Improved Source Localization of Auditory Evoked Fields using Reciprocal BEM-FMM

Abstract

Precise localization of auditory evoked fields (AEFs) from magnetoencephalography (MEG) data is very important for the functional understanding of the auditory cortex in medicine and cognitive neuroscience. The numerical solution of the field equations in the human head using the boundary element method (BEM) is a powerful tool for achieving this. The spatial resolution of the BEM is crucial for the achievable accuracy of localized neural sources. However, in classical BEM (as implemented, e.g., in MNE-Python), very high resolutions are impractical due to the associated prohibitive computational effort. In contrast, our recently introduced reciprocal boundary element fast multipole method (reciprocal BEM-FMM) allows for hitherto unprecedented spatial resolution of forward models. In this work, we employ reciprocal BEM-FMM to construct high-resolution forward models to localize AEFs. Simulated AEFs were generated using a direct BEM-FMM approach on realistic 40-tissue Sim4Life segmentations. Comparative analyses from simulated data demonstrate that high-resolution BEM-FMM forward models yield statistically superior source estimates relative to the 3-layer BEM. We also compare BEM-FMM forward models with source dipole resolution varying from 25,000 to 3,200,000 sources, and find that resolutions above 200,000 sources are sufficient for achieving accurate, high-resolution source estimates. We therefore recommend using the high-resolution reciprocal BEM-FMM to utilize high spatial anatomical precision for the modeling of neural activity.

Supplementary Information

The online version contains supplementary material available at 10.1007/s10548-026-01190-x.

Introduction

The objective of source localization (or source estimation, source reconstruction) (Knösche and Haueisen 2022) is to estimate the location of neural activation from non-invasive recordings such as electroencephalography (EEG) or magnetoencephalography (MEG) (Hämäläinen et al. 1993). In both the EEG and MEG forward problem, the quasi-static approximation to Maxwell’s Equations is sufficient (Plonsey 1972; Bosetti et al. 2008). This implies that, upon prescribing a set of possible source locations, known as source space, any activation pattern which is a linear combination of activation in different locations of the source space will have, as sensor measurements, the corresponding linear combination of the measurements of the individual source space locations. Ultimately, this implies that all information relative to activation along the source space can be obtained from the lead-field matrix, which has size , where is the number of sensors, and is the number of sources, and whose (i, j)-th entry is the measurement at sensor i elicited by source j. The calculation of the lead-field matrix given a head model—i.e. a collection of surface or volumetric meshes approximating the head geometry, together with a selection of conductivity values—and a source space is the forward modeling step of source localization.

Once the lead-field matrix is computed, the inverse step follows by producing estimates based on the lead-field matrix itself and no longer involves the forward solvers. The inverse problem has infinitely many solutions (Helmholtz 1853) so, typically, it is solved by means of a minimum norm estimation (MNE), an Occam’s razor approach aiming to give a “best explanation” given the measured data, in this case a solution with the smallest norm.

The most common approach to solve the forward step when using surface meshes is the boundary element method (BEM). When using volumetric meshes, the typical approach is the finite element method (FEM). Both BEM and FEM reduce to the computation of a forward system matrix , which solves for the measurements given activation at a location by solving the linear system . The major disadvantage of BEM is that its system matrix is dense, so the solution of the linear system already becomes computationally infeasible when the number of elements in the head model is 10,000. On the other hand the system matrix of FEM is sparse, which allows models involving millions of surface elements, however many more elements are needed in a volumetric mesh in order to capture fine geometric details.    

More recently, the boundary-element fast multipole method (BEM-FMM) has appeared as an alternative to BEM for models involving surface meshes (Makarov et al. 2018, 2020). This method is based on the coupling of the fast multipole method (Greengard and Rokhlin 1987) with the charge-based formulation of the BEM (Barnard et al. 1967; Ponasso 2024). The major advantage of BEM-FMM is that it can compute solutions with surface element meshes involing several million elements, and since less elements are required on surface meshes to capture fine details, the resolution of the models that FEM can compute pales in comparison to BEM-FMM. A recent notable example of an application of BEM-FMM is the largest map to date of extracellular electric field distributions of samples of mouse visual cortex at the microscopic level (Qi et al. 2025).

In Ponasso (2025b) we introduced BEM-FMM to the problem of source localization by combining it with the reciprocal approach based on Lorentz’s reciprocity theorem. This approach is based on the computation of cortical basis functions by means of a fictitious current injected through each MEG sensor coil, similar to forward solutions in transcranial magnetic stimulation (TMS). The use BEM-FMM allows us to avoid computing the dense system matrices of the traditional surface-potential BEM (Geselowitz 1967, 1970), and as a result our method enables source localization using multiple high resolution meshes. In Ponasso (2025b) we localized somatosensory evoked fields and obtained good results compared to the standard BEM as implemented by MNE-Python with 3-layer models (Gramfort et al. 2013, 2014).

The objective of this work is to investigate the performance of high-resolution models in contrast to lower resolution models in the context of auditory evoked fields, which are more challenging from a source reconstruction perspective than the somatosensory evoked fields: on the one hand, the binaural auditory stimulation elicits activation on both hemispheres, with two approximately symmetric sources at the far-apart left and right temporal lobes; on the other hand, the head anatomy near the temporal lobes is more complicated than near the parietal lobes, and it is harder to represent accurately using downsampled 3-layer head models. As a result, high-resolution models will be able to provide more accurate source localizations—this was already suggested in the error maps of Ponasso (2024).

Auditory evoked fields (AEFs) are magnetic signals elicited by auditory stimuli. They are localized in the auditory cortex, having components in the Heschl gyri (primary auditory cortex) and in the planum temporale as a secondary area (Pelizzone et al. 1987; Liégeois-Chauvel et al. 1994; Yoshiura et al. 1995; Godey et al. 2001; Henderson et al. 2023). There are three main components of auditory evoked fields: 1) Middle-latency components, e.g. P1m (< 50 ms) which originates in regions medial and lateral to the Heschl gyri; 2) N1m (100 ms), consistently localized in the lateral Heschl gyri and planum temporale; and 3) P2m (200 ms), which has been found to have origins near the N1m source locations although with much greater interindividual variability (Lütkenhöner and Steinsträter 1998).

The source localization of AEFs is relevant from an applications standpoint as they can be used clinically for presurgical mapping in the temporal lobe epilepsy (Godey et al. 2001), in the identification of functional reorganization in patients with lesions (Nakasato and Yoshimoto 2000), and as a biomarker for language and developmental disorders (Cardy et al. 2008; Park et al. 2022). The sources of AEF are also very important due to their correlations with mental health and psychological disorders (Kim 2021; Fitzgerald 2024; Dib et al. 2024). Given these important clinical contexts, it is essential to establish that BEM-FMM forward models provide accurate source estimates of AEFs.

