For the reconstruction of electrical activity in the brain from EEG signals, FindSources3D (FS3D) relies on a classical model of current transmission in conductive media of the head. It is made of 3 nested spherical layers (scalp, skull, brain), each of constant conductivity coefficient. Current sources are assumed to be pointwise and dipolar, located within the brain.
Together with the geometry and the conductivity values, FS3D's inputs consist in available pointwise values of the electrical potential (taken by electrodes or numerically computed) on the scalp. From these data, the pointwise dipolar sources (position and moment) are estimated in the innermost ball (brain) as follows, by a series of consecutive stages.
- Cortical mapping:
The first step is the transmission of the available pointwise values of the electrical potential (taken by electrodes) from the scalp to the cortex (brain surface). This is done in the two outermost layers where the potential is a harmonic function.
On the innermost sphere (cortex), the electrical potential is the sum of a harmonic function in the inner ball (brain) and of a singular function whose singularities within the ball coincide with the pointwise dipolar sources. The cortical mapping step actually consists in estimating this singular part of the potential on the cortex from the set of pointwise data given on the scalp from electrodes measurements. It's expansion on a number of elements of the spherical harmonic basis is the outupt of the present step and the input of the second step below.
- 2D Scans, lines of singularities:
Once the singular part of the potential is computed there, the inner sphere (brain) is sliced into families of up to 20 parallel planar sections, in up to 13 different directions.
Consider one of these directions. On the boundary of every corresponding disk (circle), the electrical potential coincides with a complex variable function, whose singularities in the disk are linked with the sources. In the situation of a unique source this function (once squared) is rational. In general, it has branched singularities (like square-roots or logarithms). In any cases, its singularities in the disk are approximated by the poles of a rational function. This rational function is estimated from the given boundary values by the algorithms of best quadratic rational approximation developped within the team APICS (software RARL2). The quantity of singularities in each disk coincides with the number of sources.
Within the sphere, the estimated singularities (poles) are superposed and draw a set of lines through the consecutive parallel sections. Each of theses lines is contained in a plane which is orthogonal to the sections direction, and also contains a source. See below a picture from an example with 2 sources, planar sections direction given by Z = constant and the corresponding 2 "lines" of poles.
The associated 2D scan shows the projection of these poles on the equator of the sphere, in the direction of the sections. See below a picture related to the above example (the plane OXY).
These 2D scans are performed in all the chosen directions.
- Scans selection:
We then select by hand (at the moment) the most representative ones that correspond to directions for which the estimated singularities are well aligned.
- 3D localisation, clusterisation, moments estimation:
The selected 2D scans are used in the last 3D localisation step, for sources estimation at the intersections of the singularities lines. This approximately determines the sources positions, that are clustered together.
Once their positions are known, the moments of the sources are linearly computed, before a last post-treatment is made against the electrodes data (dipole fitting).
- Pairing and error calculation:
Once the estimated sources are placed, and if true the sources are known, the Fs3d will compare the known sources with the estimated ones. The errors displayed are always relative errors, the full procedure is described on this page: Error Calculation
References:
M. Clerc, J. Leblond, J.-P. Marmorat, T. Papadopoulo, «Source localization using rational approximation on plane sections », Inverse Problems, 28, 2012.
L. Baratchart, J. Leblond, J.-P. Marmorat, « Inverse source problem in a 3D ball from best meromorphic approximation on 2D slices », Electronic Transactions on Numerical Analysis, 25, 2006.
