^{*}

The authors have declared that no competing interests exist.

Conceived and designed the experiments: YP. Performed the experiments: YP. Analyzed the data: YP. Contributed reagents/materials/analysis tools: YP. Wrote the paper: YP.

EEG/MEG source localization based on a “distributed solution” is severely underdetermined, because the number of sources is much larger than the number of measurements. In particular, this makes the solution strongly affected by sensor noise. A new way to constrain the problem is presented. By using the anatomical basis of spherical harmonics (or spherical splines) instead of single dipoles the dimensionality of the inverse solution is greatly reduced without sacrificing the quality of the data fit. The smoothness of the resulting solution reduces the surface bias and scatter of the sources (incoherency) compared to the popular minimum-norm algorithms where single-dipole basis is used (MNE, depth-weighted MNE, dSPM, sLORETA, LORETA, IBF) and allows to efficiently reduce the effect of sensor noise. This approach, termed Harmony, performed well when applied to experimental data (two exemplars of early evoked potentials) and showed better localization precision and solution coherence than the other tested algorithms when applied to realistically simulated data.

The EEG (electro-encephalography) method is based on amplifying and recording weak electrical currents produced by an active brain. Compared to other brain-imaging methods EEG is truly non-invasive and inexpensive. EEG, along with its ‘magnetic’ cousin MEG, is the only non-invasive brain-imaging method that has high-enough temporal resolution to track the full dynamics of brain events. Because the method has a limited spatial resolution EEG was not considered to be an ‘imaging’ technique until recently. Skull has low conductivity compared to adjacent head tissues which strongly diffuses electrical currents generated by brain activity. The problem of reconstructing brain activity from its blurred image recorded by sensors positioned outside of the head is an example of the inverse problem termed “source localization”.

Source-localization includes the two steps: creating a head model which describes how the head volume conducts electrical current, and fitting the model into the recorded data. Since the introduction of the MRI technique anatomical aspects of head modeling have became more manageable (tissue conductivities still remain a topic of debate as discussed in chs. 4 and 6 of

The localization results also strongly depend on what technique is used to fit the observed data. The focus of this study is on the ‘distributed’ or ‘nonparametric’ type of the inverse solution, where thousands of current dipoles (sources) at fixed locations are used to fit the data using the minimum-norm (L2) metric. The purpose was not to propose a new golden standard, but simply to show how choosing the source basis made of globally smooth functions improves source reconstruction within the L2 norm framework. Such choice can be considered as the first step in more complex source localization algorithms (see the

Here a new approach to EEG/MEG source localization, termed Harmony, that improves on some of the existing techniques is described. In addition to showing superior performance the proposed approach has a number of practical advantages: it easily interpolates the solution onto any cortical mesh and describes brain activity in the form of a spatial spectrum, analogous to the widely used temporal (Fourier) spectrum. The goal of this study was to show that source reconstruction in a small basis set comprising global smooth functions, such as spherical harmonics or spherical splines, significantly improves the source reconstruction quality as compared to the commonly used basis set of tens of thousands discrete cortical dipoles. To make it a fair comparison, algorithms with this commonly used basis set and a bare minimum of extra features (depth-weighting and spatial smoothing) were compared to Harmony: MNE (minimum-norm), WMNE (weighted minimum norm), IBF (Informed Basis Functions)

The idea of using a small (sparse) basis set to fit EEG/MEG data was previously investigated in the contexts of both the ‘distributed’ and ‘parametric’ algorithms, where, eventually, a small set of source dipoles was sought. For the ‘distributed’ approach, iterative schemes where source dipole weights at a given iteration are chosen based on the results of the previous iteration were used to truncate the resulting number of active sources to a small number, e.g., the FOCUSS algorithm

Results of source reconstructions for simulated and real EEG data are presented in

The results are shown on the inflated cortices: the 37-dipole source patches are shown by the green patches, dipole orientations for the sources and the solutions were constrained to be orthogonal to the cortical surfaces in this case. Color indicates amplitude and direction of cortical currents: inward – cold, outward – hot. Numbers underneath each panel give AUC measure of the reconstruction quality as given by (23).

The localization error is plotted along the x-axis. The percentage of sources reconstructed with localization error smaller than a given x-value is plotted along the y-axis. Results for different source configurations (illustrated by the insets) are shown in the corresponding panels.

