Analytic expansion of the EEG lead field for realistic volume conductors.

INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 50 (2005) 3807–3823 doi:10.1088/0031-9155/50/16/010
Analytic expansion of the EEG lead field for realistic
volume conductors
Guido Nolte1,2 and George Dassios3
1 Human Motor Control Section, NINDS, NIH, Bethesda, MD, USA
2 Fraunhofer Gesellschaft First, Berlin, Germany
3 Division of Applied Mathematics, Department of Chemical Engineering,
University of Patras and ICEHT/FORTH, Greece
E-mail: nolte@first.fhg.de and gdassios@chemeng.upatras.gr
Received 3 March 2005, in final form 3 May 2005
Published 28 July 2005
Online at stacks.iop.org/PMB/50/3807
Abstract
EEG forward calculation in realistic volume conductors using the boundary
element method suffers from the fact that the solutions become inaccurate for
superficial sources. Here we propose to correct an analytical approximation
of the respective lead fields with series of spherical harmonics with respect to
multiple expansion points. The necessary correction depends very much on
the chosen analytical approximation. We constructed the latter such that the
correction can be modelled adequately within the chosen basis. Simulations
for a 3-shell prolate spheroid demonstrate the accurate modelling of the lead
fields. Explicit comparison with analytically known solutions was done for the
3-shell spherical volume conductor showing that relative errors are mostly far
below 1% even for the most superficial sources placed directly on the innermost
surface.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
In electroencephalography (EEG), electric potentials are measured on a scalp of a subject and
are interpreted as a reflection of neural activity inside the brain. To properly localize the neural
centres of activity, the physical relationship between neural source and induced potential on the
head surface must be known. The relevant equations to solve depend on the electric properties
of the various tissues within the head which are usually termed ‘volume conductor’ (Sarvas
1987). Analytic solutions to these equations exist only for special volume conductors among
which a set of N concentric spheres, each with constant and scalar conductivity, is the most
prominent (Frank 1952, Geselowitz 1967). Analytical solutions also exist for slightly more
complicated volume conductors, namely the prolate or oblate spheroid (de Munck 1988), the
0031-9155/05/163807+17$30.00 © 2005 IOP Publishing Ltd Printed in the UK 3807

3808 G Nolte and G Dassios
general ellipsoid (Kariotou 2004) and the three confocal ellipsoidal shells model (Giapalaki
and Kariotou 2005).
For volume conductors of realistic shape, the problem can only be solved numerically.
Assuming that the volume conductor is piecewise homogeneous and isotropic, i.e., it consists
of compartments like brain, skull and scalp each with constant and scalar conductivities, the
most prominent technique is the boundary element method (BEM). In BEM, the relevant
differential equations are transformed to integral equations defined only on the surfaces of
the compartments (Kybic et al 2005, Goncalves et al 2003, Tissari and Rahola 2003, Fuchs
et al 1998, Frijns et al 2000, Mosher et al 1999, de Munck 1992). For numerical solutions,
the surfaces have to be triangulated and the respective grid-size sets an intrinsic limitation on
the accuracy: solutions become inaccurate if the distance of the neuronal activity to the surface
is of the order of the grid-size. Since the most important neural sources for EEG arise in the
cortex, which almost touches the skull, BEM solutions might be insufficiently accurate for a
relevant class of possible source locations.
The EEG forward problem can be formulated in two different ways. Most commonly,
one directly solves the equations for the electric potential induced by a current dipole. In the
‘reciprocal’ approach, one solves the equations for the electric ‘lead fields’. A lead field is
associated with a pair of electrodes and indicates how sensitive the difference of the electric
potentials at these two electrodes is to dipolar sources at a specific location (Malmivuo and
Plonsey 1995). An advantage of the reciprocal approach is that the singularities occur at fixed
locations, the electrodes, rather than at variable source locations as is the case for the direct
approach. The equations to solve are of the same type as for the direct approach and can also
be handled, e.g., with BEM (Riera 1998, Fletcher 1995), or with the finite element method
(Laarne et al 2000, Weinstein et al 2000), leading, however, to the same problems for very
superficial sources.
Here we propose an analytic expansion of the lead fields with respect to a suitable basis
such that the lead field is well behaved within the entire volume conductor. The principles
of this approach were already presented for the magnetic case (Nolte 2003), which does not
contain any singularities, because magnetic sensors are not placed directly on the volume
conductor, and which is hence much simpler than the electric case. An additional difficulty
arises because for EEG forward calculations, in contrast to the MEG case, the outer shells
must be included in modelling the volume conductor.
The paper is organized as follows. In section 2.1, we derive the relevant equations for
the lead fields. While these equations are well known, the presented derivation is, to our
knowledge, new. In section 2.2, we sketch the principles of our approach by recalling the
method for the magnetic case. The central theoretical part of this paper is given in section 2.3
where we analyse the singularities in detail and present our solution to it. Most of the
mathematical details for this solutions are found in the appendices. In section 2.4, we present
the full algorithm. This section deliberately contains some minor redundancies with the
preceding one in order to be understandable on its own. Numerical examples are given in
section 3, and our results are finally summarized and discussed in section 4.
2. Theory
2.1. The electric lead field
We are interested in the electric potential V (r1) measured at location r1 and induced by an
‘impressed current’ JI (r). An overview of the relevant physical laws is, e.g., given by Sarvas
(1987). The impressed current is always accompanied by a volume current JV such that the

