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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 52, NO. 2, FEBRUARY 2005249

A Meshless Method for Solving the EEG

Forward Problem

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-

staticMaxwellequationsandcomputetheelectroencephalography

(EEG) forward problem solution. More generally, we develop a

computationally efficient method to obtain the electric potential

distributiongeneratedbyasourceofelectricactivityinsideathree-

dimensionalbodyofarbitraryshapeandlayersofdifferentelectric

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.

I. INTRODUCTION

E

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-

ters,i.e.,knownsourceintensity,orientation,andlocation,isthe

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:

nellen@ieee.org, ellenrie@ing.unlp.edu.ar).

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 [1]. 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 [1]–[5].

Very often spherical approximations to the head surfaces are

used to reduce the computational load of these methods [6].

Once the geometry of the problem is known it is easy to se-

lectthelocationsofthenodes.Butthegenerationofameshcon-

nectingthemtodefinetheelementsisamoredifficult,time-con-

sumingjob,especiallyforFEM.AlthoughinBEMthemeshcan

be generated automatically [7], 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

bedesired.Anumberofmethodsthatdonotrequireameshcon-

necting the nodes have been developed for a wide range of ap-

plications [8]–[17]. 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

withthewell-knownBEM.Finally,inSectionVIwediscussthe

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-

triclayers

ofdifferentconductivities

layer, representing the tissues. The surfaces

and

th layers are assumed to be smooth. In the common

three-layersmodel,theyrepresentthebrain,skull,andscalp,see

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.

,constantwithineach

between the th

0018-9294/$20.00 © 2005 IEEE

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250IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 52, NO. 2, FEBRUARY 2005

Fig. 1.

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 [18]. Let

dipolepositionandmoment(intensity andorientation

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:

andbe the

),re-

(1)

where

Laplacian operator and

potentialat

is located is

. For any point

sequence inside

sequence outside

challengingduetospatiallyrapidvariationsofthemathematical

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

problemrestatementhasbeenusedinFEMformulations[5].By

linearity, the electric potential can be found as the superposition

of the two terms

given by

is any point in the domainis the

is the gradient operator. The electric

,theelectricconductivityofthelayerwhere

, 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

is

, where each term is

(2)

(3)

The term

ated by a dipole source in an infinite homogeneous medium of

conductivity

.Thistermhasasingularitysincetheelectric

potential is not defined at the dipole’s position, but it presents

no difficulty because it can be solved analytically, yielding the

following expression:

corresponds to the electric potential gener-

(4)

The term

Neumann boundary conditions [19], 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

an interface.

From now on this procedure will be referred to as the singu-

larityseparationmethod(SSM).Thefinitepointsmixedmethod

(FPMM)presentedintheSectionIIIwillbeusedforsolvingthe

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-

tionofshapefunctions,eachoneofthemassociatedtoonenode.

Thecoefficientsofthisglobalapproximationarechosentosolve

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 [20].

First, we show how to obtain the shape functions that will

be used to construct the global approximation of the electric

potential,thenweexplainhowtheforwardproblemissolvedby

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., [17], but

we include it here to introduce the notation and our choice of

local basis and weight functions.

Considerasetofnodes

inthedomain

, of any layer

component

of the electric potential is approximated in the

neighborhood of any point

andonthesurface

. The

assuming a simple variation

(5)

where

vector of

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,

is the

, and the basis is

(6)

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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

nodes

.Wewillshowthatthispotentials

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

The vector

associated with the point

minimizing the functional

of nodesaround. Let

arethe

in (5), which contains

.

is obtained then by

(7)

i.e.,

potentials

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

with radius

, the size of the of nodes near

number of nodes participating in the determination of the co-

efficients

. The smoothness of the resulting approximation

is related to the smoothness of the weight functions [9], 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 [10], [17]. 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

(8)

where

Equation (7) poses a moving (due to its dependence on

) least-squares problem for the vector

and

where

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.

. Defining

,

(9)

where

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-

tials

, 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

(10)

The function

. 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 [9]. Note that since

, this method only approximates the electric potential of

the nodes but does not interpolate their value, as BEM or FEM

do.

