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Accuracy of dipole source reconstruction in the 3-layer
BEM model against the 5-layer BEM-FMM model
Guillermo Nuñez Ponasso1∗Ryan C. McSweeney1William A. Wartman1
Peiyao Lai1Jens Haueisen2Burkhard Maess3Thomas R. Knösche3
Konstantin Weise3Gregory M. Noetscher1Tommi Raij4Sergey N. Makaroff1,4
Technical Note — May 2024
Objective: To compare cortical dipole fitting spa-
tial accuracy between the widely used yet highly
simplified 3-layer and modern more realistic 5-
layer BEM-FMM models with and without adap-
tive mesh refinement (AMR) methods.
Methods: We generate simulated noiseless 256-
channel EEG data from 5-layer (7-compartment)
meshes of 15 subjects from the Connectome
Young Adult dataset. For each subject, we test
four dipole positions, three sets of conductiv-
ity values, and two types of head segmentation.
We use the boundary element method (BEM)
with fast multipole method (FMM) acceleration,
with or without (AMR), for forward modeling.
Dipole fitting is carried out with the FieldTrip
MATLAB toolbox.
Results: The average position error (across all
tested dipoles, subjects, and models) is ∼4 mm,
with a standard deviation of ∼2 mm. The orien-
tation error is ∼20◦on average, with a standard
deviation of ∼15°. Without AMR, the numerical
inaccuracies produce a larger disagreement be-
tween the 3- and 5-layer models, with an average
position error of ∼8 mm (6 mm standard devia-
tion), and an orientation error of 28◦(28◦stan-
∗corresponding author: gcnunez@wpi.edu. 1Dept. of Elec-
trical and Computer Engineering, Worcester Polytechnic Insti-
tute, Worcester, MA, USA, 2Technische Universität Ilmenau, Il-
menau, Germany, 3Max Plank Insititute for Human Cognitive
and Brain Sciences, Leipzig, Germany, 4Athinoula A. Martinos
Center for Biomedical Imaging, Massachusetts General Hospi-
tal, Harvard Medical School, Boston, MA, USA
dard deviation).
Conclusions: The low-resolution 3-layer mod-
els provide excellent accuracy in dipole localiza-
tion. On the other hand, dipole orientation is
retrieved less accurately. Therefore, certain ap-
plications may require more realistic models for
practical source reconstruction. AMR is a criti-
cal component for improving the accuracy of for-
ward EEG computations using a high-resolution
5-layer volume conduction model.
Significance: Improving EEG source reconstruc-
tion accuracy is important for several clinical ap-
plications, including epilepsy and other seizure-
inducing conditions.
Index Terms – electroencephalography (EEG),
EEG source analysis, EEG dipole reconstruc-
tion, head modeling, boundary element method
(BEM), fast multipole method (FMM), adapta-
tive mesh refinement (AMR)
I. Introduction
EEG source reconstruction, or source localiza-
tion, consists of locating the neural activity within
the brain from EEG-recorded measurements
(Knösche and Haueisen 2022). Many excellent
open-source software packages implement EEG
source localization, including Brainstorm (Tadel
et al. 2011), FieldTrip (Oostenveld et al. 2011),
1
MNE (Gramfort et al. 2014), and EEGLab (De-
lorme and Makeig 2004).
All four packages include BEM-based dipole
source localization, typically using three layers ex-
tracted from the subject’s MRI: scalp, outer skull,
and inner skull. For most practical applications,
the resolution of these layers is limited to less than
10 000 triangles per layer. One of the main reasons
for this limitation is that these packages use the
classical potential-based formulation of the BEM
(Geselowitz 1967, Kybic et al. 2005, Ponasso 2023),
which involves dense system matrices and as such
is unable to compute lead fields of large high-
resolution models. Complicating this issue is the
fact that the intracortical compartments typically
involve very narrow gaps between layers, which
leads to numerical inaccuracies that typically
need to be resolved with refinement techniques
such as adaptive mesh refinement (AMR), which
is also known as h-refinement (Wartman et al.
2024; Weise et al. 2022), or the less common
p-refinement (Partheymüller, Białecki, and Kuhn
1994), which consists of adaptively increasing the
polynomial order of local approximations of the
variable of interest (potential or charge) on the
mesh triangles.
Yet, current automated human-head segmenta-
tion tools (FreeSurfer, Fischl 2012; SPM12, Nielsen
et al. 2018) can provide us with much higher
resolution skin, skull, cerebrospinal fluid (CSF),
grey matter (GM), and white matter (WM) layers,
among other tissues. Modern charge-based BEM
techniques with FMM acceleration, see for exam-
ple (Makarov et al. 2018), can compute forward
solutions for high-resolution models involving
tens of millions of facets, therefore removing the
∼10 000 triangle limitation of conventional BEM.
How would the source reconstruction results
change if we assume the high-resolution models
are ground truth? We will investigate this for
the canonical EEG problem — single dipole fitting.
There are many studies considering the in-
fluence of different uncertainty factors on EEG
source localization: head model complexity (Vor-
werk, Cho, et al. 2014), conductivity uncertainty
(Vorwerk, Aydin, et al. 2019), and white mat-
ter anisotropy (Güllmar, Haueisen, and Reichen-
bach 2010; Bangera et al. 2010) among others. In
all these studies, the finite element method (FEM)
was used for forward modeling, i.e. for generat-
ing electrode voltages given a dipole source in the
brain. Here, we instead use a high-resolution BEM
model, coupled with FMM acceleration, to com-
pute the forward solution. We do this for the fol-
lowing reasons
(i) Studies of dipole localization accuracy using
BEM (Akalin Acar and Makeig 2013; Fuchs,
Wagner, and Kastner 2007), are relatively less
common than those using FEM. The use of
BEM-FMM allows us to test much higher
mesh resolutions than the ones using conven-
tional BEM.
