Sparsity of Lead Field Matrix

Discussions
1

Hi all,

This is my first comment in this forum, and I am an novice user of Brainstorm software. I was under the impression from several papers, that the lead field matrix should be by nature a sparse and redundant matrix. However, the LFM gain matrix generated using 3- shell sphere method is a full matrix with all elements as non-zero. Am I wrong in the assumption that the LFM would have sparsity and redundancy ? I am sorry, I know, the question is not up to the mark for this forum. But I am desperately in need of understanding. Thank you.

2

Dear Aritra,

You may be confusing the development of the boundary or finite element method (BEM/FEM) solution for the “head model” with the overall calculation of the “forward model,” which we also call the “lead-field model.” Simplifying, in FEM, the volume of the brain is modeled using tetrahedra. Each tetrahedron can be labeled as being in the brain, the skull, or the scalp. The four faces of a tetrahedron are modeled as to their interaction ONLY with the immediately surrounding tetraheda. Thus the FEM matrix can be enormous, but very sparse. But in BEM, only the major boundaries (again, the same skull and scalp) are modeled as triangles, and using boundary equations, each triangle is now modeled with respect to its interaction with ALL other triangles. The BEM matrix is much smaller than the FEM, but generally the matrix is very full.

Both the FEM and BEM matrices are square in the number of elements (triangles or tetrahedra) that describe the shape of the head. Most users never see this intermediate head model calculation.

Once the BEM/FEM solutions are found to the head model, the so-called “lead-field matrix” (LFM) is generated that relates the forward model of each dipole in the source grid (either volume or cortical surface) to each designated sensor location (EEG or MEG). The number of rows in this matrix is equal to the number of channels (sensors), and the number of columns equal to the number of sources. It is generally always full. The term “lead-field” is derived from the rows of the matrix, which are samples of the “field” that a channel (in EEG, a pair of “leads”) is theoretically capable of generating on the sample space. By the Theorem of Reciprocity, this is also the solution to the forward problem.

The 3-shell EEG and spherical MEG head models do not employ a BEM/FEM approach, but rather are based on analytic or approximate formulas that solve directly for the spherical shape. Thus the intermediate “head model” space of the BEM/FEM comprising “elements” is completely bypassed, and we go straight to the generation of the LFM.

For more technical details, I invite you to see our Publications page, http://neuroimage.usc.edu/brainstorm/Pub, in particular

Mosher JC, Leahy RM, Lewis PS (1999)
“EEG and MEG: forward solutions for inverse methods,” IEEE Trans Biomed Eng, 46(3):245-59

Hope this brief discussion helps.

-John

3

Dear Sir,

First of all, Thanks a lot, for your answer. I think I have a basic picture now. So, ideally when we use openMEEG BEM or 3 shell sphere, I wouldn’t even encounter the forward model, and wouldn’t have access to the lead field model. I would be very grateful if you could suggest me any open source for downloading a standard/unique Lead field model, and any way to perform FEM myself, so that I could generate such models. I would also like to ask a question regarding the LFM(matrix; not model) structure. Is it coherent, and should it have a similar distribution of gain values in most vertices and very less spread of said distribution ? I hope I am not being too arrogant, or too inquisitive !! :slight_smile:

4

Dear Sir,

First of all, Thanks a lot, for your answer. I think I have a basic picture now. So, ideally when we use openMEEG BEM or 3 shell sphere, I wouldn’t even encounter the forward model, and wouldn’t have access to the lead field model. I would be very grateful if you could suggest me any open source for downloading a standard/unique Lead field model, and any way to perform FEM myself, so that I could generate such models. I would also like to ask a question regarding the LFM(matrix; not model) structure. Is it coherent, and should it have a similar distribution of gain values in most vertices and very less spread of said distribution ? I hope I am not being too arrogant, or too inquisitive !!
Aritra Chaudhuri

5

My apologies for the confusing terminology, which is not universal and can be synonymous. The “forward model” and the “lead-field model” are synonymous terms here. They model the signal at a sensor generated by a source in the brain. The lead-field matrix (or forward matrix or “gain” matrix) is the discrete representation of the model. It has rows equal to the number of channels (signals recorded), and it has columns equal to the number of dipolar sources. For simplicity, let’s call it here the “forward” model, that tells us how one dipole creates a signal at one sensor.

The forward model itself can be broken into two parts, the “head” model and the “sensor” model. The sensor model is usually relatively basic, describing the integration of flux and weight balancing of the gradiometer coils, for instance, in an MEG channel. The head model is far more complex, relating how the current and potentials of a single current dipole result in secondary volume currents that flow throughout the brain, and result in boundary potentials at the scalp and other major boundaries.

In the case of a spherical head model, we have analytic solutions for the head model, both EEG and MEG, albeit the EEG analytic solution itself is usually solved via analytic approximations. We therefore fill out the forward matrix, point by point, calculating directly how each dipole generates a signal at each sensor. In the Brainstorm software, you’ll see this matrix as the “Gain” matrix in the headmodel structure.

For more general head shapes, we rely first on a discretization of the head into either triangles at the boundaries or tetrahedra throughout the volume. These elements are then coded into a square head model matrix that is either very full (Boundary Elements) or much larger but sparse (Finite Elements), the intent of your original question. The goal is to solve the von Neuman or Dirichlet conditions that prescribe how a current dipole interacts with all of these boundaries.

Our 1999 paper laid out many different methods of solving these equations, and in earlier versions of Brainstorm, we also developed numerical codes to implement them. In recent years, however, we have opted to used Gramfort’s OpenMEEG BEM software to handle the many practical issues of the head model. Brainstorm then uses OpenMEEG’s head model results to again fill out the forward matrix point by point.

See also http://neuroimage.usc.edu/brainstorm/Tutorials/HeadModel, with its References and Additional Documentation sections.

6

Dear Sir,

Thanks a lot… I think I got some things… Can you give me some links for the analytic solutions that are used for EEG ? I would love to see how they work… Thanks again…