Localization of Brain Signals by Alternating Projection

Abstract

A popular approach for modeling brain activity in MEG and EEG is based on a small set of current dipoles, where each dipole represents the combined activation of a local area of the brain. Here, we address the problem of multiple dipole localization with a novel solution called Alternating Projection (AP). The AP solution is based on minimizing the least-squares (LS) criterion by transforming the multi-dimensional optimization required for direct LS solution, to a sequential and iterative solution in which one source at a time is localized, while keeping the other sources fixed. Results from simulated, phantom, and human MEG data demonstrated the high accuracy of the AP method, with superior localization results than popular scanning methods from the multiple-signal classification (MUSIC) and beamformer families. In addition, the AP method was more robust to forward model errors resulting from head rotations and translations, as well as different cortex tessellation grids for the forward and inverse solutions, with consistently higher localization accuracy in low SNR and highly correlated sources.

I. Introduction

Localization of brain signals collected from an array of magnetoencephalography (MEG) or electroencephalography (EEG) sensors is essential for elucidating the neural basis of cognitive processes [1]. The localization problem can be formulated as estimating the location and moment of a small set of dipoles that best match the MEG/EEG measurements in a least-squares (LS) sense [2]. This multiple dipole localization approach avoids the ill-posedness inherent in imaging methods, such as minimum-norm [3], which seek to estimate a current density map throughout the cortex and thus assume a much higher number of unknown sources (in a discrete grid) than the number of sensors.

Multiple dipole localization methods are divided into dipole fitting and scanning methods. Dipole fitting methods solve the optimization problem directly using techniques that include the Nelder-Mead simplex algorithm, gradient descent, multistart, simulated annealing, and genetic algorithms [4]–[7]. However, these methods have not been widely adopted because they are too computationally expensive or converge to a suboptimal local optimum. Dipole scanning methods use adaptive spatial filters to search for optimal dipole positions throughout a discrete grid representing the source space [8]. Source locations are then determined as those for which a metric (localizer) exceeds a given threshold. While these approaches do not lead to true least squares solutions, they can be used to initialize a local least squares search. The most popular scanning methods are MUSIC [2] and beamformers [9], [10], both widely used for electromagnetic brain mapping, but they assume linearly independent and uncorrelated sources, respectively. When the sources are highly correlated (in the extreme case, fully correlated sources are referred to as coherent sources) the performance of MUSIC and beamformers deteriorates and the estimated time courses are distorted. Several multi-source extensions have been proposed for coherent sources [11]–[19], however they require some a-priori information on the location of the coherent sources, are limited to the localization of pairs of coherent sources, or are limited in their performance.

We present a novel solution to the problem of multiple dipole localization based on the LS estimation criterion. As is well known, jointly fitting all dipoles simultaneously yields a multidimensional nonlinear and non-convex minimization problem, making it extremely hard to avoid undesirable local minima [20]. To tackle this problem and effectively ameliorate the issue of undesirable local minima, we propose the use of the Alternating Projection (AP) algorithm, which is well known in sensor array signal processing community [21]. The AP method transforms the multidimensional problem to an iterative process involving only one-source maximization problems that are computationally much simpler. In addition, it has an effective initialization scheme which is key to its good global convergence.

The AP method is conceptually close to recursive scanning methods (e.g. Recursively Applied and Projected Multiple Signal Classification (RAP-MUSIC) [22], Truncated RAP-MUSIC (TRAP-MUSIC) [23], RAP-Beamformer [1]), in that it estimates the source locations iteratively at each step, projecting out the topographies of the previously found dipoles before forming the localizer for the current step. One notable characteristic of all these methods (including AP) is the absence of formal stopping criteria to determine the number of sources. Consequently, additional estimation procedures, such as the Akaike information criterion (AIC) [24] and the minimum description length criterion (MDL) [25], are typically employed. However, while recursive methods terminate after detecting the last source based on the determined model order, the AP algorithm continues refining the detected sources across several iterations. In our implementation, we terminate the iterations of the AP algorithm after achieving convergence, which is determined when the location of any of the sources no longer undergoes any changes. This iterative refinement process distinguishes the AP method from recursive approaches, as it allows for further optimization of the localized sources.

Another distinguishing aspect is that the AP method does not require explicit computation of the signal subspace, as compared to the MUSIC-based solutions. Instead, it directly utilizes the data covariance matrix, enabling significantly improved performance in localizing highly correlated and coherent sources.

