Conventional and Reciprocal Approaches to the Forward and Inverse Problems of Electroencephalography

Le problème inverse en électroencéphalographie (EEG) est la localisation de sources de courant dans le cerveau utilisant les potentiels de surface sur le cuir chevelu générés par ces sources. Une solution inverse implique typiquement de multiples calculs de potentiels de surface sur le cuir chevelu, soit le problème direct en EEG. Pour résoudre le problème direct, des modèles sont requis à la fois pour la configuration de source sous-jacente, soit le modèle de source, et pour les tissues environnants, soit le modèle de la tête. Cette thèse traite deux approches bien distinctes pour la résolution du problème direct et inverse en EEG en utilisant la méthode des éléments de frontières (BEM): l’approche conventionnelle et l’approche réciproque. L’approche conventionnelle pour le problème direct comporte le calcul des potentiels de surface en partant de sources de courant dipolaires. D’un autre côté, l’approche réciproque détermine d’abord le champ électrique aux sites des sources dipolaires quand les électrodes de surfaces sont utilisées pour injecter et retirer un courant unitaire. Le produit scalaire de ce champ électrique avec les sources dipolaires donne ensuite les potentiels de surface. L’approche réciproque promet un nombre d’avantages par rapport à l’approche conventionnelle dont la possibilité d’augmenter la précision des potentiels de surface et de réduire les exigences informatiques pour les solutions inverses. Dans cette thèse, les équations BEM pour les approches conventionnelle et réciproque sont développées en utilisant une formulation courante, la méthode des résidus pondérés. La réalisation numérique des deux approches pour le problème direct est décrite pour un seul modèle de source dipolaire. Un modèle de tête de trois sphères concentriques pour lequel des solutions analytiques sont disponibles est utilisé. Les potentiels de surfaces sont calculés aux centroïdes ou aux sommets des éléments de discrétisation BEM utilisés. La performance des approches conventionnelle et réciproque pour le problème direct est évaluée pour des dipôles radiaux et tangentiels d’excentricité variable et deux valeurs très différentes pour la conductivité du crâne. On détermine ensuite si les avantages potentiels de l’approche réciproquesuggérés par les simulations du problème direct peuvent êtres exploités pour donner des solutions inverses plus précises. Des solutions inverses à un seul dipôle sont obtenues en utilisant la minimisation par méthode du simplexe pour à la fois l’approche conventionnelle et réciproque, chacun avec des versions aux centroïdes et aux sommets. Encore une fois, les simulations numériques sont effectuées sur un modèle à trois sphères concentriques pour des dipôles radiaux et tangentiels d’excentricité variable. La précision des solutions inverses des deux approches est comparée pour les deux conductivités différentes du crâne, et leurs sensibilités relatives aux erreurs de conductivité du crâne et au bruit sont évaluées. Tandis que l’approche conventionnelle aux sommets donne les solutions directes les plus précises pour une conductivité du crâne supposément plus réaliste, les deux approches, conventionnelle et réciproque, produisent de grandes erreurs dans les potentiels du cuir chevelu pour des dipôles très excentriques. Les approches réciproques produisent le moins de variations en précision des solutions directes pour différentes valeurs de conductivité du crâne. En termes de solutions inverses pour un seul dipôle, les approches conventionnelle et réciproque sont de précision semblable. Les erreurs de localisation sont petites, même pour des dipôles très excentriques qui produisent des grandes erreurs dans les potentiels du cuir chevelu, à cause de la nature non linéaire des solutions inverses pour un dipôle. Les deux approches se sont démontrées également robustes aux erreurs de conductivité du crâne quand du bruit est présent. Finalement, un modèle plus réaliste de la tête est obtenu en utilisant des images par resonace magnétique (IRM) à partir desquelles les surfaces du cuir chevelu, du crâne et du cerveau/liquide céphalorachidien (LCR) sont extraites. Les deux approches sont validées sur ce type de modèle en utilisant des véritables potentiels évoqués somatosensoriels enregistrés à la suite de stimulation du nerf médian chez des sujets sains. La précision des solutions inverses pour les approches conventionnelle et réciproque et leurs variantes, en les comparant à des sites anatomiques connus sur IRM, est encore une fois évaluée pour les deux conductivités différentes du crâne. Leurs avantages et inconvénients incluant leurs exigences informatiques sont également évalués. Encore une fois, les approches conventionnelle et réciproque produisent des petites erreurs de position dipolaire. En effet, les erreurs de position pour des solutions inverses à un seul dipôle sont robustes de manière inhérente au manque de précision dans les solutions directes, mais dépendent de l’activité superposée d’autres sources neurales. Contrairement aux attentes, les approches réciproques n’améliorent pas la précision des