(ii) BEM models can handle a much larger mesh
resolution than FEM. This is because FEM
requires volumetric meshes, whereas BEM
only requires a triangular mesh for the tis-
sue boundaries. On the other hand, the major
disadvantage of BEM against FEM is that, al-
though it is possible to incorporate anisotropy
to BEM (Olivi, Papadopoulo, and Clerc 2011),
most implementations of BEM lack this fea-
ture.
(iii) Prescribing arbitrary point dipoles — which
are a common mathematical model for the si-
multaneous firing of a large number of neu-
rons (Hämäläinen et al. 1993) — is an easy
task in BEM. On the other hand, modeling
point dipoles in FEM is a more challeng-
ing task: several approaches are available
(Schimpf, Ramon, and Haueisen 2002), but
using them would involve an additional error
estimation— see also the St. Venant approach
used in (Buchner et al. 1997; Vorwerk, Aydin,
et al. 2019).
Our paper’s organization is as follows: Section
II describes the materials and methods used in
2
our numerical experiments, Section III summa-
rizes our results, and Section IV includes a brief
discussion and interpretation. In Supplement A,
we include additional tables of results.
II. Materials and Methods
Outline of methodology. We set up the follow-
ing numerical experiment, which is carried out for
a total of 15 subjects, 3 conductivity sets, 2 mesh
segmentations, and 4 dipole locations:
(i) We place a dipole at a location p0approxi-
mately halfway between the
CSF
-
GM
and
GM
-
WM
tissue interfaces of our subject, and with an
orientation q0normal to the
CSF-GM
interface.
(ii) We simulate a single time sample of noise-
less EEG data using the charge-based formu-
lation of the boundary element method with fast
multipole method acceleration (BEM-FMM) and
adaptive mesh refinement (AMR), over a high-
resolution 5-layer (7-compartment) head vol-
ume conduction model.
(iii) We use the simulated EEG data to perform
source reconstruction with a low-resolution
3-layer head model, similar to the ones of
widespread use in EEG source reconstruction.
Doing so, we find the best fit for location p1,
and orientation q1provided by the FieldTrip
Toolbox’s (Oostenveld et al. 2011) source lo-
calization procedures.
(iv) We compute the distance ∥p0−p1∥2between
both locations, and the angle between q0and
q1, to measure the error of the fit.
7-compartment models. We used the SimNIBS
package (Saturnino et al. 2019) to obtain two
segmentations from the T1- and T2-weighted MRI
data from 15 subjects of the Connectome Young
Adult dataset (Van Essen et al. 2012). One segmen-
tation was obtained using the option
mri2mesh
,
which performs segmentation based on FreeSurfer
(Fischl 2012), and the other using
headreco
,
which is based on SPM/CAT (Nielsen et al. 2018).
The FreeSurfer and Headreco segmentations
generate five main layers: skin, skull, CSF, grey
matter, and white matter.
Each skin, skull, and CSF mesh from the
FreeSurfer segmentations contains approximately
120 000 triangles, whereas the grey and white
matter meshes contain circa 250 000 triangles.
Headreco segmentations comprise approximately
150 000 triangles for skin and skull, 100 000 trian-
gles for CSF, 300 000 triangles for grey matter, and
350 000 triangles for white matter.
Both segmentations include two additional
tissue meshes, in the case of FreeSurfer, these are
the cerebellum and ventricles; for Headreco these
are the eyes and ventricles. One may argue that this
should limit comparability between segmenta-
tions to some degree, however, the location of the
two additional compartments is distant from most
electrode positions. More details on the MRI data
acquisition, spatial coregistration, segmentation,
and mesh generation for this dataset can be found
on (Htet et al. 2019) and (Van Essen et al. 2012).
We tested three tissue conductivity sets for each
subject, which we labeled
IT'IS7
(Hasgall et al.
2022; Gabriel 1996),
VWB7
(Vorwerk, Wolters, and
Baumgarten 2024; Vorwerk, Aydin, et al. 2019),
and
SimNIBS7
(Saturnino et al. 2019). Its values
can be found in Table 1, for an additional reference
on the conductivity of living tissues see (Knösche
and Haueisen 2022).
Each of the chosen conductivity sets is opti-
mized for different scenarios: The
IT'IS7
set
provides conductivity values for frequencies
up to 1 MHz,
SimNIBS7
specifies conductivities
adequate for TMS and TES, see also (Opitz et al.
2015), and (Wagner et al. 2004), and
VWB7
has
been used in several studies on the uncertainty of
EEG source localization.
The only set that provides conductivity values
for all the layers is
IT'IS7
(Hasgall et al. 2022),
for
VWB7
there are no standard cerebellum con-
ductivities given. We chose a 40-60 % weighted
3
VWB7 IT'IS7 SimNIBS7
Skin 0.430 0.147 0.465
Skull 0.010 0.0179 0.010
CSF 1.790 1.880 1.654
GM 0.330 0.419 0.275
WM 0.140 0.348 0.126
Cerebellum 0.216 0.577 0.126
Ventricles 1.790 1.880 1.654
Eyes 1.790 1.880 1.654
Table 1: Conductivity values (S m−1) for the 7-
compartment models.
average between the conductivities of grey matter
and white matter as a conductivity value for the
cerebellum.
IT'IS7
assigns the same conductivity
value as CSF for the eye’s vitreous humor. Hence,
we assign the corresponding CSF conductivity for
the eye layer in each Headreco conductivity set.