In the following subsections, we will show how to compute

thevector

andobtaintheglobalapproximationtothesolution

of the EEG forward problem.

is called the shape function [11] of the node

is a

, and the coefficients of this

.

for

A. Boundary Conditions

Most of the known meshless methods solve the Neumann

boundary conditions problems with null normal flux across the

surface [10], [15]–[17]. 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 [20]. Since we

only know a set of points on the surfaces, it is natural to use a

point collocation technique [16]. The solution of the weak inte-

gralformulationoftheboundaryconditionsbypointcollocation

is equivalent to evaluating the boundary conditions of (3) on the

surfacenodes.From(10)anexpressionisfoundfortheapproxi-

mation to the normal component of the potential gradient at any

point

on a surface

(11)

Now (3) can be evaluated for every surface node by means of

(10) and (11). For any node this involves the coefficients

theneighboringnodes,yieldingthefollowing linearsystemsfor

each layer:

of

(12)

(13)

where

and

,

is the total number of nodes,andare rectangular

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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

combinationofsmoothshapefunctions,and cannotexactlyrep-

resent this jump in the normal component of the gradient. There

aresomemethodstoovercomethisdifficulty[11],[21],butthey

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-

script

, must be at the same position than the internal surface

nodes of the following layer, denoted with the superscript

For each layer

the matrices

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.

Thefollowingsystemsareassembledtoimposetheboundary

conditions of (3), for

internal nodes in layer . The

.

and are

is

(14)

where

, 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

(15)

where the coefficient vector is

of the matrix

and

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

on (10)

(16)

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

notexactlysatisfytheLaplacedifferentialequationinthewhole

does

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

systems

, we get the linear

(17)

Combining them, we obtain the block diagonal system

(18)

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-

tionsandthedifferentialequationinthedomainarenotsatisfied

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

nodes.Onepossiblesolutionisfindingthevectorthatminimizes

the Laplace equation at the nodes, constrained by the boundary

conditions, i.e.,

(19)

There exist several algorithms to solve this well-known con-

strained optimization problem [22]. 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

(20)

where

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

matrix

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 [23], [24]. Note that it is

notconvenienttocomputeexplicitlytheproduct

sparsity could be lost. Also, the value of

too small in order to avoid numerical instabilities, we obtained

good results with

.

, with.

sincethe

in (20) should not be

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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-

is then

(21)

with

.

IV. VARIATIONS

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

vector

is assumed constant in the neighborhood of . We can

write then from (9) and (10)

(22)

where

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

(23)

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.

andis

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.

Let

be thematrix formed by the rows of

to internal nodes, then the coefficients

that correspond

are obtained as

(24)

WewilldenotethisunconstrainedversionofthemethodasSLS.

C. Singularity Separation

Compared with the other layers the skull has a low electric

conductivity. This causes a shielding effect, hence the potential

intheouterlayershasamuchlowermagnitudethanthepotential

generatedbythesamesourceinaninfinitehomogeneousmedia.

The SSM of Section II consisted in computing the relatively

small electric potential

as the sum of the relatively large

andthenumericallycalculated

opposite signs and tends to cancel

errors in

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

electric potential

for

on (3) change to

.Hencethelatterhas

. Then small relative

. Thus,

th layers only, the

for and

. Then the boundary conditions

(25)

(26)

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),

the term

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

magnitudesmallerthanofitsneighboringlayers.Thisimportant

discontinuity causes some numerical instability when solving

the forward problem. The numerical stability of the results ob-

tainedwithBEMimprovesconsiderablyusingtheisolatedskull

approach (ISA) [25]. 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.

FortheFPMM,firstweassumetheskullhasnullconductivity

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

and

, where

are

(27)

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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

boundary conditions

(28)

(29)

Let

forward problem solution is given by

be the solution of this full-size problem, then the EEG

(30)

V. RESULTS

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 [6] we can compare our numerical

resultsto.Asareference,wealsocompareourmeshlessmethod

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 [25] and are defined by

(31)

(32)

where

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

orientationandposition.SincetheNRDMmeasureinvolvesfive

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

equal,

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 [2]. 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.

Randomlocationsforthenodes,withafixeddensity,wherealso

tested and do not affect the results significantly.

isavectorofanalyticallycomputedpotentialsand

Fig. 2.

sphere. Results are shown for three variations of our method: LD, MD, and

LD-SLS.

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-

rameter

of the weight function was set to

InFig.2,weshowacomparisonbetweendifferentversionsof

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

NRDMasafunctionofthedistancefromthesourcetothecenter

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

whensolvedwithaSLS.Theperformanceofthisversionisvery

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

radii

used because a series is known for the analytical solution [6].

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

fortheISAversionoftheFPMMarenotshownbecausetheyare

almost identical to the results obtained when SSM is used only

in the innermost layer. This could be expected because the SSM

alreadyexploitstheprincipleonwhichtheISAmethodisbased.

Compared with BEM, the performance of FPMM is very

good for deep sources, but for sources near the surface of the

.