positions dipolaires comparativement aux approches conventionnelles. Cependant, des exigences informatiques réduites en temps et en espace sont les avantages principaux des approches réciproques. Ce type de localisation est potentiellement utile dans la planification d’interventions neurochirurgicales, par exemple, chez des patients souffrant d’épilepsie focale réfractaire qui ont souvent déjà fait un EEG et IRM.The inverse problem of electroencephalography (EEG) is the localization of current sources within the brain using surface potentials on the scalp generated by these sources. An inverse solution typically involves multiple calculations of scalp surface potentials, i.e., the EEG forward problem. To solve the forward problem, models are needed for both the underlying source configuration, the source model, and the surrounding tissues, the head model. This thesis treats two distinct approaches for the resolution of the EEG forward and inverse problems using the boundary-element method (BEM): the conventional approach and the reciprocal approach. The conventional approach to the forward problem entails calculating the surface potentials starting from source current dipoles. The reciprocal approach, on the other hand, first solves for the electric field at the source dipole locations when the surface electrodes are reciprocally energized with a unit current. A scalar product of this electric field with the source dipoles then yields the surface potentials. The reciprocal approach promises a number of advantages over the conventional approach, including the possibility of increased surface potential accuracy and decreased computational requirements for inverse solutions. In this thesis, the BEM equations for the conventional and reciprocal approaches are developed using a common weighted-residual formulation. The numerical implementation of both approaches to the forward problem is described for a single-dipole source model. A three-concentric-spheres head model is used for which analytic solutions are available. Scalp potentials are calculated at either the centroids or the vertices of the BEM discretization elements used. The performance of the conventional and reciprocal approaches to the forward problem is evaluated for radial and tangential dipoles of varying eccentricities and two widely different skull conductivities. We then determine whether the potential advantages of the reciprocal approach suggested by forward problem simulations can be exploited to yield more accurate inverse solutions. Single-dipole inverse solutions are obtained using simplex minimization for both the conventional and reciprocal approaches, each with centroid and vertex options. Again, numerical simulations are performed on a three-concentric-spheres model for radial and tangential dipoles of varying eccentricities. The inverse solution accuracy of both approaches is compared for the two different skull conductivities and their relative sensitivity to skull conductivity errors and noise is assessed. While the conventional vertex approach yields the most accurate forward solutions for a presumably more realistic skull conductivity value, both conventional and reciprocal approaches exhibit large errors in scalp potentials for highly eccentric dipoles. The reciprocal approaches produce the least variation in forward solution accuracy for different skull conductivity values. In terms of single-dipole inverse solutions, conventional and reciprocal approaches demonstrate comparable accuracy. Localization errors are low even for highly eccentric dipoles that produce large errors in scalp potentials on account of the nonlinear nature of the single-dipole inverse solution. Both approaches are also found to be equally robust to skull conductivity errors in the presence of noise. Finally, a more realistic head model is obtained using magnetic resonance imaging (MRI) from which the scalp, skull, and brain/cerebrospinal fluid (CSF) surfaces are extracted. The two approaches are validated on this type of model using actual somatosensory evoked potentials (SEPs) recorded following median nerve stimulation in healthy subjects. The inverse solution accuracy of the conventional and reciprocal approaches and their variants, when compared to known anatomical landmarks on MRI, is again evaluated for the two different skull conductivities. Their respective advantages and disadvantages including computational requirements are also assessed. Once again, conventional and reciprocal approaches produce similarly small dipole position errors. Indeed, position errors for single-dipole inverse solutions are inherently robust to inaccuracies in forward solutions, but dependent on the overlapping activity of other neural sources. Against expectations, the reciprocal approaches do not improve dipole position accuracy when compared to the conventional approaches. However, significantly smaller time and storage requirements are the principal advantages of the reciprocal approaches. This type of localization is potentially useful in the planning of neurosurgical interventions, for example, in patients with refractory focal epilepsy in whom EEG and MRI are often already performed