We must remark, however, that the conductivity
values of these additional layers are mostly irrel-
evant in the EEG problem since these are located
far away from the scalp electrodes.
3-layer models.
From each FreeSurfer 7-compartment head model,
we created decimated (downsampled) 3-layer
models using the software MeshLab (Cignoni
et al. 2008). Namely, we create three new meshes
labeled
SKIN
,
SKULL
, and
BRAIN
from the high-
resolution skin,skull, and CSF FreeSurfer layers of
each subject.
To obtain these new meshes, we first apply a
screened Poisson surface reconstruction filter
(Kazhdan and Hoppe 2013), followed by a
quadratic edge collapse decimation filter with
14 000 triangles as a target. After this is done, we
apply a Taubin smoothing filter (Taubin 1995).
We used these 3-layer meshes to carry out dipole
source localization using the FieldTrip toolbox
(Oostenveld et al. 2011). In other words, we use
the 3-layer model as our inverse volume conduc-
tion model. For every choice of conductivity for
the forward 5-layer model we have the corre-
VWB3 IT'IS3 SimNIBS3
SKIN 0.430 0.147 0.465
SKULL 0.010 0.0179 0.010
BRAIN 0.330 0.375 0.330
Table 2: Conductivity values (S m−1) for the 3-
layer models.
sponding 3-layer conductivities labeled
IT'IS3
,
VWB3
, and
SimNIBS3
, see Table 2.
Source dipole placement. We tested a total of
four dipole positions for each subject. The dipole
centers were chosen roughly halfway between the
grey and white matter layers, and their orientation
has been taken to be perpendicular to the closest
grey matter region to their centers. The following
are the locations chosen:
(i)
dip1
— Posterior wall of the central sulcus, so-
matosensory cortex, tangential dipole, Figure
1.
(ii)
dip2
— M1HAND region, primary motor cor-
tex, radial dipole, Figure 2
(iii)
dip3
— Temporal lobe, along the Heschl’s gyri
or transverse temporal gyri, Figure 3
(iv)
dip4
— Medio-temporal region, Figure 4
The position
dip3
is at the superior part of the
temporal cortex and has been chosen to study
both auditory and language areas located at the
posterior or anterior part around the same depth.
The position
dip4
is relevant in epileptogenic
analyses (Pittau et al. 2014).
Electrode Placement. We placed 256 electrodes
in each subject’s FreeSurfer, Headreco, and deci-
mated FreeSurfer skin meshes, see Figure 5. The
high-density 256 dry electrode system that we
used is described in (Fiedler et al. 2022), see also
(Graichen et al. 2015). Since our dataset (Htet
et al. 2019) does not include data on the three
anatomical landmarks of nasion,left preauricular
point (LPA), or right preauricular point (RPA),
4
Figure 1: Dipole placement in the transverse, coronal, and sagittal planes, respectively, for
dip1
(pos-
terior wall of the central sulcus) in subject 110411.
Figure 2: Dipole placement in the transverse, coronal, and sagittal planes, respectively, for
dip2
(M1HAND) in subject 130013.
Figure 3: Dipole placement in the transverse, coronal, and sagittal planes, respectively, for
dip3
(tem-
poral lobe, Heschl’s gyrus) in subject 117122.
Figure 4: Dipole placement in the transverse, coronal, and sagittal planes, respectively, for
dip4
(medio-
temporal lobe) in subject 122620.
5
Figure 5: 256-electrode placement on skin layer of
Connectome subject 110411.
we did a semi-manual electrode placement on
each subject’s skin layer using the FieldTrip func-
tion
ft_electrode_placement
, first using the
interactive
option, followed by the
projection
option. In this way, each electrode is assigned to a
triangle in the skin layer. We can then retrieve the
electrode voltages from the BEM-FMM calculation
of the skin potential. This is done by taking the
value of the potential at the mesh triangle of each
electrode.
Forward solution with BEM-FMM We used the
charge-based formulation of the BEM (Gelernter
and Swihart 1964; Barnard, Duck, and Lynn 1967),
together with FMM acceleration (Greengard and
Rokhlin 1987).
In this formulation of the BEM forward problem,
the total electric field consists of two components,
an impressed electric field Ei, and a secondary elec-
tric field Es. The secondary electric field arises
from the surface charge density induced by the pri-
mary field on each surface of discontinuity of the
conductivity. The boundary condition is that the
normal component of the total electric current is
constant across the surfaces of discontinuity. Ap-
plying the boundary condition to the total electric
field E=Ei+Es, one derives the following inte-
gral equation for the surface charge density ρover
the surfaces of discontinuity S
ρ
2ε0
−Kn(r)·ZS
ρ(r′)
4πε0
r−r′
|r−r′|3ds(r′)
=KEi(r)·n(r),
(1)
where ε0is the permittivity of free space, n(r)is
the outward normal vector to Sat r, and K=
(σ−−σ+)/(σ−+σ+)is the conductivity contrast
at r∈S. For more details on the derivation of this
equation, see (Ponasso 2023). For a model with a
total of Mfacets, the discrete version of Equation
(1) at the m-th facet is
ρm
2ε0
−Kmnm·
M
X
n=1; n=m
ρn
4πε0
rm−rn
|rm−rn|3
=KmEi(rm)·nm.
(2)
The summation term is an n-body computation
that we accelerated using the FMM (Rokhlin
1985; Greengard and Rokhlin 1987; Beatson and
Greengard 1997). In our computations, we used
the FMM3D library (Askham et al. 2024).