Expansion of the EEG lead field 3809
total current
J = JI + JV (1)
is divergence free
∇ · J = 0. (2)
The volume current is assumed to be ohmic
JV = −σ∇V (3)
where V is the electric potential and σ is, in general, a space-dependent conductivity tensor.
For each pair of electrodes at locations r1 and r2, we can associate a lead field L(r) such
that
V (r1) − V (r2) =
∫
d3r L(r) · JI (r) (4)
for any impressed current.
Most commonly, the equation for the lead field is derived using Helmholtz’ reciprocity
theorems (Malmivuo and Plonsey 1995). Analogous to the derivation for the magnetic case
presented in Nolte (2003), here we present an alternative way which we believe is much
simpler.
We first note that closed loop impressed currents do not induce an electric potential and
hence, with Stoke’s theorem, the electric lead field must be curl-free and can be written as
L = −∇U (5)
where U is called the lead potential. Second, from (1)–(3) it follows that:
∇ · JI = ∇ · σ∇V. (6)
We will now manipulate the right-hand side of (4) as follows: integrate over the whole volume
and assume that all fields vanish sufficiently fast at infinity so that boundary terms of partial
integrations disappear4.
Then
∫
L · JI = −
∫
∇U · JI =
∫
U∇ · JI =
∫
U∇ · σ∇V =
∫
(∇ · σT ∇U)V (7)
where σT is the transpose of σ . Rewriting the left-hand side of (4) as
V (r1) − V (r2) =
∫
d3r(δ(r − r1) − δ(r − r2))V (r) (8)
and equating this with (7) leads to
∫
(∇ · σT ∇U − (δ(r − r1) − δ(r − r2)))V = 0. (9)
This is fulfilled for any V if we choose
∇ · σT ∇U(r) = δ(r − r1) − δ(r − r2). (10)
This equation determines the lead field completely. It corresponds to the electric field due to
a unit current inserted at the reference and extracted at the electrode.
For a piecewise homogeneous and isotropic volume conductor, one can derive the usual
boundary conditions: the tangential components of L are continuous at a boundary (because L
is curl-free), and the normal component of σL is continuous (because σL is divergence-free).
4 Completely analogous to the magnetic case (Nolte 2003) one can also derive the same equations in a more
complicated way using integrals only over the volume conductor avoiding ill-defined equations in regions outside the
volume conductor. It is also justified to think of the outside of the volume conductor as having a tiny but nonvanishing
conductivity considering the limit of vanishing conductivity at the end of the calculations.

3810 G Nolte and G Dassios
2.2. A quasi-analytic solution for the magnetic lead field
Our proposed concept of solving the electric lead field equation is best illustrated with the
relatively simple magnetic case (Nolte 2003). For a one-compartment volume conductor, one
can show that the magnetic lead field LM can be written as
LM = LS − ∇UM (11)
where LS is the lead field for a spherical volume conductor (which is known analytically in
closed form) and UM is a harmonic function such that the normal component of LM vanishes
at the surface.
For an approximate solution one can expand UM as
UM(r) =
K
∑
k=1
akU
M
k (r) (12)
where
(
UMk
)
is a set of fixed harmonic functions corresponding to the real and imaginary parts
of
rpYpq(θ, φ) (13)
where Ypq are spherical harmonics with coordinates originating in the centre of the volume
conductor.
For a given set of surface points ri and normals ni for i = 1, . . . , N , one can determine
the unknown coefficients ak for each magnetic sensor by minimizing the error in the boundary
condition
∑
i
(ni ·LM(ri ))2 (14)
which leads to a linear equation for the unknown coefficients ak .
For the typical MEG forward problem, one considers only the inner shell of the volume
conductor. Since MEG sensors are relatively far away from the volume conductor, the
correction to the spherical volume conductor, expressed by UM , is smooth, provided that
the volume conductor itself is fairly smooth. An expansion up to p � 10 in (13) leads to
highly accurate solutions within the entire volume conductor including the boundary itself.
Furthermore, the coefficients ak , which need to be calculated only once, can be computed
within seconds, and the actual forward calculation for each dipole is only about four times
slower than that for the spherical volume conductor.
2.3. Problem formulation for the electric case
While the concept for solving the lead field equation is in principle applicable both to the
electric and the magnetic case, the electric case introduces severe additional practical problems.
First, the electrodes are placed directly on the volume conductor and the equation to solve
becomes singular. Second, the outer shells (typically skull and scalp) have to be included in
the modelling. This second problem is more difficult in conjunction with the first: since the
electrodes are placed on the scalp, the lead field on the skull in the vicinity of the electrode is
not singular but still highly ‘non-smooth’.
To fully understand the first problem, the singularity of the lead field at the electrodes, we
will at first consider only a single-shell volume conductor. The general lead potential can be
written as
U = U0 + Uc (15)