Review on solving the forward problem in EEG source analysis

Background. The aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes. Methods. While other reviews give an extensive summary of the both forward and inverse problem, this review article focuses on different aspects of solving the forward problem and it is intended for newcomers in this research field. Results. It starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidal neurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumann and Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and white matter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last two decades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images, which requires numerical methods. The following methods are compared with each other: the boundary element method (BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conducting compartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization. It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for each dipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterative methods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method. Conclusion. Solving the forward problem has been well documented in the past decades. In the past simplified spherical head models are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of the head model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of the tissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivity values is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters of the head model and of the numerical techniques on the solution of the forward problem.peer-reviewe

State-space solutions to the dynamic magnetoencephalography inverse problem using high performance computing

Determining the magnitude and location of neural sources within the brain that are responsible for generating magnetoencephalography (MEG) signals measured on the surface of the head is a challenging problem in functional neuroimaging. The number of potential sources within the brain exceeds by an order of magnitude the number of recording sites. As a consequence, the estimates for the magnitude and location of the neural sources will be ill-conditioned because of the underdetermined nature of the problem. One well-known technique designed to address this imbalance is the minimum norm estimator (MNE). This approach imposes an $L^2$ regularization constraint that serves to stabilize and condition the source parameter estimates. However, these classes of regularizer are static in time and do not consider the temporal constraints inherent to the biophysics of the MEG experiment. In this paper we propose a dynamic state-space model that accounts for both spatial and temporal correlations within and across candidate intracortical sources. In our model, the observation model is derived from the steady-state solution to Maxwell's equations while the latent model representing neural dynamics is given by a random walk process.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS483 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Review on solving the inverse problem in EEG source analysis

In this primer, we give a review of the inverse problem for EEG source localization. This is intended for the researchers new in the field to get insight in the state-of-the-art techniques used to find approximate solutions of the brain sources giving rise to a scalp potential recording. Furthermore, a review of the performance results of the different techniques is provided to compare these different inverse solutions. The authors also include the results of a Monte-Carlo analysis which they performed to compare four non parametric algorithms and hence contribute to what is presently recorded in the literature. An extensive list of references to the work of other researchers is also provided

A finite difference forward model for MEG and EEG

Thesis (M. Eng. and S.B.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, September 2006.Includes bibliographical references (p. 45-50).In this thesis, I designed and implemented a model of the electromagnetic signals generated by the human brain as seen on magnetoencephalography and electroencephalography machines. The model makes a novel use of the principle of reciprocity combined with Ohm's Law and the Biot Savart Law to build a model of the human magnetoencephalogram that is much faster to compute than the current state of the art. The model uses an existing finite difference model for electroencephalography and modifies it to incorporate the Bi6t Savart Law into its geometry. I tested the model against a spherical model to show that it is highly sensitive to approximations made of the Bi6t Savart Law for finite plane-bounded elements, but that further refinements of the model could make it as accurate as regular finite element models for magnetoencephalography.by Omri Schwarz.M.Eng.and S.B

Development of a subcortical emphasized physical human head model for evaluation of deep brain source contribution to scalp EEG

Ph.DDOCTOR OF PHILOSOPH

Quantifying the effect of demixing approaches on directed connectivity estimated between reconstructed EEG sources

Electrical activity recorded on the scalp using electroencephalography (EEG) results from the mixing of signals originating from different regions of the brain as well as from artifactual sources. In order to investigate the role of distinct brain areas in a given experiment, the signal recorded on the sensors is typically projected back into the brain (source reconstruction) using algorithms that address the so-called EEG inverse problem. Once the activity of sources located inside of the brain has been reconstructed, it is often desirable to study the statistical dependencies among them, in particular to quantify directional dynamical interactions between brain areas. Unfortunately, even when performing source reconstruction, the superposition of signals that is due to the propagation of activity from sources to sensors cannot be completely undone, resulting in potentially biased estimates of directional functional connectivity. Here we perform a set of simulations involving interacting sources to quantify source connectivity estimation performance as a function of the location of the sources, their distance to each other, the noise level, the source reconstruction algorithm, and the connectivity estimator. The generated source activity was projected onto the scalp and projected back to the cortical level using two source reconstruction algorithms, linearly constrained minimum variance beamforming and Exact' low-resolution tomography (eLORETA). In source space, directed connectivity was estimated using multi-variate Granger causality and time-reversed Granger causality, and compared with the imposed ground truth. Our results demonstrate that all considered factors significantly affect the connectivity estimation performance