Adaptive mesh refinement. Adaptive mesh
refinement (AMR) is a general technique that im-
proves the convergence of the discretized solution
to the FMM problem to the true analytic solution
(Feischl et al. 2015). The method of AMR in the
context of BEM-FMM was introduced in (Weise
et al. 2022), see also (Wartman et al. 2024).
AMR consists of subdividing mesh triangles ac-
cording to a cost function until a stopping crite-
rion, based on the convergence of the solution over
a region of interest, is reached. In our case, the
cost function for the m-th triangular facet is Cm=
|ρm| · Am, where ρmis the charge density in the
facet, and Amis its area. In other words, our cost
function is the total charge magnitude in each tri-
angle. The triangles with the top 1% cost over all
triangles are subdivided into four congruent tri-
angles. After this, the surface charges are recom-
puted, and a new AMR step begins. The refine-
ment process is terminated when the change in
6
electrode voltages is less than 1% relative to the
previous AMR step: Let Vn∈R256 be the mea-
sured electrode potentials at the n-th AMR step,
then we stop the iterative procedure at the n-th
step provided that
∥Vn−Vn−1∥2/∥Vn−1∥2<0.01,(3)
where ∥ · ∥2is the 2-norm in R256 . When we
perform AMR, all layers of the model are refined
except for the skin layer, which is our region of
interest and it is used to compare the relative
change from one AMR step to the next. This
method is the one utilized and described in
(Wartman et al. 2024). In this paper, the reader
can find additional details on the AMR method
we used and its integration with BEM-FMM. It
also validates the method’s accuracy for the EEG
dipole forward problem.
EEG source localization For EEG source local-
ization, we used the FieldTrip MATLAB Toolbox
(Oostenveld et al. 2011). We used the functions
ft_prepare_headmodel
with the 3-layer deci-
mated meshes and the conductivity values in
Table 2. A single-dipole source localization (with
free dipole position and orientation) was carried
out using non-linear optimization (Scherg 1990)
via the FieldTrip function
ft_dipolefitting
. The
forward solver used in the dipole fitting procedure
was
bemcp
(Phillips 2001), which implements the
classical surface potential formulation of the
EEG forward problem (Geselowitz 1967; Sarvas
1987). We note that this forward solution does
not include AMR. The reported best fit for the
dipole is the least-squares approximation of the
“true” EEG data by EEG data generated with the
fitted dipole. The accuracy of the dipole fit is thus
measured in terms of residual variance (RV) which
is the variance of EEG data unexplained by the
dipole fit. Therefore, smaller values of residual
variance indicate a better fit.
We carry out two dipole fit strategies, and in ev-
ery case, report the predicted dipole that has the
smallest residual variance. In the first method, we
start a non-linear fit with the true source dipole
position as an initial value for the inverse problem
(Scherg 1990). In the second method, we prepare
a grid with a resolution of 5 mm using the func-
tion
ft_prepare_sourcemodel
, and a grid search
is performed from every grid position within the
skull. The best initial position in the grid is then
used as the initial point of a non-linear fit.
III. Results
Summary of results. When AMR is used, the av-
erage distance error in source localization (across
all models, subjects, and dipole positions we
tested) caused by the discrepancy in charge-based
BEM-FMM and potential-based BEM is ∼1 mm.
To this error, an average of ∼3 mm distance error
is added by incorporating intracortical compart-
ments. Finally, if AMR is not used, an average of
∼4 mm distance error is added due to numerical
inaccuracies. Below, we elaborate on these results.
3-layer model comparison. FieldTrip does not
use the charge-based BEM-FMM in its forward
engine, instead, it uses the classical potential-
based formulation of the BEM (Geselowitz 1967;
Sarvas 1987). Because of this, we made a model
comparison to estimate the baseline accuracy of
FieldTrip’s potential-based 3-layer inverse model
(without AMR) under the assumption that the
forward charge-based BEM-FMM (with and with-
out AMR) solution calculated using the 3-layer
model is ground truth. This is a necessary step
since there is no analytic solution available, and
knowing the discrepancy due to model differences
we will be able to better assess the decrease in
accuracy of dipole fitting due to the inclusion of
additional layers.
First, we calculated the potentials in the entire
SKIN
layer using a forward 3-layer model with
charge-based BEM-FMM and AMR. Then, we re-
trieved readings for the electrode potentials from
this calculation using the potential value at the
SKIN
triangle corresponding to each electrode po-
sition. With these electrode voltages, we generated
a raw EEG data set (compatible with FieldTrip
software) consisting of a single time sample.
7
Finally, we performed EEG source reconstruc-
tion from the simulated EEG using a FieldTrip
3-layer volume conduction model (which does
not use AMR in the forward solution) with the
same conductivity values used in the forward
model. See Figure 6 for a diagrammatic repre-
sentation of the 3-layer model comparison process.
When AMR is used, the FieldTrip inverse model
retrieves the source dipole position with an av-
erage distance error of ∼1 mm (over all dipoles,
subjects, and models), and an average angle
error of 0.75◦. If AMR is not used in the forward
solution, then the average distance error of the
fit is approximately 2.5 mm, with an orientation
error of ∼2◦. Detailed tables with 3-layer model
comparison values can be found in Supplement
A.
5-layer vs 3-layer results. We generated a forward
BEM-FMM solution from each dipole location
and each 5-layer model with AMR and with-
out AMR and assumed that these solutions are
ground truth. Then, we performed two dipole
fits (one for the AMR solution and one for the
non-AMR solution) using the 3-layer model of the
corresponding conductivity as an inverse model
(e.g. when using
IT'IS7
as the conductivity set
for our forward computations, we use
IT'IS3
as
the conductivity set for the inverse computation),
see Figure 7. We measured the distance error and
orientation error in each case.