In this paper, we present the AP method and compare it against popular MUSIC and beamformer variants in a variety of simulations involving different SNR, inter-source correlation levels, and number of active dipole sources. We also demonstrate the robustness of the AP method to forward model errors resulting from head rotations and translations and different cortex tessellation grids for the forward and inverse solutions. Finally, we validate the AP method and characterize its localization properties with experimental MEG phantom and human data. We note that, while our evaluation uses MEG data, the AP method is also directly applicable to EEG. An early part of this work was presented at the IEEE 19th International Symposium of Biomedical Imaging (ISBI), 2022 [26].

II. Materials and Methods

A. Problem formulation

The primary source of MEG and EEG signals is widely believed to be the electrical currents flowing through the apical dendrites of pyramidal neurons in the cerebral cortex [20]. Clusters of thousands of synchronously activated pyramidal cortical neurons can be modeled as an equivalent current dipole (ECD). The current dipole is therefore the basic element used to represent neural activation in MEG and EEG localization methods.

Here we briefly review the notations used to describe the measurement data, forward matrix, and sources, and formulate the problem of estimating current dipoles. Consider an array of $M$ MEG or EEG sensors that measures data from a finite number $Q$ of equivalent current dipole (ECD) sources emitting signals ${\left\{s_q\left(t\right)\right\}}_{q=1}^Q$ at locations ${\left\{p_q\right\}}_{q=1}^Q$. The measurement signal $y\left(t\right)$ received by the array is a $M\times 1$ vector given by:

$$y\left(t\right)=\sum _{q=1}^{Q}l\left(p_q\right)s_q\left(t\right)+n\left(t\right),$$ (1)

where $l\left(p_q\right)$ is the topography of the dipole at location $p_q$ and $n\left(t\right)$ is the additive noise. The topography $l\left(p_q\right)$, is given by:

$$l\left(p_q\right)=L\left(p_q\right)q,$$ (2)

where $L\left(p_q\right)$ is the $M\times 3$ forward matrix at location $p_q$ and $q$ is the 3 × 1 vector of the orientation of the ECD source. Depending on the problem, the orientation $q$ may be known, referred to as fixed-oriented dipole, or it may be unknown, referred to as freely-oriented dipole.

Assuming that the array is sampled $N$ times at $t_1,\dots ,t_N$, the matrix $Y$ of the sampled signals can be expressed as:

$$Y=AS+N,$$ (3)

where $Y$ is the $M\times N$ matrix of the sampled signals $Y=\left[y\left(t_1\right),\dots ,y\left(t_N\right)\right]$ and $A$ is the $M\times Q$ mixing matrix of the topography vectors at the $Q$ locations $P=\left[p_1,\dots ,p_Q\right]$:

$$A=\left[l\left(p_1\right),\dots ,l\left(p_Q\right)\right],$$ (4)

Note, even though the mixing matrix depends on the source locations, $A\left(P\right)$, we drop $P$ to ease notation. $S=\left[s\left(t_1\right),\dots ,s\left(t_N\right)\right]$ is the $Q\times N$ matrix of the sources, with $s\left(t\right)={\left[s_1\left(t\right),\dots ,s_Q\left(t\right)\right]}^T$, and $N=\left[n\left(t_1\right),\dots ,n\left(t_N\right)\right]$ is the $M\times N$ matrix of noise.

We further make the following assumptions regarding the emitted signals and the propagation model:

• A1: The number of sources $Q$ is known and obeys $Q<M$.

• A2: The forward matrix $L\left(p\right)$ is known for every location $p$ (computed by the forward model).

• A3: All $Q$ topography vectors ${\left\{l\left(p_q\right)\right\}}_{q=1}^Q$ are linearly independent, i.e., $\text{rank}A=Q$.

• A4: The emitted signals are unknown and arbitrarily correlated, including the case that a subset of the sources or all of them are perfectly coherent.

Assumptions A1–A3 are shared by existing methods, such as RAP-MUSIC. Assumption A1 can be addressed using approaches that estimate the model order, including the Akaike information criterion (AIC) [24], and the minimum description length criterion (MDL) [25]. Assumption A4 does not impose any restrictions to the activation time courses.

We can now state the problem of localization of brain signals as follows: Given the received data $Y$, estimate the $Q$ locations, orientations, and amplitudes of the sources ${\left\{p_q\right\}}_{q=1}^Q$.