Expansion of the EEG lead field 3811
where U0 is a special (singular) solution of (10) but in general violating the boundary
conditions, and Uc is a (hopefully regular) correction which fulfils the harmonic equation
(i.e., right-hand side of (10) set to zero) and is chosen such that the total U fulfils the correct
boundary conditions. In the MEG analogy, the field U0 corresponds, mutatis mutandis, to the
spherical solution and Uc to the expanded correction to it.
It is now tempting to set U0 to the corresponding solution of a spherical volume conductor.
The problem is that, in general, any spherical solution leaves an error in the boundary condition
which is still singular and cannot be corrected with a regular Uc. To see this, we work within
a coordinate system such that the electrode is in its centre and the surface normal at the
electrode points to the z-axis. (It is implicitly understood that the reference electrode is treated
likewise.) In this coordinate system, one has for the spherical volume conductor with radius
R and constant conductivity σ0
2πσ0U0(r) = −
1
r
+
1
2R
ln(r − z). (16)
It is straightforward to check that the lead field
2πσ0L0(r) = −2πσ0∇U0(r) = −
r
r3
−
1
2R
r/r − (0, 0, 1)T
r − z
(17)
has a constant normal component on the surface which cancels with the same constant coming
from the reference electrode.
To derive this we temporarily return to a coordinate system placed in the centre of the
volume conductor. Denoting by R the coordinate of the electrode the normal component of
the lead field (scaled with 2πσ0) reads on the surface (r = R)
r
r
· ∇
(
−
1
|r − R|
+
1
2R
ln
(
|r − R| −
(r − R) ·R
R
))
=
1
2R
. (18)
Returning to the coordinate system having the electrode in its centre U0 fulfils the Laplace
equation except at r = 0 for the first term on the right-hand side of (16) and on the line
x = y = 0 with z � 0 for the second term. The two singularities of L0 at r = 0 potentially
correspond to sources added to the system. Note that the singularity along the line z > 0 does
not add sources because this line is completely outside the volume conductor. The quadratic
singularity in L0 arising from the 1/r-term in U0 properly describes the source (i.e., the delta
function at the origin) because
−
∫
B(�)
dV ∇ · σ(r)∇1/r =
∫
∂B(�)
dn · σ(r)r/r3 =
1
�2
∫
∂B(�)
|dn|σ(r) �→0−→ 2πσ0 (19)
for integrals over a ball of radius � around the origin. Note that in the limit � → 0 only half
of the ball is inside the volume conductor.
The linear singularity arising from the ln(r − z)-term adjusts the boundary conditions
without ‘spoiling’ the source because the divergence is too weak, i.e.,
∣
∣
∣
∣
∫
B(�)
dV ∇ · σ(r)∇f (r)
∣
∣
∣
∣
=
∣
∣
∣
∣
∫
∂B(�)
dn · σ(r)∇f (r)
∣
∣
∣
∣
� 4π�2 max
r∈∂B(�)
(|σ(r)∇f (r)|)
�→0
−→ 0
(20)
for any f (r) such that ∇f (r) ∼ 1/r .
Let us now write an arbitrary surface after an appropriate rotation in the x–y plane most
generally as
z = g(x, y) = ax2 + by2 + h.o. (21)