When AMR is used, we obtain an average distance
error among (across all dipole positions and
subjects) of ∼4 mm (∼1 mm std.) using FreeSurfer
meshes and an average distance error of ∼4 mm
(∼2.5 mm std.) using Headreco meshes. The error
in dipole orientation is more pronounced, with
∼15◦average (∼10◦std.) for FreeSurfer models,
and ∼20◦average (∼20◦std.) for Headreco
models. See Table 3 for dipole-wise AMR results.
On the other hand, if AMR is not used, the forward
solution has a large numerical error which causes
worse dipole fitting results. The average distance
Table 3: AMR results per dipole position for all 5-
layer models. The column AMR denotes the total
number of AMR steps in the forward solution.
Dipole mm deg RV AMR
dip1
4.13 8.71 6.90 ×10−46.17
dip2
4.97 14.32 8.54 ×10−45.65
dip3
2.41 14.44 9.83 ×10−44.58
dip4
5.42 15.08 1.787 ×10−38.26
(a) Averages
Dipole mm deg RV AMR
dip1
1.65 4.70 6.38 ×10−43.17
dip2
1.78 8.73 5.43 ×10−42.09
dip3
1.11 7.23 5.70 ×10−42.49
dip4
3.81 8.30 1.602 ×10−34.56
(b) Standard Deviations
error per dipole is ∼8 mm (∼5.5 mm std.), and the
average orientation error is ∼28◦(∼28◦std.), see
Table 4
For additional tables containing data related to the
3-layer model comparison, and model-wise results
for 3-layer versus 5-layer models with and without
AMR, see Supplement A.
IV - Discussion
Summary of main findings. Dipole fitting with
a low-resolution 3-layer model gives excellent
results with noiseless EEG data simulated using
a BEM-FMM 5-layer model with AMR. The addi-
tional accuracy in dipole fitting that may be gained
by using a forward 5-layer model in the dipole
fitting algorithm would be lost if the numerical
error in the BEM solution calculation cannot be
controlled. This requires additional techniques,
such as the AMR method described in this paper.
Dipole fittings. The average worst dipole location
reconstruction result is that of dipole
dip4
, which
is perhaps to be expected since its location in the
medio-temporal region (Figure 4) is the deepest
8
Figure 6: A flowchart for the 3-layer model comparison process. The blue shade represents geometric
information, the pink shade represents electromagnetic information. Given a single source dipole and
a 3-layer model (consisting of meshes and a tissue conductivity set), we produce two dipole fits: one
corresponding to an AMR forward solution, and one corresponding to a non-AMR one.
Figure 7: A flowchart for the performance study of 3-layer inverse models using 5-layer forward models.
Given a single source dipole and a 5-layer model, we produce two dipole fits using an inverse 3-layer
model: one corresponding to an AMR forward solution, and one corresponding to a non-AMR one.
9
Table 4: Non-AMR results per dipole for all 5-layer
models
Dipole mm deg RV
dip1
9.47 24.84 6.129 ×10−3
dip2
6.88 19.63 2.544 ×10−3
dip3
3.97 18.69 2.668 ×10−3
dip4
10.04 50.04 6.500 ×10−3
(a) Averages
Dipole mm deg RV
dip1
8.50 17.92 5.543 ×10−3
dip2
4.60 30.58 2.612 ×10−3
dip3
2.94 11.99 3.004 ×10−3
dip4
5.84 50.73 5.788 ×10−3
(b) Standard deviations
of all those we considered. More surprisingly,
the position of
dip2
(a radial dipole located
in the M1HAND region) was the one that gave
the second-to-worst results. The reason may be
that the 3-layer models do not include CSF, and
the closer proximity of the dipole to the highly
conductive CSF layer creates a larger discrepancy
between the 5- and 3-layer models. However, one
should not over-interpret these results, as more
tests for radial dipoles with varying thicknesses
of neighboring CSF layers may be needed before a
conclusion can be made.
Advantages and shortcomings of the 3-layer
model. The residual variances we obtain (circa
0.0015 on average) are extremely small, which
indicates that the nonlinear dipole fitting using
the 3-layer model explains most of the EEG
data variance with the predicted positions. This
indicates that these are good-quality fits. In
particular, when we use AMR, the discrepancies
to the original dipole positions and orientations
are only caused by the difference in model com-
plexity (5-layers vs 3-layers). If AMR is not used,
then the errors are due to the combined effect of
model-complexity difference and the numerical
error caused by the short distance between layers,
or the proximity of the dipole singularity to the
triangle meshes.
Given this, we find that the 3-layer model retrieves
our chosen dipole positions with a very reason-
able accuracy of approximately 1-5 mm. However,
the reconstruction of the dipole orientation is not
as precise. Since the particular dipole location
arises as a "center of mass" from the region of
activity, the orientation may be more useful for
determining the precise location of activity in the
cortex in practical scenarios, see (Ebersole and
Hawes-Ebersole 2007).
We conclude that the 3-layer model remains a
very powerful tool for source localization, but
we advise that end users of dipole localization
software exercise caution when interpreting the
results. The effects of noise and the possibility
of multiple dipoles being the source of the signal
were not explored in the present study.
Advantages and challenges of the 5-layer model
Inclusion of the grey and white matter tissues is a
major factor in the variation of dipole orientation,
see Figure 8, this is in agreement with the findings
in (Vorwerk, Aydin, et al. 2019).
