A meshless method for solving the EEG forward problem.
ABSTRACT We present a numerical method to solve the quasistatic Maxwell equations and compute the electroencephalography (EEG) forward problem solution. More generally, we develop a computationally efficient method to obtain the electric potential distribution generated by a source of electric activity inside a three-dimensional body of arbitrary shape and layers of different electric conductivities. The method needs only a set of nodes on the surface and inside the head, but not a mesh connecting the nodes. This represents an advantage over traditional methods like boundary elements or finite elements since the generation of the mesh is typically computationally intensive. The performance of the proposed method is compared with the boundary element method (BEM) by numerically solving some EEG forward problems examples. For a large number of nodes and the same precision, our method has lower computational load than BEM due to a faster convergence rate and to the sparsity of the linear system to be solved.
- [Show abstract] [Hide abstract]
ABSTRACT: Volume integral equations (VIEs) are indispensable for solving inhomogeneous or anisotropic electromagnetic (EM) problems by integral equation approach. The solution of VIEs strongly relies on the discretization of volume integral domains, and tetrahedral elements in discretization are usually preferred for arbitrary geometric shapes. Unlike discretizing a surface domain, discretizing a volume domain is very inconvenient in practice, and special commercial software is needed in general even for a simple and regular geometry. To release the burden of descretizing volume domains, especially to remove the constraint of mesh conformity in the traditional method of moments (MoM), we propose a novel meshfree scheme for solving VIEs in this paper. The scheme is based on the transformation of volume integrals into boundary integrals through the Green-Gauss theorem when integral kernels are regularized by excluding a small cylinder enclosing the observation node. The original integral domain represented by the object is also expanded to a cylindrical domain circumscribing the object to facilitate the evaluation of boundary integrals. The singular integrals over the small cylinder are specially handled with singularity subtraction techniques. Numerical examples for EM scattering by inhomogeneous or anisotropic objects are presented to illustrate the scheme, and good results are observed.IEEE Transactions on Antennas and Propagation 09/2012; 60(9):4249-4258. · 2.33 Impact Factor
- Progress In Electromagnetics Research Letters 01/2013; 36:143-153.
- Revista Mexicana de Ingeniería Biomédica. 04/2013; 34(1).
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 52, NO. 2, FEBRUARY 2005249
A Meshless Method for Solving the EEG
Nicolás von Ellenrieder*, Student Member, IEEE, Carlos H. Muravchik, Senior Member, IEEE, and
Arye Nehorai, Fellow, IEEE
Abstract—We present a numerical method to solve the quasi-
(EEG) forward problem solution. More generally, we develop a
computationally efficient method to obtain the electric potential
conductivities. The method needs only a set of nodes on the surface
and inside the head,but nota mesh connecting thenodes. This rep-
resents an advantage over traditional methods like boundary ele-
ments or finite elements since the generation of the mesh is typi-
cally computationally intensive. The performance of the proposed
method is compared with the boundary element method (BEM) by
numerically solving some EEG forward problems examples. For
a large number of nodes and the same precision, our method has
lower computational load than BEM due to a faster convergence
rate and to the sparsity of the linear system to be solved.
Index Terms—EEG, EEG forward problem, layered media,
meshless method, moving least squares approximation, numerical
solution, volume conductor.
has attracted much interest in neurology. These sources of ac-
tivity generate an electric potential distribution in the head that
is noninvasively measured with a suitable array of electrodes
on the scalp. The computation of the electric potential distribu-
tion generated by current density sources with known parame-
so-called EEG forward problem. Estimation of the source pa-
rameters based on the electric potential measurements is known
as the EEG inverse problem. This paper essentially deals with
the forward problem.
To get an accurate solution to the EEG forward problem it is
necessary to correctly model the shape of the head and its elec-
STIMATION of the intensity, location, and orientation of
sources of neuronal activity in the brain from EEG signals
Manuscript received October 2, 2003; revised June 27, 2004. The work of
N. von Ellenrieder was supported in part by the Consejo Nacional de Inves-
tigaciones Científicas y Técnicas (CONICET). The work of C. Muravchik was
supported in partby the Comisión de Investigaciones Científicas de la Provincia
de Buenos Aires (CICPBA). The work of A. Nehorai was supported in part by
the National Science Foundation under Grant CCR-0105334 and Grant CCR-
0330342. Asterisk indicates corresponding author.
*N. von Ellenrieder is with the Laboratorio de Electrónica Industrial, Con-
trol e Instrumentación, Departamento de Electrotecnia, Facultad de Ingeniería,
Universidad Nacional de La Plata, C.C. 91, 1900 La Plata, Argentina (e-mail:
C. H. Muravchik is with the Laboratorio de Electrónica Industrial, Control e
Instrumentación, Departamento de Electrotecnia, Facultad de Ingeniería, Uni-
versidad Nacional de La Plata, 1900 La Plata, Argentina.
