 Research
 Open access
 Published:
Rejuvenating classical brain electrophysiology source localization methods with spatial graph Fourier filters for source extents estimation
Brain Informatics volumeÂ 11, ArticleÂ number:Â 8 (2024)
Abstract
EEG/MEG source imaging (ESI) aims to find the underlying brain sources to explain the observed EEG or MEG measurement. Multiple classical approaches have been proposed to solve the ESI problem based on different neurophysiological assumptions. To support clinical decisionmaking, it is important to estimate not only the exact location of the source signal but also the extended source activation regions. Existing methods may render overdiffuse or sparse solutions, which limit the source extent estimation accuracy. In this work, we leverage the graph structures defined in the 3D mesh of the brain and the spatial graph Fourier transform (GFT) to decompose the spatial graph structure into subspaces of low, medium, and highfrequency basis. We propose to use the lowfrequency basis of spatial graph filters to approximate the extended areas of brain activation and embed the GFT into the classical ESI methods. We validated the classical source localization methods with the corresponding improved version using GFT in both synthetic data and real data. We found the proposed method can effectively reconstruct focal source patterns and significantly improve the performance compared to the classical algorithms.
1 Introduction
EEG/MEG are noninvasive measurement modalities with high temporal resolution up to 1Â ms. EEG/MEG signals can be collected noninvasively through electrodes or sensors on the scalp. Importantly, EEG/MEG are the direct measurement methods to detect the instantaneous electrophysiological activities of the brainÂ [1]. EEG and MEG are recognized as powerful tools for capturing realtime brain functions by measuring neuronal processes with broad clinical and neuroscience applicationsÂ [2]. The EEG/MEG source localization aims to solve an inverse problem to reconstruct the underlying electrophysiological brain activities given the measured EEG/MEG signal and the brain forward modelÂ [3]. The EEG/MEG source localization is also referred to as Electrophysiological Source Imaging (ESI) [4]. However, the number of EEG/MEG channels is far less than that of the brain sources, which makes the ESI an illposed problem.
In the past decades, numerous algorithms have been developed with different assumptions on the configuration of the source signal. One seminal work is minimum norm estimate (MNE) where \(\ell _2\) norm is used as a regularization [5], which is to explain the observed signal using a potential solution with the minimum energy. Different variants of the MNE algorithm include dynamic statistical parametric mapping (dSPM) [6] and standardized lowresolution brain electromagnetic tomography (sLORETA) [7]. The \(\ell _2\)normbased methods tend to render spatially diffuse source estimation. To promote sparse solutions, Uutela et al. [8] introduced the \(\ell _1\)norm, known as minimum current estimate (MCE). Rao and KreutzDelgado proposed an affine scaling method [9] for a sparse ESI solution. The focal underdetermined system solution (FOCUSS) proposed by Gorodnitsky et al. encourages a sparse solution by introducing the \(\ell _{p}\)norm regularization [10]. Besides, Bore et al. proposed to use the \(\ell _{p}\)norm regularization (\(p < 1\)) on the source signal and the \(\ell _1\) norm on the data fitting error term to achieve sparsity [11]. Babadi et al. demonstrated that sparsely distributed solutions to eventrelated stimuli could be found using a greedy subspacepursuit algorithm [12]. Wipf et al. proposed a unified Bayesian learning method that can automatically calculate the hyperparameters for the inverse problems under an empirical Bayesian framework, and the sparsity of the solution is also guaranteed [13]. To mitigate the high level of noise when estimating source signal, Hashemi et al. applied a hierarchical Bayesian (typeII maximum likelihood) model for joint estimation of latent variables for both source and noise [14]. An innovative robust empirical Bayesian framework proposed by Ghosh et al. can estimate the lowrank noise covariance and sparse brain source activity at the same time [15]. Wan et al. reformulate the ESI problem into a graph search problem by exploiting the graph neighborhood information in the brain source space and the optimal solution can be theoretically guaranteed under high noise circumstances [16].
