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Open AcceReviewReview on solving the forward problem in EEG source analysisHans Hallez*1, Bart Vanrumste*2,3, Roberta Grech4, Joseph Muscat6, Wim De Clercq2, Anneleen Vergult2, Yves D'Asseler1, Kenneth P Camilleri5, Simon G Fabri5, Sabine Van Huffel2 and Ignace Lemahieu1
Address: 1ELIS-MEDISIP, Ghent University, Ghent, Belgium, 2ESAT, K.U.Leuven, Leuven, Belgium, 3Katholieke Hogeschool Kempen, Geel, Belgium, 4Department of Mathematics, University of Malta Junior College, Malta, 5Faculty of Engineering, University of Malta, Malta and 6Department of Mathematics, University of Malta, Malta

Email: Hans Hallez* - [email protected]; Bart Vanrumste* - [email protected]; Roberta Grech - [email protected]; Joseph Muscat - [email protected]; Wim De Clercq - [email protected]; Anneleen Vergult - [email protected]; Yves D'Asseler - [email protected]; Kenneth P Camilleri - [email protected]; Simon G Fabri - [email protected]; Sabine Van Huffel - [email protected]; Ignace Lemahieu - [email protected]

* Corresponding authors

AbstractBackground: The aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG wavesof interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electricalsource and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which isdefined as finding brain sources which are responsible for the measured potentials at the EEG electrodes.

Methods: While other reviews give an extensive summary of the both forward and inverse problem, this review article focuseson different aspects of solving the forward problem and it is intended for newcomers in this research field.

Results: It starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidalneurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumannand Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and whitematter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last twodecades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images,which requires numerical methods. The following methods are compared with each other: the boundary element method(BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conductingcompartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization.It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for eachdipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterativemethods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method.

Conclusion: Solving the forward problem has been well documented in the past decades. In the past simplified spherical headmodels are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of thehead model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of thetissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivityvalues is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters ofthe head model and of the numerical techniques on the solution of the forward problem.

Published: 30 November 2007

Journal of NeuroEngineering and Rehabilitation 2007, 4:46 doi:10.1186/1743-0003-4-46

Received: 5 January 2007Accepted: 30 November 2007

This article is available from: http://www.jneuroengrehab.com/content/4/1/46

© 2007 Hallez et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Journal of NeuroEngineering and Rehabilitation 2007, 4:46 http://www.jneuroengrehab.com/content/4/1/46

IntroductionSince the 1930s electrical activity of the brain has beenmeasured by surface electrodes connected to the scalp [1].Potential differences between these electrodes were thenplotted as a function of time in a so-called electroencepha-logram (EEG). The information extracted from these brainwaves was, and still is instrumental in the diagnoses ofneurological diseases [2], mainly epilepsy. Since the1960s the EEG was also used to measure event-relatedpotentials (ERPs). Here brain waves were triggered by astimulus. These stimuli could be of visual, auditory andsomatosensory nature. Different ERP protocols are nowroutinely used in a clinical neurophysiology lab.Researchers nowadays are still searching for new ERP pro-tocols which may be able to distinguish between ERPs ofpatients with a certain condition and ERPs of normal sub-jects. This could be instrumental in disorders, such as psy-chiatric and developmental disorders, where there is oftena lack of biological objective measures.

During the last two decades, increasing computationalpower has given researchers the tools to go a step furtherand try to find the underlying sources which generate theEEG. This activity is called EEG source localization. It con-sists of solving a forward and inverse problem. Solving theforward problem starts from a given electrical source con-figuration representing active neurons in the head. Thenthe potentials at the electrodes are calculated for this con-figuration. The inverse problem attempts to find the elec-trical source which generates a measured EEG. By solvingthe inverse problem, repeated solutions of the forwardproblem for different source configurations are needed. Areview on solving the inverse problem is given in [3].

In this review article several aspects of solving the forwardproblem in EEG source localization will be discussed. It isintended for researchers new in the field to get insight inthe state-of-the-art techniques to solve the forward prob-lem in EEG source analysis. It also provides an extensivelist of references to the work of other researchers.

First, the physical context of EEG source localization willbe elaborated on and then the derivation of Poisson'sequation with its boundary conditions. An analyticalexpression is then given for a three-shell spherical headmodel. Along with realistic head models, obtained frommedical images, numerical methods are then introducedthat are necessary to solve the forward problem. Severalnumerical techniques, the Boundary Element Method(BEM), the Finite Element Method (FEM) and the FiniteDifference Method (FDM), will be discussed. Also aniso-tropic conductivities which can be found in the whitematter compartment and skull, will be handled.

The reciprocity theorem used to speed up the calculations,is discussed. The electric field that results at the dipolelocation within the brain due to current injection andwithdrawal at the surface electrode sites is first calculated.The forward transfer-coefficients are obtained from thescalar product of this electric field and the dipolemoment. Calculations are thus performed for each elec-trode position rather than for each dipole position. Thisspeeds up the time necessary to do the forward calcula-tions since the number of electrodes is much smaller thanthe number of dipoles that need to be calculated.

The number of unknowns in the FEM and FDM can easilyexceed the million and thus lead to large but sparse linearsystems. As the number of unknowns is too large to solvethe system in a direct manner, iterative solvers need to beused. Some popular iterative solvers are discussed such assuccessive over-relaxation (SOR), conjugate gradientmethod (CGM) and algebraic multigrid methods (AMG).

The physics of EEGIn this section the physiology of the EEG will be shortlydescribed. In our opinion, it is important to know theunderlying mechanisms of the EEG. Moreover, forwardmodeling also involves a good model for the generators ofthe EEG. The mechanisms of the neuronal actionpoten-tials, excitatory post-synaptic potentials and inhibitorypost-synaptic potentials are very complex. In this sectionwe want to give a very comprehensive overview of theunderlying neurophysiology.

NeurophysiologyThe brain consists of about 1010 nerve cells or neurons.The shape and size of the neurons vary but they all possessthe same anatomical subdivision. The soma or cell bodycontains the nucleus of the cell. The dendrites, arisingfrom the soma and repeatedly branching, are specializedin receiving inputs from other nerve cells. Via the axon,impulses are sent to other neurons. The axon's end isdivided into branches which form synapses with otherneurons. The synapse is a specialized interface betweentwo nerve cells. The synapse consists of a cleft between apresynaptic and postsynaptic neuron. At the end of thebranches originating from the axon, the presynaptic neu-ron contains small rounded swellings which contain theneurotransmitter substance. Further readings on the anat-omy of the brain can be found in [4] and [5].

One neuron generates a small amount of electrical activ-ity. This small amount cannot be picked up by surfaceelectrodes, as it is overwhelmed by other electrical activityfrom neighbouring neuron groups. When a large group ofneurons is simultaneously active, the electrical activity islarge enough to be picked up by the electrodes at the sur-face and thus generating the EEG. The electrical activity



can be modeled as a current dipole. The current flowcauses an electric field and also a potential field inside thehuman head. The electric field and potential field spreadsto the surface of the head and an electrode at a certainpoint can measure the potential [2].

At rest the intracellular environment of a neuron is nega-tively polarized at approximately -70 mV compared withthe extracellular environment. The potential difference isdue to an unequal distribution of Na+, K+ and Cl- ionsacross the cell membrane. This unequal distribution ismaintained by the Na+ and K+ ion pumps located in thecell membrane. The Goldman-Hodgkin-Katz equationdescribes this resting potential and this potential has beenverified by experimental results [2,6,7].

The neuron's task is to process and transmit signals. Thisis done by an alternating chain of electrical and chemicalsignals. Active neurons secrete a neurotransmitter, whichis a chemical substance, at the synaptical side. The syn-apses are mainly localized at the dendrites and the cellbody of the postsynaptic cell. A postsynaptic neuron has alarge number of receptors on its membrane that are sensi-tive for this neurotrans-mitter. The neurotransmitter incontact with the receptors changes the permeability of themembrane for charged ions. Two kinds of neurotransmit-

ters exist. On the one hand there is a neurotransmitterwhich lets signals proliferate. These molecules cause aninflux of positive ions. Hence depolarization of the intra-cellular space takes place. A depolarization means that thepotential difference between the intra- and extracellularenvironment decreases. Instead of -70 mV the potentialdifference becomes -40 mV. This depolarization is alsocalled an excitatory postsynaptic potential (EPSP). On theother hand there are neurotransmitters that stop the pro-liferation of signals. These molecules will cause an out-flow of positive ions. Hence a hyperpolarization can bedetected in the intracellular volume. A hyperpolarizationmeans that the potential difference between the intra- andextracellular environment increases. This potential changeis also called an inhibitory postsynaptic potential (IPSP).There are a large number of synapses from different pres-ynaptic neurons in contact with one postsynaptic neuron.At the cell body all the EPSP and IPSP signals are inte-grated. When a net depolarization of the intracellularcompartment at the cell body reaches a certain threshold,an action potential is generated. An action potential thenpropagates along the axon to other neurons [2,6,7].