A. Nehorai is with the Department of Electrical and Computer Engineering,
University of Illinois at Chicago, Chicago, IL 60608 USA.
Digital Object Identifier 10.1109/TBME.2004.840499
tric properties . A prevalent model consists on layers of dif-
ferent electric conductivities, constant within layers. The shape
of the layers is obtained for example by magnetic resonance
imaging (MRI) or X-ray tomography.
The irregular geometry of the head model requires the
problem to be solved numerically. Frequently employed nu-
merical techniques, such as boundary element method (BEM)
or finite element method (FEM), require nodes on or inside the
surfaces of the head layers, and a mesh partitioning the domain
of the problem into a set of small two-dimensional (2-D)
(BEM) or three-dimensional (3-D) (FEM) elements. A simple
variation (constant or linear) is assumed for the electric poten-
tial over each element, and the resulting differential or integral
equations are solved to get the potential at the nodes –.
Very often spherical approximations to the head surfaces are
used to reduce the computational load of these methods .
Once the geometry of the problem is known it is easy to se-
be generated automatically , it still requires dedicated soft-
ware and computing time. To overcome these complications, a
method that uses only the locations of the surface points would
necting the nodes have been developed for a wide range of ap-
plications –. In this paper, we present a meshless method
for solving the EEG forward problem.
In Section II, we describe the head and source of brain elec-
tric activity models, and state the mathematical expression for
the EEG forward problem. The proposed meshless method is
described in detail in Section III, and some of its possible vari-
ations in Section IV. In Section V, we present some numerical
results obtained with our method and compare the performance
results of the method and state the future work.
II. THE EEG FORWARD PROBLEM
In this section, we describe the head and source models and
state the mathematical EEG forward problem. As is typical in
EEG, the head is modeled as a body formed by nearly concen-
layer, representing the tissues. The surfaces
th layers are assumed to be smooth. In the common
Fig. 1. In general,
layers could be used for a more accurate
representation of the head including cerebrospinal fluid, differ-
ences between gray and white matter, etc.
between the th
0018-9294/$20.00 © 2005 IEEE
250IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 52, NO. 2, FEBRUARY 2005
of constant conductivities ? separated by smooth surfaces ? ?? ? ?????. An
electric current dipole located at the point ? ? ? is the source of activity.
Schematic electromagnetic model of the head. Concentric layers ?
A group of active neurons is a source of electric activity in
the brain, and can be described in many instances of interest
for EEG as an electric current dipole . Let
spectively. Since the electric activity of the neurons takes place
in the brain, the source will be located in the inner layers, i.e.,
, usually withor 2. Hence the problem, in local
or differential form, is the following:
Laplacian operator and
is located is
. For any point
expression for the electric potential in the neighborhood of a
dipole, with values up to
. Any numerical approximation to
the analytic solution of the potential near this singularity will
necessarily be very poor.
To avoid this difficulty it is possible to remove the singu-
larity using the linearity of the Laplacian operator; this forward
linearity, the electric potential can be found as the superposition
of the two terms
is any point in the domainis the
is the gradient operator. The electric
, andis the outward normal to the surface
is the limit point of a spatial
, andis the limitpoint of a spatial
. Solving the forward problem using (1) is
, where each term is
ated by a dipole source in an infinite homogeneous medium of
potential is not defined at the dipole’s position, but it presents
no difficulty because it can be solved analytically, yielding the
corresponds to the electric potential gener-
Neumann boundary conditions , and must be computed nu-
merically because it is related to the irregular geometry of the
head model. All the information regarding the source in this
term is present in the boundary conditions, as can be seen in
(3).This termhas no singularity,so itssolution is smootherthan
the solution of the original problem, improving the accuracy of
any numerical approximation. Of course, there is an unavoid-
able numerical difficulty when the dipole source is very close to
From now on this procedure will be referred to as the singu-
numerical part of the SSM, i.e., the solution for
in (3) corresponds to a Laplace equation with
III. FINITE POINTS MIXED METHOD
We propose a FPMM that uses a collocation technique to dis-
cretize the governing equation(3). The electric potential is re-
placed by a global approximation written as a linear combina-
the discretized EEG forward problem. The shape functions re-
sult from building a moving least squares local approximation
to the electric potential about the nodes in terms of a basis of
simple functions. The method is said to be mixed because the
solution satisfies exactly neither the differential equation in the
domain nor the boundary conditions .
First, we show how to obtain the shape functions that will
be used to construct the global approximation of the electric
computing the coefficients of the global approximation, which
we will call
. The moving least squares approximation is a
standard procedure in many meshless methods, e.g., , but
we include it here to introduce the notation and our choice of
local basis and weight functions.