Incorporation of anisotropic conductivities in EEG source analysis

The electroencephalogram (EEG) is a measurement of brain activity over a period of time by placing electrodes at the scalp (surface EEG) or in the brain (depth EEG) and is used extensively in the clinical practice. In the past 20 years, EEG source analysis has been increasingly used as a tool in the diagnosis of neurological disorders (like epilepsy) and in the research of brain functionality. EEG source analysis estimates the origin of brain activity given the electrode potentials measured at the scalp. This involves solving an inverse problem where a forward solution, which depends on the source parameters, is fitted to the given set of electrode potentials. The forward solution are the electrode potentials caused by a source in a given head model. The head model is dependent on the geometry and the conductivity. Often an isotropic conductivity (i.e. the conductivity is equal in all directions) is used, although the skull and white matter have an anisotropic conductivity (i.e. the conductivity can differ depending on the direction the current flows). In this dissertation a way to incorporate the anisotropic conductivities is presented and the effect of not incorporating these anisotropic conductivities is investigated. Spherical head models are simple head models where an analytical solution to the forward problem exists. A small simulation study in a 5 shell spherical head model was performed to investigate the estimation error due to neglecting the anisotropic properties of skull and white matter. The results show that the errors in the dipole location can be larger than 15 mm, which is unacceptable for an accurate dipole estimation in the clinical practice. Therefore, anisotropic conductivities have to be included in the head model. However, these spherical head models are not representative for the human head. Realistic head models are usually made from magnetic resonance scans through segmentation and are a better approximation to the geometry of the human head. To solve the forward problem in these head models numerical methods are needed. In this dissertation we proposed a finite difference technique that can incorporate anisotropic conductivities. Moreover, by using the reciprocity theorem the forward calculation time during an dipole source estimation procedure can be significantly reduced. By comparing the analytical solution for the dipole estimation problem with the one using the numerical method, the anisotropic finite difference with reciprocity method (AFDRM) is validated. Therefore, a cubic grid is made on the 5 shell spherical head model. The electrode potentials are obtained in the spherical head model with anisotropic conductivities by solving the forward problem using the analytical solution. Using these electrode potentials the inverse problem was solved in the spherical head model using the AFDRM. In this way we can determine the location error due to using the numerical technique. We found that the incorporation of anisotropic conductivities results in a larger location error when the head models are fully isotropical conducting. Furthermore, the location error due to the numerical technique is smaller if the cubic grid is made finer. To minimize the errors due to the numerical technique, the cubic grid should be smaller than or equal to 1 mm. Once the numerical technique is validated, a realistic head model can now be constructed. As a cubic grid should be used of at most 1 mm, the use of segmented T1 magnetic resonance images is best suited the construction. The anisotropic conductivities of skull and white matter are added as follows: The anisotropic conductivity of the skull is derived by calculating the normal and tangential direction to the skull at each voxel. The conductivity in the tangential direction was set 10 times larger than the normal direction. The conductivity of the white matter was derived using diffusion weighted magnetic resonance imaging (DW-MRI), a technique that measures the diffusion of water in several directions. As diffusion is larger along the nerve fibers, it is assumed that the conductivity along the nerve fibers is larger than the perpendicular directions to the nerve bundle. From the diffusion along each direction, the conductivity can be derived using two approaches. A simplified approach takes the direction with the largest diffusion and sets the conductivity along that direction 9 times larger than the orthogonal direction. However, by calculating the fractional anisotropy, a well-known measure indicating the degree of anisotropy, we can appreciate that a fractional anisotropy of 0.8715 is an overestimation. In reality, the fractional anisotorpy is mostly smaller and variable throughout the white matter. A realistic approach was therefore presented, which states that the conductivity tensor is a scaling of the diffusion tensor. The volume constraint is used to determine the scaling factor. A comparison between the realistic approach and the simplified approach was made. The results showed that the location error was on average 4.0 mm with a maximum of 10 mm. The orientation error was found that the