. The

, and 642 nodes

and

. The spherical geometry was

. For BEM two

for the

for the remaining layers. The

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VON ELLENRIEDER: MESHLESS METHOD FOR SOLVING THE EEG FORWARD PROBLEM 255

Fig. 3.

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.

Fig. 4.

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

regularlydistributedoverasphereofradius

convergence rate is at least

gence tests, FPMM also showed

other meshless methods [13], 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

Wealsocomparedtheconvergenceratebetweenthemeshless

method and ISA-BEM, in Fig. 6. It shows the NRDM as a func-

tionofthenumberofnodespersurface,fortangentiallyoriented

sources placed on spheres at depth

thattheconvergencerateforFPMMishigherthanforBEM.We

emphasize that this comparison was made for the same number

andlocationofsurfacenodes.Thetotalnodesare246,486,966,

and 1926 for BEM and 805, 1700, 3575, and 7560 for FPMM.

.Theobserved

. In 1-D and 2-D conver-

convergence. Unlike

in (7).

and. We can see

Fig. 5.

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

full line.

Convergence of the FPMM. NRDM as a function of the separation

Fig. 6.

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

[23], [24]. 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 [26], [27].

The total number of nodes

tween neighboring nodes as

in the worst case (when

the number of boundary nodes negligible compared to the total

numberofnodes).Ifwewanttoreduce

, and the operation

for the assembling of

is related to the distance

for BEM, and for FPMM

is small enough as to make

be-

byafactortwo,FPMM

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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

thesolutionoftheforwardproblemwouldbereducedbyafactor

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.

VI. DISCUSSION

We presented a numerical FPMM for solving the EEG for-

ward problem in bodies of piecewise constant conductivity and

arbitraryshape.Themethodusesbothsurfaceandvolumenodes

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

needsonlysurfacenodes,roughlyloweringthedimensionofthe

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 [7], 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

[28].

ThesolutionoftheforwardproblemwithourFPMMinvolves

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

searchforthebestmethod.Forinstance,sinceourFPMMshares

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 [24] 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.

REFERENCES

[1] 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.

[2] 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.

[3] 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.

[4] 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.

[5] S. P. van den Broek, H. Zhou, and M. J. Peters, “Computation of neuro-

magneticfieldsusingfinite-elementsmethodandBiot-Savartlaw,”Med.

Biol. Eng. Comput., vol. 34, pp. 21–26, Jan. 1996.

[6] 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.

[7] 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.

[8] 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.

[9] I. Babu˘ ska and J. M. Melenk, “The partition of unity method,” Int. J.

Numer. Meth. Eng., vol. 40, pp. 727–758, 1997.

[10] T. Belytschko, P. Krysl, and Y. Krongauz, “A three-dimensional explicit

element-free Galerkin method,” Int. J. Numer. Meth. Fluids, vol. 24, pp.

1253–1270, 1997.

[11] 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.

[12] 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.

[13] G. E. Fasshauer, “Solving differential equations with radial basis func-

tions: Multilevel methods and smoothing,” Adv. Comput. Math., vol. 11,

pp. 139–159, 1999.

[14] Y. X. Mukherjee and S. Mukherjee, “The boundary node method for

potential problems,” Int. J. Numer. Meth. Eng., vol. 40, pp. 797–815,

1997.

[15] 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.

[16] 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.

3839–3886, 1996.

[17] 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.

[18] 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.

1988.

[19] J. D. Jackson, Classical Electrodynamics, 2nd ed.

1975.

[20] 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.

[21] C. Hèrault and Y. Marèchal, “Boundary and interface conditions in

meshless methods,” IEEE Trans. Magn., vol. 35, no. 3, pp. 1450–1453,

May 1999.

[22] C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, ser.

Automatic Computation. Upper Saddle River: Prentice-Hall, 1974.

[23] A.GeorgeandJ.W.H.Liu,ComputerSolutionsofLargeSparsePositive

Definite Systems.Upper Saddle River: Prentice-Hall, 1981.

[24] 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,

Page 9

VON ELLENRIEDER: MESHLESS METHOD FOR SOLVING THE EEG FORWARD PROBLEM257

[25] 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.

[26] N. Krzebek and S. Sauter, “Fast Cluster Techniques for BEM,” Univ.

Zurich, Zurich, Switzerland, [Online]. Available: http://www.math.

unizh.ch/fileadmin/math/preprints/04-02rev.pdf, 2002.

[27] 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.

[28] 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

PlataandamemberofitsIndustrialElectronics,Con-

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

since 1996.