Figure 1 illustrates the excitatory and inhibitory postsyn-aptic potentials. It also shows the generation of an action

Excitatory and inhibitory post synaptic potentialsFigure 1Excitatory and inhibitory post synaptic potentials. An illustration of the action potentials and post synaptic potentials measured at different locations at the neuron. On the left a neuron is displayed and three probes are drawn at the location where the potential is measured. The above picture on the right shows the incoming exitatory action potentials measured at the probe at the top, at the probe in the middle the incoming inhibitory action potential is measured and shown. The neuron processes the incoming potentials: the excitatory action potentials are transformed into excitatory post synaptic potentials, the inhibitory action potentials are transformed into inhibitory post synaptic potentials. When two excitatory post synaptic poten-tials occur in a small time frame, the neuron fires. This is shown at the bottom figure. The dotted line shows the EPSP, in case there was no second excitatory action potential following. From [2].

excitatory presynaptic activity

inhibitory presynaptic activity

postsynaptic activity

EPSP IPSP

t

t

t

0mV

-60

0

-60

mV

0

-60

mV



potential. Further readings on the electrophysiology ofneurons can be found in [2,6].

The generators of the EEGThe electrodes used in scalp EEG are large and remote.They only detect summed activities of a large number ofneurons which are synchronously electrically active. Theaction potentials can be large in amplitude (70–110 mV)but they have a small time course (0.3 ms). A synchronousfiring of action potentials of neighboring neurons isunlikely. The postsynaptic potentials are the generators ofthe extracellular potential field which can be recordedwith an EEG. Their time course is larger (10–20 ms). Thisenables summed activity of neighboring neurons. How-ever their amplitude is smaller (0.1–10 mV) [3,8].

Apart from having more or less synchronous activity, theneurons need to be regularly arranged to have a measura-ble scalp EEG signal. The spatial properties of the neuronsmust be so that they amplify each other's extracellularpotential fields. The neighboring pyramidal cells areorganized so that the axes of their dendrite tree are parallelwith each other and normal to the cortical surface. Hence,these cells are suggested to be the generators of the EEG.

The following is focused on excitatory synapses and EPSP,located at the apical dendrites of a pyramidal cell. Theneurotransmitter in the excitatory synapses causes aninflux of positive ions at the postsynaptic membrane asillustrated in figure 2(a) and depolarizes the local cellmembrane. This causes a lack of extracellular positive ionsat the apical dendrites of the postsynaptic neuron. A redis-

tribution of positively charged ions also takes place at theintracellular side. These ions flow from the apical dendriteto the cell body and depolarize the membrane potentialsat the cell body. Subsequently positive charged ionsbecome available at the extracellular side at the cell bodyand basal dendrites.

A migration of positively charged ions from the cell bodyand the basal dendrites to the apical dendrite occurs,which is illustrated in figure 2(a) with current lines. Thisconfiguration generates extracellular potentials. Othermembrane activities start to compensate for the massiveintrusion of the positively charged ions at the apical den-drite, however these mechanisms are beyond the scope ofthis work and can be found elsewhere [2,9,10].

A simplified equivalent electric circuit is presented in fig-ure 2(b) to illustrate the initial activity of an EPSP. At rest,the potential difference between the intra- and extracellu-lar compartments can be represented by charged capaci-tors. One capacitor models the potential difference at theapical dendrites side while a second capacitor models thepotential difference at the cell body and basal dendriteside. The potential difference over the capacitors is 60 mV.The neurotransmitter causes a massive intrusion of posi-tively charged ions at the postsynaptic membrane at theapical dendrite side. In the equivalent circuit, this is mod-eled by a switch that is closed. The capacitor at the cellbody side discharges causing a current flow over the extra-cellular resistor Re and the intracellular resistor Ri. Therepolarization of the cell membrane at the apical side orthe initiation of the action potential are not modeled withthis simple equivalent electrical circuit.

The capacitors and the switch, in figure 2(b), represent amodel of the electrical source at the initial phase of thedepolarization of the neuron. They could also be replacedby a time dependent current source, however this repre-sentation is not ideal. The capacitor representation, for theinitial phase of depolarization, fits closer the occurringphysical phenomena. The impedance of the tissue in thehuman head has, for the frequencies contained in theEEG, no capacitive nor inductive component and is hencepure resistive. More advanced equivalent electrical circuitscan be found elsewhere [10]. The fact that a current flowsthrough the extracellular resistor indicates that potentialdifferences in the extracellular space can be measured.

A simplified electrical model for this active cell consists oftwo current monopoles: a current sink at the apical den-drite side which removes positively charged ions from theextracellular environment, and a current source at the cellbody side which injects positively charged ions in theextracellular environment. The extracellular resistance Recan be decomposed in the volume conductor model in

equivalent circuit for a neuronFigure 2equivalent circuit for a neuron. An excitatory post syn-aptic potential, an simplified equivalent circuit for a neuron, and a resistive network for the extracellular environment. A neuron with an excitatory synapse at the apical dendrite is presented in (a). From [2]. A simplified equivalent circuit is depicted in (b). The extracellular environment can be repre-sented with a resistive network as illustrated in (c).



which the active neuron is embedded, as illustrated in fig-ure 2(c). For further reading on the generation of the EEGone can refer to [11] and [9].

Poisson's equation, boundary conditions and dipolesIn the previous sections we saw that the generators of theEEG are the synaptic potentials along the apical dendritesof the pyramidal cells of the grey matter cortex. It is impor-tant to notice that the EEG reflects the electrical activity ofa subgroup of neurons, especially pyramidal neuron cells,where the apical dendrite is systematically orientedorthogonal to the brain surface. Certain types of neuronsare not systematically oriented orthogonal to the brainsurface. Therefore, the potential fields of the synaptic cur-rents at different dendrites of neurons van cancel eachother out. In that case the neuronal activity is not visibleat the surface. Moreover, that actionpotentials, propagat-ing along the axons, have no influence on the EEG. Theirshort timespan (2 ms) make the chance of generatingsimultaneous actionpotentials very small [6,12]. In thissection, a mathematical approach on the generation of theforward problem is given.

Quasi-static conditionsIt is shown in [13] that no charge can be piled up in theconducting extracellular volume for the frequency rangeof the signals measured in the EEG. At one moment intime all the fields are triggered by the active electric source.Hence, no time delay effects are introduced. All fields andcurrents behave as if they were stationary at each instance.These conditions are also called quasi-static conditions.They are not static because the neural activity changeswith time. But the changes are slow compared to the prop-agation effects.

Applying the divergence operator to the current densityPoisson's equation gives a relationship between thepotentials at any position in a volume conductor and theapplied current sources. The mathematical derivation ofPoisson's equation via Maxwell's equations, can be foundin various textbooks on electromagnetism [6,10,14]. Pois-son's equation is derived with the divergence operator. Inthis way the emphasis is, in our opinion, more on thephysical aspect of the problem. Furthermore, the conceptsintroduced in [10,14], such as current source and currentsink, are used when applying the divergence operator.

DefinitionThe current density is a vector field and can be representedby J(x, y, z). The unit of the current density is A/m2. Thedivergence of a vector field J is defined as follows:

The integral over a closed surface ∂G represents a flux or acurrent. This integral is positive when a net current leavesthe volume G and is negative when a net current enters thevolume G. The vector dS for a surface element of ∂G witharea dS and outward normal en, can also be written asendS. The unit of ∇·J is A/m3 and is often called the currentsource density which in [15] is symbolized with Im. Gen-erally one can write:

∇·J = Im. (2)

Applying the divergence operator to the extracellular current densityFirst a small volume in the extracellular space, whichencloses a current source and current sink, is investigated.The current flowing into the infinitely small volume, mustbe equal to the current leaving that volume. This is due tothe fact that no charge can be piled up in the extracellularspace. The surface integral of equation (1) is then zero,hence ∇·J = 0.

In the second case a volume enclosed by the current sinkwith position parameters r1(x1, y1, z1) is assumed. The cur-rent sink represents the removal of positively charged ionsat the apical dendrite of the pyramidal cell. The integral ofequation (1) remains equal to -I while the volume G inthe denominator becomes infinitesimally small. Thisgives a singularity for the current source density. This sin-gularity can be written as a delta function: -Iδ(r - r1). Thenegative sign indicates that current is removed from theextracellular volume. The delta function indicates thatcurrent is removed at one point in space.

For the third case a small volume around the currentsource at position r2(x2, y2, z2) is constructed. The currentsource represents the injection of positively charged ionsat the cell body of the pyramidal cell. The current sourcedensity equals Iδ(r - r2). Figure 3 represents the currentdensity vectors for a current source and current sink con-figuration. Furthermore, three boxes are presented corre-sponding with the three cases discussed above.