B. The Alternating Projection solution

We first solve the problem for the fixed-oriented dipoles and then extend it to freely-oriented dipoles. The least-squares estimation criterion, from (3), is given by

$${\{\hat{P},\hat{S}\}}_{\text{ls}}=\text{arg\hspace{0.17em}}\underset{P,S}{\text{min}}\parallel Y-AS{\parallel }_F^2.$$ (5)

where $F$ denotes the Frobenius norm. To solve this minimization problem, we first eliminate the unknown signal matrix $S$ by expressing it in terms of $P$. To this end, we equate the derivative of (5) to zero with respect to $S$, and solve for $S$:

$$\hat{S}={\left(A^{T}A\right)}^{-1}{A}^{T}Y.$$ (6)

Defining ${\Pi }_A=A{\left(A^{T}A\right)}^{-1}{A}^T$ the projection matrix on the column space of $A$, and using the rotation property of the trace operator and the idempotence property $\left({\Pi }^2=\Pi \right)$ of the projection operator, equations (5), and (6) yield:

$$\begin{gathered} \hat{P}=\text{arg\hspace{0.17em}}\underset{P}{\text{min}}{\Vert \left(I-{\Pi }_A\right)Y\Vert }_F^2 \\ =\text{arg\hspace{0.17em}}\underset{P}{\text{max}}{\Vert {\Pi }_{A}Y\Vert }_F^2 \\ =\text{arg\hspace{0.17em}}\underset{P}{\text{max}}\text{\hspace{0.17em}tr}\left({\Pi }_{A}C{\Pi }_A\right) \\ =\text{arg\hspace{0.17em}}\underset{P}{\text{max\hspace{0.17em}}}\text{tr}\left({\Pi }_{A}C\right) \end{gathered}$$ (7)

where tr( ) denotes the trace operator, subscript $F$ denotes the Frobenius norm, $C=Y{Y}^T$ is the data covariance matrix, and $I$ denotes the identity matrix.

This is a nonlinear and nonconvex $Q$-dimensional maximization problem. The AP algorithm [21] solves this problem by transforming it to a sequential and iterative process involving only one-dimensional maximization problems. The transformation is based on the projection-matrix decomposition formula. Let $B$ and $D$ be two matrices with the same number of rows, and let ${\Pi }_{\left[\text{B},\text{D}\right]}$ denote the projection-matrix onto the column space of the augmented matrix $\left[B,D\right]$. Then:

$${\Pi }_{\left[\text{B},\text{D}\right]}={\Pi }_B+{\Pi }_{{\Pi }_B^{\perp }\text{D}},$$ (8)

where ${\Pi }_{\text{B}}^{\perp }$ is the projection onto the orthogonal complement of the span of $B$, given by ${\Pi }_B^{\perp }=\left(I-{\Pi }_B\right)$.

The AP algorithm exploits this decomposition to transform the multidimensional maximization (7) into a sequential and iterative process involving a maximization over only a single parameter at a time, with all the other parameters held fixed at their pre-estimated values. More specifically, let $j$ denote the current iteration number, and let $q$ denote the current source to be estimated (q is sequenced from 1 to $Q$ in every iteration). The other sources are held fixed at their pre-estimated values: ${\left\{{\hat{p}}_i^j\right\}}_{i=1}^{q-1}$, which have been pre-estimated in the current iteration, and ${\left\{{\hat{p}}_i^{j-1}\right\}}_{i=q+1}^Q$, which have been pre-estimated in the previous iteration. With this notation, let $A_q^j$ denote the $M\times \left(Q-1\right)$ matrix of the topographies corresponding to these values (note that the $q$-th topography is excluded), given by:

$$A_q^j=\left[l\left({\hat{p}}_1^j\right),\dots ,l\left({\hat{p}}_{q-1}^j\right),l\left({\hat{p}}_{q+1}^{j-1}\right),\dots ,l\left({\hat{p}}_Q^{j-1}\right)\right]$$ (9)

By the projection matrix decomposition (8), we have:

$${\Pi }_{\left[A_q^j,l\left(p_q\right)\right]}={\Pi }_{A_q^j}+{\Pi }_{{\Pi }_{A_q^j}^{\perp }l\left(p_q\right)}.$$ (10)

Substituting (10) into (7), and ignoring the contribution of the first term since it is not a function of $p_q$, we get:

$${\hat{p}}_q^j =\text{arg\hspace{0.17em}}\underset{p_q}{\text{max }}\text{\hspace{0.17em}tr}\left({\Pi }_{{\Pi }_{A_q^j}^{\perp }l\left(p_q\right)}C\right)$$ (11)

$$=\text{arg\hspace{0.17em}}\underset{p_q}{\text{max}}\text{\hspace{0.17em}tr}\left({\Pi }_{Q_q^{j}l\left(p_q\right)}C\right),$$ (12)

where

$$Q_q^j={\Pi }_{A_q^j}^{\perp }=\left(I-{\Pi }_{A_q^j}\right),$$ (13)

is a projection matrix that projects out all but the $q$-th source at the $j$-th iteration.