Model-wise, the conductivity set that produces
the best agreement between the 3-layer and 5-
layer models is
IT'IS
. A possible explanation is
that this set has the lowest conductivity contrast
between grey matter and CSF. We also observe
(see Supplement A) that the dipole fits for the
forward data generated with Headreco models
have higher variability than those fits using the
data from FreeSurfer models (both in terms of
location and orientation). The use of AMR should
eliminate the discrepancies caused by the differ-
ent number of triangles in the initial layers, so a
possible explanation for the different variabilities
is that Headreco meshes use a more realistic skull
layer than the FreeSurfer meshes. However, a for-
mal assessment of the effect of mesh uncertainty
would require a more detailed analysis using
techniques similar to those in (Vorwerk, Aydin, et
10
al. 2019; Vorwerk, Wolters, and Baumgarten 2024).
The results of Table 3 contrasted to Table 4 indicate
that it is necessary to incorporate some form of
AMR to have accurate forward computations and,
in turn, an accurate dipole fitting based on 5-layer
BEM models. Certain dipole positions require
some layers to be refined up to 12 times before we
can reduce the electrode error sufficiently. For this
reason, a practical implementation of dipole fitting
using BEM-FMM would require an improvement
in AMR speed. This is in contrast to 3-layer
models: the lack of intracortical compartments
implies that the dipolar sources will be typically
well-separated from the model meshes, and thus
the error caused by the lack of AMR is much lower.
Modeling EEG noise can be achieved by placing
dipoles at random positions, and with random
strengths, all over the space between the
CSF-GM
and
GM-WM
interfaces. This task can be easily
achieved with FMM-LU (Sushnikova et al. 2023)
in the 3-layer model. However, FMM-LU is in-
compatible with AMR, and severe numerical inac-
curacies can arise in the 5-layer model due to the
proximity of the dipolar sources to the mesh trian-
gles of the white and grey matter layers. Previous
studies have suggested that the addition of noise
mainly influences the source localization accuracy
of deep sources (Whittingstall et al. 2003)
Acknowledgements
The authors would like to thank Prof. Dr.
Carsten Wolters for his help regarding conduc-
tivity values for the 3-layer model. GNP, RMcS,
WAW, PL, GMN, and SNM were supported by
the NIBIB grant R01EB035484, and NIMH grant
R01MH130490. TR was partially supported by the
NINDS grant 1R01NS126337. KW was partially
supported by the BMBF grant: 01GQ2201. JH re-
ceived funding from the German Federal Ministry
of Education and Research (BMBF) grant Dry-
Pole (01GQ2304A) and the Free State of Thuringia
(2018 IZN 004), co-financed by the European
Union under the European Regional Development
Fund (ERDF).
References
Akalin Acar, Zeynep and Scott Makeig (2013). “Ef-
fects of Forward Model Errors on EEG Source
Localization”. In: Brain Topogr. 26. doi:
10.1007/
s10548-012-0274-6
.
Askham, Travis et al. (2024). FMM3D: A fast
multipole method library for three-dimensional
problems.url:
https : / / github . com /
flatironinstitute/FMM3D
.
Bangera, Nitin B. et al. (2010). “Experimental
validation of the influence of white matter
anisotropy on the intracranial EEG forward so-
lution”. In: J. Comput. Neurosci. 29, pp. 371–387.
doi:
10.1007/s10827-009-0205-z
.
Barnard, A.C., I.M Duck, and M.S. Lynn (1967).
“The application of electromagnetic theory to
electrocardiology. I. Derivation of the integral
equations”. In: Biophys. J. 7, pp. 443–462. doi:
10.
1016/S0006-3495(67)86598-6
.
Beatson, Rick and Leslie Greengard (1997). A short
course on fast multipole methods.url:
https:/ /
math.nyu.edu/~greengar/shortcourse_fmm.
pdf
.
Buchner, Helmut et al. (1997). “Inverse localiza-
tion of electric dipole current sources in finite el-
ement models of the human head”. In: Electroen-
cephalogr. clin. neurophysiol. 102, pp. 267–278. doi:
10.1016/S0013-4694(96)95698-9
.
Cignoni, Paolo et al. (2008). “MeshLab: an
Open-Source Mesh Processing Tool”. In: Eu-
rographics Italian Chapter Conference. Ed. by
Vittorio Scarano, Rosario De Chiara, and
Ugo Erra. The Eurographics Association. doi:
10 . 2312 / LocalChapterEvents / ItalChap /
ItalianChapConf2008/129-136
.
Delorme, Alain and Scott Makeig (2004).
“EEGLAB: an open source toolbox for analysis
of single-trial EEG dynamics including indepen-
dent component analysis”. In: J. Neurosci. Meth-
ods. doi:
10.1016/j.jneumeth.2003.10.009
.
Ebersole, John S. and Susan Hawes-Ebersole
(2007). “Clinical Application of Dipole Models
in the Localization of Epileptiform Activity”. In:
J. Clin. Neurophysiol. 24. doi:
10 . 1097 / WNP .
0b013e31803ece13
.
11
Potential (mV) in transverse plane
-40 -20 0 20
Distance x, mm
-50
-40
-30
-20
-10
0
10
20
Distance y, mm
-2.0
-0.82
-0.33
-0.13
0.041
-0.005
0.02
0.08
0.21
0.53
-40 -20 0 20
Distance x, mm
10
20
30
40
50
60
70
80
Distance z, mm
-0.72
-0.30
-0.12
-0.046
-0.015
-0.00
0.008
0.03
0.08
0.2
0.5
Potential (mV) in coronal plane
-40 -20 0 20
Distance y, mm
10
20
30
40
50
60
70
80
Distance z, mm
-1.00
-0.41
-0.17
-0.064
-0.02
-0.002
0.01
0.04
0.11
0.28
Potential (mV) in sagittal plane
Figure 8: Volumetric plots in logarithmic scale of the electric potential (µV) around the source dipole
dip1
in subject 110411 —transverse, coronal, and sagittal views. Observe the symmetry break with
respect to the axes of the dipole already happening in the intracortical compartments.