Uniting the three cases given above, one obtains:

∇·J = Iδ(r - r2) - Iδ(r - r1). (3)

Ohm's law, the potential field and anisotropic/isotropic conductivitiesThe relationship between the current density J in A/m2 andthe electric field E in V/m is given by Ohm's law:

J = σE, (4)

with σ(r) ∈ �3×3 being the position dependent conductiv-ity tensor given by:∇ ⋅ =

→ ∂∫J J SlimG GG d01 (1)



and with units A/(Vm) = S/m. There are tissues in thehuman head that have an anisotropic conductivity. Thismeans that the conductivity is not equal in every directionand that the electric field can induce a current densitycomponent perpendicular to it with the appropriate σ inequation (4).

At the skull, for example, the conductivity tangential tothe surface is 10 times [16] the conductivity perpendicularto the surface (see figure 4(a)). The rationale for this isthat the skull consists of 3 layers: a spongiform layerbetween two hard layers. Water, and also ionized parti-cles, can move easily through the spongiform layer, butnot through the hard layers [17]. Wolters et al. state thatskull anisotropy has a smearing effect on the forwardpotential computation. The deeper a source lies, the moreit is surrounded by anisotropic tissue, the larger the influ-ence of the anisotropy on the resulting electric field.Therefore, the presence of anisotropic conducting tissuescompromises the forward potential computation and as aconsequence, the inverse problem [18].

White matter consists of different nerve bundles (groupsof axons) connecting cortical grey matter (mainly den-drites and cell bodies). The nerve bundles consist of nerve

fibres or axons (see figure 4(b)). Water and ionized parti-cles can move more easily along the nerve bundle thanperpendicular to the nerve bundle. Therefore, the conduc-tivity along the nerve bundle is measured to be 9 timeshigher than perpendicular to it [19,20]. The nerve bundledirection can be estimated by a recent magnetic resonancetechnique: diffusion tensor magnetic resonance imaging(DT-MRI) [21]. This technique provides directional infor-mation on the diffusion of water. It is assumed that theconductivity is the highest in the direction in which thewater diffuses most easily [22]. Authors [23-25] haveshowed that anisotropic conducting compartmentsshould be incorporated in volume conductor models ofthe head whenever possible.

In the grey matter, scalp and cerebro-spinal fluid (CSF)the conductivity is equal in all directions. Thus the placedependent conductivity tensor becomes a place depend-ent scalar σ, a so-called isotropic conducting tissue. Theconductivity of CSF is quite accurately known to be 1.79S/m [26]. In the following we will focus on the conductiv-ity of the skull and soft tissues. Some typical values of con-ductivities can be found in table 1.

The skull conductivity has been subject to debate amongresearchers. In vivo measurements are very different fromin vitro measurements. On top of that, the measurementsare very patient specific. In [27], it was stated that the skullconductivity has a large influence on the forward prob-lem.

It was believed that the conductivity ratio between skulland soft tissue (scalp and brain) was on average 80 [20].Oostendorp et al. used a technique with realistic headmodels by which they passed a small current by means of2 electrodes placed on the scalp. A potential distribution

σσ σ σσ σ σσ σ σ

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

11 12 13

12 22 23

13 23 33

, (5)

Anisotropic conductivity of the brain tissuesFigure 4Anisotropic conductivity of the brain tissues. The ani-sotropic properties of the conductivity of skull and white matter tissues. The anisotropic properties of the conductiv-ity of skull and white matter tissues. (a) The skull consists of 3 layers: a spongiform layer between two hard layers. The conductivity tangentially to the skull surface is 10 times larger than the radial conductivity. (b) White matter consist of axons, grouped in bundles. The conductivity along the nerve bundle is 9 times larger than perpendicular to the nerve bun-dle.

The current density and equipotential lines in the vicinity of a dipoleFigure 3The current density and equipotential lines in the vicinity of a dipole. The current density and equipotential lines in the vicinity of a current source and current sink is depicted. Equipotential lines are also given. Boxes are illus-trated which represent the volumes G.

5

4

32

2

1

1

1

0 0

1

1

1

2

2

3

3

4

5



is then generated on the scalp. Because the potential val-ues and the current source and sink are known, only theconductivities are unknown in the head model and equa-tion (4) can be solved toward σ. Using this technique theycould estimate the skull-to-soft tissue conductivity ratio tobe 15 instead of 80 [28]. At the same time, Ferree et al. dida similar study using spherical head models. Here, skull-to-soft tissue conductivity was calculated as 25. It wasshown in [29] that using a ratio of 80 instead of 16, couldyield EEG source localization errors of an average of 3 cmup to 5 cm.

One can repeat the previous experiment for a lot of differ-ent electrode pairs and an image of the conductivity canbe obtained. This technique is called electromagneticimpedance tomography or EIT. In short, EIT is an inverseproblem, by which the conductivities are estimated. Usingthis technique, the skull-to-soft tissue conductivity ratiowas estimated to be around 20–25 [30,31]. However in[30], it was shown that the skull-to-soft tissue ratio coulddiffer from patient to patient with a factor 2.4. In [32],maximum likelihood and maximum a posteriori tech-niques are used to simultaneously estimate the differentconductivities. There they estimated the skull-to-soft tis-sue ratio to be 26.

Another study came to similar results using a differenttechnique. In Lai et al., the authors used intracranial andscalp electrodes to get an estimation of the skull-to-softtissue ratio conductivity. From the scalp measures theyestimated the cortical activity by means of a cortical imag-ing technique. The conductivity ratio was adjusted so thatthe intracranial measurements were consistent with theresult of the imaging from the scalp technique. Theyresulted in a ratio of 25 with a standard deviation of 7.One has to note however that the study was performed onpediatric patients which had the age of between 8 and 12.Their skull tissue normally contains a larger amount ofions and water and so may have a higher conductivitythan the adults calcified cranial bones [33]. In a moreexperimental setting, the authors of [34] performed con-ductivity measures on the skull itself in patients undergo-ing epilepsy surgery. Here the authors estimated the skullconductivity to be between 0.032 and 0.080 S/m, which

comes down to a soft-tissue to skull conductivity of 10 to40.

Poisson's equationThe scalar potential field V, having volt as unit, is nowintroduced. This is possible due to Faraday's law beingzero under quasi-static conditions (∇ × E = 0) [35]. Thelink between the potential field and the electric field isgiven utilizing the gradient operator,

E = -∇V. (6)

The vector ∇V at a point gives the direction in which thescalar field V, having volt as its unit, most rapidlyincreases. The minus sign in equation (6) indicates thatthe electric field is oriented from an area with a highpotential to an area with a low potential. Figure 3 alsoillustrates some equipotential lines generated by a currentsource and current a sink.

When equation (2), equation (4) and equation (6) arecombined, Poisson's differential equation is obtained ingeneral form:

∇·(σ∇(V)) = -Im. (7)

For the problem at hand, equation (3), equation (4) andequation (6) are combined yielding:

∇·(σ∇(V)) = -Iδ(r - r2) + Iδ(r - r1). (8)

In the Cartesian coordinate system equation (8) becomesfor isotropic conductivities:

and for anisotropic conductivities:

∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

= − − − −

+x

Vx y

Vy z

Vz

I x x y y z z

I

( ) ( ) ( ) ( ) ( ) ( )σ σ σ δ δ δ2 2 2

δδ δ δ( ) ( ) ( )x x y y z z− − −1 1 1(9)

Table 1: The reference values of the absolute and relative conductivity of the compartments incorporated in the human head.

compartments Geddes & Baker (1967) Oostendorp (2000) Gonçalves (2003) Guttierrez (2004) Lai (2005)

scalp 0.43 0.22 0.33 0.749 0.33skull 0.006 – 0.015 0.015 0.0081 0.012 0.0132

cerebro-spinal fluid - - - 1.79 -brain 0.12 – 0.48 0.22 0.33 0.313 0.33

σscalp/σskull 80 15 20–50 26 25



The potentials V are calculated with equations (8), (9) or(10) for a given current source density Im, in a volumeconductor model, e.g. in our application, the humanhead. Compartments in which all conductivities areequal, are called homogeneous conducting compart-ments.

Boundary conditionsAt the interface between two compartments, two bound-ary conditions are found. Figure 5 illustrates such an inter-face. A first condition is based on the inability to pile upcharge at the interface. All charge leaving one compart-ment through the interface must enter the other compart-ment. In other words, all current (charge per second)leaving a compartment with conductivity σ1 through theinterface enters the neighboring compartment with con-ductivity σ2:

where en is the normal component on the interface.

In particular no current can be injected into the air outsidethe human head due to the very low conductivity of theair. Therefore the current density at the surface of the headreads:

Equations (11) and (12) are called the Neumann bound-ary condition and the homogeneous Neumann boundarycondition, respectively.

The second boundary condition only holds for interfacesnot connected with air. By crossing the interface thepotential cannot have discontinuities,

V1 = V2. (13)

This equation represents the Dirichlet boundary condi-tion.