3812 G Nolte and G Dassios
where h.o. denotes higher order terms in x and y. The zeroth-order term vanishes because the
origin of the coordinate system is on the surface and the first-order terms vanish because we
assume that the surface normal points into the z-direction. Since z − g(x, y) is constant on
the surface, the unit surface normal is given by
n(r) =
∇(z − g(x, y))
|
∇(z − g(x, y))|
. (22)
For the surface given in (21), the quadratically singular term in the spherical lead field
leads to a linearly singular normal component of the lead field
−
r
r3
·
∇(z − g(x, y))
|
∇(z − g(x, y))|
=
a¯ + δ cos(2φ)
r
+ O(r0) (23)
with a¯ = (a + b)/2 and δ = (a − b)/2. The parameters a¯ and δ correspond to local axial
symmetric and ‘antisymmetric’ characteristics of the surface, respectively. For example, a
sphere of radius R has a¯ = −1/2R and δ = 0 while a ‘perfect’ saddle point has a¯ = 0 and
δ �= 0. For a plane one has both a¯ = 0 and δ = 0.
For a spherical volume conductor, this singularity is cancelled by the singularity in the
second term of (17).
−
1
2R
r/r − (0, 0, 1)T
r − z
· (0, 0, 1)T + O(r0) =
1
2Rr
+ O(r0) = −
a¯
r
+ O(r0). (24)
Note that we only include singular terms in this analysis. Therefore, it is sufficient to
approximate the surface normal in the last equation by the leading (zeroth order) term (0, 0, 1)T
because this part of the lead field is only linearly divergent. In contrast, in (23) the surface
normal has to be approximated up to first order as (−2ax,−2by, 1)T because that part of the
lead field is quadratically divergent.
The problem is now to find a more general solution to the Laplace equation which includes
surfaces which are not axially symmetric (δ �= 0) with respect to any axis which corresponds
to a surface normal. In principle, such a solution is given by the solution for a prolate
or oblate spheroid. However, these solutions are not given in closed form and the series
expansions do not converge for points on the surface. To solve this problem, we constructed
a function with the following properties: (a) it fulfils the Laplace equation apart from a line
pointing outward from the electrode, (b) the corresponding contribution to the lead field is only
linearly divergent at the origin and (c) in the vicinity of the electrode, the normal component
approximates −δ cos(2φ)/r . Such a solution is given by
f (r) =
δ
2
x2 − y2
(r − z)2
. (25)
We leave to the reader to check that indeed all properties are fulfilled. While this solution
looks simple it is very difficult to guess. Its construction is indeed quite involved and is given
in the appendices.
Given this solution, we can now construct the full U0 for arbitrary surfaces as
2πσ0U0(r, a¯, δ) = −
1
r
− a¯ ln(r − z) +
δ
2
x2 − y2
(r − z)2
. (26)
To recall, this solution fulfils equation (10) in the inside and on the volume conductor, and
leaves a non-singular error in the boundary condition.
It should be noted that, in spite of all the effort to get here, the error defined as a function
of the surface coordinates, although not singular, still contains a discontinuity at the electrode.
A hypothetical correction to next order would leave a continuous but not differentiable error.
It is desirable to have a theory which corrects up to arbitrary orders. However, this is beyond
the scope of this paper.

Expansion of the EEG lead field 3813
2.4. Proposed ansatz for the electric lead field
For K shells and for each electrode at position R1 with reference electrode at R2, we assign
a lead potential Uk(r) for k = 1, . . . , K (from inside to outside) according to the following
form:
Uk(r) = U
mi
k (r) + U
ma
k (r) + U
me
k (r). (27)
The superscripts mi,ma and me denote microscopic, macroscopic and mesoscopic potentials,
respectively, referring to the special purpose of these terms as explained in the following.
Microscopic potential. The lead field is singular at the electrode and reference. It is tempting
to try to describe this singularity by an analytic solution for the half-space or sphere. As
explained in the preceding subsection, this is in general insufficient: if curvature radii differ
at the electrode/reference location, any spherical solution still has a divergent error at these
locations. The length scale of this singularity is zero and hence the notion ‘microscopic’.
We perform a coordinate transformation such that R1 is in the centre of the coordinate
system and the positive z-axis coincides with the outward normal at the electrode location.
The x–y plane is rotated such that in the vicinity of the electrode the surface can be described
with the new coordinates denoted as r1 locally as
z1 = a¯1
(
x21 + y
2
1
)
+ δ1
(
x21 − y
2
1
)
. (28)
Accordingly the transformation is also done for the reference electrode resulting in coordinates
r2 and surface parameters a¯2 and δ2.
The microscopic potential is non-vanishing only for the outermost shell (k = K) and
reads
UmiK (r) = U0(r1(r), a¯1, δ1) − U0(r2(r), a¯2, δ2) (29)
where 2πσKU0(r, a¯, δ) is given by the right-hand side of (26). Note that apart from electrode
location and surface normal the microscopic potential has only three free parameters for each
channel which are found directly from the volume conductor: a¯, δ and implicit rotation angle
such that the surface can be locally parametrized as in (28).
Macroscopic potential. Typically, volume conductors can be regarded as smooth
deformations of the spherical volume conductor. Errors are also very smooth but are distributed
over the whole volume conductor surfaces. The length scale of this error is given by the length
scale of these deformations.
We expand the lead potential in a basis constructed from spherical harmonics centred at
the centre of the volume conductor:
Umak (r) =
∑
pq
(
akpqr
pYpq(�, ) + b
k
pq
1
rp+1
Ypq(�, )
)
(30)
with bkpq = 0 for the innermost shell (k = 1). The centre is found from a fit of K concentric
spheres to the volume conductor. We emphasize that it is sufficient that the chosen centre is
reasonable. The only mathematical requirement is that the surfaces are starlike with respect
to this centre. This set of basis functions is smooth and can properly model relatively
smooth deformations of the surfaces. The order of spherical harmonics is taken to be 25.
In general, for K shells and order P of spherical harmonics the number of unknowns is
K(2(P + 1)2 − 1) − (P + 1)2.

