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Electromagnetic Brain Activity: A Novel Mathematical Approach

Final Activity Report Summary - BRAIN (Electromagnetic brain activity: a novel mathematical approach)

The goal of the project BRAIN was the investigation of the electric and magnetic activity of the human brain. Most of the work in this area was focused on a spherical model of the head-brain system, because the highly symmetric properties of the sphere reduce considerably the underlying mathematical analysis. But in reality, almost no head has spherical symmetry. A realistic head model is that of an ellipsoid with fixed semi-axes. Nevertheless, the mathematical difficulties appearing in the analysis of boundary value problems in ellipsoidal geometry is very complicated and in most cases impossible. This project, which achieved results beyond its initial plan, is focused exactly on the analysis of Electroencephalography (EEG) and Magnetoencephalography (MEG) in ellipsoidal geometry.

Some of the major qualitative and quantitative results obtained are : a) for the case of spherical geometry, there is no part of the neuronal current that contributes both to the EEG and MEG fields, b) this property of complementarity, described in a) is not true if the geometry of the domain is not spherical, c) if the Helmholtz decomposition is used to decompose the neuronal current into irrotational and solenoidal fields within a conductor of any shape, then the electric potential (and therefore EEG) depends only on the scalar potential, while the magnetic field (and therefore MEG) depends on both the scalar potential and the radial component of the vector potential, d) for the case of the sphere and the Hansen decomposition of the current, the electric potential depends exactly on the scalar potentials that the magnetic field is missing, e) analytic algorithms have been obtained that can identify a single dipolar current, a finite number of dipoles, or the location of a localised current, f) for the case of a dipole inside any ellipsoidal conductor the octapolic term of the multipole expression is obtained in closed form. Many more analytic results were obtained, some of which belong to different areas of applied mathematics, which found nevertheless their solution as a consequence of the work performed for this project.

As a concluding statement we can declare that we have now a much better understanding of the behaviour of the electric and magnetic response of the brain in realistic geometries, and a strict set of limitations identifying what can and what cannot be done within the present level of the mathematical modelling of brain activity.