Error analysis for nonconfocal ellipsoidal systems in the forward problem of electroencephalography

Abstract

Electroencephalography (EEG) remains the utmost important technique recording brain activity. The present investigation examines the sensitivity of analytic algorithms employed in order to evaluate EEG data with respect to different brain template models. These algorithms are based on mathematical models built upon certain geometrical and/or physical assumptions regarding the head‐brain shape or the neuronal source. While the ellipsoidal head model provides a realistic approach regarding the interpretation of EEG signals in view of diverse geometries, its eccentricities influence strongly the results. The present study quantifies the deviation of the computed electric potentials when nonconfocal ellipsoids are used to model a homogeneous head conductor. To this end, a correspondence is proposed between points of the different ellipsoids, in view of the Gauss map. The investigation demonstrates that the introduction of nonconfocality imports a highly elevated error rate when EEG recordings are misinterpreted by arriving from different head‐brain models, including ellipsoidal vs spherical ones. Moreover, evidence is presented concerning the location of extrema regarding the errors in the upper brain hemisphere, which could lead to a more precise and accurate protocol regarding the placement of sensors. Although the present work refers to the forward EEG problem, its results may be used under other approaches as well. In particular, it provides the error in the solution of an elliptic boundary value problem with transmission conditions under small perturbations of the eccentricities of its ellipsoidal domain.