Colored spots mark tested locations, spot colors indicate the localization error as given by the colorbar. Cyan colored spots (cyan is not represented on the color bar) mark sources which could not be localized significantly above chance level.

The dashed line indicates the true amplitude ratio = 2 between the two simulated sources. Results for different source configurations (illustrated by the insets) are shown in the corresponding panels.

Results for different source configurations (illustrated by the insets) are shown in the corresponding panels. Solid bars show results for sources in different cortical hemispheres, hashed bars - results for both sources in the same hemisphere.

Results for different source configurations (illustrated by the insets) are shown in the corresponding panels.

Results for different source configurations (illustrated by the insets) are shown in the corresponding panels.

Results for different source configurations (illustrated by the insets) are shown in the corresponding panels. Solid bars show results for sources in different cortical hemispheres, hashed bars - results for both sources in the same hemisphere.

Solid bars indicate results for sources on opposite hemispheres, hashed bars for sources on the same hemisphere. nHRM and nSPL denote Harmony solutions in the basis of spherical harmonics and spherical splines normalized by the standard deviation of the solution for each current dipole, i.e., dSPM-like.

From left to right: (1) A snapshot of the observed ERP potentials interpolated and shown on a flattened scalp. EEG sensor positions are marked with dark dots, the overlay shows ERP time course at the scalp location marked with a white dot. The red arrow indicates the time point at which the displayed scalp potentials were recorded (N1: 97 msec for visual stimulation, 86 msec for tactile stimulation). (2) The expected cortical activations: labels refer to the expected sites of cortical activation: V1 – primary visual cortex, and S1 – primary somatosensory cortex. (3) Harmony reconstruction results. (4) MNE reconstruction results.

The two Harmony solutions (spherical harmonics, HRM, and spherical splines, SPL) clearly stand out. Harmony produced coherent smooth solutions located close to the sources and without an apparent surface bias. The spherical harmonics and spherical splines solutions look very similar. The splines solution appears to be slightly ‘tighter’ but also slightly less coherent: it shows some artefacts of the spline ‘mesh’, where the centers of several neighboring splines appear as ‘grains’ inside the left hemisphere hotspot.

The remaining algorithms produced visibly incoherent solutions characterized by stripes of activity aligned along gyri with alternating ‘hot’ and ‘cold’ colors corresponding to alternating outward and inward cortical currents. Note that these artefacts would not be apparent if the solutions were displayed on folded cortical surfaces using a colormap indicating only the absolute values of the current and not its direction, which is a common practice. The local alternations in the current's direction similarly exist in orientation-free reconstructions but are less apparent in this case because the dipole magnitudes may vary fairly smoothly even as their directions vary abruptly across cortical surface.

The stripy appearance is a well known tendency of distributed solutions

Next to each reconstruction the corresponding

Six measures defined in the

The localization error (

Both the spherical harmonics and spherical splines Harmony algorithms demonstrated superior performance compared to the other tested algorithms. The advantage was particularly large for the extended patch configurations and for those sources which could be localized with better than 3 cm accuracy. This is important given that practical significance of reconstructions with larger localization errors is questionable.

Removing the orientation constraint somewhat decreased localization errors of all algorithms except IBF. The improvements were particularly significant for WMNE. Although the unconstrained solution had less localization error it was characterized by a significantly larger surface bias (

From the practical point of view it is useful to know where the reconstructions were not significantly better than chance. This statistical measure was computed by comparing localization results for the simulated signal (plus the added sensor noise) to localization results for noise-only data. For the latter case the simulated signal was subtracted from the dataset and only the added sensor noise was present prior to source reconstruction. The

Reconstructions for a given source location were considered to be significantly better than chance if the signal+noise

The amplitude ratio measure

The surface bias measure

More surprisingly, both Harmony algorithms showed relatively small surface bias even though no depth-weighting prior was used in this case. Apparently, Harmony reduces the surface bias by constraining the solution to be smooth - much more so than IBF and LORETA. This forces the solution to extend deep into the sulci. Depth-weighting can increase localization error instead of decreasing it (compare MNE and WMNE results in

Comparing the middle and right panels in

The coherency measure

Removing the orientation constraint produced more coherent solutions overall (compare right and left panels in

The congruency measure

The

Comparing

The MNE solutions have a strong surface bias which can be reduced by depth-weighting (