In this study, we analyze the source localization of auditory evoked fields elicited by binaural auditory stimuli for a cohort of 7 patients with drug-resistant depression, whose data were provided by the authors of the study of Dib et al. (2024). We construct forward models using the standard 3-layer BEM implemented by MNE-Python and using reciprocal BEM-FMM on models of 7 tissues, which we refer to as the BEM and BEM-FMM forward models, respectively. We simulate noiseless N1m peaks of binaural auditory stimulation for each patient using a direct BEM-FMM approach on the Sim4Life 40-tissue segmentation by the IT’IS Foundation (Sim4life 2024, 2025). We compute source localization solutions from simulated data of varying levels of signal-to-noise ratio, and analyze the effects of inverse operator regularization on the accuracy of source estimates. We then compute solutions from experimental N1m peaks and draw comparisons between estimated sources using the BEM and BEM-FMM forward models. Additionally, we compare BEM-FMM forward models of varying dipole resolution, from 25,000 to 3.2 million sources, and on 3- and 4-layer head models.

Materials and Methods

Source Localization Forward Modeling Via Reciprocity

Reciprocal computation of the BEM-FMM forward model. Source localization is a type of mathematical inverse problem where measured MEG signals are related to cortical dipoles strengths by a forward model. In this work, all forward models analyzed are linear forward models, i.e. lead-field matrices, in which the relationship between dipole strengths and MEG signals is assumed linear. We construct forward models using the reciprocal BEM-FMM approach (Ponasso 2025b), which hinges on the Lorentz reciprocity theorem relating the primary electric field [] induced by an induction coil outside the cortical surface and the magnetic flux density [] induced by a current dipole source at location [] within the cortical surface (Plonsey 1972; Heller and van Hulsteyn 1992; Nummenmaa et al. 2013; Ponasso 2025b):

1

where [] is the dipole moment of the current dipole source located at , is the boundary of the induction coil with unit normal , and [] is the amplitude of the current flowing through the coil. Assuming the dipole moment is oriented normal to the cortical surface, we can extend Equation 1 to induction coils and current dipoles, yielding

2

where denotes the lead-field matrix whose entries are given by

3

where is the induced electric field of the -th induction coil, is the strength of the -th current dipole, and is the total magnetic flux through the -th induction coil.

Equation 2 is the template for MEG source localization forward models, where each of the coils can now be considered as MEG sensors, with the entries of indicating the total measured output of each sensor, and denoting the unknown strengths of each of the current source dipoles (Knösche and Haueisen 2022).

In the BEM approach to forward modeling, Eq. 2 is typically solved directly, where simulated MEG signal outputs are computed independently for each of the dipoles (Knösche and Haueisen 2022). That is, the entries of are computed column-wise. In the reciprocal BEM-FMM approach, Eq. 2 is solved reciprocally, where the electric field induced by the -th coil is computed across all dipoles. That is, the entries of are computed row-wise. Further details on the computation of the BEM and BEM-FMM forward models utilized in this study are found in Section Computation of Forward Models.

Inverse Solutions Using Minimum Norm Estimation

Minimum norm estimation. We compute the maximum a posteriori solution to the inverse problem of solving for source strengths p given noisy MEG data (Knösche and Haueisen 2022). Such solutions are found by using minimum norm estimation to invert Eq. 3. This is done using the inverse operator (Hämäläinen and Ilmoniemi 1994)

4

where is the prior source covariance, is the noise covariance, and is a regularization parameter. In this way, the minimum norm estimated solution to Eq. 2 is .

For numerical stability, it is useful to whiten both the MEG signal and lead-field matrix with a whitener . The whitened signals possess unit variance, and the whitened lead-field matrix is weighted to bias solutions away from regions of significant noise. In this manner, the whitened inverse operator is

5

where is the identity matrix. There are infinitely many possible choices for whiteners (Kessy et al. 2018); we opt for the most commonly used whitener .

To correct for superficial sources appearing on the cortical gyri nearest the MEG sensors, we use a depth weighted source covariance matrix which is a diagonal matrix whose diagonals are the norms of the corresponding lead-field matrix column (Knösche and Haueisen 2022),

6

Solution normalization with dSPM. We can further bias our solutions towards the most statistically relevant dipoles using dynamic statistical parametric mapping (dSPM) (Dale et al. 2000). The dSPM solution is obtained from normalizing the reconstructed sources by

7

Then is the vector of depth-corrected reconstructed source strengths. The reader may note that we excluded the depth-weighting parameter in the exponent of the source covariance . Since we are only considering dSPM normalized solutions in this work, the effect of the depth exponent is negligible (Lin et al. 2006). Therefore, we assume that this exponent is 1, and then the source covariance remains as it is shown in Eq. 6.

Automatic regularization. Typically, the regularization parameter is chosen as the inverse of the signal-to-noise ratio, , since this will yield a model with a theoretical minimal relative model error under the ideal assumption of zero-mean Gaussian noise and source covariance (Knösche and Haueisen 2022). However, in practical scenarios one cannot know the optimal value of , and thus it serves as a model hyperparameter.

By inspecting Eq. 5, one can see that serves the role of improving the condition number (the ratio between the maximum and minimum singular values) of by nudging the diagonal. Therefore, we may automatically determine a sufficient choice for by iteratively ‘nudging’ the diagonal of by until the condition number is within a tolerable size (e.g. ). This procedure removes zero eigenvalues from the matrix, and thus guarantees a numerically stable computation of (Belsley et al. 1980; Dennis and Schnabel 1996). We take this approach of regularization for all inverse solutions in this study unless stated otherwise.

It is worth noting that our approach here is akin to applying Tikhonov regularization (a.k.a. ridge regression) or truncated singular value decomposition (TSVD) to compute the whitened inverse operator . In Tikhonov regularization, one uses a technique such as cross-validation or the L-curve method to determine the optimal choice of . In TSVD, the forward operator dimensions associated to near-zero singular values are projected off. Both of these approaches are concerned with improving the conditioning of the forward operator and have been used for MEG source localization (Golub et al. 1979; Gencer and Williamson 1998; Knösche and Haueisen 2022). The key distinction between these methods and our approach of automatic regularization is that we seek the numerically stable inversion of without directly modifying the forward operator.

Auditory Response Data Processing

Data acquisition. Data were provided to us by the authors of Dib et al. (2024). These data were collected from 9 study participants with major depressive disorder. Out of these 9 participants (labeled N01– N09), two of them (N03 and N05) did not have MRI data available, so we restricted our study to the remaining 7. Participants were recruited from the University of New Mexico Mental Health Center to participate in a study focused on the loudness dependence of auditory evoked potentials (LDAEP) in pre- and post- electroconvulsive therapy (ECT). However, for this paper we only analyze the pre-treatment data.