, of any layer
of the electric potential is approximated in the
neighborhood of any point
assuming a simple variation
we choose a complete monomial basis of order 2 with
elements, in three dimensions
is the electric potential approximation,
functions of the approximation basis. In this paper,
, and the basis is
VON ELLENRIEDER: MESHLESS METHOD FOR SOLVING THE EEG FORWARD PROBLEM251
There exist a certain number
be the the value of the electric potential on the
coefficients of a global approximation of the electric potential.
To do this we must relate the vector
the coefficients of the local approximation, to the potentials
associated with the point
minimizing the functional
of nodesaround. Let
in (5), which contains
is obtained then by
The weight function
, evaluated at , is responsible for the local character
of the approximation. The support of the weight function, also
known as the influence region of the th node, is a spherical
region centered at
, the size of the of nodes near
number of nodes participating in the determination of the co-
. The smoothness of the resulting approximation
is related to the smoothness of the weight functions , thus,
the higher order derivatives of the approximation have disconti-
nuities at the border of the influence regions of the nodes when
radial basis functions or truncated Gaussian functions are used
as weight functions , . We propose to use smooth, i.e.,
, weight functions in order to obtain a smooth approxima-
tion. The smooth functions we use are
is the sum of the weighted approximation errors to the
on the nodes
associated with the node
. In this way, for a given
determines , i.e., the
Equation (7) poses a moving (due to its dependence on
) least-squares problem for the vector
is a diagonal matrix with the given elements
on the main diagonal, we getthe following well-knownsolution
controls the rate the weight function tends to zero.
squares solution, the point
region of at least
certain special patterns such as a plane or spherical surface, to
preserve the full column rank of the matrix
Note that the vector
, whose elements are the poten-
, is not known a priori and has to be computed.
However, by assuming it was known we were able to get
expression (9) for computing
constructed a global approximation for the potential let
. In order to obtain a unique least-
must be contained in the influence
nodes. These nodes cannot be arranged in
in (5). To show that we
then, from (5) and
(9) we get
. We see then that the electric potential approximation at
linear combination of the shape functions associated with nodes
whose influence region enclose
linear combination are contained in the vector
The shape functions have the same support as the weight
functions, they are also smooth, and any linear combination of
the functions used as the basis of the local potential approxima-
tion can be represented globally without error as a linear combi-
nation of the shape functions . Note that since
, this method only approximates the electric potential of
the nodes but does not interpolate their value, as BEM or FEM
In the following subsections, we will show how to compute
of the EEG forward problem.
is called the shape function  of the node
, and the coefficients of this
A. Boundary Conditions
Most of the known meshless methods solve the Neumann
boundary conditions problems with null normal flux across the
surface , –. This allows imposing the boundary
conditions by forcing the shape functions to satisfy them.
The resulting approximation is a linear combination of the
shape functions and satisfies the boundary conditions too. This
kind of methods are called domain methods. But the use of
singularity separation explained in Section II means that the
normal component of the electric potential gradient across the
surfaces changes when the parameters of the source vary. Since
we plan to use the method for solving the inverse problem, it is
important to keep the shape functions independent of the source
parameters. With such purpose in mind we introduce in the rest
of the section our FPMM, since existing meshless methods do
not handle this situation satisfactorily.
The method is mixed because neither the boundary condi-
tions nor the differential equation over the domain are satisfied
exactly, in fact only a weak weighted residuals integral formula-
tion of the boundary conditions can be satisfied . Since we
only know a set of points on the surfaces, it is natural to use a
point collocation technique . The solution of the weak inte-
is equivalent to evaluating the boundary conditions of (3) on the
mation to the normal component of the potential gradient at any
on a surface
Now (3) can be evaluated for every surface node by means of
(10) and (11). For any node this involves the coefficients
is the total number of nodes,andare rectangular
252IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 52, NO. 2, FEBRUARY 2005
matrices, because the boundary conditions only apply to nodes
on the surfaces, a subset of the total nodes.
The true normal component of the electric potential gradient
has a jump across theinterfaces betweenlayers of different con-
ductivity. But the electric potential approximation is a linear
resent this jump in the normal component of the gradient. There
assume a certain extra knowledge of the surface in addition to
the location of the nodes. We propose to deal with each layer
separately, then combine them by imposing the boundary con-
ditions. This approach increases the size of the linear system to
be solved, but it leads to a better approximation.
Suppose there are
layers of different conductivities, with
nodes on interface and
external surface nodes of every layer, denoted with the super-
, must be at the same position than the internal surface
nodes of the following layer, denoted with the superscript
For each layer
computed as described in previous paragraphs. Note that
an empty matrix for the first layer has no inner surface, and
is necessary for the computation of the potential approximation
on the scalp surface, not to impose boundary conditions.
conditions of (3), for
internal nodes in layer . The
, and for
is the normal component of the flux at the
the th surface.
