization, without the risk of ionizing radiation associated with computed tomography (CT) scans.

iElectrodes: A Comprehensive Open-Source Toolbox for Depth and Subdural Grid Electrode Localization.

The localization of intracranial electrodes is a fundamental step in the analysis of invasive electroencephalography (EEG) recordings in research and clinical practice. The conclusions reached from the analysis of these recordings rely on the accuracy of electrode localization in relationship to brain anatomy. However, currently available techniques for localizing electrodes from magnetic resonance (MR) and/or computerized tomography (CT) images are time consuming and/or limited to particular electrode types or shapes. Here we present iElectrodes, an open-source toolbox that provides robust and accurate semi-automatic localization of both subdural grids and depth electrodes. Using pre- and post-implantation images, the method takes 2-3 min to localize the coordinates in each electrode array and automatically number the electrodes. The proposed pre-processing pipeline allows one to work in a normalized space and to automatically obtain anatomical labels of the localized electrodes without neuroimaging experts. We validated the method with data from 22 patients implanted with a total of 1,242 electrodes. We show that localization distances were within 0.56 mm of those achieved by experienced manual evaluators. iElectrodes provided additional advantages in terms of robustness (even with severe perioperative cerebral distortions), speed (less than half the operator time compared to expert manual localization), simplicity, utility across multiple electrode types (surface and depth electrodes) and all brain regions.

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.

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.

Surface-source modeling and estimation using biomagnetic measurements.

We propose a number of electric source models that are spatially distributed on an unknown surface for biomagnetism. These can be useful to model, e.g., patches of electrical activity on the cortex. We use a realistic head (or another organ) model and discuss the special case of a spherical head model with radial sensors resulting in more efficient computations of the estimates for magnetoencephalography. We derive forward solutions, maximum likelihood (ML) estimates, and Cramér-Rao bound (CRB) expressions for the unknown source parameters. A model selection method is applied to decide on the most appropriate model. We also present numerical examples to compare the performances and computational costs of the different models and illustrate when it is possible to distinguish between surface and focal sources or line sources. Finally, we apply our methods to real biomagnetic data of phantom human torso and demonstrate the applicability of them.

Parametric surface-source modeling and estimation with electroencephalography.

Electroencephalography (EEG) is an important tool for studying the brain functions and is becoming popular in clinical practice. In this paper, we develop four parametric EEG models to estimate current sources that are spatially distributed on a surface. Our models approximate the source shape and extent explicitly and can be applied to localize extended sources which are often encountered, e.g., in epilepsy diagnosis. We assume a realistic head model and solve the EEG forward problem using the boundary element method. We present the source 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. In order to evaluate the applicability of the proposed models, we first compare their estimation performances with the dipole model's using several known source distributions. We then discuss the conditions under which we can distinguish between the proposed extended sources and the focal dipole using the generalized likelihood ratio test. We also apply our models to the electric measurements obtained from a phantom body in which an extended electric source is imbedded. We observe that the proposed model can capture the source extent information satisfactorily and the localization accuracy is better than the dipole model.

MEG forward problem formulation using equivalent surface current densities.

We present a formulation for the magnetoencephalography (MEG) forward problem with a layered head model. Traditionally the magnetic field is computed based on the electric potential on the interfaces between the layers. We propose to express the effect of the volumetric currents in terms of an equivalent surface current density on each interface, and obtain the magnetic field based on them. The boundary elements method is used to compute the equivalent current density and the magnetic field for a realistic head geometry. We present numerical results showing that the MEG forward problem is solved correctly with this formulation, and compare it with the performance of the traditional formulation. We conclude that the traditional formulation generally performs better, but still the new formulation is useful in certain situations.

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.

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.

Line-source modeling and estimation with magnetoencephalography.