T1-weighted structural MRI images were recorded using the 3-Tesla Siemens Trio MRI scanner at the Mind Research Network of Albuquerque, New Mexico. Raw MEG recordings were collected using the Elekta Neuromag VectorView 306, which possesses 102 magnetometers and 204 gradiometer pairs. Participants were stimulated with binaural auditory tones at a frequency, with each tone lasting . The stimulation was performed using tones of varying loudness; however, for this paper we only consider the tones. Further details of patient data acquisition are described in Dib et al. (2024).

MEG signal pre-processing. MEG data pre-processing steps for the Max Planck Institute pipeline were performed in MNE-Python 1.8.0 (Gramfort et al. 2013). Measurement data were run through a FIR low-pass filter with the upper pass-band edge at , processed with Maxwell filtering—spatiotemporal signal space separation (tSSS), see Taulu and Simola (2006)—and finally high-pass filtered with a lower pass-band edge. Maxwell filtering was done with the default recommended expansion settings of and , a spatiotemporal buffer duration of , and a spatiotemporal correlation setting of 0.9. The applicability of the default expansion settings was evaluated for a single patient’s data and found adequate. N1m peaks for each participant were determined at a mean latency of .

Computation of Forward Models

Tissue layers. The BEM and BEM-FMM forward models were constructed using different tissue segmentations. For the BEM models, participant MRI images were segmented using the FreeSurfer pipeline (Fischl 2012). Three layer BEM models were created in MNE-Python via the ico-5 option, which creates the scalp, outer skull, and inner skull tissue layers from the FreeSurfer segmentation. The BEM-FMM forward models utilize tissue layers segmented using the SimNIBS headreco pipeline (Nielsen et al. 2018), with the BEM-FMM using the full seven layer model comprising of the scalp, skull, cerebrospinal fluid (CSF), gray matter (GM), white matter (WM), ventricles, and eyes. The conductivity values assigned to each tissue are listed in Table 1.

Table 1.

Conductivity values of each tissue layer used in the BEM and BEM-FMM forward models

Tissue	Conductivity []	Reference
(a) Tissue layers used in the BEM-FMM forward models, their associated conductivities, and corresponding references
Scalp	0.25	(Truong et al. 2012)
Skull	0.01	(Wagner et al. 2004)
CSF	1.654	(Wagner et al. 2004)
GM	0.275	(Wagner et al. 2004; Thielscher and Wichmann 2009)
WM	0.126	(Wagner et al. 2004; Thielscher and Wichmann 2009)
Ventricles	1.654	(Wagner et al. 2004)
Eyes	0.5	(Gabriel et al. 1996; Opitz et al. 2015)
(b) Tissue layers in the BEM forward models and their corresponding conductivities and references.
Scalp	0.25	(Truong et al. 2012)
Outer Skull	0.01	(Wagner et al. 2004)
Inner Skull	0.33	(Vorwerk et al. 2019)

The conductivity value of the ventricles is chosen to coincide with that of the CSF

In addition to these tissue segmentations, we employed the Sim4Life 40-tissue head model from the IT IS Foundation (Sim4life 2024, 2025) to generate simulated auditory signals. This segmentation delineates 40 distinct tissue classes, encompassing cortical and subcortical regions, cranial bones, meninges, vasculature, and sensory organs. The anatomical detail provided by this model enables the generation of more physiologically realistic synthetic auditory signals. The procedure for computing these synthetic signals is described in Section Generation of Synthetic Data for the Evoked N1m Peak. The full list of tissues may be found in the Supplemental Tables, and the tissue conductivities used in the simulations may be found in the IT’IS tissue conductivity database (Baumgartner et al. 2025).

Reciprocal BEM-FMM forward models of varying dipole resolution. Dipole resolution in BEM models is limited due to the computational constraints of computing large system matrices and computing the lead-field matrix directly. As the reciprocal BEM-FMM method does not posses these constraints, we are better suited to examine the effects of extremely high dipole resolution on localized sources. Therefore, we compute BEM-FMM forward models using resolutions of approximately 25, 50, 100, 200, 800, and 3200 thousand dipoles.

For each BEM-FMM forward model, we take as a source space the SimNIBS WM mesh (which initially possesses approximately 200,000 triangles for each participant) with dipoles normally oriented to the WM. The models of various resolution are achieved by decimating and refining this initial WM mesh. Mesh decimation was performed in MeshLab (Cignoni et al. 2008) using the Quadric Edge Collapse Decimation at varying levels. Mesh refinement was performed using a simple barycentric triangular subdivision routine.

Additionally, on models with 200, 800, and 3200 thousand dipoles, we apply zero, one, or two iterations of Laplacian smoothing (Lalancette 2018) on the dipole orientations, to examine effects of a smoother dipole vector field on these extremely high resolution models. Table 2 shows the parameter values used in the computation of the reciprocal BEM-FMM forward models. Additional details regarding this computation are found in Ponasso (2025b). Because of the variety of BEM-FMM forward models analyzed in this work, the reader may assume that “BEM-FMM forward model” refers to the 200,000 dipole model unless stated otherwise.

Table 2.

Parameter values used in the computation of the BEM-FMM forward models

Parameter	Value
FMM precision
Region of interest radius	[]
Neighbor integral correction radius	[]
Characteristic of the coil pulse waveform:	1[]

The symbol  denotes the average triangle area in the mesh

Reciprocal BEM-FMM forward models on three and four tissue layers. To assess the impact of tissue complexity on the localized sources, we computed two additional BEM-FMM forward models with three and four tissue layers. Both models were derived from the SimNIBS tissue segmentation described above. The three-layer model included the scalp, bone, GM, and WM tissues, while the four-layer model additionally incorporated the CSF tissue. Each model was evaluated using the 200,000 dipole source space without Laplacian smoothing.

BEM forward models. Because MNE-Python is limited in its capacity to build high resolution forward models, we cannot draw precise comparisons from these forward models with more than approximately 20,000 dipoles. Despite this, it is useful to analyze the differences in localized sources computed from various forward model methodologies.

We computed BEM forward models using MNE-Python with approximately 20,000 dipoles (the upper limit of the software’s capabilities). The source spaces for the BEM models are found via the ico-5 option of MNE-Python, which creates a uniform sampling of 10242 sources (for each hemisphere) on the interface between the FreeSurfer-segmented WM and GM meshes. The resulting LFMs were exported from MNE-Python into our in-house source localization software.

Head model coregistration. Participant head-to-MEG coregistration was performed on the SimNIBS scalp mesh. First, the sensor digitization points were manually transformed such that the anatomical points (left/right preauricular and nasion) were in the approximately correct locations. Then an iterative closest points (ICP) algorithm (Bergström and Edlund 2014) is applied so that the digitization points are better mapped to the scalp. Final small manual adjustments were made to the transformation as needed to ensure visually correct coregistration. The coregistration transformation obtained was then used for each of the BEM and BEM-FMM forward model computations. Errors in the digitization and coregistration can propagate into the localized sources; we assume that these errors are negligible when compared to model errors or signal measurement errors. For detailed analysis of how coregistration errors can affect source localization, please see (Troebinger et al. 2013; Sonntag et al. 2018; Chella et al. 2019).