In addition to using optimization approaches, deep learning methods have become increasingly popular in recent years. Hecker et al. came up with a CNN model with a special design of the input matrices of EEG and MEG signal, termed as ConvDip, for source localization which achieved good performance [17]. Jiao et al. proposed to use of a graph Fourier transform based bidirectional longshortterm memory, termed as GFTBiLSTM, which can reduce the output dimensions and can construct an extended brain region activation. The GFTBiLSTM method achieves good performance localizing seizure onset zoneÂ [18]. To take advantage of both the high spatial resolution of fMRI and the high temporal resolution of EEG, Liu et al. proposed a hierarchical deep transcoding model for fusing simultaneous EEGfMRI data to estimate the brain source activity with high spatiotemporal resolution [19, 20]. Another multimodal deep learning model that fuses both MEG and EEG information is shown to have high accuracy compared to using a single modality of EEG or MEG in the context of deep learning approach [21].
As the brain sources are not activated discretely due to its volume conductivity property, a compact and extended area of source estimation is preferred [4, 22, 23], and it has been used for multiple applications, such as somatosensory cortical mapping [24], and epileptogenic zone in focal epilepsy patients [25, 26]. To estimate an extended area of source activation, we propose to use the GFT technique, which has been shown with an improved accuracy in reconstructing extended source activations [18, 27]. GFT plays a pivotal role in understanding the spectral characteristics of the signals on a graph, analogous to how the Fourier Transform reveals the frequency content of a signal in the time domain [28]. By decomposing graph data into its spectral components, it provides a unique perspective on the underlying structure and connectivity/continuity patterns within the graph. Numerous interesting applications can be found in various domains. For example, in their seminal work, Defferrard et al. proposed to use GFT with deep learning and came up with ChebNet, which highlighted the use of GFT in neural networks for graphstructured data [29]. GFT has been used to characterize the spectrum properties of the brain connectivity networks [30], Brahim and Farrugia showed that using GFTbased structural connectivity and functional connectivity analysis can provide a more accurate classification for patients with Autism Spectrum Disorders [31].
In this work, we propose to improve the classical ESI methods using GFT and its lowfrequency components as source space subspaces, and we highlight the importance of using GFT for an extended area of source activation before applying the classical source imaging methods. A preliminary version has been published in the Brain Informatics Proceedings [32].
2 Method
In this section, we start with the introduction of the ESI inverse problem, which is followed by the presentation of the GFT and the improved version of classical methods using GFT.
2.1 EEG/MEG source imaging
The source imaging forward problem can be described in the formula as \(Y = KS + E\), where \(Y\in \mathbb {R}^{C\times T}\) is the EEG or MEG measurements, C is the number of EEG or MEG channels, T represents the time sequence length, \(K\in \mathbb {R}^{C\times N}\) is the leadfield matrix which is a linear mapping from the brain sources to the EEG/MEG sensors, N is the number of brain source, \(S\in \mathbb {R}^{N\times T}\) represents the brain source signal in N source locations for all the T time points, and E is the measurement noise. The inverse problem is to estimate S given Y and K. Since the source dimension N is much larger than the number of electrodes C, making the ESI an illposed inverse problem, it is challenging to obtain a unique solution. Various regularization terms were designed based on the prior assumptions of the spatial and temporal structure of the source signal to find a unique solution. The inverse problem of ESI can be formulated as below:
where \({\Vert \cdot \Vert _F}\) is the Frobenius norm, and S can be obtained by solving the minimizing problem. The first term in Eq.Â (1) is the data fitting term trying to explain the recorded EEG/MEG measurements. The second term is called the regularization term, which is imposed to find a unique solution using sparsity or other neurophysiologyinspired regularizations. For example, if R(S) is a \(\ell _2\) norm, the problem is called minimum norm estimate (MNE).
2.2 Graph Fourier transform in brain source space
Consider an undirected graph \(G = \left\{ \mathcal {V}, A\right\} \) generated from the 3D mesh of brain cortex, where \(\mathcal {V} = \left\{ v_1, v_2, \ldots , v_N \right\} \) is the set of N nodes, A is the weighted adjacent matrix with entries given by the edge weights \(a_{ij}\) that represents the connection strength between node i and node j. The graph Laplacian matrix is defined as \(L = D  A\), where D is the degree matrix with \(D_{ii} = \sum _{j\ne i}A_{ij}\). Since L is a positive semidefinite matrix, its eigenvalues are all greater or equal to 0, and the associated eigenvectors \(U = [u_1, u_2, \ldots , u_N]\), \(U \in \mathbb {R}^{N \times N}\) can be regarded as the basis vectors of GFT where any signal in the graph can be approximated as the linear combinations of basis. Thus, the GFT for a signal S can be defined as \(\tilde{S}= U^{T}S\), whereas the inverse GFT is given as \(S = U\tilde{S}\). Here we define normalized graph frequency (NGF) [27] as
where Tr(L)is the trace of L, and \(f_s(u_i)\) is defined as
where \(\mathcal {N}(m)\) represents all neighbors of node m, and \(\mathbb {I}(\cdot )\) is the indicator function which equals 1 if the values of \(u_i\) on node m and n have different signs and 0 otherwise. The number of sign flips at time t indicates how many zero crossings of a signal are within a bounded region at t.