12
Feischl, Michael et al. (2015). “Adaptive Boundary
Element Methods”. In: Arch. Computat. Methods
Eng. 22. doi:
10.1007/s11831-014-9114-z
.
Fiedler, Patrique et al. (2022). “A high-density 256-
channel cap for dry electroencephalography”.
In: Human Brain Mapp. 43. doi:
10.1002/ hbm.
25721
.
Fischl, B. (2012). “FreeSurfer”. In: NeuroImage 62.2,
pp. 774–781. doi:
/10 . 1016 / j . neuroimage .
2012.01.021
.
Fuchs, Manfred, Michael Wagner, and Joern Kast-
ner (2007). “Development of Volume Conductor
and Source Models to Localize Epileptic Foci”.
In: J. Clin. Neurophysiol. 24. doi:
10.1097/WNP.
0b013e318038fb3e
.
Gabriel, Camelia (1996). “Compilation of the Di-
electric Properties of Body Tissues at RF and Mi-
crowave Frequencies.” In: N.AL/OE-TR- 1996-
0037, Occupational and environmental health
directorate, Radiofrequency Radiation Division,
Brooks Air Force Base, Texas (USA).
Gelernter, H.L. and J.C. Swihart (1964). “A
Mathematical-Physical Model of the Genesis of
the Electrocardiogram”. In: Biophys. J 4, pp. 285–
301. doi:
10.1016/s0006-3495(64)86783-7
.
Geselowitz, David B. (Jan. 1967). “On Bioelectric
Potentials in an Inhomogeneous Volume Con-
ductor”. In: Biophys J. 7.1, pp. 1–11. doi:
10 .
1016/S0006-3495(67)86571-8
.
Graichen, Uwe et al. (2015). “SPHARA - A Gener-
alized Spatial Fourier Analysis for Multi-Sensor
Systems with Non-Uniformly Arranged Sen-
sors: Application to EEG”. In: PLOS ONE.doi:
10.1371/journal.pone.0121741
.
Gramfort, A. et al. (Feb. 2014). “MNE software for
processing MEG and EEG data”. In: NeuroImage
86, pp. 446–460. doi:
10.1016/j.neuroimage.
2013.10.027
.
Greengard, Leslie and Jr. Rokhlin Vladimir (1987).
“A fast algorithm for particle simulations”. In: J.
Comp. Phys. 73, pp. 325–348. doi:
10.1016/0021-
9991(87)90140-9
.
Güllmar, Daniel, Jens Haueisen, and Jürgen R. Re-
ichenbach (2010). “Source analysis of EEG data
is an important tool in scientific and clinical ap-
plications. This is the first study on EEG source
analysis using the BEM. BEM can utilize true
point-dipole sources, as opposed to past stud-
ies using FEM, which can only use approxi-
mate point-dipole sources”. In: NeuroImage 51,
pp. 145–163. doi:
10 . 1016 / j . neuroimage .
2010.02.014
.
Hämäläinen, M. et al. (Apr. 1993). “Magnetoen-
cephalography—theory, instrumentation, and
applications to noninvasive studies of the work-
ing human brain”. In: Rev. Mod. Phys. 65,
pp. 413–497. doi:
10.1103/RevModPhys.65.413
.
Hasgall, P.A. et al. (Feb. 2022). IT’IS Database for
thermal and electromagnetic parameters of biological
tissues. Version 4.1. doi:
10. 13099/VIP21000 -
04-1
.
Htet, Aung Thu et al. (2019). “Collection of CAD
human head models for electromagnetic simu-
lations and their applications”. In: Biomed. Phys.
Eng. Express 5. doi:
10 . 1088 / 2057 - 1976 /
ab4c76
.
Kazhdan, Michael and Hugues Hoppe (2013).
“Screened Poisson surface reconstruction”. In:
ACM Transactions on Graphics (TOG) 32.3, p. 29.
Knösche, Thomas R. and Jens Haueisen (2022).
EEG/MEG Source Reconstruction. Textbook for
Electro-and Magnetoencephalography. Springer
Cham. isbn: 978-3-030-74916-3. doi:
10 . 1007 /
978-3-030-74918-7
.
Kybic, J. et al. (2005). “A common formalism for
the Integral formulations of the forward EEG
problem”. In: IEEE Transactions on Medical Imag-
ing 24.1, pp. 12–28. doi:
10. 1109/ TMI.2004.
837363
.
Makarov, Sergey N. et al. (2018). “A Quasi-Static
Boundary Element Approach With Fast Multi-
pole Acceleration for High-Resolution Bioelec-
tromagnetic Models”. In: IEEE. Trans. Biomed.
Eng. 65, pp. 2675–2683. doi:
10 . 1109 / TBME .
2018.2813261
.
Nielsen, Jesper D. et al. (2018). “Automatic skull
segmentation from MR images for realistic vol-
ume conductor models of the head: Assess-
ment of the state-of-the-art”. In: NeuroImage 174,
pp. 587–598. issn: 1053-8119. doi:
https://doi.
org/10.1016/j.neuroimage.2018.03.001
.
13
Olivi, Emmanuel, Théodore Papadopoulo, and
Maureen Clerc (2011). “Handling white-matter
anisotropy in BEM for the EEG forward prob-
lem”. In: 2011 IEEE International Symposium on
Biomedical Imaging: From Nano to Macro.doi:
10.