Using the properties of the projection and trace operators, the $j$-th iteration (12) can be written as:

$${\hat{p}}_q^j=\text{arg\hspace{0.17em}}\underset{p_q}{\text{max}}\frac{l^T\left(p_q\right)Q_q^{j}C{Q}_q^{j}l\left(p_q\right)}{l^T\left(p_q\right)Q_q^{j}l\left(p_q\right)},q=1,\dots ,Q.$$ (14)

The maximization is carried out by an exhaustive search over the grid of dipole locations.

The initialization of the algorithm is straightforward. First we solve (7) for a single source:

$${\hat{p}}_1^0=\text{arg\hspace{0.17em}}\underset{p_1}{\text{max}}\frac{l^T\left(p_1\right)Cl\left(p_1\right)}{l^T\left(p_1\right)l\left(p_1\right)}.$$ (15)

Then, we add one source at a time and solve for the $q$-th source:

$${\hat{p}}_q^0=\text{arg\hspace{0.17em}}\underset{p_q}{\text{max}}\frac{l^T\left(p_q\right)Q_q^{0}C{Q}_q^{0}l\left(p_q\right)}{l^T\left(p_q\right)Q_q^{0}l\left(p_q\right)},q=2,\dots ,Q,$$ (16)

where $Q_q^0$ is the projection matrix that projects out the previously estimated $q-1$ sources:

$$Q_q^0=\left(I-{\Pi }_{A\left({\hat{P}}_q^0\right)}\right),$$ (17)

with $A_q^0$ being the $M\times \left(q-1\right)$ matrix:

$$A_q^0=\left[l\left({\hat{p}}_1^0\right),\dots ,l\left({\hat{p}}_{q-1}^0\right)\right].$$ (18)

Once the initial location of the $Q$-th source has been estimated, subsequent iterations, described by (14), refine the estimate by sweeping consecutively through all sources multiple times. The iterations continue until achieving convergence, which is determined when the location of any of the sources no longer undergoes any changes. The algorithm is bound to converge because a maximization is performed in every iteration, so the value of the maximized function can never decrease. Depending on the initial condition, convergence may be at a local or global maximum [21].

In the case of freely-oriented dipoles, it follows from (2) that the solution (14) becomes:

$${\hat{p}}_q^j=\text{arg\hspace{0.17em}}\underset{p_q,q}{\text{max}}\frac{q^T{L}^T\left(p_q\right)Q_q^{j}C{Q}_q^{j}L\left(p_q\right)q}{q^T{L}^T\left(p_q\right)Q_q^{j}L\left(p_q\right)q},q=1,\dots ,Q.$$ (19)

whose solution is given by

$${\hat{q}}^j=\text{arg\hspace{0.17em}}\underset{p_q}{\text{max}}{v}_1\left(F_q^j,G_q^j\right),$$ (20)

$${\hat{p}}_q^j=\text{arg\hspace{0.17em}}\underset{p_q}{\text{max\hspace{0.17em}}}{\lambda }_1\left(F_q^j,G_q^j\right),$$ (21)

where

$$F_q^j=L^T\left(p_q\right)Q_q^{j}C{Q}_q^{j}L\left(p_q\right),$$ (22)

$$G_q^j =L^T\left(p_q\right)Q_q^{j}L\left(p_q\right),$$ (23)

with ${\lambda }_1\left(F_q^j,G_q^j\right)$ and $v_1\left(F_q^j,G_q^j\right)$ denoting the maximum generalized eigenvalue and its corresponding generalized eigen-vector, derived as solutions to the generalized eigenvalue problem of the matrix pencil $\left(F_q^j,G_q^j\right)$:

$$F_q^{j}v=\lambda G_q^{j}v.$$ (24)

C. Performance evaluation with simulations

We evaluated the performance of the AP method for the fixed-oriented dipoles model against RAP-MUSIC [22], RAP-beamformer [1], and Truncated RAP-MUSIC [23]. Comparisons included different source configurations of varying levels of SNR and inter-source correlation (denoted by $\rho$).