Generation of Synthetic Data for the Evoked N1m Peak

Generation of noiseless data. We simulated noiseless data for the N1m peak of binaural auditory stimulation. This is similar to the simulation of evoked somatosensory data described in Ponasso (2025b), with the difference that to model activation on the primary auditory cortex we used multiple dipoles distributed over a patch of the region rather than a single dipole. We generate this simulated data in two manners, the first using MNE-Python’s SourceSimulator functionality, and the second using direct BEM-FMM on the 40-tissue Sim4Life head models.

To generate the BEM-FMM simulated data, we manually segmented the Heschl gyri of each participant from their T1-weighed MRI data by selecting nearby points in MRI coordinates. The nearest WM mesh triangles were tagged, and a dipole was placed directly above each of the tagged triangle centers between the WM and GM interfaces. The orientations of each of these dipoles were chosen to be the average normal to the nearest GM triangles. Once the dipoles were placed, we used the method of BEM-FMM with b-refinement described in Wartman et al. (2025), to calculate the electric potential on each tissue interface; see also Ponasso (2024). Once the potential is calculated, we computed the B-field on 64 observation points for each magnetometer coil and each individual coil the gradiometer pairs. With this, the integral of the normal component of the B-field is computed to generate the sensor outputs of the magnetometers, and in the case of gradiometers we oriented the unit normals in opposite ways for each gradiometer coil and divided the integral by the distance separating the coils of each gradiometer pair, cf. Knösche and Haueisen (2022). This process of simulation of the evoked N1m is illustrated in Fig. 1.

Fig. 1.

Simulation of the magnetometer signals for the evoked N1m peak on participant N01. (a) Sagittal view of the MRI and head model using the Sim4Life 40-tissue segmentation. The full list of Sim4Life tissues with corresponding edge colors is listed in Table 1 of the Supplemental Tables. (b) A patch of WM points (green) at the right Heschl gyrus is manually segmented from MRI data; finite-length current dipoles are placed directly above and oriented normal to the cortical surface, with the red and blue points denoting the isotropic current sources and sinks, respectively. As such, the direction of each dipole moment is given by the vector from the blue point to the red point. The WM mesh has been smoothed for visualization purposes. (c) The potential on each tissue (smooth and defaced skin plotted) is calculated with BEM-FMM and b-refinement, the B-field is subsequently computed at 64 observation points per magnetometer coil. The skin has been smoothed for visualization and anonymization. (d) A surface integral is computed to obtain the sensor outputs; here the interpolated and projected topography of the simulated N1m peak is shown.

The sensor outputs were simulated independently for the dipoles on the Heschl gyri of each hemisphere. The strength of the dipoles was normalized by the number of placed dipoles to make the total strength of the dipoles placed on each hemisphere exactly . To obtain the sensor outputs for the binaural N1m peak, we simply added the outputs of the activation of the left and right hemisphere—this process is valid by the quasi-static approximation to Maxwell’s equations (Plonsey 1972; Ponasso 2024).

Generation of noisy signals. Noisy synthetic signals were generated following the procedure outlined in Ponasso (2025b), which we summarize here. First, 2000 dipoles were distributed along the surface between the GM and the WM. For each of these dipoles, the simulated MEG sensor outputs were computed using BEM-FMM in the same manner described above. From these solutions, we generate a noise vector via a linear combination the simulated sensor outputs with each dipole having a random strength (sampled from the standard normal distribution). The noisy signal will be of the form

8

where is the simulated auditory signal described above, and is a parameter which determines the signal-to-noise ratio. Particularly, for a desired SNR, is calculated by minimizing

9

over values of between 0 and 1. Then the noisy signal will have the desired signal-to-noise ratio. We repeat this procedure to produce 1000 different noisy signals of the same SNR, and this is done for SNR values 1.5, 5, 10, 25, and 50, for a total of 5000 noisy signals per participant.

Source Reconstruction

All source localization solutions shown in this paper were found using dSPM. For source localization of synthetic N1m peaks, the noise covariance was set to be the identity matrix. For localization using the experimental data, we created a noise covariance matrix from the pre-baseline signals at the window.

To visualize reconstructed sources from BEM and BEM-FMM forward models in a same mesh, we mapped the BEM source spaces to the SimNIBS headreco mesh using a rotation and translation provided by the ICP algorithm. For each source (i.e. triangle center) on the headreco WM mesh, we computed its source strength by averaging the estimated source strengths of those sources of the BEM source models within a sphere of radius from the triangle center.

Model Analysis Using ROC Curves

To analyze model performance, we compute the receiver operating characteristic (ROC) curve for each of the source localization solutions using the simulated data. The ROC curve is a useful statistical tool for evaluating the performance of a binary classification model (Hastie et al. 2009). The ROC curve is computed for source localization by treating the source strengths as binary classification variables across varying activation thresholds (Knösche and Haueisen 2022). Specifically, we define each of the triangle faces denoting the Heschl gyrus found in Section Generation of Synthetic Data for the Evoked N1m Peak as true dipoles, and all other triangle faces as false dipoles. Given the normalized dSPM solution of source localization, consider a reconstructed source or normalized strength and a threshold from 0 to 1. If the source lies on the tagged Heschl gyrus, we call the source a true positive (TP) if , a false negative (FN) if . Similarly, if the source does not lie on the Heschl gyrus, we call the source a false positive (FP) if , a true negative (TN) if . The ROC curve is defined as the curve plotting the false positive rate, , versus the true positive rate, , across all thresholds .

The area under the ROC curve (AUC) can be used as a scalar metric to evaluate the performance of the solutions. In particular, if , then source localization perfectly captures the true dipoles tagged on the Heschl gyrus. If , then the source localization is no better than a random chance solution.

We perform ROC-AUC analysis on the BEM and BEM-FMM models for every participant using the noiseless simulated data described in Section Generation of Synthetic Data for the Evoked N1m Peak. The smoothing that is done on the source estimations from BEM forward models may negatively bias the ROC curves. Thus, when computing ROC curves for these solutions, we only consider the sparse BEM model solution and define the true dipoles as those on the sparse white matter mesh that are within of the triangle centers of the tagged Heschl gyrus. The TP, FN, FP, and TN sources are then computed in the same manner as described above.