The current dipoleCurrent source and current sink inject and remove thesame amount of current I and they represent an activepyramidal cell at microscopic level. They can be modeledas a current dipole as illustrated in figure 6(a). The posi-tion parameter rdip of the dipole is typically chosen halfway between the two monopoles.

The dipole moment d is defined by a unit vector ed (whichis directed from the current sink to the current source) anda magnitude given by d = ||d|| = I·p, with p the distancebetween the two monopoles. Hence one can write:

d = I·ped. (14)

It is often so that a dipole is decomposed in three dipoleslocated at the same position of the original dipole andeach oriented along one of the Cartesian axes. The magni-tude of each of these dipoles is equal to the orthogonalprojection on the respective axis as illustrated in figure6(b). one can write:

d = dxex + dyey + dzez, (15)

with ex, ey and ez being the unit vectors along the threeaxes. Furthermore, dx, dy and dz are often called the dipolecomponents. Notice that Poisson's equation (8) is linear.

σ σ σ σ σ σ11 22 33 12 13 232

2

2

2

2

22

2 2 2∂

∂+ ∂

∂+ ∂

∂+ ∂

∂ ∂+ ∂

∂ ∂+ ∂V

x

V

y

V

z

Vx y

Vx z

VVy z

x y zVx x

∂ ∂

⎛

⎝⎜⎜

⎞

⎠⎟⎟

+ ∂∂

+ ∂∂

+ ∂∂

⎛

⎝⎜

⎞

⎠⎟∂∂

+ ∂∂

+ ∂∂

σ σ σ σ σ11 12 13 12 22yy z

Vy x y z

Vz

I x x

+ ∂∂

⎛

⎝⎜

⎞

⎠⎟∂∂

+ ∂∂

+ ∂∂

+ ∂∂

⎛

⎝⎜

⎞

⎠⎟∂∂

=

− −

σ σ σ σ

δ

23 13 23 33

( 22 2 2 1 1 1) ( ) ( ) ( ) ( ) ( ).δ δ δ δ δy y z z I x x y y z z− − + − − −

(10)

J e J e

e e1 2⋅ = ⋅

∇ ⋅ = ∇ ⋅n n

n nV V

,

( ) ( ) ,σ σ1 1 2 2(11)

J e

e1 ⋅ =

⋅∇ ⋅ =n

nV

0

01 1

,

( ) .σ(12)

The boundary between two compartmentsFigure 5The boundary between two compartments. The boundary between two compartments. The boundary between two compartments, with conductivity σ1 and σ2. The normal vector en to the interface is also shown.



Due to a dipole at a position rdip and dipole moment d, apotential V at an arbitrary scalp measurement point r canbe decomposed in:

V(r, rdip, d) = dxV(r, rdip, ex) + dyV(r, rdip, ey) + dzV(r, rdip, ez).(16)

A large group of pyramidal cells need to be more or lesssynchronously active in a cortical patch to have a measur-able EEG signal. All these cells are furthermore orientedwith their longitudinal axis orthogonal to the cortical sur-face. Due to this arrangement the superposition of theindividual electrical activity of the neurons results in anamplification of the potential distribution. A large groupof electrically active pyramidal cells in a small patch ofcortex can be represented as one equivalent dipole onmacroscopic level [36,37]. It is very difficult to estimatethe extent of the active area of the cortex as the potentialdistribution on the scalp is almost identical to that of anequivalent dipole [38].

General algebraic formulation of the forward problem

In symbolic terms, the EEG forward problem is that offinding, in a reasonable time, the scalp potential g(r, rdip,

d) at an electrode positioned on the scalp at r due to a sin-gle dipole with dipole moment d = ded (with magnitude d

and orientation ed), positioned at rdip. This amounts to

solving Poisson's equation to find the potentials V(r) on

the scalp for different configurations of rdip and d. For

multiple dipole sources, the electrode potential would be

. In practice,

one calculates a potential between an electrode and a ref-erence (which can be another electrode or an average ref-erence).

For N electrodes and p dipoles:

where i = 1,...,p and j = 1,...,N. Here V is a column vector.

For N electrodes, p dipoles and T discrete time samples:

where V is now the matrix of data measurements, G is thegain matrix and D is the matrix of dipole magnitudes atdifferent time instants.

More generally, a noise or perturbation matrix n is added,

V = GD + n.

In general for simulations and to measure noise sensitiv-ity, noise distribution is a gaussian distribution with zeromean and variable standard deviation. However in reality,the noise is coloured and the distribution of the frequencydepends on a lot of factors: patient, measurement setup,pathology,....

A general multipole expansion of the source modelSolving the inverse problem using multiple dipole modelrequires the estimation of a large number of parameters, 6for each dipole. Given the use of a limited number of EEGelectrodes, the problem becomes underdetermined. Inthis case, regularization techniques have to be applied,but this leads to oversmoothed source estimations. Onthe other hand, the use of a limited number of dipoles(one, two or three) leads to very simplified sources, whichare very often ambiguous and cause errors due to simpli-fied modelling. The dipole model as a source is a goodmodel for focal brain activity.

V g g ddip ii

dip ii

i i i( ) ( , , ) ( , , )r r r d r r ed= =∑ ∑

V

r

r

r r e r r ed d

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

V

V

g g

N

dip dipp p( )

( )

( , , ) ( , , )1 1 11 1

gg g

d

dN dip N dip pp p( , , ) ( , , )r r e r r ed d1 1

1⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥=

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

G r r ed({ , , })j dip

p

i i

d

d

1

V

r r

r r

G r r

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=

V V T

V V TN N

j dipi

( , ) ( , )

( , ) ( , )

({ , ,

1 11

1

ee G r r e Dd di i i

d d

d d

T

p p T

j dip}) ({ , , }), ,

, ,

1 1 1

1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

(17)

The dipole parametersFigure 6The dipole parameters. (a) The dipole parameters for a given current source and current sink configuration. (b) The dipole as a vector consisting of 6 parameters. 3 parameters are needed for the location of the dipole. 3 other parameters are needed for the vector components of the dipole. These vector components can also be transformed into spherical components: an azimuth, elevation and magnitude of the dipole.

Y

Z

X

�

�dY

dX

dZ

(a) (b)



A multipole expansion is an alternative (first introducedby [39]), which is based on a spherical harmonic expan-sion of the volume source, which is not necessarily focal.It provides the added model flexibility needed to accountfor a wide range of physiologically plausible sources,while at the same time keeping the number of estimationparameters sufficiently low. In fact, The zeroth-order andfirst-order terms in the expansion are called the monopoleand dipole moment, respectively. A quadrupole is ahigher order term and is generated by two equal andoppositely oriented dipoles whose moments tend toinfinity as they are brought infinitesimally close to eachother. An octapole consists of two quadrupoles broughtinfinitesimally close to each other and so on. It can beshown that if the volume G containing the active sourcesIsv(r') is limited in extent, the solution to Poisson's equa-tion for the potential V may be expanded in terms of amultipole series:

V = Vmonopole + Vdipole + Vquadrupole + Voctapole + Vhexadecpole + ...(18)

where Vquadrupole is the potential field caused by the quad-rupole. In practice, a truncated multipole series is used upto a quadrupole, because the contribution to the electrodepotentials by a octapole or higher order sources rapidlydecreases when the distance between electrode and sourceis increasing. The use of quadrupoles can sound plausiblein the following case: A traveling action potential causes adepolarization wave through the axon, followed by arepolarization wave. These two phenomenon producetwo opposite oriented dipoles very close to each other[40]. In sulci, pyramidal cells are oriented toward eachother, which makes the use of quadrupole also reasona-ble. However, the skull causes a strong attenuation of theelectrical field created by the source. Therefore, even aquadrupole has low contribution to the electrode poten-tials of the EEG, created by the volume current in theextracellular region. In EEG and ECG multiple dipoles ofdipole layers are preferred over a multipole. Multipolesare popular in magnetoencephalographic (MEG) sourcelocalization, because of its low sensitivity to the skull con-ductivity [6,10,41,42].

Solving the forward problemDipole field in an infinite homogeneous isotropic conductorThe potential field generated by a current dipole withdipole moment d = ded at a position rdip in an infinite con-ductor with conductivity σ, is introduced. The potentialfield is given by:

with r being the position where the potential is calculated.Assume that the dipole is located in the origin of the Car-tesian coordinate system and oriented along the z-axis.Then it can be written:

where θ represents the angle between the z-axis and r andr = ||r||. An illustration of the electrical potential fieldcaused by dipole is shown in figure 7.

Equation (20) shows that a dipole field attenuates with 1/r2. It is significant to remark that V, from equation (19),added with an arbitrary constant, is also a solution ofPoisson's equation. A reference potential must be chosen.One can choose to set one electrode to zero or one can optfor average referenced potentials. The latter result in elec-trode potentials that have a zero mean.

