Abstract
The inverse problem of magnetoencephalography (MEG) seeks the neuronal current within the conductive brain that generates a measured magnetic flux in the exterior of the brain-head system. This problem does not have a unique solution, and in particular, it is not even possible to identify the support of the current if it extends over a three-dimensional set. However, a localized current supported on a zero-, one- or two-dimensional set can in principle be identified. In the present work, we demonstrate an analytic algorithm that is able to recover a one-dimensional distribution of current from the knowledge of the exterior magnetic flux field. In particular, we consider a neuronal current that is supported on a small line segment of arbitrary location and orientation in space, and we reduce the identification of its characteristics to a nonlinear algebraic system. A series of numerical tests show that this system has a unique real solution. A special case is easily solved via the use of trivial algebraic operations.
Highlights
The brain is a conducting material and every generated neuronal current is accompanied by an induction current
When we measure the magnetic flux density outside the head we measure the effects of both the neuronal as well as the induction current. This is the main problem with the inverse problem of magnetoencephalography, the fact that the induction current “hides” somehow the primary neuronal excitation
A hundred and sixty years ago Helmholtz [3] showed that it is not possible to recover an electric current within a conductor from knowledge of the magnetic flux generated outside the conductor
Summary
The brain is a conducting material and every generated neuronal current is accompanied by an induction current. When we measure the magnetic flux density outside the head we measure the effects of both the neuronal as well as the induction current. Albanese and Monk [5] proved that such localization is not possible More precisely they showed that it is impossible to find the support of the current if the current occupies a threedimensional subset of the brain. If the current is distributed over a surface, which is a two-dimensional subset, a curve, which is a one-dimension subset, or on isolated points, which form zero-dimensional subsets, it is possible to identify it. We consider a dipolar current distribution over a small line segment, and we develop an algorithm that reduces the identification of the position, the length and the orientation of the line segment, as well as the average dipolar moment of the current, to the solution of a nonlinear algebraic system.
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