To assess the validity of the automatic regularization procedure introduced in Section 2.2, we conducted ROC AUC analyses on localized sources derived from noiseless data across varying regularization parameters. Particularly, we vary from 0.1 to 100, 000, with the increase in corresponding to a reduction in regularization (see Eq. 4). The resulting AUC values were then compared against those obtained using the automatic regularization procedure via box plots for each parameter choice, aggregated across participants. We performed this analysis using the BEM forward models, as well as the BEM-FMM forward models on 20 and 200 thousand sources. In this case, the 20 thousand source model was obtained by selecting the source dipoles on the SimNIBS WM mesh nearest to the 20 thousand dipoles obtained from MNE-Python using k-nearest neighbors, and truncating the 200 thousand dipole LFMs.

Additionally, we evaluate the robustness of the BEM-FMM forward models to noise by performing ROC-AUC analyses with the simulated data across varying SNR levels described in Section Generation of Synthetic Data for the Evoked N1m Peak. Particularly, for each tested SNR, we compute source localizations for each of the 1000 generated noisy signals, repeated for every participant. In each of these scenarios, the automatic regularization method described in Section Inverse Solutions Using Minimum Norm Estimation is employed. This analysis is not applied to the BEM forward models, as the noise robustness of these low resolution, 3-layer models has been well studied (Goldenholz et al. 2009).

Quantifying Solution Focality Via Dispersion

We can quantify the focality of the source estimation solutions by obtaining a measurement of solution dispersion. This is accomplished by constructing a probability density function from the normalized source estimates . Specifically, we compute where is a constant so that . In this manner, is a probability density function with Cartesian coordinates treated as random variables. We then measure the dispersion of the estimated sources by computing the standard deviation of the distribution across X, Y, and Z coordinates. A smaller standard deviation, and thus a smaller dispersion, indicates a more focal solution and vice versa. Of course, this measurement depends on anatomical structure, and therefore cannot be used for precise comparison across participants. But we may make use of this to compare the focality of different solutions on the same participant.

Each white matter is aligned with the Cartesian axis, with the lateral, anterior-posterior, and superior-inferior axes corresponding to the X, Y, and Z axes, respectively. The binaural stimulation of the experimental data is therefore expected to yield symmetric activation across the X direction (i.e. across the left and right hemispheres). It follows that the distribution would be bimodal in the X direction. In this case, the standard deviation would misrepresent the focality of the source estimates. Because of this, we construct two distributions in the same manner described above, each corresponding to the left and right hemispheres. Focality of the source estimates can then be measured across each hemisphere separately.

Comparison of BEM-FMM Models

We analyze how the localized sources from the BEM-FMM forward models of varying resolution compare to each other by their correlation and relative error with the solutions from the most refined and smoothed model. Denote by the source localization solution from the forward model with 3200 thousand dipoles and two Laplacian smoothing steps, and the solutions from every other BEM-FMM forward model discussed in Section Computation of Forward Models. We map onto the nearest triangle faces of the reference model. Then, using as a reference, we compute the correlation between and using the Pearson linear correlation (Hastie et al. 2009)

10

where and are the standard deviations of the solutions and , respectively. Additionally, we compute the relative error between and

11

where is the standard Euclidean norm. Each of these metrics will provide insight into the effect of spatially refining and smoothing the BEM-FMM forward models, with particular relevance to identifying the limits of refinement for source localization.

To compare the localized sources between the 3- and 4- tissue models, we compute the relative difference percentage (RPD) of these models against the 7-tissue model (on 200,000 dipoles) as a reference:

12

Results

Figure 2 shows source localization results for the simulated N1m peak data on participant N01. Figure 2a shows the dSPM solutions found from simulated magnetometer and gradiometer data using the BEM and BEM-FMM forward models. Figure 2b shows the sensor topography for data simulated from the Sim4Life 40-tissue model. Figure 2c shows the ROC curves and corresponding AUC values for participant N01 across each model. Figure 3 illustrates the effects of regularization on the recovered AUC using the BEM and BEM-FMM forward models. Figure 3a shows box plots of AUC values as a function of the regularization parameter, obtained via sources localized from noiseless simulated data and aggregated across participants. Figure 3b shows the condition numbers of the forward and inverse operators of each model, aggregated across participants. Figure 4 shows the achieved AUC of participant N01 from noiseless simulated data across models of varying dipole resolutions and varying regularization parameters. Table 3 shows the computed AUC values corresponding to each participants source localization solutions obtained using the automatic regularization described in Section Inverse Solutions Using Minimum Norm Estimation.

Fig. 2.

Source localization and ROC curve analysis for participant N01 using synthetic data. (a) The source localization solutions using the simulated N1m magnetometer (first row) and gradiometer (second row) peak data on each of the BEM and BEM-FMM forward models. Perspectives are shifted to achieve a better view of the Heschl gyri. (b) The gradiometer topography of the simulated N1m peak data is shown for data types. (c) The ROC curves of each source reconstruction are shown, with dashed lines corresponding to reconstructions using the BEM forward models, and solid lines corresponding to reconstructions using BEM-FMM forward models. The red and blue curves correspond to magnetometer and gradiometer solutions, respectively.

Fig. 3.

Evaluation of the effect of regularization on the AUC values of source localization solutions from simulated noiseless data using BEM forward models with 20,000 sources (red) and BEM-FMM forward models of 20,000 (cyan) and 200,000 (blue) sources. (a) Box plots of the achieved AUC values using varying regularization parameters. Box plots are computed from AUC values achieved for each participant model, and shown for regularization parameters varying from 0.1 to 100, 000. The first and last columns in each plot are computed from AUC values achieved via the automatic regularization discussed in Section Inverse Solutions Using Minimum Norm Estimation and no regularization (), respectively. Plots are separated by AUC values achieved from simulated magnetometer and gradiometer data. (b) On the left, box plots of the condition numbers of the forward operator for each model are shown, aggregated across all participants. On the right, condition numbers of the inverse operator of each model, averaged across each participant, are plotted as a function of the regularization parameter, with error bars denoting the standard deviation. Additional data tables can be found in the Supplemental Tables.

Fig. 4.

Inverse regularization versus recovered AUC of participant N01 using simulated noiseless N1m peaks on models of varying dipole resolution. Achieved AUC values from simulated magnetometer and gradiometer signals are shown on the left and right plots, respectively. AUC values are computed for regularization parameters varying from 0.1 to 100, 000. The first row shows results from the 3- and 4-tissue layer models with 200 thousand dipoles. The second row shows results from models of 25, 50, and 100 thousand dipoles with no Laplacian smoothing. The third, fourth, and fifth rows show results from the 200, 800, and 3200 thousand dipole models, respectively, with zero, one, and two Laplacian smoothing steps. Details of the model resolutions tested are described in Section Computation of Forward Models.

Table 3.

The AUC values of the BEM and BEM-FMM source localization models at the simulated noiseless N1m peak across all participants, corresponding to the Automatic Regularization column of Fig. 3a.

