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Neuronal Current Decomposition via Vector Surface Ellipsoidal Harmonics

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Inverse Problems

Electroencephalography (EEG) and Magnetoencephalography (MEG) provide the two most efficient imaging techniques for the study of the functional brain, because of their time resolution. Almost all analytical studies of EEG and MEG are based on the spherical model of the brain, while studies in more realistic geometries are restricted to numerical treatments alone. The human brain can best approximated by an ellipsoid with average semi-axes equal to 6, 6.5 and 9 centimeters. An analytic study of the brain activity in ellipsoidal geometry though, is not a trivial problem and a complete closed form solution does not seems possible for either EEG or MEG . In the present work we introduce vector surface ellipsoidal harmonics, we discuss their peculiar orthogonality properties, and finally we use them to decompose the neuronal current within the brain into the part that is detectable by the EEG and that is detectable by the MEG measurements. The decomposition of a vector field in vec tor surface ellipsoidal harmonics leads to three subspaces R, D and T, depending on the character of the surface harmonics that they span this subspaces. We see that both, the electric field obtained from EEG and the magnetic field obtained from MEG , have no T-component. Furthermore, the T-component of the neuronal current does not influence the EEG recordings, while the MEG recordings depend on all three components of the current.

This talk is part of the Isaac Newton Institute Seminar Series series.

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