George Dassios

Electric and Magnetic Activity of the Brain in Spherical and Ellipsoidal Geometry

Understanding the functional brain is one of the top challenges of contemporary science. The challenge is connected with the fact that we are trying to understand how an organized structure works and the only means available for this task is the structure itself. Therefore an extremely complicated scientific problem is combined with a hard philosophical problem.

Estimates for the electric field in the presence of adjacent perfectly conducting spheres

In this paper we prove that, unlike the two-dimensional case, the electric field in the presence of closely adjacent spherical perfect conductors does not blow up even though the separation distance between the conducting inclusions approaches zero.

Laplace’s equation in the exterior of a convex polygon. The equilateral triangle

A general method for studying boundary value problems for linear and for integrable nonlinear partial differential equations in two dimensions was introduced in Fokas, 1997. For linear equations in a convex polygon (Fokas and Kapaev (2000) and (2003), and Fokas (2001)), this method: (a) expresses the solution q(x, y) in the form of an integral (generalized inverse Fourier transform) in the complex κ-plane involving a certain function q(κ) (generalized direct Fourier transform) that is defined as an integral along the boundary of the polygon, and (b) characterizes a generalized Dirichlet-to-Neumann map by analyzing the so-called global relation. For simple polygons and simple boundary conditions, this characterization is explicit. Here, we extend the above method to the case of elliptic partial differential equations in the exterior of a convex polygon and we illustrate the main ideas by studying the Laplace equation in the exterior of an equilateral triangle. Regarding (a), we show that whereas q(κ) is identical with that of the interior problem, the contour of integration in the complex κ-plane appearing in the formula for q(x, y) depends on (x, y). Regarding (b), we show that the global relation is now replaced by a set of appropriate relations which, in addition to the boundary values, also involve certain additional unknown functions. In spite of this significant complication we show that, for certain simple boundary conditions, the exterior problem for the Laplace equation can be mapped to the solution of a Dirichlet problem formulated in the interior of a convex polygon formed by three sides.

Stokes flow in ellipsoidal geometry

Particle-in-cell models for Stokes flow through a relatively homogeneous swarm of particles are of substantial practical interest, because they provide a relatively simple platform for the analytical or semianalytical solution of heat and mass transport problems. Despite the fact that many practical applications involve relatively small particles (inorganic, organic, biological) with axisymmetric shapes, the general consideration consists of rigid particles of arbitrary shape. The present work is concerned with some interesting aspects of the theoretical analysis of creeping flow in ellipsoidal, hence nonaxisymmetric domains. More specifically, the low Reynolds number flow of a swarm of ellipsoidal particles in an otherwise quiescent Newtonian fluid, that move with constant uniform velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity, is analyzed with an ellipsoid-in-cell model. The solid internal ellipsoid represents a particle of the swarm. The external ellipsoid con...

The octapolic ellipsoidal term in magnetoencephalography

The forward problem of magnetoencephalography (MEG) in ellipsoidal geometry has been studied by Dassios and Kariotou [“Magnetoencephalography in ellipsoidal geometry,” J. Math. Phys. 44, 220 (2003)] using the theory of ellipsoidal harmonics. In fact, the analytic solution of the quadrupolic term for the magnetic induction field has been calculated in the case of a dipolar neuronal current. Nevertheless, since the quadrupolic term is only the leading nonvanishing term in the multipole expansion of the magnetic field, it contains not enough information for the construction of an effective algorithm to solve the inverse MEG problem, i.e., to recover the position and the orientation of a dipole from measurements of the magnetic field outside the head. For this task, the next multipole of the magnetic field is also needed. The present work provides exactly this octapolic contribution of the dipolar current to the expansion of the magnetic induction field. The octapolic term is expressed in terms of the ellipso...

Neuronal currents and EEG–MEG fields

In a recent paper by the author, Fokas and Hadjiloizi proved that a neuronal current within a spherical homogeneous conductor can be split into two orthogonal components in such a way that one component provides the electroencephalography (EEG)-related fields and the other component provides the fields related to magnetoencephalography (MEG). Hence, in spherical geometry, the EEG and MEG measurements contain no overlapping information about the current. In the present work, we utilize a new integral representation for the magnetic potential, introduced recently by Fokas, Kariotou and the author, to prove that this elegant property is not true once the highly symmetric spherical environment is abandoned. It seems that any ambiguity concerning overlapping information coming from EEG and MEG measurements has its origin in the fact that in most clinical applications the spherical model is used although the actual data never come from a perfect sphere.

On growth of ellipsoidal tumours

The existing mathematical models for tumour growth, to a large extent, are new and not well established as of today. This is mainly due to the fact that there are many known and unknown factors that enter the process of malignant tumour development, and no convincing arguments about their relative importance are generally established. As a consequence of this search for a credible model, almost every tumour model that has been investigated so far refers to the highly symmetric case of the spherical geometry, where the curvature is a global invariant over its outer surface. Hence, no information about the effects of the local curvature upon the shape of the exterior proliferating boundary is available. In this presentation, we discuss first the standard Greenspan model for a spherical tumour, where the basic ideas are presented, and then we extend the model to that of triaxial ellipsoidal geometry. In this way, we elevate fundamental qualitative characteristics of the growth process that are invisible in spherical geometry. One such thing is the effect of the local mean curvature on the development of the outer boundary of the tumour, as it is governed by the Young-Laplace law, which controls the interface between two non-mixing fluids. A second advantage of the ellipsoidal model is due to the way the confocal system is generated. Indeed, in contrast to the spherical system which springs out of a central point, the confocal ellipsoidal system starts out as an inflated focal ellipse which, if it is interpreted as a biological membrane, provides a much more realistic candidate for tumour genesis. Nevertheless, the investigation of the ellipsoidal model of a tumour growth is by no means completed, and a lot of further study needs to be done before final conclusions on the effects of curvature variations are drawn.