Because dSPM and sLORETA are popular approaches to source localization a comparison between these methods and Harmony is presented here. To this end, Harmony solutions (spherical harmonics basis and spherical splines basis) were normalized dSPM-style, i.e., by the standard deviation of the solution's noise estimated for each current dipole. Variance of the solution was calculated as given by (13). The results for the 37-patch orientation-constrained condition are shown in

The localization accuracy (top left panel) was significantly improved by normalization (compare with

The Harmony algorithm was also tested with real EEG data. The EEG recordings were done using HydroCell GSN 128-channel system (EGI Inc.). Visual and tactile sensory stimulation were used in two separate experiments. The stimuli were: a full-screen grating contrast reversing at 1 Hz for the visual experiment and a 2 Hz vibrotactile stimulation of the left index finger for the tactile experiment. ERP epoch duration was 1000 msec and 500 msec respectively. The tactile pressure was applied for the first 100 msec of the epoch and then the pressure was released. 200 stimulation epochs were recorded for each experiment, the epochs were blocked into 10 epoch trials separated by no-stimulus intervals 3 seconds long. Epoch start markers were recorded along with the EEG data using a DIN signal generated by an in-house stimulation software. About 5% of epochs were rejected as too noisy based on potential thresholding at the preprocessing stage. The remaining epochs were averaged. Two different subjects participated in these experiments. Their BEM head models were constructed from high-resolution anatomical MRI scans as described in

Experimental data and source reconstructions are shown in

A new approach to EEG/MEG source reconstruction based on choosing a parsimonious subset of basis functions for the source space was presented. Two basis sets were tested: spherical harmonics and spherical splines. The main advantage of this approach is the explicit way in which the basis set for the solution is chosen. It allows making the solution spatially smooth reducing surface bias and effects of sensor noise. The method's performance was evaluated based on simulated and real EEG data and was compared with performance of several popular source-localization algorithms.

The proposed method, called Harmony, produced realistic source reconstructions when applied to EEG data collected in two different experiments. The reconstructions were based on individual BEM head models derived from anatomical MRI data. The obtained solutions were smooth and coherent, showed little surface bias, and were located over the expected cortical sources (

The algorithm was also tested with carefully simulated data. Two simultaneous cortical sources were used in the simulations. The sources had unequal amplitudes and were positioned at 66 uniformly spaced locations each, both on the opposite and on the same cerebral hemispheres. Scalp potentials produced by the two sources were calculated using a BEM head model derived from group-averaged anatomical MRI data provided by FreeSurfer. Noise recorded in a real EEG experiment was added to the simulation and was also used to estimate the statistical significance of solutions. Harmony solutions were compared with solutions for other algorithms: IBF, MNE, WMNE, dSPM, sLORETA, and LORETA. A special emphasis was made on choosing the regularization parameter individually for each of the tested algorithms using OCV. It was a significant factor for the comparison given that the OCV-estimated parameter varied by the factor of 20 among the algorithms. OCV was shown to provide near-optimal regularization in terms of the localization error and the solution width (

The curve was obtained by averaging the measures over all 2-source simulations.

The algorithms were compared on the following six measures: localization error (

Harmony solutions had surface bias significantly weaker than that for MNE, about the same as IBF, LORETA, and WMNE solutions (

The algorithms were tested with and without orientation constraint. When the orientation constraint was used, only dipoles orthogonal to the cortical surface were allowed in the solution. Arbitrary dipole orientations were allowed for unconstrained solutions. Although unconstrained solutions somewhat decreased the localization error, increased coherency, and

The algorithms were separately tested for source configurations comprising two sources located in opposite cortical hemispheres and in the same hemisphere. Only the

MNE solution normalization (dSPM and sLORETA) significantly reduced localization error and surface bias. Adversely, the solutions' congruency was also strongly reduced indicating that the spatial extent of dSPM and sLORETA reconstructions was less indicative of the localization accuracy compared to MNE. This result is in agreement with the increased source dispersion metric between MNE and dSPM/sLORETA

The high coherence (low scatter) of the Harmony solutions is a unique feature of this approach.