3814 G Nolte and G Dassios
–15 –10 –5 0 5 10 15
–15
–10
–5
0
5
10
15
Volume conductor
mesoscopic expansion
points for outer shell
electrode
macroscopic expansion point
x/cm
z/
cm
Figure 1. 3-shell prolate spheroid. We expand the lead field of a given electrode, e.g., for the
outermost shell in three sets of spherical harmonics originating in the centre and in two close-by
expansion points, respectively. The two rectangles correspond to the areas shown in figure 4.
Mesoscopic potential. Superficial shells are relatively thin. A meaningful expansion must
take into account that the lead field deviates significantly from simple functions e.g. on an
inner shell in the vicinity of the electrode/reference. This scale is typically of the order of
1 cm. Furthermore, our treatment of the singularity leaves a regular but discontinuous error
which can also be corrected sufficiently well with the mesoscopic potential.
For each shell, we add two expansion points: one is about 2 cm below the inner surface
and the other is about 2 cm above the outer surface of that compartment. Both expansion
points are on the line which connects the electrode to the centre. For the innermost shell,
we have only an outer expansion point. Each electrode including the reference has separate
expansions. In figure 1 these two expansion points are shown for the outermost compartment
of a 3-shell volume conductor.
For each expansion point and for both the electrode and the reference, we add a mesoscopic
potential of the following form:
Umek (r) =
∑
pq
ckpq
1
rp+1
Ypq(�, ) (31)
where r now denotes the coordinates centred at the respective expansion point.
The introduction of this mesoscopic potential corresponds to a local grid refinement in
the vicinity of the source as was done by Frijns et al (2000) for BEM. The major conceptual
difference is that using lead fields this refinement can be done in the vicinity of the electrodes
which are at fixed locations. In general, for K shells and order P of spherical harmonics the
number of unknowns for each channel is (2K − 1)(P + 1)2.

Expansion of the EEG lead field 3815
Putting it together. The above ansatz contains as unknowns the coefficients for multiple
expansions in spherical harmonics. The unknowns are estimated from minimizing the error
in the boundary condition. In general, the cost-function H(U) is defined here as
H(U) =
K
∑
k=1
∑
i
(nik · (σ˜kLk(rik) − σ˜k+1Lk+1(rik)))2
+
K−1
∑
k=1
∑
i
[(sik · (Lk(rik) − Lk+1(rik)))2 + (tik · (Lk(rik) − Lk+1(rik)))2] (32)
where σ˜k = σk/σK are the relative conductivities, rik is the ith point on the kth surface with
surface normal nik and two orthogonal tangential vectors sik and tik . Note that σK+1 = 0, and
the unspecified lead field outside the volume conductor (LK+1) is irrelevant.
Minimization is done in two steps. First, we minimize the cost function for microscopic
and mesoscopic potentials only and for each electrode separately ignoring the reference which
is treated as another electrode. This minimization is done only locally in the vicinity of the
respective electrode (distance � 4 cm) on a very fine grid (grid-size ≈ 1 mm). The grid is
calculated from a model as explained below. The remaining error is not small but smooth. In
the second step, we minimize the error for the full lead potential keeping the parameters for
the mesoscopic potentials fixed.
In this algorithm, we need an analytic model for the volume conductor to (a) estimate the
local surface curvature parameters a¯ and δ and (b) extrapolate from a given surface grid to a
finer one in order to properly describe the highly non-smooth lead field in the vicinity of the
electrodes. As was done by Purcell et al (1991), we model each surface by an angle-dependent
radius as
r(�, ) =
N
∑
n=0
N
∑
m=−N
an,mYnm(�, ) (33)
with a∗n,m = an,−m to have a real valued radius. Choosing N = 12 typically results in errors
less than 1 mm. To avoid inconsistencies between fine and coarser grids, we regard the model
as the true volume for all calculations.
For a 58-channel EEG and a 3-shell volume conductor, the estimate of the unknown
expansion parameters takes about 3 h on a 2.5 GHz PC. The forward calculation takes about
30 ms for each dipole.
3. Examples
3.1. Prolate spheroid
As a first example, we assume the volume conductor to be a 3-shell prolate spheroid as shown
in figure1. The short and long axes of the outer surface have lengths of 10 cm and 15 cm,
respectively, corresponding to a relatively large deviation from the spherical case. The short
axes for the two inner shells have lengths of 9 cm and 8 cm, respectively. The long axes are
scaled accordingly. In this example, we put an electrode at top (x = y = 0 and z = 10 cm ) and
reference at x = 0 and −y = z = 7.07 cm. The conductor values are set to σ1 = σ3 = 50σ2.
Given the conductivity ratios, the external potential is then simply proportional to 1/σ1 which
is not specified here.
The effect of microscopic, mesoscopic and macroscopic potentials can be seen in figures 2
and 3 where we show the errors in the boundary conditions for the outer and middle surfaces,
respectively. The inner surface is similar to the middle one and is not shown. Furthermore,