Magnetometers	N01	N02	N04	N06	N07	N08	N09	Avg.
BEM	0.83	0.83	0.83	0.82	0.87	0.75	0.84	0.82
BEM-FMM	0.79	0.91	0.88	0.87	0.90	0.91	0.85	0.87

Gradiometers	N01	N02	N04	N06	N07	N08	N09	Avg.
BEM	0.82	0.87	0.84	0.79	0.84	0.79	0.83	0.83
BEM-FMM	0.85	0.95	0.92	0.90	0.92	0.93	0.89	0.91

Additional source localization plots and ROC curves may be found in the Supplemental Figures. Inter-participant averaged AUC values are highlighted in bold in the right-most column.

Table 4 shows the ROC-AUC values achieved from source localizations of the BEM FMM 200,000-dipole models using simulated N1m auditory data across varying SNR levels. The reported means and standard deviations were obtained by averaging AUC values across the 1,000 noisy signal realizations per SNR per participant (see Section Generation of Synthetic Data for the Evoked N1m Peak). Table 4a and b shows results from simulated magnetometer and gradiometer data, respectively. The estimated sources and corresponding ROC curves for every participant may be found in the Supplemental Figures.

Table 4.

Source localization ROC-AUC values for each participant’s BEM-FMM 200,000 dipole forward model from simulated N1m auditory signals

SNR	N01	N02	N04	N06	N07	N08	N09	Avg.
(a) Average AUC values achieved for simulated N1m magnetometer data.
1.5	0.76	0.89	0.87	0.87	0.86	0.89	0.84	0.85
5	0.86	0.96	0.92	0.91	0.94	0.94	0.90	0.92
10	0.89	0.97	0.93	0.92	0.94	0.94	0.91	0.93
25	0.91	0.97	0.94	0.93	0.95	0.95	0.92	0.94
50	0.92	0.97	0.95	0.93	0.95	0.95	0.92	0.94
(b) Average AUC values achieved for simulated N1m gradiometer data.
1.5	0.80	0.89	0.86	0.86	0.87	0.87	0.83	0.85
5	0.88	0.95	0.93	0.93	0.95	0.94	0.91	0.93
10	0.90	0.96	0.94	0.94	0.95	0.94	0.92	0.94
25	0.91	0.96	0.94	0.94	0.95	0.95	0.93	0.94
50	0.92	0.97	0.95	0.95	0.96	0.94	0.93	0.94

Each entry shows the mean and standard deviation AUC achieved across 1000 simulated noisy signals at each SNR value. The average AUC value across participants for each SNR value is highlighted in bold in the right-most column.

Figure 5 shows the source reconstruction results for the measured N1m peak data of participant N01 using BEM and BEM-FMM forward models, with Fig. 5a showing the evoked gradiometer signals and topography, and Fig. 5b showing the dSPM solutions for each model. Figure 6 shows detailed source reconstruction results for participant N01 using the N1m magnetometer peaks with the BEM-FMM forward model, with Fig. 6a and b showing the normalized source strengths on the left and right white matter hemispheres, respectively, and Fig. 6c showing the corresponding probability distributions and focality measurements. Fig. 7 shows the dSPM source reconstructions for all participants using BEM and BEM-FMM forward models on the N1m gradiometer peaks. Please see Fig. 5 in the Supplemental Figures for corresponding solutions from magnetometer peaks. Table 5 shows the measured dispersion of the source localizations for every participant using the BEM and BEM-FMM forward models. Fig. 8 shows a comparison of source localization results for participant N01 across varying dipole resolutions, with Fig. 5a showing localized sources from N1m gradiometer peaks on BEM-FMM models with 25,000 and 3,200,000 dipoles, and Fig. 8b showing the plot of correlation versus relative norm percentages for each model against the model with 3,200,000 dipoles after two Laplacian smoothing steps. Figure 9 shows the relative percent difference between localized sources on the 7-tissue BEM-FMM forward model and the 3- and 4-tissue models. Additional figures showing the probability density functions for every participant may be found in the Supplemental Figures.

Fig. 5.

Source reconstruction results of participant N01 using the gradiometer data with the BEM and BEM-FMM forward models. (a) The evoked gradiometer sensor data is shown on the left, along with the signal RMS. The N1m peak is marked with the dotted black line, and the sensor topography at this peak is shown on the right. (b) The normalized dSPM solutions are shown, with the BEM solution shown on the top and the BEM-FMM solution shown on bottom. The left and right columns show focus perspectives of the left and right hemispheres.

Fig. 6.

Detailed source reconstruction results for participant N01 using the magnetometer sensor data with the BEM-FMM forward model. (a) To the left, the magnetometer topography at the N1m peak is shown. To the right, the reconstructed source strengths are plotted on the white matter surface, with visual focus emphasizing the left hemisphere. The Sylvian fissure is outlined with the blue curve. In the magnified view, a portion of the Heschl gyrus is highlighted in blue. (b) The same reconstructed source strengths are shown, with visual focus now on the right hemisphere, and the Heschl gyrus highlighted again in blue. (c) The source estimate distributions along the Cartesian coordinates. The distribution from the left and right hemispheres are shown in blue and pink, respectively. The corresponding dispersion measures for each distribution are shown on the right.

Fig. 7.

Source localization results of all participants at their respective auditory evoked N1m gradiometer peaks using reciprocal BEM-FMM and MNE-Python. For each participant, white matter perspectives are chosen to focus on the temporal lobe and Heschl gyrus in the top and bottom rows, respectively, on both the left and right hemispheres. The color map indicates the normalized current density, with blue and red colors indicating 0 and 1 values, respectively.

Table 5.

Dispersion measurements [] of the estimated sources of every participant using the BEM and BEM-FMM forward models for both magnetometer and gradiometer N1m peaks.

	Dispersion []
	Mag.	N01	N02	N04	N06	N07	N08	N09
BEM	LH Avg.	18	20	17	18	21	22	19
	RH Avg.	22	18	22	22	20	22	19
BEM-FMM	LH Avg.	20	20	18	20	22	22	20
	RH Avg.	22	18	24	23	21	24	21

	Grad.	N01	N02	N04	N06	N07	N08	N09
BEM	LH Avg.	17	20	16	18	20	22	19
	RH Avg.	20	17	21	21	19	22	18
BEM-FMM	LH Avg.	20	21	19	22	23	24	21
	RH Avg.	23	19	24	21	22	24	22

The dispersion is measured by treating the normalized source estimates as a probability density function and computing the standard deviation. Dispersion measurements are computed for each Cartesian coordinate direction, and averaged for both the left and right hemispheres. Tables denoting the dispersion measurements across all Cartesian directions may be found in the Supplemental Tables.

Fig. 8.