The geometry of the sensor array was based on the whole-head Megin Triux MEG system (306-channel probe unit with 204 planar gradiometer sensors and 102 magnetometer sensors). The geometry of the MEG source space was modeled with the cortical manifold extracted from a real adult human subject’s MR data using Freesurfer [27]. Simulated sources were restricted to approximately 15,000 grid points over the cortex, and reconstructed sources were estimated on a different grid of 50,000 points over the cortex. The simulation and reconstruction grids did not have overlapping points (average distance of 0.7 mm between neighboring points) to avoid the inverse crime of using identical parameters to synthesize and invert data in an inverse problem [28], [29]. The forward matrix for both grids was estimated using the boundary element method as implemented in OpenMEEG [30] in BrainStorm [31]. To evaluate localization performance under realistic head model errors, we introduced translations and rotations to the reconstruction grid before computing the forward matrix.

In our study, we comprehensively assessed the localization accuracy across a wide range of inter-source correlation values, SNR levels, and varying numbers of sources. As the number of active sources increased, the localization problem posed increasing complexity. To ensure robustness, we conducted extensive simulations involving 500 Monte Carlo random samples in each case.

The source time courses were modeled using 50 time points, represented as mixtures of sinusoidal signals. Each source had random frequencies ranging from 10 Hz to 30 Hz. Sources were simulated in random locations throughout the cortex and with equal amplitude. The orientation of each source was normal to the cortex at its respective location, in accordance with the fixed-oriented dipoles model. Source time courses were then modified using the Cholesky decomposition method to assign specific inter-source correlation values. Following the mapping of sources to sensor measurements, we introduced Gaussian white noise to the MEG sensors to emulate instrumentation noise at specific SNR levels. To quantify the SNR, we used the ratio of the Frobenius norm of the signal-magnetic-field spatiotemporal matrix to that of the noise matrix for each trial, as described in [32]. We terminated the iterations of the AP algorithm after convergence, that is, when the location of the sources would not change.

D. Performance evaluation with a real phantom

We assessed the performance of the AP localization method using the phantom data provided in the phantom tutorial¹ of the Brainstorm software [31]. The phantom device was provided with the Megin Neuromag system and contained 32 artificial dipoles in locations dispersed in four quadrants. The phantom dipoles were energized using an internal signal generator and an external multiplexer box was used to connect the signal to the individual dipoles. Only one dipole could be activated at a time, therefore we combined data trials from different dipoles to simulate the concurrent activation of multiple sources. All dipoles were activated with the same sinusoidal signal and were thus perfectly coherent.

The phantom data was collected using a Megin Neuromag MEG system (306-channel probe unit with 204 planar gradiometer sensors and 102 magnetometer sensors). For evaluation, we utilized the freely-oriented AP source localization method in volume space. This decision was driven by the fact that the phantom dipoles are distributed within a volume and do not conform to a specific cortical surface. The reconstruction source space comprised the volume of a sphere, centered within the MEG sensor array with 64.5 mm radius, sampled with a regular (isotropic) grid of points with 2.5 mm resolution yielding a total of 56,762 grid points. The forward matrix was estimated using BrainStorm [31] based on a single sphere head model.

E. Performance evaluation with multimodal sensory human MEG data

We collected MEG data in a multimodal sensory experiment presenting somatosensory and auditory stimuli to a single human subject. The subject gave a written informed consent, and the study was approved by the local institutional review board (Committee on the Use of Humans as Experimental Subjects at the Massachusetts Institute of Technology, protocol number 1102004309, date of approval 11/14/2019) and conducted according to the principles of the Declaration of Helsinki.

The somatosensory stimuli were delivered with electrical stimulation to the right median nerve. The auditory stimuli were binaural sounds (beeps) delivered with tubal-insert earphones. The stimuli produce well-known brain responses localized in the left primary somatosensory cortex and bilateral primary auditory cortex, respectively. Fifty trials were recorded for each stimulus type with an interstimulus interval 500 ms for the somatosensory stimuli and 1200 ms for the auditory stimuli. The data was collected with a MEGIN Triux MEG system (306-channel probe unit comprising 102 magnetometers and 204 planar gradiometers). The source space was restricted to approximately 15,000 grid points over the subject-specific cortex extracted with Freesurfer [27]. For the localization process, we assumed the sources to be fixed-oriented and oriented normal to the cortex. The raw data was pre-processed with the Maxfilter software (Elekta, Stockholm) with default parameters, a standard procedure to suppress environmental and instrumentation noise with spatiotemporal filters in MEGIN MEG systems [33]. The data was then whitened using a regularized noise covariance matrix in the BrainStorm software [31]. Regularization added an identity matrix that was scaled to 10% of the noise covariance matrix’s largest eigenvalue. Note, data whitening is standard practice in MEG localization procedures, and necessary when the data comprise different types of sensors that need to be combined, as in our case (magnetometers in units of Tesla, and gradiometers in units of Tesla/meter).