These equations can be combined in a single system
surface nodes of
where the coefficient vector is
of the matrix
boundary conditions for the multilayered problem.
. The size
is , with
. Equation (15) summarizes the
B. Laplace Equation
In this section, we deal with the Laplace differential equation
for the electric potential in the domain representing the head. If
we compute the Laplacian of the potential approximation based
it can be seen that since the Laplacian of the shape functions
is not zero, neither is the Laplacian from the proposed approx-
imation. Even though the potential approximation
domain, we can still force it to satisfy a weighted residual inte-
gral formulation. Using again a point collocation technique, the
evaluation of this integral formulation is equivalent to the eval-
uation of the first equation of (3) at every node. Writing (16)
for all the nodes in each layer
, we get the linear
Combining them, we obtain the block diagonal system
where the matrix
is of size.
C. The Complete Set of Linear Equations System
In the previous two sections we saw that the boundary condi-
exactly, but through a weighted residual integral formulation.
This is the reason why the method is called mixed. However,
with this formulation (15) and (18) can not be satisfied simulta-
neously by any vector
because there are more equations than
the Laplace equation at the nodes, constrained by the boundary
There exist several algorithms to solve this well-known con-
strained optimization problem . Since the electric poten-
tial approximation is computed at each point based only on
the nearest nodes, the resulting linear systems are represented
by sparse matrices. A good algorithm to solve the numerical
problem should take advantage of the matrices’ sparsity, for this
decreases the computational burden. One possibility is to solve
the linear problem
Since the EEG forward problem has Neumann boundary
conditions, it has an infinite number of solutions differing in a
constant value. The MLS scheme used in our FPMM ensures
that any linear combination of the functions used as the basis
of the local potential approximation can be represented glob-
ally without error. Hence the constant function is represented
without error in the dicretized version and the rank of the
is, with a null singular value associated to
the constant eigenvector.
It is possible to avoid this rank deficiency by imposing a new
restriction on the optimization. This restriction should fix the
potential at some point, or the mean potential of any subset of
points. The equation for this restriction is obtained from (10)
evaluated at the points of interest. Under these conditions (20)
is a symmetric positive definite linear system, and a wide range
of iterative methods exist for solving it , . Note that it is
sparsity could be lost. Also, the value of
too small in order to avoid numerical instabilities, we obtained
good results with
in (20) should not be
VON ELLENRIEDER: MESHLESS METHOD FOR SOLVING THE EEG FORWARD PROBLEM253
Once the coefficient vector
sion for the electric potential approximation at any point. The
solution to the EEG forward problem at a point
is known, (10) gives an expres-
In this section, we analyze some variations of our method
in order to improve certain aspects of its behavior. The perfor-
mance of the different versions is analyzed in Section V.
A. Moving or Local Differentiation
Two different approaches can be taken when computing the
derivatives of the shape functions needed in (11) and (16). A
local approach assumes that the proposed approximation for the
electric potential behaves according to (5) where the coefficient
is assumed constant in the neighborhood of . We can
write then from (9) and (10)
We call this approach local differentiation (LD) method as
opposed to a moving differentiation (MD) approach that arises
when the dependence of vector
given by (9)
withis considered, as
Comparing (22) with (23) we can see that the LD version is
simpler and hence the computation of the matrices
faster for the LD version.
B. Square Linear System (SLS)
In the previous section we solved the forward problem as a
linearly constrained optimization, by minimizing the Laplacian
of the approximation subject to the boundary conditions.
It is also possible to write a SLS for the forward problem.
This can be done simply by forcing the Laplacian to be null
only at the internal nodes. In this way, we get an equation for
each node; the Laplace equation for the internal nodes and the
boundary conditions for the surface nodes.
be thematrix formed by the rows of
to internal nodes, then the coefficients
are obtained as
C. Singularity Separation
Compared with the other layers the skull has a low electric
conductivity. This causes a shielding effect, hence the potential
The SSM of Section II consisted in computing the relatively
small electric potential
as the sum of the relatively large
opposite signs and tends to cancel
may cause large relative errors in
for the layers enclosing the skull it may be better to directly
compute the potential distribution rather than to use SSM.
If the SSM is applied from the first to
on (3) change to
. Then small relative
th layers only, the
. Then the boundary conditions
These changes in the boundary conditions affect only the vector
in (15), while the matrix remains unchanged. Also, when
computing the solution to the EEG forward problem with (21),
should be added only to points located in the layers
D. Isolated Skull Approach
The electric conductivity of the skull is almost two orders of
discontinuity causes some numerical instability when solving
the forward problem. The numerical stability of the results ob-
approach (ISA) . We discuss how to apply this approach to
our meshless method.
