The existing functional imaging techniques for the brain have excellent space resolution, but their time resolution is of the order of 1s, which is too slow for the study of the brain. The only non-invasive techniques with adequate time resolution are magnetoencephalography (MEG) and electroencephalography (EEG). The main reason that MEG is not yet established as an acceptable clinical technique is the lack of uniqueness, namely there exist many currents, which give rise to the same magnetic field.
However, it has been recently shown that for the case that the brain can be approximated by a sphere, the current is uniquely determined from the measurements, provided that one assumes that it minimizes the integral of its square. Furthermore, the numerical implementation of the minimizing current has been tested successfully with real data. The basic limitation of the above result is that the brain is not a sphere. A realistic approximation of the brain is provided by a tri-axial ellipsoid and the main objective of the project is to extend the above result from the spherical to ellipsoidal geometry and to take into consideration the cerebrospinal fluid. Mathematical techniques to be used include expansion in terms of ellipsoidal harmonics and the technique of generalized transforms, which has been recently introduced in the literature. Graduate students and young researchers will be trained on these topics.
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