The spherical head modelThe first volume conductor models of the human headconsisted of a homogeneous sphere [43]. However it wassoon noticed that the skull tissue had a conductivitywhich was significantly lower than the conductivity ofscalp and brain tissue. Therefore the volume conductormodel of the head needed further refinement and a three-shell concentric spherical head model was introduced. Inthis model, the inner sphere represents the brain, theintermediate layer represents the skull and the outer layer

Vdip

dipdip( , , )

( )

|| ||,r r d

d r r

r r=

⋅ −

−4 3πσ(19)

V dd

rz( , , )

cos,r 0 e = θ

πσ4 2(20)

The equipotential lines of a dipoleFigure 7The equipotential lines of a dipole. The equipotential lines of a dipole oriented along the z-axis. The numbers cor-respond to the level of intensity of the potential field gener-ated of the dipole. The zero line divides the dipole field into two parts: a positive one and a negative one.

y

z

1

2

345

54

3

2

1

0 0



represents the scalp. For this geometry a semi-analyticalsolution of Poisson's equation exists. The derivation isbased on [44,45]. Consider a dipole located on the z-axisand a scalp point P, located in the xz-plane, as illustratedin figure 8. The dipole components located in the xz-planei.e. dr the radial component and dt the tangential compo-nent, are also shown in figure 8. The component orthogo-nal to the xz-plane, does not contribute to the potential atscalp point P due to the fact that the zero potential planeof this component traverses P. The potential V at scalppoint P for the proposed dipole is given by:

with gi given by:

Where:

dr is the radial component (3 × 1-vector in meters),

dt is the tangential component (3 × 1-vector in meters),

R is the radius of the outer shell (meters),

S is the conductivity of scalp and brain tissue (Siemens/meter),

X is the ratio between the skull and soft tissue conductivity(unitless),

b is the relative distance of the dipole from the centre(unitless),

θ is the polar angle of the surface point see figure 8 (radi-ans),

Pi(·) is the Legendre polynomial,

is the associated Legendre polynomial,

i is an index,

i1 equals 2i + 1,

r1 is the radius of the inner shell (in meters),

r2 is the radius of the middle shell (in meters),

f1 equals r1/R (unitless) and

f2 equals r2/R (unitless).

Equation (21) gives the scalp potentials generated by adipole located on the z-axis, with zero dipole moment inthe y direction. To find the scalp potentials generated byan arbitrary dipole, the coordinate system has to berotated accordingly. Typical radii of the outer boundariesof the brain, skull and scalp compartments are equal to 8cm, 8.5 cm and 9.2 cm, respectively [46]. An illustrationof a typical spherical head model is shown in figure 8.These radii can be altered to fit a sphere more to thehuman head. The infinite series of equation (21) is oftentruncated. If the first 40 terms are used, the maximumscalp potential obtained with the truncated series, devi-ates less than 0.1% from the case where 100 terms areapplied, for dipoles with a radial position smaller than95% of the maximum brain radius.

There are also semi-analytical solutions available for lay-ered spheroidal anisotropic volume conductors [47-49].Here the conductivity in the tangential direction can bechosen differently than in the radial direction of thesphere. Analytic solutions also exist for prolate and oblatespheroids or eccentric spheres [50-52].

VSR

X igi i i

b id P d Pi r i t ii

= ++

+−

=

∞1

4 22 1 3

11 1

1πθ θ( )

( )[ (cos ) (cos )],∑∑

(21)

g i X iiXi

X i X i f f i Xii i= + +

++ + − + + − − −[( ) ][ ] ( )[( ) ]( ) ( ) (1

11 1 1 11 2

21 1 ff f i1 2 1/ ) .

(22)

Pi1( )⋅

The three-shell concentric spherical head modelFigure 8The three-shell concentric spherical head model. The dipole is located on the z-axis and the potential is measured at scalp point P located in the xz-plane.

8.5cm

9.2cm

8.0cm

q

X

Z

Pdr

dt



Variants of the three-shell spherical head model, such asthe Berg approximation [53], in which a single-spheremodel is used to approximate a three- (or four-) layersphere, have also been used to improve further the com-putational efficiency of multi-layer spherical models.

Recently however, it is becoming more apparent that theactual geometry of the head [54-56] together with the var-ying thickness and curvatures of the skull [57,58], affectsthe solutions appreciably. So-called real head models arebecoming much more common in the literature, in con-junction with either boundary-element, finite-element, orfinite-difference methods. However, the computationalrequirements for a realistic head model are higher thanthat for a multi-layer sphere.

An approach which is situated between the spherical headmodel approaches and realistic ones is the sensor-fittedsphere approach [59]. Here a multilayer sphere is fitted toeach sensor located on the surface of a realistic headmodel.

The boundary element methodThe boundary element method (BEM) is a numerical tech-nique for calculating the surface potentials generated bycurrent sources located in a piecewise homogeneous vol-ume conductor. Although it restricts us to use only iso-tropic conductivities, it is still widely used because of itslow computational needs. The method originated in thefield of electrocardiography in the late sixties and made itsentrance in the field of EEG source localization in the lateeighties [60]. As the name implies, this method is capableof providing a solution to a volume problem by calculat-ing the potential values at the interfaces and boundary ofthe volume induced by a given current source (e.g. adipole). The interfaces separate regions of differing con-ductivity within the volume, while the boundary is theouter surface seperating the non-conducting air with theconducting volume.

In practice, a head model is built from surfaces, eachencapsulating a particular tissue. Typically, head modelsconsist of 3 surfaces: brain-skull interface, skull-scalpinterface and the outer surface. The regions between theinterfaces are assumed to be homogeneous and isotropicconducting. To obtain a solution in such a piecewisehomogenous volume, each interface is tesselated withsmall boundary elements.

The integral equations describing the potential V(r) at anypoint r in a piecewise volume conductor V were describedin [61-63]:

where σ0 corresponds to the medium in which the dipolesource is located (the brain compartment) and V0(r) is the

potential at r for an infinite medium with conductivity σ0

as in equation (19). and are the conductivities of

the, respectively, inner and outer compartments dividedby the interface Sj. dS is a vector oriented orthogonal to a

surface element and ||dS|| the area of that surface element.

Each interface Sj is digitized in triangles, (see figure

9) and in each triangle centre the potentials are calculatedusing equation (23). The integral over the surface Sj is

transformed into a summation of integrals over traingleson that surface. The potential values on surface Sj can be

written as

where the integral is over , the j-th triangle on the

surface Sj, R is the number of interfaces in the volume. An

exact solution of the integral is generally not possible,

therefore an approximated solution on surface Sk

may be defined as a linear combination of simpleba-

sis functions

Vk k

Vj j

k k

V d( ) ( )_

( )|| ||

r r rr r

r rS= −+ +

+− +

−+ +′ ′−

′−

2 0 12 30

σ

σ σ π

σ σ

σ σjj

r′∈=∫∑ S

j

R

j1

,

(23)

σ j− σ j

+

NS j

Vr r

Vk k

r rV d( ) ( )

_( )

|| ||r r r

r r

r rS= −+ +

+− +

−+ +′ ′−

′−

2 0 12 30

σ

σ σ π

σ σ

σ σkk

ΔSk j

Sk

j

N

k

R

,

,∫∑∑== 11

(24)

ΔS kj ,

V k( )r

NSk

V V hk ik

i

i

NSk

( ) ( ).r r==∑

1

(25)

Example mesh of the human head used in BEMFigure 9Example mesh of the human head used in BEM. Trian-gulated surfaces of the brain, skull and scalp compartment used in BEM. The surfaces indicate the different interfaces of the human head: air-scalp, scalp-skull and skull-brain.



The coefficients represent unknowns on surface Sk

whose values are determined by constraining to sat-

isfy (24) at discrete points, also known as collocationpoints. Moreover, equation (24) can be rewritten as

This equation can be transformed into a set of linear equa-tions:

V = BV + V0, (27)

where V and V0 are column vectors denoting at every nodethe wanted potential value and the potential value in aninfinite homogeneous medium due to a source, respec-tively. B is a matrix generated from the integrals, whichdepends on the geometry of the surfaces and the conduc-tivities of each region.

Determination of the elements of the matrix B is compu-tationally intensive and there exist different approachesfor their computation. The integral in equation (23) isalso often called the solid angle [62,64,65]. The basisfunctions hi(r) can be defined in several ways. The "con-stant-potential" approach for triangular elements usesbasis functions defined by

where Δi denotes the ith planar triangle on the tesselatedsurface. The collocation points are typically the centroidsof the surface elements and the unknown potentials V arethe potentials at each triangle [66]. The "linear potential"approach uses basis functions defined by

where ri, rj, rk are the nodes of the triangle and the triplescalar product is defined as [ri rj rk] = det(ri, rj, rk). The nota-tion Δi(jk) is used to indicate any triangle for which one ver-tex is defined by the vector ri, the remaining two verticesdenoted as rj and rk. The function hi(r) attains a value ofunity at the ith vertex and drops linearly to zeros at theopposite edge of all triangles to which ri is a vertex. In thiscase, the collocation points are the vertices of the elements[66]. The approaches can be expanded into higher-orderelements [67]. Gençer and Tanzer investigated quadratic

and cubic element types and concluded that these gavesuperior results to models with linear elements [68].

