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Edited by: Sergey M. Plis, Mind Research Network, United States

Reviewed by: Rey Ramirez, University of Washington, United States; Hermann Sonntag, Max Planck Institute for Human Cognitive and Brain Sciences (MPG), Germany

*Correspondence: Maria Carla Piastra

This article was submitted to Brain Imaging Methods, a section of the journal Frontiers in Neuroscience

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In Electro- (EEG) and Magnetoencephalography (MEG), one important requirement of source reconstruction is the forward model. The continuous Galerkin finite element method (CG-FEM) has become one of the dominant approaches for solving the forward problem over the last decades. Recently, a discontinuous Galerkin FEM (DG-FEM) EEG forward approach has been proposed as an alternative to CG-FEM (Engwer et al.,

Together with electroencephalography (EEG), magnetoencephalography (MEG) is a technique used to investigate brain activity. EEG and MEG are devoted to detect the electric potential distribution and the magnetic field generated by the brain, respectively, with a unique time resolution (Brette and Destexhe,

In this work, we introduce the first application of DG-FEM for solving the MEG forward problem. As MEG solutions depend on EEG solutions, we implement a setup where the same method (CG- or DG-FEM) is adopted for both modalities, allowing for a combined EEG and MEG source reconstruction study. We analyze the accuracy of DG-FEM results in comparison to the CG-FEM results and investigate the propagation of the effects of conservation properties on the MEG results, as these effects have been proven to play an important role for EEG forward simulations in leaky scenarios. We will show that, in contrast to the EEG, the accuracy of the forward solution for the MEG is basically not affected by skull leakage effects. In fact, the accuracy of DG-FEM forward modeling is in the same range as for standard CG-FEM. However, because of the advantages on the EEG side, DG-FEM is an interesting new approach for combined MEG/EEG source reconstruction scenarios.

In this section, after a summary of the EEG and MEG background, the theory of Discontinuous Galerkin- (DG-) FEM for solving the MEG forward problem will be presented. As the MEG forward problem is build on the solution of the EEG forward problem, a brief section will be about the latter.

Following Hämäläinen et al. (_{0},

related to the electrical part, and

related to the magnetic part.

In Equation (2a)

where ^{p} is the so called primary current, ^{s} the secondary or volume current and ^{3}. In neuromagnetism, the primary current is widely represented as a

where ^{3} stands for the dipolar moment and δ is the Dirac delta distribution, centered in the dipole position

holds true (Ohm's law), where σ indicates the conductivity profile of the conductive medium. While, for the mathematical point dipole, the primary current is present only at the source position, the secondary current flows passively everywhere in the medium.

To derive the EEG forward problem, we have to consider Equations (1a) and (2a). From Equation (1a) we deduce that there exists a potential

so that Equation (5) can be written as

Applying the divergence to Equation (2a), we obtain

Combining Equations (3), (7), and (8), we get an inhomogeneous Poisson equation that, together with the homogeneous Neumann boundary condition, models the EEG forward problem:

where Ω is the volume conductor and

A fundamental physical property of the EEG forward problem is the conservation of charge:

where ^{p} and

For FEMs this property carries over to the discrete solution only if the test space contains the characteristic function, which is one in

The solution of the MEG forward problem consists in the computation of the magnetic induction (flux), Φ, generated by a dipolar source in the brain. The magnetic flux is computed from the magnetic field

where

Furthermore, following Biot-Savart's law, the B-field at a point ^{3} outside the domain Ω can be computed as

(Hämäläinen et al.,

When combining Equations (3), (13), and (4), one obtains (Hämäläinen et al.,

Namely, the B-field can be split into two contributions as well, the primary B-field ^{p}, which is calculated analytically for a mathematical point dipole in Equation (13), and the secondary B-field ^{s}, which has to be computed numerically when the electrical potential is computed numerically (since it depends on the electrical potential

In simplified geometries, similarly to the EEG forward problem (Brette and Destexhe,

where _{0},

Ilmoniemi (

From Equation (15), one can deduce three important features of the analytical MEG solution for a multi-layer homogeneous sphere model and a point outside the model:

^{s}(

In this section we will recall the concepts of CG- and DG-FEM for the EEG forward problem that are then needed in section 2.3 for the derivation of the two FEM based MEG forward approaches.