Comparison of source localization results of participant N01 auditory N1m peaks using BEM-FMM forward models of varying levels of refinement and smoothness. (a) Localized sources of N1m gradiometer peaks. The top plot shows results on the left and right hemispheres on the 25,000 dipole model with no Laplacian smoothing applied. The bottom plot shows the results on the 3,200,000 dipole model with two iterations of Laplacian smoothing. (b) Plot of the correlation versus relative norm of every BEM-FMM source localization solution against those of the 3,200,000 dipole model. Models of particular dipole resolution are denoted by each shape, with each shade denoting the number of Laplacian smoothing steps in the model, and color signifying the sensor type used in the reconstruction.

Fig. 9.

The relative percent difference (RPD) of the localized sources from auditory N1m peaks using BEM-FMM on the 3- and 4-tissue models is shown for magnetometer (top row) and gradiometer (bottom row) solutions. RPD is calculated with respect to the solutions using the full 7-tissue model. Color scales for the RPD are different for each model, as denoted by the color bars at the bottom.

Discussion

Source estimation of simulated data. Noiseless simulated data was recovered more accurately with BEM-FMM forward models over the BEM forward models, as depicted in Fig. 2 and Table 3, with participant averaged AUC yielding a 5 point improvement with magnetometer data, and an 8 point improvement with gradiometer data. In Fig. 3a, we observe that the variability of the AUC values between participants is smaller when using the BEM-FMM forward models. We also note how the automatic regularization procedure yields quite good AUC values across all participants and models. This suggests that the condition number threshold described in Section Inverse Solutions Using Minimum Norm Estimation is indeed sufficient for numerically stable and accurate source localization.

In Fig. 3a, we also see a trend in the recovered AUC as regularization is varied. When the models are over-regularized (i.e. is decreased) we see a reduction in the achieved AUC values, with this reduction being slightly more pronounced in the BEM forward models. As expected for simulated MEG signals with no noise, when regularization is reduced (i.e. is increased), the recovered AUC values increase and stabilize when using the BEM-FMM forward models, with slight improvements when using BEM-FMM forward models of 200,000 sources over those of 20,000 sources. However, perhaps unexpected, we see the opposite trend occur when using the BEM forward models. This may be in part due to the poor conditioning of the BEM forward operators as compared to the BEM-FMM forward operators, as shown in Fig. 3b. We see that the condition numbers of BEM-FMM forward models are almost an order of magnitude smaller than those of the BEM forward models, on average, with the BEM-FMM models of 200,000 sources being slightly better conditioned than the models of 20,000 sources. In Fig. 3a, we do see a slight decrease in the recovered AUC from BEM-FMM forward models when using gradiometer signals, particularly for the models on 20,000 sources. When comparing to the trend from the magnetometer signals, this difference may also be explained by the conditioning of the forward operator in Fig. 3b. Here, it is clear that the condition numbers of the magnetometer forward models are much better than those of the gradiometer forward models, which also holds true for each model type. Note that the conditioning of the inverse operator is likely not the source of this trend in recovered AUC, as depicted by Fig. 3b. We see that, although the inverse operators from the BEM-FMM models are generally much better conditioned than those from the BEM models, there is a trend in the condition number as regularization is decreased, in which each model type seem to follow.

Additionally, we see in Fig. 4 that the BEM-FMM forward models achieve high AUC values across a wide range of regularization. From these plots, we see that the magnetometer forward models recover AUC more consistently than the gradiometer forward models across varying model resolutions. This may again be a result of the generally improved condition number for magnetometer forward models. Further, we see that models with 200,000 sources perform slightly better than the other models. Notably, the AUC recovery of the 3- and 4-tissue models closely match those of the 7-tissue models. This is perhaps expected, as MEG measurements are relatively insensitive to additional anatomical detail in the head model (Lau et al. 2012; Piastra et al. 2020). (We might expect different behavior for EEG signals, where signal propagation is more strongly affected by tissue geometry and conductivity contrasts Lau et al. 2012; Vorwerk et al. 2019.)

The improved performance of the BEM-FMM forward models is additionally seen through the localization of noisy simulated data, as demonstrated by Table 4. Within, we see an average achieved AUC of 0.85 for both the magnetometer and gradiometer signals of highest tested noise (), demonstrating the robustness of BEM-FMM forward models for localizing signals within a tolerable amount of noise.

In Fig. 2 (as well as Fig. 4 in the Supplemental Figures), we note a visual improvement in the focality of the recovered solutions using the BEM-FMM forward models, with the localized sources from BEM forward models appearing more spread around the temporal lobe. The estimated sources from the BEM-FMM forward models are distributed almost exclusively along the Heschl gyri, with peaks occurring on one or both of these gyri; whether peaks occur on one or both gyri is mostly explained by head model coregistration to the MEG sensors. The source reconstructions from BEM forward models show instead spurious regions of activation in cortical regions away from the Heschl gyri, and a much less focal distribution; nevertheless the peaks remain located near the Heschl gyri with an identical pattern of peak asymmetry observed in the BEM-FMM reconstructions. The distinction between solutions from the two models likely arises from three factors: first, the improved forward operator conditioning of BEM-FMM forward models over those of BEM; second, differences in dipole resolution between the BEM and BEM-FMM forward models; and third, the more realistic tissue models employed by reciprocal BEM-FMM compared to the simplified tissue models used in MNE-Python.

These results suggest the clear influence of dipole resolution and number of tissues on source localization accuracy. However, more mysterious is the nature of what underpins the conditioning of the forward operators. Figs. 3 and 4 show that the BEM-FMM forward models are much less sensitive to smaller regularization parameters than the BEM forward models. Although this could be in part due to the number of sources and tissue layers, as indicated by Fig. 4, Fig. 3b suggests there may be some additional factor influencing the worse conditioning of the BEM forward operators. One possibility is that the BEM source space created by MNE-Python has less smooth normal vectors than the BEM-FMM source space. This may be unlikely, however, since the source space created in MNE-Python is sampled from the FreeSurfer meshes, which have quite smooth normals. Another possibility is the propagation of ‘hidden’ numerical errors in the creation of the BEM forward operator. Using BEM requires a dense kernel matrix, and there are a variety of approaches that can be taken in its construction (Mosher et al. 1999); we are unaware of the precise details of how the kernel is built in MNE-Python. In contrast, charge based BEM-FMM is a matrix-less method, the parameters of which we use are shown in Table 2. There could be a propagation of numerical errors in both methods, but the errors from BEM may be more pronounced. Exploring the underlying cause of the discrepancy shown in Fig. 3b would be a valuable direction for future investigation.

Given that real experimental data will be more similar to the high-resolution forward models, these results indicate that the BEM-FMM forward models will have better performance than the BEM forward models on experimental data. This is confirmed in the following subsection.