Barnard et al. [64] showed that the potentials in equation(27) are only defined up to an additive constant. Hence,equation (27) has no unique solution. This ambiguity canbe removed by deflation, which means that B must bereplaced by

where e is a vector with all its N (the total number ofunknowns) components equal to one. The deflated equa-tion

V = CV + V0, (31)

possesses a unique solution which is also a solution to theorignal equation (27). If I denotes the N × N identitymatrix and A represents I - C then

V = A-1V0. (32)

This equation can be solved using direct or iterative solv-ers. Direct solvers are especially usefull when the matrix Ais relatively small because of a coarse grid. If one wants touse a fine grid, then iterative methods should be used. Theuse of multiple deflations during the iterations can signif-icantly increase the convergence time to the solution ofequation (31) [69].

A typical head model for solving the forward probleminvolves 3 layers: the brain, the skull and the scalp. Theconductivity of the skull is lower than the conductivity ofbrain and scalp. If β is defined as the ratio of the skull con-ductivity to the brain conductivity Meijs et al. showed thatan accurate solution of equation (23) is difficult to obtainfor small β (β < 0.1). The large difference between the con-ductivities will cause an amplification of the numericalerrors in the calculation. To solve this problem, the Iso-lated Problem Approach (IPA) can be used (also calledIsolated Skull Approach), which was introduced byHämäläinen and Sarvas [70]. Assume the labeling of thecompartments as Cscalp, Cskull and Cbrain and Sscalp as theouter interface, Sskull as the interface between Cskull andCbrain and Sbrain as the interface between Cbrain and Cskull. TheIPA uses the following decomposition of the potential val-ues:

V(r) = V'(r) + V''(r) (33)

where V'' is the potential on surface Sbrain when the head isa homogeneous brain region, thus omitting the skull and

Vik

V( )x

Vr r

Vk k

r rV hi

k

i

N

i

Sk

( ) ( )_

( )|

r r rr r= −+ +

+− +

−+ +′−

=∑2 0 120

1

σ

σ σ π

σ σ

σ σ || ||.

, ′−∫∑∑ == r r Sk311 dSk jSk

j

N

k

R

Δ

(26)

hii

i( )r

r

r=

∈∉

⎧⎨⎩

1

0

ΔΔ

(28)

h

j k

i j kii jk

i jk

( )

[ ]

[ ] ( )

( )

r

r r r

r r rr

r

=∈

∉

⎧

⎨⎪

⎩⎪

Δ

Δ0

(29)

C B ee= − 1N

T , (30)



scalp compartments. V' is the correction term. When V iswritten like above, equation (32) can be written as

Because V'' is zero on the interfaces Sscalp and Sskull, V' con-

tains the potential values on the outer surface. The IPA isbased on the more accurate solution of the right-hand side

term . An accurate solution can be obtained by setting

to the following

where , and are the potentials at respectively

the brain-skull surface, the skull-scalp surface and the

outer surface. This imposes that has to be calculated.

This can be done by solving the potentials at Sbrain with the

scalp and skull compartments omitted. The increase inaccuracy comes at a small cost of computational speed. Aweighted IPA approach was developed by Fuchs et al.[71]. The IPA approach was extended to multi-spheremodels by Gençer and Akahn-Acar [72]. The calculationof the forward problem involves every node on the mesh,making it very computation intesive. Accelerated BEMcomputes the node potentials on a small subset of nodescorresponding to the electrode positions [73].

To improve the localization accuracy, one can locallyrefine the mesh. Yvert et al. showed that if the dipole is at2 cm below the surface, a mesh of 0.5 triangles/cm2 isneeded to have acceptable results. However, for shallowdipoles (between 2 mm and 20 mm below the brain sur-face) a mesh density of 2–6 triangles per cm2 is needed toobtain comparable results. Of course, the area in the meshthat has to be refined, has to be defined.

A main disadvantage using BEM in the EEG forward prob-lem is that in all aforementioned implementations theprecision drops when the distance of the source to one ofthe surfaces becomes comparable to the size of the trian-gles in the mesh. Kybic et al. presented a new frameworkbased on a theorem that characterizes harmonic functionsdefined on the complement of a bounded smooth surface

[74]. Using this framework, they proposed a symmetricformulation. The main benefit of this approach is that theerror increases much less dramatically when the currentsources approach a surface where the conductivity is dis-continuous. In another paper by the same authors, a fastmultipole acceleration was used to overcome the com-plexity of the symmetric formulation [75]. A recent articleof the same authors demonstrates that the frameworkallows the use of more realistical head models, whichdon't have to be nested. In nested head models, an innerinterface is completely enveloped by an outer interface.Non-nested compartments are compartments that are notpart of the brain, but part of the head (such as eyes,sinuses,...) [76].

The finite element methodAnother method to solve Poisson's equation in a realistichead model is the finite element method (FEM). TheGalerkin approach [77] is used to equation (7) withboundary conditions (11), (12), (13). First, equation (7)is multiplied with a test function φ and then integratedover the volume G representing the entire head. UsingGreen's first identity for integration:

in combination with the boundary conditions (12), yieldsthe 'weak formulation' of the forward problem:

If (v, w) = ∫Gv(x, y, z)w(x, y, z)dG and a(u, v) = -(∇v, σ∇u),this can be written as:

a(V, φ) = (Im, φ) (38)

The entire 3D volume conductor is digitized in small ele-ments. Figure 10 illustrates a 2D volume conductor digi-tized with triangles.

The computational points can be identified with

the vertices of the elements (n is the number of vertices).The unknown potential V(x, y, z) is given by

where denotes a set of test functions also called

basis functions. They have a local support, i.e. the area in

′ + ′′ =

′ = − ′′

′ = ′

−

−

−

V V A V

V A V AV

V A V

10

10

10

( )

.

(34)

′V0′V0

′ =

′

′

′

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟=

−+

′′

⎛

⎝

⎜⎜⎜⎜

V

V

V

V

V

V

V V

0

21

01

02

03

01

02

03

03

β

β

β ββ

⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

, (35)

V01 V0

2 V03

′′V03

∇ ⋅ ∇ = ∇ ⋅ − ∇ ∇∫ ∫ ∫∂φ σ φσ φ σ( ) ( ) ,V dG V V dGG G GdS(36)

− ∇ ⋅ ∇ =∫ ∫φ σ φ( ) .V dG I dGG mG (37)

{ }Vi in=1

V x y z V x y zi ii

n

( , , ) ( , , ),==∑ φ

1

(39)

{ }φi in=1



which they are non-zero is limited to adjacent elements.Moreover, the basis functions span a space of piecewisepolynomial functions.

Furthermore, they have the property that they are eachequal to unity at the corresponding computational pointand equal to zero at all other computational points. Sub-stituting (39) in equation (38) produces n equations in nunknowns V = [V1...Vn]T ∈ �n×1:

Due to the local support of the basis function, each equa-tion consists only of a linear combination of Vi's and itsadjacent computational points. Hence the system A ∈�n×n, Aij = a(φi, φj) is sparse. In matrix notation one canobtain:

A·V = I, (42)

with I ∈ �n×1 being the column vector of the source termsobtained by the right hand side of equation (41).

An important consideration in finite-element methods ishow to represent a dipole source in the model.

• The obvious direct method is to represent a dipole usinga pair of fixed voltage conditions of opposite polarityapplied to two adjacent nodes [78].

• Another method is to embed a dipole source in the ele-ment basis functions. When the dipole lies along the edgeof an element, this approach reduces to the simple idea ofusing two concentrated sources at either end of that edge[78].

• A third formulation is to separate the field in two parts– one part is a standard field produced by an ideal dipolein an infinite homogeneous domain and the other part isa solution in the closed sourceless domain under bound-ary conditions that correct the current movement acrossboundaries between regions of different conductivity[78].

• In the Laplace formulation, a small volume containingthe dipole is removed and fixed boundary conditions areapplied at all nodes on the surface of the removed vol-ume. This can be interpreted as replacing current sourcesby an estimate of the equivalent voltage sources [78].

• A fifth formulation is the blurred dipole model, wheresource and sink monopoles are divided over the neigh-bouring nodes. In most cases the source and sink monop-oles do not coincide with nodes of the FEM-mesh.Therefore a way to represent the dipole is by a summationof monopoles placed at neighbouring nodes [79].

A comparison of the resulting surface potentials using thefirst four methods with the exact analytical solution usingideal dipoles (with an infinitesimal separation betweenthe two poles, an infinite total current exiting one poleand entering the other, and a finite dipole moment, whichis the product of the current and separation) in a 4-layerconcentric sphere was made in [78]. It was found that thethird formulation gives the best performance for bothtransverse and radial dipoles (followed by the Laplace for-mulation for radial dipoles).