The mathematical point dipole model introduces a singularity on the right hand side of the PDE in Equation (9) that can be treated with the so-called subtraction approach (Bertrand et al., ^{∞} around the source in _{0} can be found with homogeneous conductivity σ^{∞}. The conductivity tensor σ is then split into two parts,

where σ^{corr} vanishes in Ω^{∞}. The potential

The so-called ^{∞} is the solution of the Poisson equation in an unbounded and homogeneous conductor with constant conductivity σ^{∞}, and it can be computed analytically. The ^{corr} becomes the unknown of a new Poisson equation:

after embedding Equations (16) and (17) in Equations (9) and (10).

The conforming weak formulation of (18) and (19) presented in Wolters et al. (

holds true, ∀_{h} ∈ _{h}. Choosing _{h} as the space of piecewise linear, continuous functions give the classical CG-FEM.

The subtraction approach is theoretically well understood. A deep numerical analysis of the subtraction approach including proofs for uniqueness and existence has been carried out in Wolters et al. (

A discontinuous Galerkin- (DG-) FEM forward modeling approach has recently been proposed for the EEG by Engwer et al. (

and the _{int} ∪ ∂Ω. Let

where ^{l} denotes the space of polynomial functions of degree _{i} and denote its value by σ_{i}.

Furthermore we recall the definition of _{e} and _{f} of the triangulation

Note that the normals _{e} and _{f} are opposing vectors, i.e., _{e} = −_{f}. In addition, the

The DG-FEM for solving Equations (18) and (19) then reads: Find

with

and

where _{γ} and

If Equation (25) has been solved toward the correction potential _{h} can be computed as

^{corr} = −∇ · σ^{corr}∇^{∞}. For ^{corr} = σ∇^{corr}.

As we will see in more detail in section 2.3.3, for the MEG problem the main quantity of interest is the electric flux, needed to compute the B-field. This flux, again, is closely related to the conservation of charge property.

In this section, CG- and DG-FEM formulations for the MEG forward problem are derived, when a conservative or a non-conservative flux expression is adopted along with the subtraction approach.

In this section, we will focus on the expression of the secondary B-field ^{s}, as the primary B-field ^{p} is analytically computable.

When inserting Equation (17) into Biot-Savart's law Equation (14), we obtain the following expression for the secondary B-field:

Both

Since we are dealing with numerical integration, both ^{∞} and ^{corr} are projected in a discrete space, _{h}, i.e.,

and

where (φ_{i})_{i} represent a basis of the discrete space _{h}, while ^{∞} and ^{corr}, respectively. Note that ^{∞} has an analytical expression. The description of the discretization process in both the CG- and DG-FEM schemes is the content of the following sections.

In a CG-FEM approach the following expression of the electric flux (

where (φ_{i})_{i} is a collection of _{h}.

The discretization of

and

respectively. Note that

If we call _{n} the center of the _{n} are:

and

respectively.

and

respectively.

An alternative treatment of

Following Equation (31), we can consider the analogous formula for the electric flux in the DG-FEM scheme, i.e.,

where (φ_{i})_{i} is a basis of

As already mentioned, in general this discrete formulation of the flux does not verify the conservation of charge property. Conversely and despite the CG-FEM case, in the DG-FEM approach we can consider another expression of the discrete electric flux, i.e.,

that verifies the conservation of charge law, as described in Remark 3.

The main idea is to embed this _{int} (Equation 21) and not in the entire volume Ω. In order to integrate _{0}). _{0} is

and _{0} as (Nédélec,

As we are considering hexahedral elements, ^{l}(^{l}(

therefore also in Equation (41), we have ^{0} = ℚ^{0}. For a regular, hexahedral mesh with edge length _{0} basis function _{k} is supported on the two hexahedral elements _{k} = _{e} ∩ _{f} with normal vector _{k} and centroid

For more insights see Fortin and Brezzi (_{k} has been visualized.

Visualization of a zeroth-order Raviart-Thomas basis function _{e} and _{f}, which are sharing the face _{k} with unit outer normal _{k}. The vector valued function is equal to 1 · _{k} on the face _{k} and it decays when reaching the other parallel faces.

For the discretization of

If (φ_{i})_{i} is a basis of

and, due to linearity, we have

If we now apply the projection Π_{RT0} into _{0} to

Finally,

If we call _{n} the center of the _{n} reads,

where

^{th} basis function of the space

_{k})_{k} form a basis of _{0} and

_{k} the centroid and the external normal of the face _{k}, respectively (see Figure

As described in the next section, MEG forward computations will be carried out for a large number of dipole sources. In order to speed up the many numerically expensive computations of the secondary B-field ^{s} for all of these sources, following Wolters et al. (

If

If we combine Equations (53) and (50), we obtain

where _{MEG} is the so-called MEG transfer matrix and allows computing

To compute

Using the symmetry of

which can be solved for each row of ^{t}).