III. Results and Discussion

A. AP method had consistently low localization error in all simulated conditions

We extensively evaluated the localization performance of the AP method against RAP-MUSIC, RAP-beamformer, and Truncated RAP-MUSIC in a range of scenarios, including different number of sources, inter-source correlation values, SNR levels, and various head model errors of translation and rotation (Figures 1, 2). The AP method had the lowest localization error in nearly all tested conditions. This performance advantage was most prominent when localization was under challenging conditions, under low SNR and high inter-source correlation values. This was true for both $Q=2$ (Figure 1) and $Q=3$ (Figure 2) sources.

Fig. 1:

Localization performance for $Q=2$ simulated sources. Panels show results for different inter-source correlation and SNR values. Localization was estimated under the following head model error conditions: (1) Ideal conditions with same grid for source simulation and reconstruction and no head model errors. All other conditions used a distinct grid for source simulation and reconstruction, and (2) 1 mm posterior translation of the head; (3) 2 mm posterior translation; (4) 1° right tilt rotation; (5) 2° right tilt rotation; (6) 1 mm right translation; (7) 2 mm right translation; (8) 1° upward rotation; (9) 2° upward rotation; (10) 1 mm upward translation; (11) 2 mm upward translation; (12) 2° right rotation. AP: Alternating Projection; RAP: RAP-MUSIC; RAP-BF: RAP-beamformer; T-RAP: Truncated RAP-MUSIC. In all cases, localization error was defined as the average distance computed when each original source was paired with the closest estimated source.

Fig. 2:

Localization performance for $Q=3$ simulated sources. Panels and head models same as in Figure 1.

As expected, the performance of all methods was best in ideal conditions, under no head model errors and with the same grid for source simulation and reconstruction. Performance decreased substantially when the grids were different (to avoid the inverse crime) and the head models had errors of translation and rotation. However, the AP method was relatively more robust to all tested head model error conditions and fared better than all other methods.

The second best performance was achieved by RAP-MUSIC, which was close to the AP method when the SNR was high (0 dB) and the inter-source correlation value was low $\left(\rho =0.1\right)$, but the performance gap was wider otherwise. RAP-Beamformer and TRAP-MUSIC yielded overall higher localization errors, but their relative performance varied considerably for the different SNR and inter-source correlation values. Examples of source time courses estimated with the AP method are shown in Figure 3.

Fig. 3:

Examples of true and estimated source time courses using the AP method. Results are for the case of $Q=3$ simulated sources, SNR = 0 dB, inter-source correlation $\rho =0.9$, and head model error of 1 mm posterior translation.

Last, we evaluated localization performance for $Q=3$ highly correlated sources (Figure 4). As expected, for very high inter-source correlation values $\left(\rho \ge 0.95\right)$ the performance of all methods deteriorated. Yet, while the AP method deteriorated only slightly, the other methods deteriorated significantly, especially for perfectly coherent sources $\left(\rho =1\right)$. Similar results were obtained for $Q=6$ highly correlated sources (Supplementary Figure 1).

Fig. 4:

Localization performance for highly correlated and for coherent $\left(\rho =1.0\right)$ sources. Top row: $Q=2$ simulated sources. Bottom row: $Q=3$ simulated sources. Head models same as in Figure 1.

It is worth noting that for all experiments, the true number of sources was provided to all localization algorithms and we have not estimated $Q$ in any case. This ensured a fair comparison, allowing us to directly assess the localization accuracy of each method without any bias introduced by source number estimation.

B. The sequential iterations of the AP method refined the localization of phantom dipoles

We assessed the performance of the AP method in localizing real experimental data collected using the MEGIN phantom. The data comprised MEG recordings collected during the sequential activation of 32 artifical dipoles. Each dipole was activated 20 times with amplitude 200 nAm and 2000 nAm, yielding a total of 20 trials in each experimental condition. Note, the 200 nAm condition is close to the range of amplitudes we can expect from inter-ictal spikes due to epilepsy, visible in raw data [34]. The 2000 nAm condition was selected to characterize the AP performance and limitations in ideal conditions. The true locations of the dipoles are shown in Figure 5a.