3816 G Nolte and G Dassios
Figure 2. Errors of the lead field on the outermost surface including microscopic, mesoscopic
and macroscopic potentials as indicated. In the lower right panel we show the ratio of normal
component and the absolute value of the tangential component for each surface point.
we only discuss the error in the normal field component. The errors in tangential components
are similar and do not provide additional insight.
In figure 2 we observe that the microscopic lead potential leaves an error on the outer
surface which is not continuous at the electrode/reference. At both locations, the curvature
radii differ from each other (δ �= 0). If we had chosen a spherical approximation for the
microscopic potential, this error would have been singular (not shown). Adding the mesoscopic
potential leads to a smoothing and to increased errors. The reason for the increase is that the
mesosopic potential mainly compensates errors on the inner surfaces which are about 200
times larger than the error on the outer surface. The error then finally decreases when all
potentials are added. The final decrease appears to be rather moderate, but the outer surface
had a relatively small error in the first place. The ‘tangentiality’ can be seen in the lower right
panel where we show the ratio of the normal component and absolute value of the tangential
component. This ratio is rarely larger than 0.001.
As can be seen in figure 3, the microscopic potential leaves a regular but very focal error
which is difficult to compensate with spherical harmonics placed in the centre of the volume
conductor. The mesoscopic potential reduces this error and, more importantly, smoothes it
such that the final error is reduced by a factor of about 2000 as compared to the microscopic
potential only.
In figure 4 we show the lead field in two areas of the outer shell for the special case
that the conductivity in the middle shell vanishes. In this case, the current flowing from the

Expansion of the EEG lead field 3817
Figure 3. Errors of the lead field on the middle surface including microscopic, mesoscopic and
macroscopic potentials as indicated.
−15 −14.5 −14 −13.5
−1
−0.5
0
0.5
1
x[cm]
z[c
m]
−0.4 −0.2 0 0.2 0.4 0.6
8.8
9
9.2
9.4
9.6
9.8
10
x[cm]
z[c
m]
Figure 4. Lead field in outer shell for vanishing skull conductivity in areas remote from (left panel)
and close to (right panel) the electrode. The two areas are shown as rectangles in figure 1.
reference to the electrode can only flow within the outer shell. We observe that the lead field
is qualitatively correct.
3.2. Comparison with spherical solutions
We now want to analyse the accuracy on a more quantitative level for the case of a 3-shell
spherical volume conductor. In some sense, this case is too simple because the error on the

3818 G Nolte and G Dassios
outer surface due to the microscopic potential alone already vanishes. On the other hand,
however, we have seen that the main cause of error, once the singularity on the outer surface
is removed, arises on inner surfaces. This error mainly depends on the thickness of the shells
and can be expected to be similar for spherical and other geometries.
We considered three different cases. The first consists of relatively thick outer shells with
radii 8 cm, 9 cm and 10 cm; the second has thinner outer shells corresponding to radii of
9 cm, 9.5 cm and 10 cm. For both cases the conductivity ratios were set to 1:0.02:.1. In
order to compare the results with existing methods, the third case was constructed to match
the parameters chosen by Kybic et al (2005) with radii of 8.7 cm, 9.2 cm and 10 cm and
conductivity ratios 1:0.0125:1. Results were calculated for three dipole orientations (radial
and two tangential) and in the third case, also for a dipole with an oblique orientation of
(1, 0, 1)/
√
(2).
The exact solutions were calculated analytically expanding the respective series of
spherical harmonics up to order 200. The potentials were calculated in 58 electrodes equally
distributed on the upper hemisphere with one of the electrodes taken as reference. The
electrode setting was taken from a real measurement extrapolated to the sphere.
In figure 5 we show the error of our approach for dipoles moved along the z-axis from the
bottom to the top of the respective inner compartment. The relative error of V is defined as