We implemented the CG-FEM and the two DG-FEM approaches [non-conservative Equation 38 and conservative flux (Equation 39)] for the MEG forward problem in the Distributed and Unified Numerics Environment (DUNE)^{1}^{2}

For numerical accuracy tests of our new CG- and DG-FEM implementations, we generated 4-layer homogeneous sphere models for which an analytical solution for the MEG exists (see section 2.1.4). We used four compartments with different conductivities in order to evaluate if, besides the analytical solution in Equation (15), also our numerical implementations show conductivity-independence of MEG in spherical volume conductors and because the four compartment model is closer to a realistic head model as shown in

Four compartment sphere model.

Brain | 78 | 0.33 | Ramon et al., |

CSF | 80 | 1.79 | Baumann et al., |

Skull | 86 | 0.01 | Dannhauer et al., |

Skin | 92 | 0.43 | Ramon et al., |

Parameters (from left to right) of the regular hexahedral meshes of the 4-layer sphere models used for validation purposes: segmentation resolution (Segm. Res.), mesh width (h), number of vertices and number of elements.

4 | 4 | 56,235 | 51,104 | |

2 | 2 | 428,185 | 407,907 | |

1 | 1 | 3,342,701 | 3,262,312 |

As only tangential orientation components produce an MEG signal in a multi-layer sphere model (section 2.1.4), we generated 8,000 dipoles with purely tangential orientations and unit strengths. The sources were uniformly distributed inside the brain compartment on spherical surfaces with 8 different logarithmically scaled eccentricities reported in Table

Source eccentricities and corresponding distances to the CSF compartment.

Distance to CSF comp. (mm) | 77.22 | 38.80 | 19.60 | 9.99 | 5.19 | 2.79 | 1.59 | 0.99 |

As the cortex has a thickness of 4 to 2 mm (Hämäläinen et al.,

With regard to the MEG sensors, we used 256 point-magnetometers outside the sphere model at a fixed radius of 110 mm (see Figure

Visualization of the 256 point-magnetometers used in the sphere model analysis. Radially ^{p}, ^{s},

We will use the two error metrics that are commonly used for validating EEG and MEG forward approaches (Meijs et al.,

and magnitude error (MAG%):

where ^{s} or the full B-field ^{p}, ^{s},

Statistical results of numerical accuracies will be visualized with mean curves and boxplots (see

The study of Engwer et al. (

Parameters (from left to right) of the regular hexahedral meshes of the 4-layer sphere models used to investigate the influence of skull leakages on the presented CG- and DG-FEM MEG approaches: segmentation resolution (Segm. Res.), mesh width (h), outer radius of the skull (mm) and number of leaky points.

2 | 2 | 86 | 0 | |

2 | 2 | 82 | 10,080 |

As a proof of concept, we computed one MEG forward solution using the DG-FEM approach in a more realistic scenario. Based on MRI recordings of a human head, a segmentation considering six tissue compartments (white matter, gray matter, cerebrospinal fluid, skull compacta, skull spongiosa, and skin) that includes realistic skull openings such as the foramen magnum and the optic nerve canal was generated. Based on this segmentation, a six-compartment realistically shaped head model was built, a hexahedral mesh of 2 mm resolution resulting in 508,412 vertices and 484,532 elements (

In this section, the results relative to the evaluation and validation in multi-layer homogeneous sphere will be presented, followed by the results of one forward computation on a realistically shaped head model.

In this section, we will validate, compare and evaluate the three developed and implemented approaches for the MEG forward problem, namely the CG-FEM and the DG-FEM with non-conservative (Equations 31, 38) and conservative flux (Equation 39), in spherical volume conductor models.