1109/ISBI.2011.5872526
.
Oostenveld, R. et al. (2011). “FieldTrip: Open
Source Software for Advanced Analysis of MEG,
EEG, and Invasive Electrophysiological Data”.
In: Comput. Intell. Neurosci. 2011. Article ID
156869. doi:
10.1155/2011/156869
.
Opitz, Alexander et al. (2015). “Determinants of
the electric field during transcranial direct cur-
rent stimulation”. In: NeuroImage 109, pp. 140–
150. doi:
https : / / doi . org / 10 . 1016 / j .
neuroimage.2015.01.033
.
Partheymüller, P., R.A. Białecki, and G. Kuhn
(1994). “Self-adapting algorithm for evalua-
tion of weakly singular integrals arising in
the boundary element method”. In: Engineering
Analysis with Boundary Elements 14.3. doi:
https:
//doi.org/10.1016/0955-7997(94)90044-2
.
Phillips, Christophe (2001). “Source localisation in
EEG: Combining anatomical and functional con-
straints”. PhD thesis. Université de Liège. url:
http://www.fil.ion.ucl.ac.uk/spm/doc/
theses/chrisp/localis.pdf
.
Pittau, Francesca et al. (2014). “The Role of Func-
tional Neuroimaging in Pre-Surgical Epilepsy
Evaluation”. In: Front. Neurol. 5. doi:
10.3389/
fneur.2014.00031
.
Ponasso, G. N. (Dec. 2023). “A survey on integral
equations for bioelectric modeling”. preprint.
url:
https://hal.science/hal-04343472
.
Rokhlin, V (1985). “Rapid solution of integral
equations of classical potential theory”. In: J.
Comput. Phys. 60. doi:
https:/ /doi.org /10.
1016/0021-9991(85)90002-6
.
Sarvas, Jukka (Jan. 1987). “Basic mathematical
and electromagnetic concepts of the biomag-
netic inverse problem”. In: Phys. Med. Biol. 32.
doi:
10.1088/0031-9155/32/1/004
.
Saturnino, G.B. et al. (2019). “SimNIBS 2.1:
A Comprehensive Pipeline for Individualized
Electric Field Modelling for Transcranial Brain
Stimulation ”. In: Brain and Human Body
Modeling: Computational Human Modeling at
EMBC 2018. Cham (CH): Springer. Chap. 1. doi:
10.1007/978-3-030-21293-3_1
.
Scherg, Michael (1990). “Fundamentals of dipole
source potential analysis”. In: Adv. Audiol. 6.40-
69, p. 25.
Schimpf, P.H., C. Ramon, and J. Haueisen (2002).
“Dipole models for the EEG and MEG”. In: IEEE
Trans. Biomed. Eng. 49. doi:
10.1109/10.995679
.
Sushnikova, Daria et al. (2023). “FMM-LU: A Fast
Direct Solver for Multiscale Boundary Integral
Equations in Three Dimensions”. In: Multiscale
Model. Simul. doi:
10.1137/22M1514040
.
Tadel, F. et al. (2011). “Brainstorm: a user-friendly
application for MEG/EEG analysis ”. In: Comput.
Intell. Neurosci. doi:
10.1155/2011/879716
.
Taubin, Gabriel (1995). “Curve and surface
smoothing without shrinkage”. In: Proceedings of
IEEE International Conference on Computer Vision,
pp. 852–857. doi:
10.1109/ICCV.1995.466848
.
Van Essen, D.C. et al. (2012). “The Human Con-
nectome Project: A data acquisition perspec-
tive”. In: NeuroImage 62.4, pp. 2222–2231. doi:
10.1016/j.neuroimage.2012.02.018
.
Vorwerk, J., J-H. Cho, et al. (Oct. 2014). “A guide-
line for head volume conductor modeling in
EEG and MEG ”. In: Neuroimage.doi:
10.1016/
j.neuroimage.2014.06.040
.
Vorwerk, Johannes, Ümit Aydin, et al. (2019). “In-
fluence of Head Tissue Conductivity Uncertain-
ties on EEG Dipole Reconstruction”. In: Front.
Neurosci. 13. doi:
10.3389/fnins.2019.00531
.
Vorwerk, Johannes, Carsten H. Wolters, and
Daniel Baumgarten (2024). “Global sensitivity
of EEG source analysis to tissue conductivity un-
certainties”. In: Front. Hum. Neurosci. 18. doi:
10.
3389/fnhum.2024.1335212
.
Wagner, T.A. et al. (2004). “Three-dimensional
head model Simulation of transcranial magnetic
stimulation”. In: IEEE Transact. Biomed. Eng. 51.9,
pp. 1586–1598. doi:
10 . 1109 / TBME . 2004 .
827925
.
Wartman, William A et al. (2024). “An adaptive
h-refinement method for the boundary element
fast multipole method for quasi-static electro-
14
magnetic modeling”. In: Phys. Med. Biol. 69,
p. 055030. doi:
10.1088/1361-6560/ad2638
.
Weise, Konstantin et al. (2022). “The effect of
meninges on the electric fields in TES and TMS.
Numerical modeling with adaptive mesh refine-
ment”. In: Brain Stimul. 15, pp. 654–663. doi:
10.
1016/j.brs.2022.04.009
.
Whittingstall, Kevin et al. (2003). “Effects of dipole
position, orientation and noise on the accuracy
of EEG source localization”. In: Biomed. Eng. On-
line 2. doi:
10.1186/1475-925X-2-14
.
15