An important question arousing in the framework of electroencephalography (EEG) is the possibility to recognize, by means of a recorded surface potential, the number of activated areas in the brain. In the present paper, employing a homogeneous spherical conductor serving as an approximation of the brain, we provide a criterion which determines whether the measured surface potential is evoked by a single or multiple localized neuronal excitations. We show that the uniqueness of the inverse problem for a single dipole is closely connected with attaining certain relations connecting the measured data. Further, we present the necessary and sufficient conditions which decide whether the collected data originates from a single dipole or from numerous dipoles. In the case where the EEG data arouses from multiple parallel dipoles, an isolation of the source is, in general, not possible.

It is known from the fundamental work of Albanese and Monk that, the recovery of the support of a three dimensional current, within a conducting medium, from measurements of the generated exterior electric potential, is not possible. However, it is possible to recover the support of any other current, which is supported on a set of dimension lower than three. Nevertheless, no algorithm for such an inversion is known. Here, we propose such an algorithm for a two dimensional current distribution, and in particular, we apply this algorithm to the inverse problem of electroencephalography in the case where the neuronal current is restricted to a small disk of arbitrary location and orientation within the brain. The solution of this inverse problem is reduced to the solution of a nonlinear algebraic system, and numerical tests show that the there exists a unique real solution to this system.

A fundamental principle of Cognitive Psychology states that the rate at which the human brain learns a certain amount of knowledge is proportional to the amount of knowledge yet to be learned. This is the so called pure memory or tabula raza law of learning. The mathematical formulation of this principle leads to a simple ordinary differential equation of the first order. Here we expand the existing mathematical model to a fractional differential equation which allows for a more realistic model having a much higher freedom to fit possible experimental data, as well as allowing for memory effects during the learning process. Two different definitions of the fractional derivative are used, one is the standard Riemann-Liouville global definition and the other is a local definition based on the choice of the unit that measures functional variation. A detailed comparison with the conventional model both at the analytical and the numerical level is included.

Albanese and Monk (2006) have shown that, it is impossible to recover the support of a three-dimensional current distribution within a conducting medium from the knowledge of the electric potential outside the conductor. On the other hand, it is possible to obtain the support of a current which lives in a subspace of dimension lower than three. In the present work, we actually demonstrate this possibility by assuming a one-dimensional current distribution supported on a small line segment having arbitrary location and orientation within a uniform spherical conductor. The immediate representation of this problem refers to the inverse problem of electroencephalography (EEG) with a linear current distribution and the spherical model of the brain-head system. It is shown that the support is identified through the solution of a nonlinear algebraic system which is investigated thoroughly. Numerical tests show that this system has exactly one real solution. Exact solutions are analytically obtained for a couple of special cases.