To recall the most important symmetry properties of the MEG forward problem in spherical volume conductor models, to prepare the numerical studies below and to enable an easier interpretation of their results, we first tested and visualized the properties of the MEG analytical solution for a multi-layer homogeneous sphere model, as reported in Remark 1. Here, we consider radial and tangential point-magnetometers, i.e., we have projected the B-field (^{p},^{s},

In Figure ^{2} norm of primary ^{p} (in pink) and secondary ^{s} (in blue) B-fields, i.e.,

We notice that the only contribution to radial point-magnetometers is given by the primary component of the B-field, ^{s}, as proven in Sarvas (

^{2} norm of the primary (^{p}, pink) and secondary (^{s}, blue) B-fields (see Equation 58) for tangentially-oriented sources at logarithmically scaled eccentricities. Values are expressed in Tesla ^{2} norm of the radial full B-field component relative to the one for the most eccentric source (see Equation 59) for tangentially-oriented sources at logarithmically scaled eccentricities.

In Figure ^{2} norm of the full B-field for radial point-magnetometers normalized to the maximum over all tested sources, which is achieved for the most eccentric source, i.e.,

We can see how the magnitude of the full B-field increases for sources with an increasing eccentricity.

In Figure ^{2} norm of the primary (in pink) and secondary (in blue) tangential B-field components, i.e.,

for tangentially-oriented sources at different eccentricities.

Analytical solutions in spherical volume conductor model for tangential point-magnetometers: ^{2} norm of the primary (^{p}, pink) and secondary (^{s}, blue) B-fields (see Equation 60) for tangentially-oriented sources at logarithmically scaled eccentricities. Values are expressed in Tesla

In this Figure we can see that, for tangential point-magnetometers, the deeper the sources are, the more the primary and secondary B-fields give identical contributions, but with opposite signs, to the full B-field, i.e., they more and more cancel each other out. Toward the sphere center, sources become more and more radial and the full B-field goes down to zero. However, as Figure ^{p} · ^{s} ·

We now turn our interest to the validation and evaluation of our implemented new numerical FEM approaches for the MEG forward problem in spherical models. We will only consider tangentially-oriented sources for the validations and evaluations in the next sections, because, as seen in section 4.1.1, radial sources do not produce any magnetic field outside spherical volume conductor models. Following Equations (28) and (14), we will from now on measure errors of the vector fields ^{s} (Figures

Accuracy comparison for secondary B-field ^{s} computation (Equation 28) between DG-FEM with non-conservative flux (Equation 38, in red) and DG-FEM with the conservative flux (Equation 39, in green) in a 4 mm hexahedral sphere model: visualized are the means

Validation and convergence analysis for secondary B-field ^{s} computation (Equation 28) of DG-FEM with conservative flux (Equation 39) in a 4 mm (green), 2 mm (red) and 1 mm (blue) hexahedral sphere model: visualized are the means

Accuracy comparison for secondary B-field ^{s} computation (Equation 28) between CG-FEM (in warm colors) and DG-FEM with the conservative flux (in cold colors), for different mesh resolutions: visualized are the means

Accuracy comparison between CG- and DG-FEM for solving the MEG forward problem, i.e., the full B-field

In this analysis the focus is on DG-FEM and the necessity of embedding the conservative flux (Equation 38) in the evaluation of the secondary B-field ^{s}. We will thus validate and compare the DG-FEM MEG forward methods with non-conservative (Equation 38) and conservative (Equation 39) flux.

The RDM% and MAG% statistical errors can be seen in Figure

Let us now discuss in more detail the eccentricity of 0.9796, i.e., 1.59 mm from the brain-CSF boundary. Higher eccentricities do not have practical importance, as already explained in section 3.3. For the eccentricity of 0.9796, the maximum difference of 20

With regard to the boxplot of the RDM%, the median values of the conservative flux case are overall smaller than the ones of the non-conservative flux. For sources with eccentricity value of 0.9796 the RDM% median difference is greater than 20 pp; the IQR difference is approximately 15 pp and the TR is constant and similar for both approaches.

In the MAG% boxplot (right column), the much better performance of the conservative flux approach is especially clearly visible. The MAG% median difference reaches 40 pp for realistic sources of eccentricity 0.9796. For the same sources, the TRs, IQRs and means are in general large, with a ratio 1:4 between conservative and non-conservative flux values. For lower eccentricities, we observe overall smaller errors.

