Michael Doschoris

Localizing Brain Activity from Multiple Distinct Sources via EEG

An important question arousing in the framework of electroencephalography (EEG) is the possibility to recognize, by means of a recorded surface potential, the number of activated areas in the brain. In the present paper, employing a homogeneous spherical conductor serving as an approximation of the brain, we provide a criterion which determines whether the measured surface potential is evoked by a single or multiple localized neuronal excitations. We show that the uniqueness of the inverse problem for a single dipole is closely connected with attaining certain relations connecting the measured data. Further, we present the necessary and sufficient conditions which decide whether the collected data originates from a single dipole or from numerous dipoles. In the case where the EEG data arouses from multiple parallel dipoles, an isolation of the source is, in general, not possible.

Limiting values of the Gegenbauer functions on the cut

Limiting values and series representations of the Gegenbauer functions of the first and second kind of general complex degree ν and order λ on the cut (−1, 1) are presented. These limits are necessary for the analysis of certain boundary value problems associated with the theory of the potential and Stokes’ flow.

An algebraic formula for the accelerated computation of the low frequency scattering coefficients

An explicit expression for the rapid computation of the low frequency scattering coefficients is presented. Although the demonstration of the methodology is restricted to the simple case of an acoustically soft spherical scatterer, the introduced main concept can be applied to a wide class of scattering problems. The procedure, based on algebraic calculations only, provides an analytic expression of the corresponding scattering coefficients in a straightforward fashion. The proposed algorithm is readily implemented, furnishing an efficient way for obtaining any order approximation of the total field in the exterior of the scatterer.

Sensitivity Analysis of the Forward Electroencephalographic Problem Depending on Head Shape Variations

A crucial aspect in clinical practice is the knowledge of whether Electroencephalographic (EEG) measurements can be assigned to the functioning of the brain or to geometrical deviations of the human cranium. The present work is focused on continuing to advance understanding on how sensitive the solution of the forward EEG problem is in regard to the geometry of the head. This has been achieved by developing a novel analytic algorithm by performing a perturbation analysis in the linear regime using a homogenous spherical model. Notably, the suggested procedure provides a criterion which recognizes whether surface deformations will have an impact on EEG recordings. The presented deformations represent two major cases: (1) acquired alterations of the surface inflicted by external forces and (2) deformations of the upper part of the human head where EEG signals are recorded. Our results illustrate that neglecting geometric variations present on the heads surface leads to errors in the recorded EEG measurements less than 2%. However, for severe instances of deformations combined with cortical brain activity in the vicinity of the distortion site, the errors rise to almost 25%. Therefore, the accurate description of the head shape plays an important role in understanding the forward EEG problem only in these cases.

Connection Formulae between Ellipsoidal and Spherical Harmonics with Applications to Fluid Dynamics and Electromagnetic Scattering

The environment of the ellipsoidal system, significantly more complex than the spherical one, provides the necessary settings for tackling boundary value problems in anisotropic space. However, the theory of Lame functions and ellipsoidal harmonics affiliated with the ellipsoidal system is rather complicated. A turning point would reside in the existence of expressions interlacing these two different systems. Still, there is no simple way, if at all, to bridge the gap. The present paper addresses this issue. We provide explicit formulas of specific ellipsoidal harmonics expressed in terms of their counterparts in the classical spherical system. These expressions are then put into practice in the framework of physical applications.

The Influence of Surface Deformations on the Forward Magnetoencephalographic Problem

A perturbational model is developed providing explicit computationally efficient solutions for the forward magnetoencephalographic problem, namely, calculating the external magnetic fields for known neuronal sources. The aim of the study is to investigate the sensitivity of the particular measurements to deformations occurring on the conductors surface. These geometric variations represent irregularities in head shapes and correspond to two major situations: (1) localized acquired injuries of the scalp-skull delivered by external forces; (2) craniofacial alterations due to natural mechanisms or defects. The presented methodology has the following advantages. First, it supports the installation of tailored functions, which individually describe aforesaid deformations. Second, it allows rapid calculation of the forward problem for superficial cerebral activity, where similar numerical methods produce large errors. Our results indicate that surface deformations can have an eminent impact on magnetoencephalo...

Mathematical Foundation of Electroencephalography

Electroencephalography (EEG) has evolved over the years to be one of the primary diagnostic technologies providing information concerning the dynamics of spontaneous and stimulated electrical brain activity. The core question of EEG is to acquire the precise location and strength of the sources inside the human brain by knowledge of an electrical potential measured on the scalp. But in what way is the source recovered? Leaving aside the biological mechanisms on the cellular level responsible for the recorded EEG signals, we pay attention to the mathematical aspects of the narrative. Our goal is to provide a brief and concise introduction of the mathematical terminology associated with the modality of EEG. We start from the very beginning, presenting step by step the mathematical formulation behind EEG in a simple and clear manner, keeping the mathematical notation to a minimum. Whilst we serve only the key relations for the described problems, we focus specifically on the limitations of each modelling approach. In this fashion, the reader can appreciate the beauty of the formulas presented and discover every single piece of information encoded within these formulas.

The influence of surface deformations on electroencephalographic recordings

The precise identification of neuronal currents via Electroencephalographic (EEG) recordings is an important aspect in clinical practice and strongly depends on the accuracy of the corresponding forward problem. In addition, the precision of the EEG forward model is closely connected to the existence of a volume conductor model as realistic as possible. In this paper, the impact of geometric variations of the head on the measured electric potential has been studied by means of a homogeneous spherical conductor. In the case where the activated region is situated in the vicinity of the deformation, the calculated potential values show a slight increase. On the other hand, for neuronal currents away from the deformation no influence upon the surface electric measurements is observed.

Analysis of relative errors and bounds in localization of dipoles in various ellipsoidal brain models in Encephalography

Electroencephalography (EEG) measures potential differences on part of the surface of the head. These measurements are directly connected with activated regions within the brain, modeled as dipoles, and are accurately interpreted if originating from a average ellipsoidal conductor with semi-axes 5.5, 6.5 and 8.5 × 10-2 m. However, the volume of modern human brains varies significantly depending on sex and age. These variations in volume could introduce a source of error affecting the location of the dipole if not incorporated in existing models. In what follows, an error estimation is established for EEG readings in the case where the average ellipsoidal brain is replaced by an ellipsoid with different volume.

An alternative method to obtain integral kernels for Lamé integral equations

Abstract In the present paper we introduce an alternative approach to obtain kernels for Lame integral equations. The introduced procedure, based solely on simple algebraic manipulations, furnishes the well known kernels of hypergeometric form in an almost trivial manner without the use of transformations. More important, it provides a set of novel nuclei for Lame integral equations in the form of Heun functions.