Source estimation of experimental data. The estimated sources recovered by the BEM and BEM-FMM forward models seen in Fig. 7 (as well as Fig. 4 in the Supplemental Figures) show very focal activation peaks on the Heschl gyri across all participants, as expected from a binaural auditory evoked N1m peak. Solutions appear to have similar focality across forward models, with BEM-FMM forward models yielding source estimates which encompass more of the Heschl gyri when compared to those of the BEM models, particularly for participants N01, N04, N06, N08, and N09. This larger coverage of the Heschl gryus is particularly easy to see in participant N01 of Fig. 5, where we see a strong activation spanning the full extent of the left Heshl gyrus, with minor activation on the right Heschl gyrus and left Heschl sulcus. By contrast, source estimates from BEM forward models on participant N07 seem more focal, and occupies a partial extent of the right Heschl gyri, with strong activation in the right Heschl sulcus. These focality observations are justified by the dispersion measurements from Table 5, where within we see that the localized sources from BEM and BEM-FMM forward models achieve similar levels of dispersion across all participants, with the BEM solutions yielding slightly more focal solutions.

In Fig. 8a, we see that the solutions from BEM-FMM forward models are localized in similar regions in both the 25 thousand and 3.2 million dipole models, as expected. We see that the 25 thousand dipole model strongly recovers sources on the medial portion of the Heschl gyrus, whereas the 3.2 million dipole model most strongly yields sources along the full extent of the Heschl gyrus. In Fig. 8b, we see how the solutions from models of varying resolution compare to the most refined and smoothed 3.2 million dipole model depicted in Fig. 8a. We see that solutions from all models are highly correlated with the solution on the reference model, with a correlation greater than 91%. This correlation is greatly increased for models of at least 200,000 dipoles, with a correlation greater than 98%. We also see the relative norm of solutions decrease substantially with increasing dipole resolution, from approximately 35% with the 25,000 dipole model, down to 15% with the 200,000 dipole model. The trend observed here indicates that BEM-FMM forward models with a resolution of 200 800 thousand dipoles can adequately approximate the solutions obtained from extremely high resolution models, at least in the context of evoked auditory signals. Interestingly, no clear trend emerges in relative norm percentage or correlation as a function of Laplacian smoothing steps. This may be due to our restricting dipoles orientations normal to the WM surface; perhaps the inclusion of arbitrarily oriented dipoles may reveal a more systematic trend.

In Fig. 9, we see that the relative percent difference between the 3- and 4-layer models against the 7-tissue model is quite small: smaller than 0.1% for both model types. This is expected, as the addition of the CSF (from 3 to 4 layers) and the eyes and ventricles (4 layer to 7 tissues) would have minimal impact on the localization MEG sources (Ramon et al. 2006; Birot et al. 2014), especially for evoked auditory signals.

In the comparative source reconstruction results of (Ponasso 2025b) between our BEM-FMM forward models and the BEM forward models of MNE-Python, our methods performed only marginally better. However, for the auditory data that we considered here, our method provides a significant improvement in localization accuracy. A possible explanation for this is that the geometry of the head near the central sulcus can be sufficiently approximated by a sphere or by a simplified classical BEM model with low-resolution shells. On the other hand, the temporal lobes cannot be accurately described with simpler models. Another possible explanation for this is that the quality of the evoked data in (Ponasso 2025b) allowed for simpler low resolution models to perform well. Conversely, the experimental auditory data used here was less than ideal—it posed a fair challenge to process and, even with sufficient filtering, it still seemed subject to moderate noise. With this in mind, we find the accurate and focal source localizations obtained from BEM-FMM forward models to be quite impressive.

With these observations, we find that head models with 200,000 source dipoles appear sufficient for localization of AEFs, with Laplacian smoothing used for visualization. Future work could assess the fit of signals from cortical patches of varying size and region, comparing high- and low-resolution models. Another interesting direction of future study is the evaluation of high-resolution multi-layer models for EEG source localization. Since EEG is more sensitive to conductivity changes in the head, this modality may benefit more from highly-realistic models. Preliminary results on EEG source localization employing highly-realistic models can be found in Ponasso (2025a)

Implications of Applying Reciprocal BEM-FMM to Auditory Source Localization. Accurate auditory source localization is critical for both clinical and research applications, as AEFs provide non-invasive markers of auditory cortical function. Validation of source localization through reciprocal BEM-FMM across varying physiological contexts ensures that high-resolution forward models can be applied reliably in translational neuroscience. For instance, in regards to the data studied in this work, analysis of AEFs was critical for determining whether LDAEP serves as a valid biomarker for major depressive disorder. Typically, analysis of loudness dependence was more common as an EEG task; application of MEG source estimation to investigate this phenomenon is relatively recent (Dib et al. 2024). In addition to psychiatric applications, AEF localization has been crucial for auditory cortical response in patients with auditory pathway lesions and neurodevelopmental disorders. AEF localization also plays an important role in pre-surgical mapping of the auditory cortex, particularly within the left temporal lobe, given its close proximity to language-processing regions (Shvarts et al. 2020). By enabling the computation of high-resolution forward models on realistic tissue models, reciprocal BEM-FMM can support further clinical and neurological research through the precise localization of auditory signals. These models could facilitate a more detailed understanding of the functional organization of the auditory cortex, and could extend to mapping across the entire cerebral cortex (Matsubara et al. 2024; Sanchez-Bornot et al. 2024).

Conclusions

Our results indicate that high-resolution BEM-FMM forward models including cortical layers significantly improve the accuracy of source estimates from AEFs compared to those of the 3-layer BEM models utilized by MNE-Python. All of our source estimations using high-resolution models indicate good quality of source localizations, with focal peaks of activation found on the Heschl gyri of each participant. This performance held true across various types of data, both simulated and experimental. Using simulated AEFs from the realistic 40-tissue Sim4Life segmentation and ROC measures, we find statistically superior results to the 3-layer BEM, with over a 5 point increase in achieved AUC from localized sources. Results from experimental data show that BEM-FMM forward models localize sources which cover more of the extent of the Heschl gyri when compared to those of the BEM forward models, while maintaining comparable solution focality. Comparison of BEM-FMM forward models of varying dipole resolution indicate that 200,000 source dipoles is sufficient for obtaining accurate, high-resolution source estimates. The high resolution source space and accurate source estimates achieved by BEM-FMM forward models indicates that our methodology could improve the applications of source localization in clinical settings, particularly in the localization of auditory evoked fields.

Supplementary Information

Below is the link to the electronic supplementary material.

Author Contributions

D.A.D., G.N.P., A.L., and Z.D. wrote the main manuscript text. D.A.D., G.N.P., S.G., and A.W. prepared all of the figures. J.L., C.C.A, and Z.D. performed data acquisition and investigative analysis. A.L., G.M.N., and B.M. performed essential data processing. D.A.D, G.N.P, and S.N.M developed underlying methodology and software. B.M., T.R.K., J.H., S.N.M., and Z.D provided project guidance and supervision. All authors reviewed the manuscript.

Data Availability

No datasets were generated or analysed during the current study.

Declarations

Competing interests

The authors declare no competing interests.