EEG forward models: from a source space to a validated leadfield

1. A head surface and an electrode montage

A subdivided icosahedron gives a nearly uniform triangulated sphere. We use it both as the conducting surface for the boundary-element solver and as a place to put electrodes (here, a handful of its vertices).

mesh <- icosphere(subdivisions = 2)      # 320 triangles, 162 vertices
c(vertices = nrow(mesh$vertices), faces = nrow(mesh$faces))
#> vertices    faces 
#>      162      320

electrodes <- mesh$vertices[1:19, ]      # 19 scalp electrodes on the unit sphere

2. A cortical source space

sphericalSourceSpace() lays sources out on an inner shell and returns the outward radial normal at each one - the orientation a surface-constrained forward model assumes.

ss <- sphericalSourceSpace(n_sources = 60, radius = 0.6)
dim(ss$positions)
#> [1] 60  3
dim(ss$normals)
#> [1] 60  3

3. The analytic single-sphere leadfield

For a homogeneous sphere there is a closed-form (Legendre-series) solution. It is fast and exact for this geometry, so it doubles as a reference when checking numerical solvers.

lf <- sphereLeadfield(electrodes, ss$positions)
dim(lf)                                   # 19 electrodes x (60 sources * 3)
#> [1]  19 180

If each source’s orientation is fixed to its surface normal, collapse the three orientation columns into one with constrainOrientation():

lf_fixed <- constrainOrientation(lf, ss$normals)
dim(lf_fixed)                             # 19 electrodes x 60 sources
#> [1] 19 60

4. A boundary-element leadfield, and validating it

bemLeadfield() solves the same problem numerically with a double-layer collocation BEM, which is what you would use for a realistic (non-spherical) head. On a sphere mesh we can score it against the analytic answer with leadfieldRDM(): the relative difference measure (RDM) captures topography error (0 = identical), while the magnitude ratio (MAG) captures amplitude.

few_sources <- rbind(c(0, 0, 0.5), c(0.3, 0, 0.2), c(0, 0.35, 0.1))

bem <- bemLeadfield(mesh, conductivities = c(0.33, 0), electrodes, few_sources)
ana <- sphereLeadfield(electrodes, few_sources)

leadfieldRDM(bem, ana)
#>       RDM       MAG 
#> 0.2172864 0.4273529

The topography (RDM) is already close and improves as the mesh is refined, while the amplitude (MAG) carries the well-known coarse-collocation bias:

rdm_at <- function(sub) {
  m <- icosphere(sub)
  e <- m$vertices
  leadfieldRDM(bemLeadfield(m, c(0.33, 0), e, few_sources),
               sphereLeadfield(e, few_sources))
}
rbind(`icosphere(1)` = rdm_at(1),
      `icosphere(2)` = rdm_at(2))
#>                    RDM       MAG
#> icosphere(1) 0.4335262 0.4546159
#> icosphere(2) 0.2178245 0.4244965

Because source localization is invariant to a global leadfield scale, RDM is the metric to trust for validation; the absolute amplitude bias does not affect the inverse solution.

Where to go next

• Published, precomputed head models (the New York Head, FreeSurfer fsaverage) are fetched on demand with fetchNYHead() / nyHeadLeadfield() and fetchFsaverage(); they are downloaded to headModelCache() rather than shipped, and the New York Head reader needs the hdf5r package.
• The leadfields and source spaces produced here feed EEG signal processing and inverse-analysis workflows in the PhysioEEG package.

Forward-model geometry, coordinate frames, conductivity assumptions, and the reference convention should always be checked against the needs of each study.