Unit:
Κατεύθυνση Εφαρμοσμένα ΜαθηματικάLibrary of the School of Science
Author:
Πάσιου Νιόβη-Παρασκευή
Supervisors info:
Καθηγητής Ιωάννης Στρατής (επιβλέπων)
Original Title:
Μαθηματική ανάλυση της Ηλεκτροεγκεφαλογραφίας και της Μαγνητοεγκεφαλογραφίας
Translated title:
Mathematical analysis of Electroencephalography and Magnetoencephalography
Summary:
The object of this Master Thesis is to study forward and inverse problems of
Electroencephalography and Magnetoencephalography in order to identify and
characterize the source. In the interior of the brain there is electromagnetic
activity that generates electric and magnetic fields. The resulting electric
and magnetic fields are measured on the surface and the exterior of the head
via the EEG and MEG, respectively. In the first part of the thesis (chapter 1)
we describe the morphology and the functionality of the human brain and we also
present an overview of the electromagnetic theory. Subsequently there is a
brief review of the main brain imaging methods such as EEG and MEG. In chapter
2, we solve the forward problem of Electroencephalography, in spherical
geometry for the case where the source is considered to be a point current
dipole in the interior of the brain. We find analytical solutions in spherical
harmonics for the electric potential in the interior and the exterior of the
conductor and we present the well-known closed form solutions. Subsequently we
interpret the solution by the Ima ge Source Method. The inverse EEG problem
(chapter 3) is analyzed in spherical geometry in the case of a continuously
distributed neuronal current. It turns out that only one of the three
components of the neuronal current is identified from the knowledge of surface
potential. This is an indication of the fact that the solution of the inverse
problem is not unique. We solve the inverse problem in spherical geometry and
in the case where a primary current is a dipole and we construct an algorithm
that computes the exact position and the moment of the dipole. We follow same
methodology for a continuously distributed neuronal current. The forward MEG
problem (chapter 4) is solved with similar techniques that used in
Electroencephalography. We also present transformation Kelvin, produce a new
representation for the magnetic potential and solve an integral equation on the
surface of a sphere. Finally we solve the inverse problem of MEG (chapter 5).
It is shown that for an arbitrary geometry MEG provides information about two
of the three components of the neuronal current. Consequently, the inverse MEG
problem is not uniquely solvable. In particular, for spherical geometry it is
not possible to determine the radial parts of the neuronal current. In the case
of the spherical conductor only one of the two components is “visible”, the one
that it is invisible for EEG.
Keywords:
Brain, Neural, Spherical conductor, EEG, MEG