The ISA solves the forward problem in two stages. First, the
problem is solved assuming that the conductivity of the skull is
null. Then, the difference between this result and the real elec-
tric potential distribution is computed. The complete solution is
the sum of these two terms. The idea is that the first problem
involves only the layers beneath the skull, for which the results
have relatively small errors. The difference with the real solu-
tion is small, hence its error is also small.
which isolates the outer layers. Hence a reduced size problem
must be solved. The vector
and the matrices
assembled from (14) and (17) with
is the skull layer. Letbe the vector that minimizes
subject to, and
254 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 52, NO. 2, FEBRUARY 2005
Then the full-size problem (19) involving all the layers is
solved, but with a different vector , given by the modified
forward problem solution is given by
be the solution of this full-size problem, then the EEG
In this section, we analyze the performance of our method by
numerically solving the forward problem for head models with
spherical geometry. For these models the forward problem has
a known analytical solution  we can compare our numerical
with the frequently used BEM.
To quantify the error we considered two different measures, a
normalized relative difference measure (NRDM) that measures
the error in the shape of the distribution of the electric potential,
and a measure of the electric potential magnitude error (MAG).
These measures are frequently used for distributions compar-
isons  and are defined by
is a vector of numerically computed potentials, both potentials
evaluated in all the nodes on the outermost surface.
From the inverse problem point of view, the MAG is related
only to the intensity of the source, whereas the NRDM to its
of the six source parameters, we focused our attention on it. The
NRDM value varies between 0 and 2, it is null if the vectors are
if the vectors are orthogonal and 2 if they are equal
but with opposite sign.
All the BEM results where obtained using the same surface
nodes as with FPMM, where the nodes are the vertices of tri-
angular elements and a linear variation of the potential over the
elements is adopted . The results shown in this section cor-
respond to tangentially oriented dipoles. As when solving with
BEM, the results for radially oriented ones are similar with a
slightly higher error. Each point shown is the mean NRDM of
240 sources uniformly distributed over a sphere at the given
depth, with a random angle in the tangential plane.
Since we have a high degree of freedom to choose the in-
ternal node positions because there is no need to form regular
elements, we chose the easiest way to place them, i.e., in a reg-
ular grid with the separation
between internal nodes constant.
tested and do not affect the results significantly.
sphere. Results are shown for three variations of our method: LD, MD, and
NRDM as a function of the source depth in a constant conductivity
In all the examples the radius of the support of each weight
function (8) was
, such that approximately 81 nodes
were included in the influence region of the node, and the pa-
of the weight function was set to
the method for a constant conductivity sphere of radius
distance between nodes was set to
were placed on the surface of the sphere. The figure shows the
of the sphere. It can be seen that the LD version performs better
than the MD version.
Fig. 2 also shows the performance of the forward problem
poor for sources near the surfaces. This is not surprising since
the Laplacian is not minimized at the surface nodes. Since the
multilayered EEG problem has thin layers the SLS version will
not perform adequately.
Theremainingresults showninthissectionwere obtainedfor
a three layered sphere with conductivities
used because a series is known for the analytical solution .
Figs. 3 and 4 show a comparison between our FPMM and
BEM based on the NRDM and MAG as a function of the source
distance to the center of the spheres relative to
results are shown, with and without ISA, in both cases for 642
nodes per surface. The meshless method has the same surface
nodes, but additional interior nodes, with
innermost layer and
total number of nodes for the three layers is near 7500.
We can see the results for two FPMM versions, when we use
theSSMup tothethird surface theerror is higher thanwhen use
ituptothefirst surface,aspredictedinSectionIV.C. Theresults
almost identical to the results obtained when SSM is used only
in the innermost layer. This could be expected because the SSM
Compared with BEM, the performance of FPMM is very
good for deep sources, but for sources near the surface of the
, and 642 nodes
. The spherical geometry was
. For BEM two
for the remaining layers. The
VON ELLENRIEDER: MESHLESS METHOD FOR SOLVING THE EEG FORWARD PROBLEM 255
Comparison between our FPMM with singularity separation up to the first
(FPMM-SSM S.1) and third (FPMM-SSM S.3) surfaces and BEM with the
isolated skull approach (BEM-ISA) and without it (BEM).
NRDM as a function of the source depth in a three layered sphere.
sphere. Comparison between our FPMM with singularity separation up to the
first (FPMM-SSM S.1) and third (FPMM-SSM S.3) surfaces and BEM with the
isolated skull approach (BEM-ISA) and without it (BEM).
MAG measure as a function of the source depth in a three layered
brain the performance of ISA-BEM is better. Both error mea-
sures show this behavior.
Fig. 5 shows the convergence of FPMM. The mean, max-
imum, and minimum NRDM values are shown for 240 sources
convergence rate is at least
gence tests, FPMM also showed
other meshless methods , convergence is achieved with a
fixed number of nonzero elements per row of the sparse ma-
trices of the method, i.e., with a constant value of
method and ISA-BEM, in Fig. 6. It shows the NRDM as a func-
sources placed on spheres at depth
emphasize that this comparison was made for the same number
and 1926 for BEM and 805, 1700, 3575, and 7560 for FPMM.