We calculated the NGF in the whole time series within firstorder neighbors, secondorder neighbors, and thirdorder neighbors, respectively. The spectrogram, which is illustrated in Fig.Â 1, reveals that the NGF is positively correlated with the order of the eigenvalue of L. Thus we can further separate U into low, medium, and highfrequency components according to NGF values, and reformat it as \(U = [U_{L}, U_{M}, U_{H}]\). The source activation patterns corresponding to eigenvectors with different NGFs are illustrated in Fig.Â 1.
2.3 Brain source imaging with spatial graph filters
The existing sparse source localization methods with \(\ell _1\)norm and \(\ell _{2,1}\)norm as regularizations usually provide oversparse solutions when estimating source extents, whereas the other nonsparsity methods commonly result in multiple foci. In this work, we propose to use the lowfrequency subspaces spanned by the lowfrequency graph filters (eigenvectors) to approximate an extended area of brain source localization. The proposed GFT technique can improve the performance of classical source localization methods by providing the lowfrequency spatial graph filters \([U_{L}] \in \mathbb {R}^{N\times P}\), thus removing the highfrequency noises to reconstruct the focally extended sources. Here P serves as the cutoff value for the set of lowfrequency components in the subspace. The intuition is that the main energy of the source signal usually lies in the lowfrequency components which are associated with the regions on the cortex with relatively large source extend area in a time series. Keeping the low graph frequency could promote a source extend area reconstruction and decrease the impact of the noise. Moreover, the reduced dimensional estimation in the inverse problem could further constrain the solution space and make the solution more easily solved and robust.
Specifically, we project graph signals S onto a subspace spanned by \(\tilde{U}\) with low NGF values. The data fitting term can be written as \(\Vert Y K\tilde{U}\tilde{U}^T S\Vert ^2_F\). Let \(\tilde{S} = \tilde{U}^T S\) and \(\tilde{K} = K \tilde{U} \), then the data fitting term can be rewritten as \(\Vert Y \tilde{K}\tilde{S}\Vert ^2_F\). Solving S is equivalent to finding parameters in the subspace \(\tilde{U}\). Our goal is to estimate \(\tilde{S}\), and all regularizations will be added to \(\tilde{S}\) instead of S.
Finally, the source estimation \(\hat{S}_{GFT}\) can be simply obtained from the inverse GFT \(\tilde{U}\hat{S}^*_{GFT}\).
We delineate the solutions for 5 classical ESI methods, including MNE [33], MCE [8], MxNE (with L21 implementation) [34], dSPM [6] and sLORETA [7], and show the solution or objective function of both original form and GFTbased form as follows.
MNE MNE is proposed by HĂ¤mĂ¤lĂ¤inen by imposing an \(\ell _2\) regularization on the source solutionÂ [33], the MNE algorithm has a closedloop solution, given as
We define \(\tilde{K}=K\tilde{U}\), then the GFTMNE solution is
sLORETA sLORETA is a method proposed by PascualMarqui [35], and the solution is also in a closed form
The GFTsLORETA solution is given as below:
dSPM dSPM is proposed by Dale et al. It has a closedform solution, and the solution is
the GFTdSPM solution can be obtained as:
MCE MCE is a method proposed by Uutela et al. with \(\ell _1\) regularizationÂ [8]. There is no closedloop solution. Multiple iterative solvers can solve this problem. The objective function is:
and the GFTMCE solution can be obtained by solving the following equation:
MxNE MxNE used a mixed norm as a regularization term, where \(\ell _2\) norm is applied to the temporal domain and \(\ell _p\) (\(p=\{0.5,1\}\) is applied in the spatial domain, which is proposed by Gramfort et al. [34]. There is no closedloop solution for this regularization. In this paper, we use \(\ell _{2,1}\) norm version for MxNE. The objective function is:
and the GFTL21 solution is given as:
where \(\Vert \cdot \Vert _{2,1}\) denotes \(\ell _2\) norm on the horizontal direction and \(\ell _1\) on the vertical dimension of a matrix.