A recent innovation [80,81] is to consider current monop-oles (point sources/sinks) instead of dipoles. Using theequivalent-current inverse solution (ECS) approach for pgrid locations, only p variables need to be determined inthe inverse problem, whereas if a dipole is placed at eachof the p grid locations, the solution space consists of 3punknown variables because each dipole has 3 directionalcomponents. This results in an advantage of using currentmonopoles instead of current dipoles as demonstrated in[81] where it is shown that the time required to calculate

a V Ii i ji

n

m j( , ) ( , ),φ φ φ=∑ =

1

(40)

a V Ii ji

n

i m j( , ) ( , )φ φ φ=∑ =

1

(41)

Example mesh in 2D used in FEMFigure 10Example mesh in 2D used in FEM. A digitization of the 2D coronal slice of the head. The 2D elements are the trian-gles.



the forward matrix in realistic finite-element head modelsusing the conventional approach may be reduced by onethird. It is further shown in [80] that ECS imaging isequivalent to the equivalent-dipole inverse solution(EDS) in that it provides source-sink distribution corre-sponding to the dipole sources, but additional informa-tion is needed to determine the current flows (bycombining ECS and EDS estimates).

In general the stiffness matrix is very big, making the com-putation of the electrode potentials very computationallyintensive. To solve equation (42), iterative solvers forlarge sparse systems are used as given in [82]. Some tech-niques have been proposed to reduce the computationalburden and increase efficiency as will be illustrated in sec-tion 5. A freely licensed software package that implementsboth FEM end BEM is NEUROFEM [79,83-85].

The finite difference methodIsotropic media (iFDM)The differential equation (9) with boundary conditions(11), (12), (13) is transformed into a linear equation uti-lizing the 'box integration' scheme [86] for the cell-cen-tered iFDM. Consider a typical node P in a cubic grid withinternode spacing h. The six neighbouring nodes are Qi (i= 1,...,6) as illustrated in figure 11.

Introducing αi and α0 as,

a finite difference approximation of (9) is obtained:

with

For volumes G, which contain a current monopole, IPbecomes I or -I. αi has the dimension of Ω-1 and corre-sponds with the conductance between P and Qi. Further-more, for IP = 0 Kirchoff's law holds at the node P. For eachnode of a cubic grid we obtain a linear equation given by(44). The unknown potentials at the n computationalpoints are represented by V ∈ �n×1. The source terms rep-resented by I ∈ �n×n are calculated in each of the n cubesutilizing equation (45). Notice that in the linear equation

(44) only the neighbouring computational points areincluded. The system matrix A ∈ �n×n has at most six off-diagonal elements and is a sparse matrix. In matrix nota-tion one can write:

A·V = I (46)

To solve this large sparse set of equations iterative meth-ods are used. A discussion of the most popular solvers willbe discussed in section. More extensive literature on thismethod can be found in [29,87-92].

The finite difference method in anisotropic media (aFDM)The differential equation (7) with Dirichlet and Neumannboundary conditions can be transformed into a set of lin-ear equations even in the case of anisotropic media. Thisapproach uses a cubic grid in which each cube (or ele-ment) has a conductivity tensor. In anisotropic tissues theconductivity tensor can vary between neighbouring ele-ments. There are two ways to have anisotropy. In general,the directions of the anisotropy can be in any direction.Using tensor transformations, the matrix representationof the concuctivity tensor can be deduced. In a more spe-cific case, the directions of the anisotropy are limitedalong the axes of the coordinate system of the headmodel.In this case, the anisotropy is orthotrophic [93]. In thenext paragraphs, the general case as shown in [94] isdepicted.

α σ σσ σ

α α

i

i

i

h ii

=+

==∑

2 00

0

1

6

,

(43)

α αi Qi

P PV V Ii=∑ − =

1

6

0 , (44)

I I x x y y z z I x x y y z z dxdydzPG

= − − − − + − − −∫ δ δ δ δ δ δ( ) ( ) ( ) ( ) ( ) ( )2 2 2 1 1 10

∫∫∫ .(45)

The computation stencil used in FDMFigure 11The computation stencil used in FDM. A typical node P in an FDM grid with its neighbours Qi (i = 1�6). The volume G0 is given by the box.

X

Y

Z

P

Q5

Q1

Q3

Q2

Q4

Q6

N1N2

N3

N4

N5

N6

(h,0,0)

(-h,0,0)

G0



If the local coordinate system coincides with the axes rep-resenting the principal directions of the anisotropy, thenthe conductivity tensor at an element can be written as adiagonal matrix. The diagonal elements represent the con-ductivities in the orthogonal directions. The matrix repre-sentation has to be transformed to the global Cartesiansystem of the head, the same for all elements. A rotationmatrix is then required to transform the principal direc-tions to a conductivity tensor in the Cartesian coordinatesystem. In the local coordinate system the conductivitytensor at an element j can be written as follows:

where are the conductivities in the principal directions

at element j, respectively. The matrix representation has tobe transformed to a global cartesian coordinate system ofthe head. Therefore a rotation matrix has to be applied.The matrix representation of the conductivity tensor atelement j in the cartesian system of the head is then given

by , where T is a rotation transfer matrix

from the local coordinate system to the global coordinatesystem [95] and T denotes the transpose of a matrix.

A finite difference method which can handle anisotropicproperties of tissues was presented by Saleheen et al. [96].Here, the authors used a transition layer technique [97] toderive a finite difference formulation for the Laplace'sequation that is valid everywhere in a piecewise inhomo-geneous anisotropic medium, where the conductivity ten-sor can have a different value in each element of the grid.This formulation is extended to Poisson's equation byHallez et al. [94].

A typical node 0 in the grid represents the intersection ofeight neighbouring cubic elements, as shown in figure 12.The finite difference formulation of equation 10 at node0, derived from Saleheen's method. From the 26 neig-bouring nodes at node 0, the formulation uses the 18nearest neighbours, with rectilinear distance ≤ 2:

where Vi is the discrete potential value at node i. ai are thecoefficients depending on the conductivity tensor of theelements and the internode distance. These coefficientsare given in [96].

For nodes at the corners of the compartments as illus-trated in figure 12, for example node 11, the boundarynormal cannot be obviously defined. Therefore, the Neu-mann boundary equations (12) and (11) contain singu-larities in spatial derivatives of the conductivities. Themethod presented in [94] and [96] has an advantage ifone wants to enforce such a Neumann boundary condi-tion: the formulation allows a discrete change or disconti-nuity in conductivity between neighbouring elements andwill automatically incorporate the boundary between twodifferent materials. In short, the boundary condition isalready implicitly formulated in equation (47).

For each node a linear equation can be written as in equa-tion (47), and for all computational points a set of linearequations is obtained: A·V = I. Due to the large size of thesystem, iterative methods have to be used.

Comparing the various numerical methodsThe three methods BEM, FEM and FDM can all be used tosolve the forward problem of EEG source analysis in arealistic head model. A summary of the comparisonbetween the BEM, FEM and FDM is given below and intable 2.

A first difference between BEM and FEM or FDM is thedomain in which the solutions are calculated. In the BEMthe solutions are calculated on the boundaries betweenthe homogeneous isotropic compartments while in theFEM and FDM the solution of the forward problem is cal-culated in the entire volume. Subsequently, the FEM andFDM lead to a larger number of computational pointsthan the BEM. On the other hand, the potential at an arbi-trary point can be determined with FEM and FDM byinterpolation of computational points in its vicinity,

σ

σ

σ

σ

j

i

i

i

=

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

1

2

3

0 0

0 0

0 0

,

σ ij

σ σheadj T j= T Tj j

a V a V Ii ii

i

i= =∑ ∑−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

=1

18

1

18

0 , (47)

The computation stencil used in FDM if anistropic conductiv-ities are incorporatedFigure 12The computation stencil used in FDM if anistropic conductivities are incorporated. The potential at node 0 can be written as a linear combination of 18 neighbouring nodes in the FDM scheme. For each node we obtain an equa-tion, which can be put into a linear system Ax = b.

1

2

34

5

6

78

910

11

12

13

14

15

1617

18

0

element 1

element 2

element 3

element 4

element 5

element 6

element 7

element 8



while for the BEM it is necessary to reapply the Barnardformula [62] and numerical integration.

Another important aspect is the computational efficiency.In the BEM, a full matrix (I - C), represented in equation(31), needs to be inverted. When the scalp potentials needto be known for another dipole, V0 in equation (31) needsto be recalculated and multiplied with the already availa-ble (I - C)-1. Hence once the matrix is inverted, only amatrix multiplication is needed to obtain the scalp poten-tials. This limited computational load is an attractive fea-ture when solving the inverse problem, where a largenumber of forward evaluations need to be performed.

