Dissertation committee:
Ομότιμος Καθηγητής Ιωάννης Στρατής-τμήμα Μαθηματικών-ΕΚΠΑ
Καθηγητής Γεράσιμος Μπαρμπάτης-τμήμα Μαθηματικών-ΕΚΠΑ
Επίκουρος Καθηγητής Παναγιώτης Σμυρνέλης-τμήμα Μαθηματικών-ΕΚΠΑ
Ομότιμος Καθηγητής Γεώργιος Δάσιος-τμήμα Χημικών Μηχανικών-Πανεπιστήμιο Πατρών
Ομότιμος Καθηγητής Χριστόδουλος Αθανασιάδης-τμήμα Μαθηματικών-ΕΚΠΑ
Kαθηγητής Αθανάσιος Γιαννακόπουλος-τμήμα Στατιστικής-Οικονομικό Πανεπιστήμιο Αθηνών
Αναπληρώτρια Καθηγήτρια Καριώτου Φωτεινή-Σχολή θετικών επιστημών και τεχνολογίας -ΕΑΠ
Summary:
The electromagnetic activity of the human brain is studied via the non invasive
methods of Electroencephalography and Magnetoencephalography.
Electroencephalography (EEG) and Magnetoencephalography (MEG) are the two
brain imaging modalities which have the necessary temporal resolution, for the study
of the functional brain.
It is well known that an electrochemically generated current in the interior of the brain
generates an electric and a magnetic field, both in the interior and exterior of the
brain.
The resulting electric and magnetic fields are measured on the surface and the exterior
of the head via the EEG and MEG, respectively.
For both of these modalities we know the direct mathematical problems of
determining the electric potential and the magnetic flux when the interior neuronal
sources, as well as the geometry of the brain-head system, are given, and the inverse
mathematical problems of identifying the sources once the measured EEG and MEG
data are obtained.
During the last three decades both the direct and the inverse problems of EEG and
MEG have been studied extensively, as it can be seen in the references of the present
work.
Obviously, the solutions of the mathematical problems of the electromagnetic brain
activity depend crucially on the geometry of the head, since this is the fundamental
domain where the relative boundary value problems are defined. Since the simplest
VIII
such model is the sphere, almost all contributions up until the 1990s were assuming
that the head-brain system had spherical symmetry.
However, the realistic geometry for these problems is that of an ellipsoid with average
semi-axes equal to 6 cm, 6.5 cm and 9 cm. Α number of relative solutions for the EEG
and MEG problems with and without inhomogeneous nested shells, with dipole as
well as with continuous source distributions in ellipsoidal geometry have been
published. In particular, the inverse problems of identifying the position and the
moment of a dipolar source for EEG and MEG in spherical and in ellipsoidal
geometry are know well known.
Our interest on the present work steams from the fact that the machinery used in EEG
and MEG practical diagnostics utilizes algorithms which are based on the assumption
that the head has spherical symmetry. Hence, they record data from an ellipsoidal
shape and interpret them as they were coming from a sphere. Then the question is:
what is the error made by this assumption on the estimation of the location and the
moment of a dipole?
Since the solutions for the inverse EEG dipole problem in spherical and ellipsoidal
geometry were known the challenge was to transfer information from one geometrical
system to the other, and anyone who has tried to do something like that immediately
recognizes the difficulty of such a task. However, as we demonstrate in this work,
after lengthy and tedious calculations these representations were achieved at least for
the needed low degree eigenfunctions. The final result is the dependence of the error
on the values of the principal eccentricities of the ellipsoid. The closer the
eccentricities are to zero the smaller the error, which vanishes when the eccentricities
approach zero, that is, as the ellipsoid degenerates to the sphere. On the other end, as
the eccentricities approach the value 1 the error becomes very large. An abrupt change
appears when the maximum eccentricity of the ellipsoid approaches values greater
than 0.9.
This work justifies the medical practice of using the spherical, instead of the
ellipsoidal, brain model since the eccentricities of the human brain belong to the
interval where no significant errors are present, that is in the neighborhood of 0.7
Since this work can be used in any other scientific or technological applications it is
IX
important to remember that things are not so easy in the case of very elongated or
very flat ellipsoids.
Keywords:
brain, neural, dipole current, primary current, passive current, Inverse Problems, spherical geometry, ellipsoidal geometry, EEG, MEG