. In 1-D and 2-D conver-
and. We can see
between neighboring internal nodes. Results correspond to dipoles located at
a constant depth in a three layered spherical head model. The maximum and
minimum error for 240 dipoles are shown as dotted lines, the mean error as a
Convergence of the FPMM. NRDM as a function of the separation
function of the number of nodes per surface in a three layered spherical head
model. The figure shows the error for sources at two different depths.
Convergence comparison between FPMM and BEM. NRDM as a
Analysis of the computational complexity of our method
indicates that the operation count for assembling the matrices is
, whereis the total number of nodes. The memory
needed to store the matrices is
count for solving (20) consists in a few iterations each of order
. Also, a proper preconditioner should be computed
, . This can involve a large number of operations, but
needs to be done only once if many forward problems must
be solved, e.g., when solving the inverse problem. Note that
for BEM the operation count is
the matrix and solution to one forward problem, although the
number of operations could be lowered according to recent
results , .
The total number of nodes
tween neighboring nodes as
in the worst case (when
the number of boundary nodes negligible compared to the total
, and the operation
for the assembling of
is related to the distance
for BEM, and for FPMM
is small enough as to make
256IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 52, NO. 2, FEBRUARY 2005
would increase 8 times the operation count for assembling the
matrices and the memory needed to store them, and the error in
4. In contrast, for BEM the number of nodes would increase 4
times, the operation count for assembling the matrix would in-
crease 16 times, just as the memory needed to store it, while the
error would be reduced by a factor smaller than FPMM’s 4.
We presented a numerical FPMM for solving the EEG for-
ward problem in bodies of piecewise constant conductivity and
but does not need a mesh connecting the nodes of the problem.
This can save the typically computationally expensive genera-
tion of these meshes for BEM and FEM in 3-D problems, espe-
cially for the intricate shapes involved in the layers for the EEG
and related problems.
From our comparisons with the widely used BEM, it is clear
that our FPMM needs more nodes to achieve solutions with
the same precision. This could have been predicted since BEM
problem by one. On the other hand, the matrix related to FPMM
is sparse, whereas the BEM matrices are full. This amounts
to important savings in time, storage requirements and in the
number of operations needed to assemble the matrices associ-
ated with the problem. Our results indicate that for a few thou-
sands nodes per surface, as in the usual descriptions of the head
surfaces used in BEM , the error of FPMM and BEM are of
the same order, whereas the operation count to solve the for-
ward problem is lower for BEM. These results also suggest that
for problems with a higher number of nodes (e.g., in surface de-
scriptions including the sulci and fissures of the cortical layer)
the FPMM would need less operations than BEM for solving
the forward problem with similar precision. These savings be-
come significant in view of the ever increasing number of nodes
being used as the demand for precision increases. When the in-
verse problem involves many consecutive computations of the
direct problem, the savings in the computational effort obtained
with our FPMM become an important advantage.
Forproblems witha limitednumber ofnodes themethodalso
produces good results and could be used in EEG related prob-
lems. In fact, our FPMM allows a simple computation of the
sensitivity of the forward problem solution to perturbations in
the geometry of the head model. We used this feature to study
the effect of perturbations in the geometry on the EEG inverse
problem. Preliminary results of this approach were presented in
a minimization with constraints. The search of the numerical
algorithm best suited to solve the problem is not the purpose
of this paper, although is a topic worth studying. In this paper,
we proposed a procedure to solve the problem taking advantage
of the sparsity of the matrices involved. However, other choices
are possible, and different alternatives may be considered in the
many characteristics (such as sparsity and the need of internal
nodes) with FEM, it would be interesting to study the use of fast
solvers designed for FEM  to solve (19).
Finally we would like to point out that although the method
was designed to solve specifically the EEG forward problem,
it could be used with little modifications to solve other partial
differential equations problems with boundaries known only at
a finite set of points.
 M. S. Hamäläinen and J. Sarvas, “Realistic conductivity geometry
model of the human head for interpretation of neuromagnetic data,”
IEEE Trans. Biomed. Eng., vol. 36, no. 2, pp. 165–171, Feb. 1989.
 J. C. de Munck, “A linear discretization of the volume conductor
boundary integral equation using analytically integrated elements,”
IEEE Trans. Biomed. Eng., vol. 39, no. 9, pp. 986–990, Sep. 1992.
 A. van Oosterom and J. Strackee, “The solid angle of a plane triangle,”
IEEE Trans. Biomed. Eng., vol. 30, no. 2, pp. 125–126, Feb. 1983.