orientation could range up to 60 degrees. The large orientation error was located at regions where the anisotropic ratio was low using the realistic approach but was 9 using the simplified approach. Furthermore, as the DW-MRI can also be used to measure the anisotropic diffusion in a gray matter voxel, we can derive a conductivity tensor. After investigating the errors due to neglecting these anisotropic conductivities of the gray matter, we found that the location error was very small (average dipole location error: 2.8 mm). The orientation error was ranged up to 40 degrees, although the mean was 5.0 degrees. The large errors were mostly found at the regions that had a high anisotropic ratio in the anisotropic conducting gray matter. Mostly these effects were due to missegmentation or to partial volume effects near the boundary interfaces of the gray and white matter compartment. After the incorporation of the anisotropic conductivities in the realistic head model, simulation studies can be performed to investigate the dipole estimation errors when these anisotropic conductivities of the skull and brain tissues are not taken into account. This can be done by comparing the solution to the dipole estimation problem in a head model with anisotropic conductivities with the one in a head model, where all compartments are isotropic conducting. This way we determine the error when a simplified head model is used instead of a more realistic one. When the anisotropic conductivity of both the skull and white matter or the skull only was neglected, it was found that the location error between the original and the estimated dipole was on average, 10 mm (maximum: 25 mm). When the anisotropic conductivity of the brain tissue was neglected, the location error was much smaller (an average location error of 1.1 mm). It was found that the anisotropy of the skull acts as an extra shielding of the electrical activity as opposed to an isotropic skull. Moreover, we saw that if the dipole is close to a highly anisotropic region, the potential field is changed reasonable in the near vicinity of the location of the dipole. In reality EEG contains noise contributions. These noise contribution will interact with the systematical error by neglecting anisotropic conductivities. The question we wanted to solve was “Is it worthwhile to incorporate anisotropic conductivities, even if the EEG contains noise?” and “How much noise should the EEG contain so that incorporating anisotropic conductivities improves the accuracy of EEG source analysis?”. When considering the anisotropic conductivities of the skull and brain tissues and the skull only, the location error due to the noise and neglecting the anisotropic conductivities is larger then the location error due to noise only. When only neglecting the anisotropic conductivities of the brain tissues only, the location error due to noise is similar to the location error due to noise and neglecting the anisotropic conductivities. When more advanced MR techniques can be used a better model to construct the anisotropic conductivities of the soft brain tissues can be used, which could result in larger errors even in the presence of noise. However, this is subject to further investigation. This suggests that the anisotropic conductivities of the skull should be incorporated. The technique presented in the dissertation can be used to epileptic patients in the presurgical evaluation. In this procedure patients are evaluated by means of medical investigations to determine the cause of the epileptic seizures. Afterwards, a surgical procedure can be performed to render the patient seizure free. A data set from a patiënt was obtained from a database of the Reference Center of Refractory Epilepsy of the Department of Neurology and the Department of Radiology of the Ghent University Hospital (Ghent, Belgium). The patient was monitored with a video/EEG monitoring with scalp and with implanted depth electrodes. An MR image was taken from the patient with the implanted depth electrodes, therefore, we could pinpoint the hippocampus as the onset zone of the epileptic seizures. The patient underwent a resective surgery removing the hippocampus, which rendered the patient seizure free. As DW-MRI images were not available, the head model constructed in chapter 4 and 5 was used. A neuroradiologist aligned the hippocampus in the MR image from which the head model was constructed. A spike was picked from a dataset and was used to estimate the source in a head model where all compartments were isotropic conducting, on one hand, and where the skull and brain tissues were anisotropic conducting, on the other. It was found that using the anisotropic head model, the source was estimated closer to the segmented hippocampus than the isotropic head model. This example shows the possibilities of this technique and allows us to apply it in the clinical practice. Moreover, a thorough validation of the technique has yet to be performed. There is a lot of discussion in the clinical community whether the spikes and epileptical seizures originate from the same origin in the brain. This question can be solved by applying our technique in patient studies