Since we have seen in the last study that the conservative flux DG-FEM approach (Equation 39) performs remarkably better than the non-conservative approach (Equation 38), for the remainder of the paper, we proceed with DG-FEM as in Equation (49). The third study proposed is about the convergence of the DG-FEM for computing the secondary B-field ^{s}, when the mesh resolution is increased, namely from the coarsest resolution of 4 mm over 2 mm to the highest resolution of 1 mm. We studied the behavior of the RDM% and MAG% errors for 8,000 tangentially oriented and randomly distributed dipoles at different eccentricities. Results can be seen in Figure

The RDM% and MAG% error mean curves (Figure

The fourth analysis performed in this work is a comparison between CG- and DG-FEM for the MEG forward computation. RDM% and MAG%s are evaluated both for the secondary B-field ^{s} (Figure

In our following result discussion, we focus on the comparison between the two methods, rather than the performance of each method alone, which has been done for DG-FEM in section 4.1.3.

With regard to the secondary B-field ^{s} results, we will first analyze the mean RDM% curve (Figure

If we focus on the 1 mm analysis, we notice a high accuracy (up to around 1.5%) for eccentricities smaller or equal to 0.9796 (i.e., 1.59 mm from the CSF compartment). Even if in our current implementation, CG-FEM achieves slightly better results, the differences to DG-FEM are below 0.5 pp, so that in summary, DG-FEM constitutes an interesting alternative to the CG-FEM approach. Also for lower mesh resolutions of 2 and 4 mm, the performance of CG- and DG-FEM are very comparable for the realistic eccentricities up to 0.9796. A similar observation can be made for the mean MAG% curve, as the general trend for the three couples of curves (i.e., CG-DG 1 mm, CG-DG 2 mm, CG-DG 4 mm) is the same as before. When focusing on sources with eccentricity value of 0.9796, the mean MAG% difference between CG- and DG-FEM remains below 0.11 pp.

As for the boxplots for 1 mm mesh resolution (

The results when focusing on the full B-field

Motivated by the EEG results of Engwer et al. (

In Figure

Accuracy comparison for secondary B-field ^{s} computation (Equation 28) between CG-FEM (in warm colors) and DG-FEM with the conservative flux (in cold colors), in two different 2 mm hexahedral sphere models:

We observe that, in contrast to the improvement that DG-FEM could achieve in the EEG case (Engwer et al., ^{s} and, since the primary B-field ^{p} is also not influenced, thereby also the full B-field and thus the MEG forward problem.

If we observe the plots in the left columns, we notice that the curves of the leaky scenarios are completely overlaying the curves of the non-leaky scenarios, both for CG- and DG-FEM and both for RDM% and MAG% mean curves.

Also in the boxplots we cannot distinguish the behavior of the RDM% and MAG% in the leaky or non-leaky scenarios.

In the last study, as a proof of concept, we simulated an auditory N1 MEG signal using the new DG-FEM method with conservative flux (Equation 39) in the 6 compartment realistically-shaped head volume conductor model. Following experimental evidence (Okamoto et al.,

Exemplary EEG and MEG forward computation for an auditory source computed using DG-FEM in a realistically shaped head model. Hexahedral mesh with 2 mm resolution, 6 compartments, sagittal slice

In this paper, we developed, implemented and evaluated one CG-FEM and two new DG-FEM approaches, a conservative and a non-conservative one, to solve the MEG forward problem. In section 2, we provided the mathematical theory for the CG-FEM and for the two new DG-FEM approaches with conservative and non-conservative discrete representation of the electrical flux. We started from the EEG formulation and continued with the MEG approaches. In section 3, we first described the implementation of the FEM-based MEG forward approaches in DUNEuro, a new modular C++ toolbox dedicated to solve partial differential equations in neuroscience. Furthermore, we presented the validation and evaluation platform for the new methods. In section 4, we presented the results of our analysis.

First, we tested and visualized the symmetry properties of the MEG analytical solution for a multi-layer homogeneous sphere model, as described in Remark 1 and as proven by Sarvas (

In a second analysis, we studied how large the influence of a conservative representation of the electrical flux in the computation of the secondary B-field is by adopting the DG-FEM is adopted. By comparing the DG-FEM with a conservative (Equation 39) and non-conservative (Equation 38) flux in a 4 mm multi-layer homogeneous sphere model, the high importance of DG-FEM with conservative flux could be worked out, outperforming the non-conservative DG-FEM scheme in all cases (Figure

Results from the third study show the convergence of the DG-FEM numerical solutions toward the analytical solution when the resolution of the meshes is increased from 4 mm over 2 mm down to 1 mm (Figure

