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FindSources3D (FS3D)
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Electroencephalography (EEG) and magnetoencephalography (MEG) are two non invasive techniques for imaging functional or clinical brain activity (electrical currents occurring naturally in the brain). This activity produces effects out of the head (electric potential and magnetic field).
EEG records the electrical potential created on the scalp using
electrodes.
MEG records components of the magnetic field using magnetometers.
FS3D is a software which aims at recovering internal currents viewing EEG or MEG records.
Head, sources, measures: spherical layered model.
FS3D relies on a simplified, but classical model of current transmission in conductive media [1, 2]. The head model is made of 3 nested spherical layers (scalp, skull, brain). Each layer admits a constant electrical conductivity. Current sources are located within the brain, and are assumed to be pointwise and dipolar.
Due to this simplification, given the sources, an explicit solution to the underlying elliptic partial differential equation (that actually comes from Maxwell’s equations in quasi-static regime) can be expressed for the electric potential, solving the so-called EEG direct problem.
FS3D aims at solving the inverse problem. Given inputs corresponding to a finite number of pointwise values of the electrical potential on the scalp, recover dipolar sources within the brain consistent with these measures.
Inverse problem, algorithm description.
We are in the spherical model. Computations are performed using expansions of all functions on a spherical harmonics basis. Dipolar sources (position and moment) are estimated in the innermost ball (brain), by a series of consecutive stages.
Cortical mapping.
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 Us whose singularities within the ball coincide with the pointwise dipolar sources. The linear map from this singular part to the EEG values is explicitly known.
The cortical mapping first step actually consists in estimating the singular part of the potential on the cortex from the set of pointwise data given on the scalp. This inverse problem is solved by minimizing a regularized quadratic error at the electrodes, a bounded extremal problem.
We obtain an approximation of the singular part Us (a finite number of its spherical spherical harmonics expansion). We can the consider the positive valued function Us2 defined on the inner sphere.
2D slicing, rational approximation in disks.
The inner sphere is sliced into families of parallel planar sections (currently up to 20) along different directions (currently up to 13).
Each 2D slice is mapped onto the unit disk in the complex plane. The trace on the boundary of each slice of Us2 defines on the unit complex circle a positive valued function which can be factorized. One factor is analytic outside the disk, and has singularities inside. This function is exactly rational in the case of one source, has branched singularities in the case of many sources [1]. We build estimation of these 2D simgularities with a rational appoximation algorithm (ARL2 simplified parameterization).
3D singularities lines, sensible points estimation.
When all 2D singularities are mapped back into the 3D inner sphere we obtain a collection 3D points. These “3D singularities” accumulate in different planes where they form lines. Each plane contains a source and the slicing direction. In each plane the line passes by the source where it atteins maximum distance to the slicing axis. The extremal point of each line is estimated, and called a sensible point (SP or source position estimation).
SP clustering and least squares fitting
The collection of SPs is partitionned in well separated clusters by a classical clustering algorithm. Each unknown dipolar source needs 6 real parameters (position, moment). The center of each cluster is considered as the initial position of a source. Etsimation of moments and tuning of the positions is achieved by a final non-linear least squares algorithm.
More about FS3D possibilities.
FS3D has evolved in several directions: time varying data, magnetometer sensors (MEG). It has also been used in non purely spherical context (non constant conductivities).
Time varying data. Independent activities.
True EEG are time varying records of the potential at electrodes. We use a simplified model of the sources. Position and moment direction of a source are assumed fixed, only the moment amplitude is governed by a unknown signal. A group of sources governed by the same signal are called synchronous and form an activity. Another release of FS3D aims to identify asynchronous activities and their time signal.
A SVD factorization is performed up to an invertible matrix, tuned for best dipolar identification.
MEG data. Simultaneous EEG+MEG.
Maxwell’s equations in quasi-static regime also model in a similar manner the magnetic field outside the scalp. A MEG records at given sensors positions one component of the magnetic field. FS3D has been adapted to process MEG data, and simultaneous EEG+MEG data.
Case studies. FreeFEM++ generated data.
Under construction.