rel =
‖V − Vexact‖
‖Vexact‖
(34)
where ‖·‖ denotes the Euclidian norm and Vexact is the exact solution. We first note that the
two tangential dipoles have slightly different errors because the electrode setting is not exactly
invariant with respect to a rotation of 90◦. The relative error is clearly much smaller for
the thick shells being mainly below 10−4 which can be considered as exact for all practical
purposes. Only near the surface does the error increase to the still small value of slightly less
than 10−3. Note that, most importantly, the solutions are highly accurate even if we put the
source directly on the surface.
For the thin shells the relative error, being typically below 0.01, is sufficiently small for
almost all practical purposes. An exception is the radial dipole in the bottom part of the
volume conductor. However, the large relative error can be traced back to the smallness of the
respective potential: the absolute error defined as

abs = ‖V − Vexact‖ (35)
is indeed almost independent of the dipole orientation as can be seen in the lower panels of
figure 5.
Kybic et al (2005) proposed a new BEM-formulation which was analysed for a 3-shell
spherical volume conductor with parameters as in our third case apart from the electrode
locations which we assume here to be of minor importance. The error was studied by these
authors for a dipole with oblique orientation for four dipole locations with radii between
4.25 cm and 8.415 cm. While ‘conventional’ BEM methods lead to errors eventually larger
than 100% for superficial dipoles, the authors reduced this error to about 1% or slightly larger
in the entire range studied. It must be noted that for these simulations, the authors chose
relatively coarse grids because of memory and time limitations. The accuracy for finer grids
is unclear.
Our corresponding result is given by the curve in the upper rightmost panel of figure 5,
which should be compared with figure 4 of Kybic’s paper. We observe that the relative error
is typically slightly above 0.01% and is clearly below 0.1% even for dipoles placed directly
on the surface of the innermost compartment. This relative improvement of about a factor
100 appears quite academic given that a relative error of about 1% is sufficient for practical

Expansion of the EEG lead field 3819
−5 0 5
10−6
10−4
10−2
tangential dipoles
radial dipole
thick shells
re
la
tiv
e
er
ro
r
−5 0 5
10−6
10−4
ab
so
lu
te
e
rr
or
[a
.u.
]
position [cm]
−5 0 5
10−3
10−2
10−1
100
tangential dipoles
radial dipole
thin shells
−5 0 5
10−4
10−3
10−2
position [cm]
−5 0 5
10−4
10−3
10−2
medium thickness
−5 0 5
10−6
10−4
position [cm]
Figure 5. Comparison of this method with analytic solution for a 3-shell spherical volume
conductor with thick shells (1 cm) and thin shells (5 mm) with conductivity ratios 1:.02.:1. In the
rightmost column, we show results with radii of 8.7 cm, 9.2 cm and 10 cm and conductivity ratios
1:.0125:1. Sources with all three orientations are placed on the z-axis and moved from the bottom
to the top end of the innermost shell. For the rightmost column, we calculated results only for a
dipole with an oblique orientation (1, 0, 1)/
√
(2). The upper row shows the relative errors and the
lower row the absolute error in arbitrary units indicating that the absolute error is fairly independent
of the dipole orientation. This curve corresponds to results shown by Kybic et al (2005).
purposes. However, as was also shown by Kybic et al, methods which are quite accurate for
a 3-shell spherical volume conductor lead to very different potentials for a 3-shell realistic
volume conductor even for extremely fine surface grids. Hence, the actual performance of the
various methods for realistic cases is largely unknown. The observed relative improvement
might be crucial for practical applications—if it persists.
4. Discussion
We have presented an algorithm to calculate the EEG lead field for multiple shells of realistic
shape. The lead field was expressed as a closed form analytic approximation which was
corrected by an expansion in a suitable basis constructed from spherical harmonics. With this
formulation the lead field always fulfils the corresponding differential equation exactly in the
inside of the compartment. The remaining error arises from the boundary conditions and can
be expressed as a set of functions defined on the surfaces.

