Abstract
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.
Highlights
Electroencephalography (EEG) has a history of almost 90 years [1]
It is an imaging modality that associates the electric potential that is generated on the surface of the head with the electric neuronal activity in the interior of the head
The measurements recorded on the surface of the head involve the effects of the primary neuronal current but those of the induction current as well
Summary
Electroencephalography (EEG) has a history of almost 90 years [1]. It is an imaging modality that associates the electric potential that is generated on the surface of the head with the electric neuronal activity in the interior of the head. Since the half of this length is much larger than the characteristic dimension of the brain-head system it follows that the Quasi-Static theory is well justified as the appropriate theory for the investigation of the electric and magnetic brain activity. As it is the case with most of real-life problems, electroencephalography involves a forward and an inverse problem. In the forward EEG problem, the neuronal current is given and we seek to calculate the electric potential on the surface of the head.
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