 Y. Yan, P. L. Nunez, and R. T. Hart, “Finite-element model of the human
head: Scalp potentials due to dipole sources,” Meth. Biol. Eng. Comput.,
vol. 29, pp. 475–481, Sep. 1991.
 S. P. van den Broek, H. Zhou, and M. J. Peters, “Computation of neuro-
Biol. Eng. Comput., vol. 34, pp. 21–26, Jan. 1996.
 J. C. de Munck and M. J. Peters, “A fast method to compute surface
potentials in the multisphere model,” IEEE Trans. Biomed. Eng., vol.
40, no. 11, pp. 1166–1174, Nov. 1993.
 D. van’t Ent, J. C. de Munck, and A. L. Kaas, “A fast method to derive
realistic BEM models for E/MEG source reconstruction,” IEEE Trans.
Biomed. Eng., vol. 48, no. 12, pp. 1434–1443, 2001.
 S. N. Atluri and S. Shen, “The meshless local Petrov-Galerkin (MLPG)
method: A simple & less-costly alternative to the finite element and
boundary element methods,” Comput. Model. Eng. Sci., vol. 3, no. 1,
pp. 11–51, 2002.
 I. Babu˘ ska and J. M. Melenk, “The partition of unity method,” Int. J.
Numer. Meth. Eng., vol. 40, pp. 727–758, 1997.
 T. Belytschko, P. Krysl, and Y. Krongauz, “A three-dimensional explicit
element-free Galerkin method,” Int. J. Numer. Meth. Fluids, vol. 24, pp.
 T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl,
“Meshles methods: An overview and recent developments,” Comput.
Method. Appl. Mech. Eng., vol. 139, pp. 3–47, 1996.
 C. A. M. Duarte and J. T. Oden, “Hp clouds—An hp meshless method,”
Numer. Meth. Partial Diff. Eqs., vol. 12, pp. 673–705, 1996.
 G. E. Fasshauer, “Solving differential equations with radial basis func-
tions: Multilevel methods and smoothing,” Adv. Comput. Math., vol. 11,
pp. 139–159, 1999.
 Y. X. Mukherjee and S. Mukherjee, “The boundary node method for
potential problems,” Int. J. Numer. Meth. Eng., vol. 40, pp. 797–815,
 B. Nayroles, G. Touzot, and P. Villon, “Generalizing the finite element
method: Diffuse aproximation and diffuse elements,” Comput. Mech.,
vol. 10, pp. 307–318, 1992.
 E. Oñate, S. Idelsohn, O. C. Zienkiewicz, and R. L. Taylor, “A finite
point method in computational mechanics. Applications to convective
transport and fluid flow,” Int. J. Numer. Meth. Eng., vol. 39, no. 22, pp.
 T. Zhu, J.-D. Zhang, and S. N. Atluri, “A local boundary integral equa-
tion (LBIE) method in computational mechanics, and a meshless dis-
cretization approach,” Comput. Mech., vol. 21, pp. 223–235, 1998.
 J. C. de Munck, B. W. van Dijk, and H. Spekreijse, “Mathematical
dipoles are adequate to describe realistic generators of human brain
activity,” IEEE Trans. Biomed. Eng., vol. 35, no. 11, pp. 960–966, Nov.
 J. D. Jackson, Classical Electrodynamics, 2nd ed.
 B. Jiang, The Least-Squares Finite Element Method. Theory and Ap-
plications in Computational Fluid Dynamics and Electromagnetics, ser.
Scientific Computation.New York: Springer, 1998.
 C. Hèrault and Y. Marèchal, “Boundary and interface conditions in
meshless methods,” IEEE Trans. Magn., vol. 35, no. 3, pp. 1450–1453,
 C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, ser.
Automatic Computation. Upper Saddle River: Prentice-Hall, 1974.
Definite Systems.Upper Saddle River: Prentice-Hall, 1981.
 I. S. Duff, “Sparse numerical linear algebra: Direct methods and pre-
conditioning,” in The State of the Art in Numerical Analysis, I. S. Duff
and G. A. Watson, Eds.Oxford, U.K.: Oxford Univ. Press, 1997, pp.
27–62. [Online]. Available: citeseer.ist.psu.edu/duff96sparse.html.
New York: Wiley,
VON ELLENRIEDER: MESHLESS METHOD FOR SOLVING THE EEG FORWARD PROBLEM257
 J. W. H. Meijs, O. W. Weier, M. J. Peters, and A. van Oosterom, “On
the numerical accuracy of the boundary element method,” IEEE Trans.
Biomed. Eng., vol. 36, no. 10, pp. 1038–1049, Oct. 1989.
 N. Krzebek and S. Sauter, “Fast Cluster Techniques for BEM,” Univ.
Zurich, Zurich, Switzerland, [Online]. Available: http://www.math.