From our comparison studies between CG- and DG-FEM regarding the secondary and the full B-fields, we first of all noticed that the accuracy for the 1 mm mesh resolution is extremely accurate for both methods: the mean RDM% is only up to ≈1.5% and the mean MAG% only up to ≈0.1% for sources with realistic eccentricities of 0.9796 (i.e., 1.59 mm to the next conductivity jump at the brain-CSF boundary) (Figures

To the best of our knowledge, not many recent studies on finite element methods applied to solve the MEG forward problem have been presented, and none of them about DG-FEM. (Van den Broek et al.,

van den Broek et al. (

A CG-FEM MEG forward modeling study in a human (and rabbit) head volume conductor model was performed by Haueisen et al. (

Another example of a CG-FEM and Biot-Savart's law scheme used to compute the electric potential and the B-field was presented by Schimpf et al. (

In Vorwerk (^{3}

In Vorwerk et al. (

Our last study was about the influence of leaky points on the computation of the secondary B-field when DG-FEM is adopted (Figure

A further important aspect to discuss is that, if a combined EEG and MEG source reconstruction is strived for (Fuchs et al.,

In this study, we did not evaluate the computational costs of the CG- and DG-FEM schemes for the computation of the MEG forward solution. Because of the higher number of degrees of freedom, DG-FEM is computationally more expensive than CG-FEM. However, the FEM transfer matrix approach (section 2.4) considerably reduces the computational costs of both approaches, so that this aspect gets less relevant for practical applications.

We now discuss possibilities for further accuracy increase that we plan to evaluate in our future work. In this study, sources were just chosen randomly, i.e., the influence of the source position relative to an element of the discretization was not yet investigated. It is well known that the combination of computing leadfields only for the most accurate sources combined with leadfield inter- and extrapolation techniques for other sources might not only speed up computations, but might also further increase numerical accuracy (Yvert et al., _{int} can reach high values when the source is relatively close to a quadrature point on the internal skeleton, because of the singularity in ∇^{∞}. Moreover, in Drechsler et al. (^{∞} was derived for isotropic and anisotropic conductivity distributions in the source space. A further future goal will thus be its implementation and use to further decrease the numerical errors in our FEM implementations, both on the CG- and DG-FEM sides. In addition, the degrees of polynomials in

Overall the newly implemented conservative flux DG-FEM scheme offers an interesting new EEG and MEG forward modeling approach. It can be used especially in leakage scenarios and, in general, for comparison purposes, not only in EEG and MEG source analysis, but also in bioelectromagnetism applications, i.e., including also the simulation of transcranial electric and/or magnetic stimulation (Miranda et al.,

We presented theory, validation and evaluation of three finite element method (FEM) approaches for the MEG forward problem, namely the continuous Galerkin FEM (CG-FEM), as well as two new approaches, the discontinuous Galerkin FEM (DG-FEM) with a conservative and a non-conservative flux implementation. All three methods have been implemented in the DUNEuro software module. Statistical validations and evaluations have been performed on multi-layer homogeneous sphere models represented via hexahedral meshes and the subtraction approach has been adopted as source model. DG-FEM with conservative flux implementation, i.e., a main feature of a DG-FEM discretization, turned out to be superior to the non-conservative flux variant. The new DG-FEM method showed proper convergence behavior with increasing mesh resolution. When compared to the CG-FEM, DG-FEM provided results that are in a comparable range of high accuracy. Furthermore, both methods are able to model realistic head volume conductor models with their tissue inhomogeneities and anisotropies. In contrast to EEG studies, the so-called skull leakage effects did not play a crucial role for MEG. However, for EEG or combined MEG/EEG source analysis scenarios, DG-FEM offers an interesting new alternative to CG-FEM, considering the importance of a high accuracy of the forward problem solution in MEG/EEG source reconstruction. Finally, the DG-FEM MEG forward simulation in a realistic head model for an auditory source resulted in EEG and MEG topographies that are in line with practical findings in the field of auditory evoked responses.

MP, CE, and CW conceived the study. MP wrote the code with the supervision of AN and CE. JV and CW provided spherical and realistic head models. MP and JV constructed the FEM models. MP performed the simulations and the analysis, produced the images, interpreted the results, and wrote the paper. All authors took part in the scientific discussion at multiple stages of the study and provided feedback from the modeling (CW, CE, HB, RO), theoretical (CE, CW, AN, JV), and technical (AN, CE) perspective. All authors reviewed the manuscript and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We wish to thank Professor Marco M. Fato and Dr. Gabriele Arnufo (University of Genova) for their support.

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