We propose a number of source models that are spatially distributed on a line for magnetoencephalography (MEG) using both a spherical head with radial sensors for more efficient computation and a realistic head model for more accurate results. We develop these models with increasing degrees of freedom, derive forward solutions, maximum-likelihood (ML) estimates, and Cramér-Rao bound (CRB) expressions for the unknown source parameters. A model selection method is applied to select the most appropriate model. We also present numerical examples to compare the performances and computational costs of the different models, to determine the regions where better estimates are possible and when it is possible to distinguish between line and focal sources. We demonstrate the usefulness of the proposed line-source models over the previously available focal source model in certain distributed source cases. Finally, we apply our methods to real MEG data, the N2O response after electric stimulation of the median nerve known to be an extended source.

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.

Electrode and brain modeling in stereo-EEG.

OBJECTIVE: To quantify the perturbation due to the presence of a measuring depth electrode on the intracranial electric potential distribution, and to study the effect of the heterogeneity and anisotropy of the brain tissues' electric conductivity. METHODS: The governing differential equations are solved with the Boundary Elements Method to compute the perturbation on the electric potential distribution caused by the presence of the measuring electrode, and with the Finite Elements Method to simulate measurements in an heterogeneous anisotropic brain model. RESULTS: The perturbation on the measured electric potential is negligible if the source of electric activity is located more than approximately 1mm away from the electrode. The error induced by this perturbation in the estimation of the source position is below 1mm in all tested situations. The results hold for different sizes of the electrode's contacts. The effect of the brain's heterogeneity and anisotropy is more important. In a particular example simulated dipolar sources in the gray matter show localization differences of up to 5mm between homogeneous isotropic and heterogeneous anisotropic brain models. CONCLUSIONS: It is not necessary to include detailed electrode models in order to solve the stereo-EEG (sEEG) forward and inverse problems. The heterogeneity and anisotropy of the brain electric conductivity should be modeled if possible. The effect of using an homogeneous isotropic brain model approximation should be studied in a case by case basis, since it depends on the electrode positions, the subject's electric conductivity map, and the source configuration. SIGNIFICANCE: This simulation study is helpful for interpreting the sEEG measurements, and for choosing appropriate electrode and brain models; a necessary first step in any attempt to solve the sEEG inverse problem.

Electromagnetic source imaging for sparse cortical activation patterns.

We propose modifications to the Automatic Relevance Determination (ARD) algorithm for solving the EEG/MEG inverse problem when the activation map of the cortex is known to be sparse. We propose to include a term to account for the background noise activity, i.e. electric activity of sources not in the cortex. Also, we prune the results of the ARD algorithm using a Model Selection criterion to get sparser results. Simulations with a realistic head model show a very important reduction of the number of sources incorrectly detected as active.

Shrinkage approach for EEG covariance matrix estimation.

We present a shrinkage estimator for the EEG spatial covariance matrix of the background activity. We show that such an estimator has some advantages over the maximum likelihood and sample covariance estimators when the number of available data to carry out the estimation is low. We find sufficient conditions for the consistency of the shrinkage estimators and results concerning their numerical stability. We compare several shrinkage schemes and show how to improve the estimator by incorporating known structure of the covariance matrix.

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.

On the EEG/MEG forward problem solution for distributed cortical sources.

In studies of EEG/MEG problems involving cortical sources, the cortex may be modeled by a 2-D manifold inside the brain. In such cases the primary or impressed current density over this manifold is usually approximated by a set of dipolar sources located at the vertices of the cortical surface tessellation. In this study, we analyze the different errors induced by this approximation on the EEG/MEG forward problem. Our results show that in order to obtain more accurate solutions of the forward problems with the multiple dipoles approximation, the moments of the dipoles should be weighted by the area of the surrounding triangles, or using an alternative approximation of the primary current as a constant or linearly varying current density over plane triangular elements of the cortical surface tessellation. This should be taken into account when computing the lead field matrix for solving the EEG/MEG inverse problem in brain imaging methods.

Capacitive electrodes in electroencephalography.

We present a forward problem formulation for computing biopotentials measured with dry or capacitive electrodes. This formulation is not quasistatic and has mixed boundary conditions. Our results show that simple approximations to the measurements based on capacitive coupling are adequate in most situations. We study the range of validity and errors committed in the EEG forward and inverse problems when using this approximation.