Harmony in the basis of spherical harmonics (HRM) produced reconstructions almost identical to those for the basis of spherical splines (SPL), which are in many aspects very different basis sets. This suggests that there is nothing special about a particular choice of the basis set functions as long as the functions share certain generic characteristics, such as smoothness and global extent. HRM significantly surpassed SPL only on the coherency score. HRM was also more parsimonious, it used 121 basis functions per hemisphere compared to 162 basis functions for SPL. Among other things, such a compact description provides an efficient way to store and share source reconstruction results. In addition to the left and right cortical meshes on which any cortex-based source reconstruction is defined, an orientation-constrained HRM solution requires to store only 242 numbers for each time sample. For example, assuming 2-bytes per data record, the whole 4D ‘movie’ of source reconstruction comprising 1 second long ERP epoch sampled at 500 Hz can be saved as a single 522 KB file: 242 KB of HRM solution + 280 KB required to describe 20,000-strong mesh of cortical nodes for a given subject).

Another interesting aspect of the HRM solution is its power spectrum, i.e., the distribution of the source power across different spatial frequencies. Although this aspect was not discussed in this study it provides a new viewpoint on brain activity. For example, it allows to correlate temporal and spatial dynamics of brain rhythms to search for traveling waves of activation

The part of the study which involved human subjects was conducted in accordance with the institutional IRB guidelines. The study was approved by the Northeastern University Institutional Review Board. A written informed consent (approved by the IRB) was obtained from all human subjects.

The forward EEG/MEG problem for the ‘distributed’ solution is defined by the gain matrix

Assuming normal distribution for

The approach presented here seeks the solution in the form of an expansion into a small set of global continuous and smooth basis functions instead of starting from the ten-thousand-strong set of discrete cortical dipoles. Because of the well-known lowpass spatial effect of the skull on electric currents high spatial frequency components of cortical sources simply cannot be reliably inferred based on scalp potentials due to the potentials being swamped by high spatial frequency sensor noise. Arguably, the proposed choice of basis functions forces the solution to better represent the available information about brain activity and reduces effects of sensor noise.

Consider a linear transformation

Writing the solution (2) for the new basis set one obtains

Because EEG/MEG sources are spread on topologically spherical cortices (ignoring the corpus callosum) it is natural to use spherical harmonics or spherical splines as the basis set functions, the method termed Harmony here. Between the two basis sets spherical harmonics produced somewhat better results, but the differences were small.

Spherical harmonics are 2D analogs of the sine and cosine functions, and the spherical harmonics expansion is a 2D analog of the conventional Fourier series expansion. The spherical harmonics define a complete orthonormal set, each vector is global (spatially extended) and is characterized by a spatial scale determined by the Harmonic's

Any square-integrable scalar function defined on a spherical mesh can be expanded into a set of spherical harmonics, as is illustrated in the top of

Bottom: left cortex displayed as a spherical mesh with a spherical harmonic computed at its nodes, then the mesh is folded into the actual cortical shape (as produced by the FreeSurfer toolbox). The colormap indicates the source sign and amplitude.

Some degree of non-orthonormality will be introduced into the basis set by the folding distortions of the spherical mesh, but this is not a concern because strict orthonormality is not a requirement for a source basis set, although non-orthonormality can potentially affect the stability of the numeric solution of (6). Moreover, the fact that the basis set of spherical harmonics produced solutions almost identical to those obtained in the basis set of spherical splines (11), which was not at all orthonormal and otherwise very different from the basis set of spherical harmonics, indicates that variations in the basis functions due to the folding mesh distortions were probably inconsequential.

Because EEG/MEG sensors sample signals at discrete spatial locations (typically, about

The proposed choice of the source space basis is by no means unique. The basis of spherical harmonics allows to fit any measured data without introducing high-frequency information not present in the data as defined by the Nyquist limit. Arguably, any other choice of a basis set with similar properties would result in the same or very similar inverse solution. Indeed, the simulation results presented here indicate that the basis set of spherical splines produced a solution very similar to the basis set of spherical harmonics, even though the basis functions of the two sets were very different: spherical harmonics are periodic, while spherical splines are centered.

The spherical splines

The

Although the described choice of

BEM head models were constructed based on high-resolution MRI data collected for two subjects and also using the FreeSurfer group averaged head (MRI data averaged over 40 subjects). The BEM head model comprised three volumes: scalp, skull, and CSF/brain; the volumes were reconstructed with the help of the FSL toolbox

Cortical surfaces and ROIs were determined by automatic segmentation with the help of the FreeSurfer toolbox

Two basis sets: spherical harmonics and spherical splines were used with the Harmony method. The spherical harmonics basis set was defined by (10).