Reduction of conductivity uncertainty propagations in the inverse problem of EEG source analysis

In computer simulations, the response of a system under study depends on the input parameters. Each of these parameters can be assigned a fixed value or a range of values within the input parameter space for system performance evaluations. Starting from values of the input parameters and a certain given model, the so-called forward problem can be solved that needs to approximate the output of the system. Starting from measurements related to the output of the system model it is possible to determine the state of the system by solving the so-called inverse problem. In the case of a non-linear inverse problem, non-linear minimization techniques need to be used where the forward model is iteratively evaluated for different input parameters. The accuracy of the solution in the inverse problem is however decreased due to the noise available in the measurements and due to uncertainties in the system model. Uncertainties are parameters for which their values are not exactly known and/or that can vary in time and/or depend on the environment. These uncertainties have, for given input parameter values, an influence on the forward problem solution. This forward uncertainty propagation leads then to errors in the inverse solutions because the forward model is iteratively evaluated for recovering the inverse solutions. Until now, it was assumed that the recovery errors could not be reduced. The only option was to either quantify the uncertain parameter values as accurate as possible or to reflect the uncertainty in the inverse solutions, i.e. determination of the region in parameter space wherein the inverse solution is likely to be situated. The overall aim of this thesis was to develop reduction techniques of inverse reconstruction errors so that the inverse problem is solved in a more robust and thus accurate way. Methodologies were specifically developed for electroencephalography (EEG) source analysis. EEG is a non-invasive technique that measures on the scalp of the head, the electric potentials induced by the neuronal activity. EEG has several applications in biomedical engineering and is an important diagnostic tool in clinical neurophysiology. In epilepsy, EEG is used to map brain areas and to receive source localization information that can be used prior to surgical operation. Starting from Maxwell’s equations in their quasi-static formulation and from a physical model of the head, the forward problem predicts the measurements that would be obtained for a given configuration of current sources. The used headmodels in this thesis are multi-layered spherical head models. The neural sources are parameterized by the location and orientation of electrical dipoles. In this thesis, a set of limited number of dipole sources is used as source model leading to a well posed inverse problem. The inverse problem starts from measured EEG data and recovers the locations and orientations of the electrical dipole sources. A loss in accuracy of the recovered neural sources occurs because of noise in the EEG measurements and uncertainties in the forward model. Especially the conductivity values of scalp, skull and brain are not well known since these values are difficult to measure. Moreover, these uncertainties can vary from person to person, in time, etc. In this thesis, novel numerical methods are developed so to provide new approaches in the improvement of spatial accuracy in EEG source analysis, taking into account model uncertainties. Nowadays, the localization of the electrical activity in the brain is still a current and challenging research topic due to the many difficulties arising e.g. in the process of modeling the head and dealing with the not well known conductivity values of its different tissues. Due to uncertainty in the conductivity value of the head tissues, high values of errors are introduced when solving the EEG inverse problem. In order to improve the accuracy of the solution of the inverse problem taking into account the uncertainty of the conductivity values, a new mathematical approach in the definition of the cost function is introduced and new techniques in the iterative scheme of the inverse reconstruction are proposed. The work in this thesis concerns three important phases. In a first stage, we developed a robust methodology for the reduction of errors when reconstructing a single electrical dipole in the case of a single uncertainty. This uncertainty concerns the skull to soft tissue conductivity ratio which is an important parameter in the forward model. This conductivity ratio is difficult to quantify and depends from person to person. The forward model that we employed is a three shell spherical head model where the forward potentials depend on the conductivity ratio. We reformulated the solution of the forward problem by using a Taylor expansion around an actual value of the conductivity ratio which led to a linear model of the solution for the simulated potentials. The introduction of this expanded forward model, led to a sensitivity analysis which provided relevant information for the reconstruction of the sources in EEG source analysis. In order to develop a technique for reducing the errors in inverse solutions, some challenging mathematical questions and computational problems needed to be tackled. We proposed in this thesis the Reduced Conductivity Dependence (RCD) method where we reformulate the traditional cost function and where we incorporated some changes with respect to the iterative scheme. More specifically, in each iteration we include an internal fitting procedure and we propose selection of sensors. The fitting procedure makes it possible to have an as accurate as possible forward model while the selection procedure eliminates the sensors which have the highest sensitivity to the uncertain skull to brain conductivity ratio. Using numerical experiments we showed that errors in reconstructed electrical dipoles are reduced using the RCD methodology in the case of no noise in measurements and in the case of noise in measurements. Moreover, the procedure for the selection of electrodes was thoroughly investigated as well as the influence of the use of different EEG caps (with different number of electrodes). When using traditional reconstruction methods, the number of electrodes has not a high influence on the spatial accuracy of the reconstructed single electrical dipole. However, we showed that when using the RCD methodology the spatial accuracy can be even more increased. This because of the selection procedure that is included within the RCD methodology. In a second stage, we proposed a RCD method that can be applied for the reconstruction of a limited number of dipoles in the case of a single uncertainty. The same ideas were applied onto the Recursively Applied and Projected Multiple Signal Classification (RAP-MUSIC) algorithm. The three shell spherical head model was employed with the skull to brain conductivity ratio as single uncertainty. We showed using numerical experiments that the spatial accuracy of each reconstructed dipole is increased, i.e. reduction of the conductivity dependence of the inverse solutions. Moreover, we illustrated that the use of the RCD-based subspace correlation cost function leads to a high efficiency even for high noise levels. Finally, in a third stage, we developed a RCD methodology for the reduction of errors in the case of multiple uncertainties. We used a five shell spherical head model where conductivity ratios with respect to skull, cerebrospinal fluid, and white matter were uncertain. The cost function as well as the fitting and selection procedure of the RCD method were extended. The numerical experiments showed reductions in the reconstructed electrical dipoles in comparison with the traditional methodology and also compared to the RCD methodology developed for dealing with a single uncertainty