For the FEM and the FDM, a direct inversion of the largesparse matrices found in (42) and (46) is not possible dueto the dimension of the matrices. Typically at least500,000 computational points are considered which leadsto system matrices of 500,000 equations with 500,000unknowns which cannot be solved in a direct mannerwith the computers now available. However matricesfound in FEM and FDM can be inverted for a given sourceconfiguration or right-hand side term, utilizing iterativesolvers such as the successive over-relaxation method, theconjugate gradient method [82], or algebraic multigridmethods [98,99] (see next section). A disadvantage of theiterative solvers is that for each source configuration thesolver has to be reapplied. The FEM and FDM would becomputationally inefficient when for each dipole an iter-ative solver would need to be used. To overcome this inef-ficiency the reciprocity theorem is used, as will beexplained in section.

When a large number of conducting compartments areintroduced, a large number of boundaries need to be sam-pled for the BEM. This leads to a large full system matrix,thus a lower numerical efficiency. In FEM and FDM mod-eling, the heterogeneous nature of realistic head modelswill make the stiffness matrix less sparse and badly condi-tioned. Moreover, the incorporation of anisotropic con-

ductivities will decrease the sparseness of the stiffnessmatrix. This can lead to an unstable system or very slowconvergence if iterative methods are used. To obtain a fastconvergence or a stable system, preconditioning shouldbe used. Preconditioning transforms the system of equa-tions Ax = b into a preconditioned system M-1Ax = M-1b,which has the same solution as the orignal system. M is apreconditioning matrix or a preconditioner and its goal isto reduce the condition number (ratio of the largest eigen-value to the smallest eigenvalue) of the stiffness matrixtowards the optimal value 1. Basic preconditioning can beused in the form of Jacobi, Gauss-Seidel, Successive Over-Relaxation (SOR) and Symmetric Successive Over-Relaxa-tion (SSOR). These are easily implemented [100]. Moreadvanced methods use incomplete LU factorization andpolynomial preconditioning [93,100].

For the FDM in contrast with the BEM and FEM, the com-putational points lie fixed in the cube centers for the iso-tropic approach and at the cube corners for theanisotropic approach. In the FEM and BEM, the computa-tional points, the vertices of the tetrahedrons and trian-gles, respectively, can be chosen more freely. Therefore,the FEM can better represent the irregular interfacesbetween the different compartments than the FDM, forthe same amount of nodes. However, the segmented med-ical images used to obtain the realistic volume conductormodel are constructed out of cubic voxels. It is straightfor-ward to generate a structured grid used in FDM from thesesegmented images. In the FEM and the BEM, additionaltessellation algorithms [101] need to be used to obtainthe tetrahedron elements and the surface triangles, respec-tively.

Finally, it is known that the conductivities of some tissuesin the human head are anisotropic such as the skull andthe white matter tissue. Anisotropy can be introduced inthe FEM [102] and in the FDM [96], but not in the BEM.

Reciprocal approachesIn the literature one finds two approaches to solve the for-ward problem. In the conventional approach, the transfer-coefficients making up the matrix G in equation (17) areobtained by calculating the surface potentials from dipolesources via Poisson's equation. The calculations are madefor each dipole position within the head model and thepotentials at the electrode positions are recorded.

In the reciprocal approach [103], Helmholtz' principle ofreciprocity is used. The electric field that results at thedipole location within the brain due to current injectionand withdrawal at the surface electrode sites is first calcu-lated. The forward transfer-coefficients are obtained fromthe scalar product of this electric field and the dipolemoment. Calculations are thus performed for each elec-

Table 2: A comparison of the different methods for solving Poisson's equation in a realistic head model is presented.

BEM FEM iFDM aFDM

Position of computational points

surface volume volume volume

Free choice of computational points

yes yes no no

System matrix full sparse sparse sparseSolvers direct iterative iterative iterativeNumber of compartments

small large large large

Requires tesselation yes yes no noHandles anisotropy no yes no yes



trode position rather than for each dipole position. Thisspeeds up the time necessary to do the forward calcula-tions since the number of electrodes is much smaller thanthe number of dipoles.

The general idea of reciprocityConsider a resistor circuit, with two clamps AB and rx asillustrated in figure 13. The clamp AB represents a pair ofscalp electrodes. The clamp rx is located in the brainregion.

First a current at clamp rx is introduced. This source

will generate a potential UAB( ) at AB as illustrated in

figure 13(a). Next, current IAB at AB is set. This will give

rise to a potential difference (IAB) at rx illustrated in fig-

ure 13(b). The reciprocity theorem in circuit analysisstates:

Mathematical treatmentA mathematical treatment for a digitized volume conduc-tor model is considered. Consider a digitized volume con-ductor model with n computational points or nodes. Ateach of the nodes the potential Vi with i = 1...n is calcu-lated for given sources which are the current monopoles Iiwith i = 1...n. Poisson's equation can then be transformedto a linear equation at each node, as illustrated for the

FEM and FDM in subsections and. This set of linear equa-tions can be written in matrix notation. The system matrixthen becomes A ∈ �n×n and has the following properties:it is sparse, symmetric and regular. One can write:

A·V = I,

with V = [V1...Vn]T ∈ �n×1 and I = [I1...In]T ∈ �n×1 and withT the transpose operator. The desired potential differenceVk - Vl between nodes k and l can be obtained for a currentsource If at node f and a current sink Ig at node g with If =-Ig. All other sources are zero. Cramer's solution for a lin-ear system then becomes:

with A*� the minor for row * and column �.

On the other hand the potential Vf and Vg for a currentsource Ik and current sink Il with Ik = -Il, are:

Furthermore, A*� is equal to A�* due to the fact that A issymmetric. Hence, (eqn.(49) - eqn.(50))/If equals(eqn.(51) - eqn.(52))/Ik. Subsequently the reciprocity the-orem is deduced:

Ik(Vk - Vl) = If(Vf - Vg).

Reciprocity for a dipole source with random orientation

Considering equation (48), a dipole can be represented astwo current monopoles, a current source and sink, provid-

ing and - , separated by a distance 2h. The dipole is

oriented from the negative to the positive current monop-ole and is assumed to be along the x-axis of the resistornetwork with node spacing h. The magnitude of the

dipole moment is then 2h . The centre r of the two

monopoles can then be seen as the dipole position. Thescalp electrodes are located sufficiently far from thesources compared with the distance 2h between thesources so that we can assume a dipole field. Equation(48) can be rewritten as:

Irx

Irx

Vrx

U I V IAB AB r rx x= . (48)

VI f

k f A fkk g Agk

k =− + + − − + +[( ) ( ) ]

det,

1 1 1 1

A(49)

VI f

l f A fll g Agl

l =− + + − − + +[( ) ( ) ]

det,

1 1 1 1

A(50)

VIk

f k Akff l Alf

f =− + + − − + +[( ) ( ) ]

det,

1 1 1 1

A(51)

VIk

g k Akgg l Alg

g =− + + − − + +[( ) ( ) ]

det.

1 1 1 1

A(52)

Irx Irx

Irx

ReciprocityFigure 13Reciprocity. A resistor network where a current source is introduced in the brain and the a potential difference is meas-ured at an electrode pair, and visa versa: (a) a current source

is introduced and the potential UAB is measured, and (b) a

current source IAB is introduced and a potential is meas-

ured.

BA

rx

h

ABI

Vrx

BA

rx

h

ABU

Irx

(a) (b)

IrxVrx



The forward problem in EEG source analysis gives thepotential UAB for a current dipole located at r and oriented

along the x-axis. Rewriting equation(53) with dx = 2h

and

gives:

In a similar way, UAB can be calculated for a dipole locatedat r oriented along the y-axis and the z-axis.

Consider a dipole at position r and with dipole compo-nents d = (dx, dy, dz)T ∈ �3×1. The potential UAB reads:

with ∇V(r) = (∂V(r)/∂x, ∂V(r)/∂y, ∂V(r)/∂z)T ∈ �3×1.

The flowchart in figure 14 shows the consecutive stepsthat are necessary in the reciprocity approach in conjunc-tion with FDM.

• A fictive current IAB of arbitrary value is introducedwhich enters the head at electrode A and leaves the headat electrode B.

• Utilizing the FDM the potentials V(hi, hj, hk) can be cal-culated with h the internode spacing and i, j, k the nodenumbers along the Cartesian axes. Figure 15 illustrates theequipotential lines and current density vectors J = -σ∇V inthe brain region, with ∇V = (∂V/∂x, ∂V/∂y, ∂V/∂z)T. Thepartial derivative ∂V/∂x is approximated by [V(h(i + 1), hj,hk) - V(h(i - 1), hj, hk)]/2h. The partial derivatives ∂V/∂y,∂V/∂z are obtained in a similar way.

• UAB the potential difference between the scalp electrodesA and B generated by the dipole at position r and dipolemoment d is obtained by applying eqn. (55). When r doesnot coincide with a node, then ∇V(r) is obtained with tri-linear interpolation [104].

By solving only one forward calculation numerically, byintroducing current monopoles at electrodes A and B, and

storing the obtained node potentials in a data structure,UAB is obtained for every dipole position and orientation.

If N sc

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