Fig. 5:

Localization performance of real phantom dipoles. Only one phantom dipole could be activated at a time, so trials from individual dipoles were combined to simulate the simultaneous activation of pairs of dipoles. (a) Location of the 32 artificial dipoles of the MEGIN phantom. (b) Average localization error for 200 nAm and 2000 nAm amplitude dipoles for the first and last iteration of the AP method. (c) Histograms of the total number of AP iterations across all the 496 possible combinations of pairs of dipoles.

To evaluate the ability of the AP method in localizing pairs of coherent dipoles, we simulated the simultaneous activation of two sources by averaging the evoked responses for each of the 496 possible combinations of pairs of dipoles before source localization. AP localization relied on the freely-oriented solution (Eq. 19). Source localization was applied on whitened data with a noise covariance matrix that was regularized by adding an identity matrix that was scaled to 10% of the largest eigenvalue of the noise covariance matrix.

The average localization error across all combinations of pairs of dipoles for both the 200 nAm and 2000 nAm conditions in shown in Figure 5. To investigate the importance of AP iterations in refining localization, we computed the error separately for the first and last iteration of the AP algorithm. We used violin plots that combine the advantages of box plots with density traces. Each “violin” contains a box-plot (white dot, vertical thick black box and thin black line), where the white dot represents the median of the distribution, the vertical thick black box indicates the inter-quartile range, and the thin black line denotes the extensions to the maximum and minimum values. The median value is also written on the top of the violin for convenience. The mean value is depicted by a horizontal line. The shaded areas surrounding the box plot show the probability density of the data across all pairs of dipoles, with individual data points plotted as colored circles.

Localization error dropped considerably from the first to the last iteration of the AP method (Figure 5a). For the 200 nAm condition, the median localization error dropped from 9.7 mm to 4.1 mm, and for the 2000 nAm condition from 8.2 mm to 2.8 mm. The number of required AP iterations were similar for the 200 nAm and 2000 nAm conditions, with an average of 3.10 and 2.97 iterations respectively, indicating fast convergence of the AP method (Figures 5b and 5c). The quality of dipole localization is exemplified in Figure 6, showing that errors are considerably reduced from first to the last AP iteration for both the 200 nAm and 2000 nAm conditions.

Fig. 6:

Example localization of phantom dipole pairs. (a) 200 nAm dipoles, first AP iteration. (b) 200 nAm dipoles, last AP iteration. (c) 2000 nAm dipoles, first AP iteration. (d) 2000 nAm dipoles, last AP iteration.

C. Characterizing the phantom performance of the AP method

We further explored the factors affecting the quality of source localization of the AP method. Localization error was higher for sources with small spatial separation, and progressively decreased as the true dipoles were further apart (Figure 7). This was expected, as nearby sources have similar topographies and are hard to dissociate with each other.

Fig. 7:

Localization error versus source separation for the phantom dipoles in the 200 nAm condition.

The iterations of the AP algorithm were terminated upon achieving convergence, which was determined when the location of any of the sources no longer underwent any changes. Notably, the final localization error was found to be independent of the required number of AP iterations, implying that the quality of source localization did not rely on the algorithm’s initialization (Figure 8a). This observation supports the notion that the algorithm’s performance is consistent regardless of the starting conditions.

Fig. 8:

Performance of AP method for the 200 nAm phantom dataset. (a) Localization error for different values of the last AP iteration. (b) Reduction of localization error (Δ Localization Error), defined as the difference in localization error between the first and the last AP iteration. (c) Localization error for the first and second detected source. (d) Reduction of localization error for the first and second detected source.

Moreover, our analysis revealed that with an increased number of AP iterations, there was a higher reduction in localization error (Δ Localization Error, defined as the difference in localization error between the first and the last AP iteration), indicating that each successive iteration progressively improved the localization performance (Figure 8b). This finding further underscores the algorithm’s capability to refine the localization estimates through successive iterations, ultimately leading to enhanced accuracy.

The final localization error was smaller for the first than the second detected source (Figure 8c). This is consistent with the initialization of the AP method, which first detects the source that achieves the best LS fit and then progressively detects other sources. Despite this difference in localization error between the two sources, the reduction of localization error over iterations was higher for the second than the first source (Figure 8d).