 G. Alléon, M. Benzi, and L. Giraud, “Sparse approximated inverse pre-
conditioning for dense linear systems arising in computational electro-
magnetics,” Numer. Algorithms, vol. 16, no. 1, pp. 1–15, 1997.
 N. von Ellenrieder, C. H. Muravchik, and A. Nehorai, “Effect of per-
turbations in the geometry on the EEG inverse problem,” in Proc. 2003
IEEE Workshop Statistical Signal Processing, Oct. 2003, pp. 273–276.
Nicolás von Ellenrieder (S’01) was born in Ar-
gentina, April 27, 1975. He graduated in 1998 as an
Electronics Engineer from the National University
of La Plata, La Plata, Argentina, where he is working
towards the Ph.D. degree.
He is an Assistant Lecturer at the Department
of the Electrical Engineering of the National Uni-
versity of La Plata and a member of its Industrial
Electronics, Control and Instrumentation Laboratory
(LEICI). His research interests include statistical
processing of biomedical signals.
Carlos H. Muravchik (S’81–M’83–SM’99) was
born in Argentina, June 11, 1951. He graduated as an
Electronics Engineer from the National University of
La Plata, La Plata, Argentina, in 1973. He received
the M.Sc. in statistics (1983) and the M.Sc. (1980)
and Ph.D. (1983) degrees in electrical engineering,
from Stanford University, Stanford, CA.
He is a Professor at the Department of the Elec-
trical Engineering of the National University of La
trol and Instrumentation Laboratory (LEICI). He is
also a member of the Comision de Investigaciones Cientificas de la Pcia. de
Buenos Aires. He was a Visiting Professor at Yale University, New Haven, CT,
in 1983 and 1994, and at the University of Illinois at Chicago in 1996, 1997,
1999, and 2003. His research interests are in the area of statistical signal and
array processing with biomedical, control, and communications applications,
and nonlinear control systems.
Since 1999, Dr. Muravchik is a member of the Advisory Board of the journal
Latin American Applied Research and is currently an Associate Editor of the
IEEE TRANSACTIONS ON SIGNAL PROCESSING.
Arye Nehorai (S’80–M’83–SM’90–F’94) received
the the B.Sc. and M.Sc. degrees in electrical engi-
neering from the Technion, Haifa, Israel, and the
Ph.D. degree in electrical engineering from Stanford
University, Stanford, CA.
After graduation he worked as a Research Engi-
neer for Systems Control Technology, Inc., in Palo
Alto, CA. From 1985 to 1989, he was Assistant
Professor and from 1989 to 1995 he was Associate
Professor with the Department of Electrical Engi-
neering at Yale University, New Haven, CT. In 1995,
he joined the Department of Electrical Engineering and Computer Science at
The University of Illinois at Chicago (UIC), as a Full Professor. From 2000 to
2001, he was Chair of the department’s Electrical and Computer Engineering
(ECE) Division, which is now a new department. In 2001, he was named
University Scholar of the University of Illinois. He holds a joint professorship
with the ECE and Bioengineering Departments at UIC. His research interests
are in signal processing, communications, and biomedicine.
Dr. Nehorai is Vice President-Publications and Chair of the Publications
Board of the IEEE Signal Processing Society. He is also a member of the Board
of Governors and of the Executive Committee of this Society. He was Ed-
itor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from January
2000 to December 2002, and is currently a Member of the Editorial Board of
Signal Processing, the IEEE Signal Processing Magazine, and The Journal of
the Franklin Institute. He is the founder and Guest Editor ofthe special columns
on Leadership Reflections in the IEEE Signal Processing Magazine. He has
previously been an Associate Editor of IEEE TRANSACTIONS ON ACOUSTICS,
SPEECH, AND SIGNAL PROCESSING, IEEE SIGNAL PROCESSING LETTERS, the
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE JOURNAL OF
OCEANIC ENGINEERING, and Circuits, Systems, and Signal Processing. He
served as Chairman of the Connecticut IEEE Signal Processing Chapter from
1986 to 1995, and a Founding Member, Vice-Chair, and later Chair of the
IEEE Signal Processing Society’s Technical Committee on Sensor Array and
Multichannel (SAM) Processing from 1998 to 2002. He was the co-General
Chair of the First and Second IEEE SAM Signal Processing Workshops held
in 2000 and 2002. He was co-recipient, with P. Stoica, of the 1989 IEEE
Signal Processing Society’s Senior Award for Best Paper, and co-author of the
2003 Young Author Best Paper Award of this Society, with A. Dogandzic. He
received the Faculty Research Award from the UIC College of Engineering in
1999 and was Adviser of the UIC Outstanding Ph.D. Thesis Award in 2001. He
was elected Distinguished Lecturer of the IEEE Signal Processing Society for
the term 2004 to 2005. He has been a Fellow of the Royal Statistical Society