A special care was taken to make the simulations as realistic as possible. Electrode locations were taken from a real EEG experiment, in which a 128-channel HydroCell GSN electrode net was used (EGI Inc.). The electrode positions were measured using a Polhemus FASTRACK digitizer.

Because, typically, several cortical sources are simultaneously activated, two simultaneously activated sources were used in the main bulk of simulations. The sources located in opposite cerebral hemispheres and in the same hemisphere were analysed separately. Two source configurations were tested: point-like and extended. For the point-like configuration, each of the two sources consisted of a single current dipole. Although this configuration is widely used for simulations, it is not very realistic. The neurologist's “rule of thumb” is that at least 6

The dipole orientations were fixed to be orthogonal to the cortical surface, which reflects the common assumption that EEG and MEG are due to synaptic currents produced by activity of cortical pyramidal cells. These currents flow along the cells axons primarily perpendicular to the cortex.

The single-dipole and hexagonal 37-dipole patches are illustrated by insets in

The reconstructed sources were thresholded based on the signal-to-noise power ratio (

Real experimental noise was added to the simulated sensor potentials. The noise was recorded in the course of a VEP (visually evoked potentials) experiment, where a contrast reversing checkerboard was occasionally replaced by empty gray background with a fixation mark. Subjects were instructed to fixate at the mark and avoid any head-muscle or eye-muscle activity during the trials. Each trial lasted for 10 seconds and comprised ten 1000 msec VEP epochs (typical epoch duration for an evoked response experiment). The whole experiment lasted for 40 minutes. The “empty screen” epochs (190 altogether) were averaged to obtain the estimate of the residual noise in the averaged VEP epoch. The noise amplitude was in the

Because the source covariance matrix

In practical terms, weak regularization results in unrealistically patchy high-amplitude solutions which, nevertheless, fit the data very well. Strong regularization results in diffuse and low-amplitude solutions which produce scalp potentials lower than the ones actually observed. As mentioned in Section “The choice of

It is important for the purpose of comparison of different source localization algorithms to choose the regularization parameter intelligently. Simply setting the regularization parameter to the same value for all algorithms would not do because it would affect different algorithms differently. Hence the parameter has to be set by some optimization scheme applied to each algorithm individually. There are many such schemes available

The OCV method uses “leaving-out-one” validation strategy, where one datapoint at a time is left out, the remaining data is fit by the model, and the misfit of the left-out datapoints is minimized. The procedure can be reduced to a single formula: the optimal regularization parameter is given by minimizing the OCV cost function

Harmony performance for simulated data was compared with the performance of the following algorithms: MNE, WMNE, dSPM, sLORETA, LORETA, and IBF. The latter method (Informed Basis Functions) claims the most informative source space basis set, i.e., the set preserving most information about the known source constraints (coherence or smoothness in our case). A Gaussian coherence matrix with

WMNE, LORETA, and IBF algorithms employ “depth weighting”. The weighting factor for the

Because the main focus of this study was on the choice of the source space for linear algorithms based on the ‘distributed’ solution (2) no comparison was made to algorithms involving various other approaches, e.g., beamformer methods or iterative prior learning methods, such as FOCUSS or MSP. Although these approaches appear promising, they address different aspects of source localization and therefore will not be discussed here. Nevertheless, it is worth mentioning that these new aspects of source localization can be easily combined with the Harmony method.

Six measures were used to quantify the quality of source reconstructions: localization error, amplitude ratio, surface bias, coherence, congruency (width – error correlation), and area under the ROC curve. These measures are explained next.

The localization error was calculated as follows. First, the location of the true source was calculated by averaging the locations of its

The amplitude ratio measure quantifies how well the relative source strengths are preserved in the solution. The measure was calculated as follows:

This measure quantifies the degree of surface bias in the solution. First, the solution's distance

The last two measures quantify two important properties characterizing the spatial extent of a solution: its coherence (how ‘scattered’ is the solution) and its congruency. The latter is defined as the correlation between the solution's width (defined below) and its localization error

The solution width is defined first. Its measure is calculated the same way as the localization error, except that the distances

Area under the ROC curve (