3820 G Nolte and G Dassios
The main theoretical achievement was to formulate an analytic approximation such
that the remaining error, which needs to be corrected, is regular for general surfaces. A
drawback is that this error has still a discontinuity at the locations of the electrodes. A more
general formulation which leaves an error which is differentiable up to nth order (or up to
all orders) is a highly challenging theoretical problem which is beyond the scope of this
paper.
We may regard this analytic approximation as an exact and closed form solution with
respect to a different volume conductor. To us, it is an open question what the shape of this
volume conductor is. However, since we can construct arbitrarily complicated lead potentials
simply by adding harmonic functions we can also construct closed form solutions to arbitrarily
complicated volume conductors. Thus, we challenge the view that the EEG forward calculation
can be solved analytically (let alone in closed form) only for very few volume conductors.
This is possible because we are seeking solutions for fixed electrode positions. The respective
volume conductors, which make the solution for each electrode exact, are in general not
identical—but can be made arbitrarily similar to the true volume conductor.
A practical problem of the EEG forward calculation is the presence of relatively thin
outer shells. The error of the analytic approximation at an inner surface in the vicinity of an
electrode is highly non-smooth and it is difficult to correct for with a set of smooth expansion
functions like spherical harmonics of relatively low order centred in the middle of the head. We
addressed this problem by introducing additional expansion points somewhat in the vicinity
of the electrodes. We admit that this part of the algorithm is rather heuristic and not very
elegant. However, it was our primary goal to have an acceptable solution for the entire volume
conductor.
Accuracy was tested rather qualitatively for a 3-shell prolate spheroid indicating that the
lead field ‘looks’ correct and showing that the errors in the boundary conditions are tiny. In
a more detailed comparison with a 3-shell spherical volume conductor, we have seen that
the error is in general below and mostly far below 1% even for the most superficial sources
placed directly on the surface of the inner shell. We regard this accuracy as sufficient for
practical applications. One may criticize the fact that the 3-shell spherical volume conductor
is too simple because the analytic solution, corresponding in this case to a 1-shell spherical
solution, leaves a vanishing (and especially continuous) error at the outer surface. However,
after removing the singularity for a general volume conductor, the remaining error is much
larger on inner surfaces than on the outer one. Correcting this error is similarly difficult for
spherical and realistic volume conductors.
Appendix A
Here we construct the function given in (25). The solution is based on virtual charges
placed outside the volume conductor. The spherical solution contains the term ln(r − z) =
ln(r(1 − cos(�)) which is singular at the origin and at a vertical line from the 0 to +∞, i.e.
on the line � = 0. We can regard this line as containing charges. To solve our problem,
we place virtual charges on lines which are tilted with respect to the vertical line by a fixed
angle �0. The second angle 0 will not be fixed, but rather we integrate over all lines with
an appropriate weight. We can then regard the final solution as induced by virtual charges
distributed on a cone with an opening angle �0.
Let us first recall the boundary condition and express it in a convenient form for this
section. We are looking for a function f (r), which fulfills the Laplace equation and for which
∇f is at most linearly divergent, such that in the vicinity of the electrode we have for the

Expansion of the EEG lead field 3821
surface normal
−(0, 0, 1)T · ∇f ≈ −δ
cos(2 )
r
. (A.1)
We first note that since ln(r − z) fulfils the Laplace equation and due to the rotational
invariance of the Laplace operator also
H(r,�0, 0) ≡ ln(r − b · r) (A.2)
fulfils the Laplace equation for any normalized vector b which we choose here to be of the
general form
b(�0, 0) = (sin �0 cos 0, sin �0 sin 0, cos �0). (A.3)
Since H fulfils the Laplace equation, so does
f (r,�0) =
∫ 2π
0
d 0W( 0)H(r,�0, 0) (A.4)
for any (non-singular) W( 0). We choose
W( 0) =
cos(2 0)
N
(A.5)
with a normalization N to be fixed below. To calculate f we substitute − 0 = x and have
cos(2 0) = cos(2 − 2x) = cos(2 ) cos(2x) + sin(2 ) sin(2x) (A.6)
leading to
f (r,�0) =
cos(2 )
N
∫ 2π
0
dx cos(2x) ln(r(1 − cos � cos �0 − sin � sin �0 cos x)) (A.7)
where we have exploited the fact that the term with sin(2x) vanishes after integration because
it is anti-symmetric. We now observe that any constant factor in the argument of the logarithm
is irrelevant because
∫ 2π
0 cos(2x) dx = 0 and hence
f (r,�0) =
cos(2 )
N
∫ 2π
0
dx cos(2x) ln(1 − A cos x) (A.8)
with
A ≡
sin � sin �0
1 − cos � cos �0
. (A.9)
The treatment of the integral is quite difficult. We will show below that it evaluates to
f (r,�0) =
π cos(2 )
4N
1 −
√
1 − A2
1 +
√
1 − A2
. (A.10)
Using
√
1 − A2 =
|cos � − cos �0|
1 − cos � cos �0
(A.11)
f becomes
f (r,�0) =
π cos(2 )
4N
1 − cos � cos �0 − |cos � − cos �0|
1 − cos � cos �0 + |cos � − cos �0|
. (A.12)
The evaluation now depends on the sign of cos � − cos �0. For � > �0, which shall include
the inside of the volume conductor, we get
f (r,�0) =
π cos(2 )
4N
(1 − cos �0)(1 + cos �)
(1 + cos �0)(1 − cos �)
(A.13)