D. AP method reliably localized human MEG data

To establish ground truth, we first fit single dipoles separately to the somatosensory and auditory data (all tested localization methods are equivalent for the single dipole case). Dipole fits were applied at the time of peak response, at 100 ms for the somatosensory responses and 40 ms for the auditory responses with respect to stimulus onset. The dipoles localized to well-known areas in the left primary somatosensory cortex and bilateral primary auditory cortex. In the auditory case, we fit data separately for the left and right sensors to estimate a dipole in the left and right primary auditory cortex, respectively (Figure 9a). This follows a well-known strategy of localizing a single dipole with a subset of sensors containing the dominant dipolar pattern while neglecting contributions from other sources [35].

Fig. 9:

Localization performance for multisensory MEG human data. (a) Ground truth dipoles from a human experiment presenting somatosensory and auditory stimuli. Somatosensory and auditory evoked responses were combined with different numbers of trials to assess localization performance in multisensory data: (b,c) AP method with 50 and 20 trials, respectively; (d,e) RAP-MUSIC with 50 and 20 trials, respectively. Results are shown in left and right lateral views of semi-transparent cortex. (f) Localization error averaged across the three dipoles.

The somatosensory and auditory data were then combined by adding the corresponding sensor measurements at 100 ms and 40 ms, respectively (the time of peak response), to yield multimodal data. For evaluation purposes, we utilized either 50 trials (SNR = 4.31 dB) or 20 trials (SNR = 0.33 dB) for each modality to assess the performance of the AP and RAP-MUSIC methods under varying SNR levels. SNR was calculated as the ratio of the Frobenius norm of the MEG data at peak response to that of baseline time for the averaged trial [34].

The AP method localized the three dipoles in close agreement with the ground truth (Fig. 9b,c,f). In contrast, the RAP-MUSIC method yielded only approximate results and was highly variable with the number of trials (Fig. 9d,e,f). Last, for completeness we present min-norm [3] and sLORETA [36] reconstruction results for the same multimodal data (Supplementary Figure 2). Both approaches yielded blurry and imprecise maps because they assume distributed solutions and as a result are not suitable priors for multiple dipole localization.

Note, the observed worse localization error for the AP method in Figure 9f), despite having more trials and a better SNR, is relatively minor. The decrease in localization performance was only by 0.94 mm, which, when compared to the average distance between two sources in the cortical grid we employed (4.6 mm), represents only a small fraction. The slightly worse localization error could be attributed to different factors, including random Initialization and local instabilities.

IV. Conclusion

We have addressed the problem of multiple dipole localization in MEG/EEG with a novel sequential and iterative solution based on the AP algorithm. Through simulation, real phantom, and human MEG data, we found that the AP method consistently outperformed the localization accuracy of RAP-MUSIC, RAP-beamformer, and Truncated RAP-MUSIC. This performance improvement was observed across a wide range of experiments involving different model errors, SNR, inter-source correlation values, and number of sources, and was particularly pronounced in the challenging scenario of highly correlated $\left(\rho \ge 0.95\right)$ sources.

The computational cost of the AP method was found to be similar to that of other competing recursive scanning algorithms based on the beamformer and MUSIC frameworks. Specifically, in our phantom experiment, the AP method converged in approximately 3 iterations, a computational cost not much greater than that of the competing recursive approaches.

In terms of limitations, in this work we assumed a priori knowledge of the number of true sources $Q$. Estimating the number of sources can be done using various techniques, such as the Akaike information criterion (AIC) [24], the minimum description length criterion (MDL) [25], and the Bayesian information criterion (BIC) [37]. But since this is outside the scope of this work, we leave it for future research to explore this problem and its impact on the different localization methods. Further, in all cases, we defined localization error as the average distance computed when each original source was paired with the closest estimated source. However, one can also use alternate measures of localization error. These include sensitivity/specificity or precision/recall available metrics, or measures such as the dipole localization error (DLE) function, which symmetrically computes the distance of original sources paired to the closest estimated source, along with the distance of each estimated source paired to the closest original source [38], [39].

Taken together, our work demonstrated the exceptional performance of the AP method in accurately localizing MEG sources. It also highlighted the importance of iterating through all sources multiple times until convergence, as opposed to recursively scanning for sources and terminating the scan after the last source is found, which is the strategy employed by existing scanning methods.

Highlights:

• A novel solution to the problem of MEG and EEG brain signals localization

• A low-complexity iterative solution is presented, involving only 1-dimensional optimization

• Performance simulation in challenging signal-to-noise ratios and head modeling errors demonstrate the superiority of the proposed solution

• Experimental study with phantom data as well as somatosensory and auditory human data demonstrate the superiority of the proposed solution

Data Availability

The data and analysis tools used in the current study are available at https://alternatingprojection.github.io/, including implementations in Matlab and MNE-Python.