3 Result
In this section, we conducted numerical experiments to validate the effectiveness of the proposed method on synthetic EEG data under different levels of neighbors (LNs) and signal Noise Ratio (SNR) settings for source localization followed by a causal time series reconstruction simulation. Lastly, we further validate it on real MEG recordings from a visualauditory test.
3.1 Simulation experiments
To validate the proposed GFTbased methods, We first conducted experiments on synthetic data with known activation patterns.
Forward model To generate synthetic EEG data, we build a realistic head model based on T1MRI. The brain tissue segmentation and source surface reconstruction were conducted using FreeSurfer [36]. Then a threelayer boundary element method (BEM) head was built based on these surfaces, given the tissue conductivity values. A 128channel BioSemi EEG cap layout was used, and the EEG channels were coregistered with the head model using Brainstorm [37] and then further validated on the MNEPython toolbox [38]. The source space contains 1026 sources in each hemisphere, with 2052 sources combined, resulting in a leadfield matrix L with a dimension of 128 by 2052.
Synthetic data generation We randomly generated 200 source locations out of 2052 locations in the source space to conduct the experiments. Furthermore, as illustrated in Fig.Â 2, we used three different neighborhood levels (1, 2, and 3level of the neighborhood) to represent different sizes of source extents, then we activated the whole â€śpatch" with different neighborhood levels at the same time. The activation strength of the 1, 2, and 3level adjacent regions was successively set to be 80%, 60%, and 40% of the central region. The scalp EEG data was generated based on the forward model under different measurement noise configurations with different Noise Ratio (SNR) set to be 40Â dB, 30Â dB, 20Â dB, and 10Â dB. SNR is defined as the ratio of the signal power \(P_{\text {signal}}\) to the noise power \(P_{\text {noise}}\): \(\text {SNR}=10\log (P_{\text {signal}}/P_{\text {noise}})\). In total, there were 12: 3 (levels of neighborhood) \(\times \) 4 (SNRs) data sets (Y and S pairs).
For causal time series reconstruction, we generated a simulated causal time series in the source space based on the Berlin Brain Connectivity Benchmark (BBCB) [39]. We slightly simplified it by using the autoregression with randomly generated 2 by 2 state transition matrix \(\varvec{\Phi }\), with the order K to be 1 and the number of activated brain regions is set to be 2, to generate the source signal, given in Eq.Â 15.
Then an independent random Gaussian noise with SNR from 10 to 40Â dB was added at each time point. Lastly, a thirdorder Butterworth filter with zero phase delay was applied with pass bandwidth [0.1Â Hz, 40Â Hz].
Experimental settings We adopted MNE [5], MCE [8], \(\ell _{2,1}\)(MxNE) [34], dSPM [6], and sLORETAÂ [7], as benchmark algorithms for comparison. We separately performed EEG source localization based on benchmark algorithms with and without the proposed GFTbased dimensionality reduction method. All the experiments were conducted on a Linux environment with CPU Intel(R) Xeon(R) Gold 6130 CPU @2.10Â GHz and 128Â GB memory.
Source localization simulation results The performance of each algorithm was quantitatively evaluated based on the following metrics:

(1)
Localization error (LE): it measures the Euclidean distance between centers of two source locations on the cortex meshes.

(2)
Area under curve (AUC): it is particularly useful to characterize the overlap of an extended source activation pattern.
Better performance for localization is expected if LE is close to 0 and AUC is close to 1. The performance comparison between the proposed methods and benchmark algorithms on LE and AUC is summarized in TableÂ 1, and the boxplot figures, as well as the difference significance test for SNRÂ =Â 30Â dB, and 10Â dB are given in Fig.Â 3. The comparison between the reconstructed source distributions with a 3level of the neighborhood and 40Â dB SNR is shown in Fig.Â 4.
From TableÂ 1 and Figs.Â 3 and 4, we can find that: (1) MNE, MCE, \(\ell _{2,1}\), sLORETA, and dSPM can only reconstruct the brain sources when the activated area is small and the SNR level is high, and even the evaluation metrics can be good in some cases, the reconstruction for source extend area is discrete or disconnected as shown in Fig.Â 4. As the source range expands and the SNR decreases, a significant increase in LE and an obvious reduction in AUC can be observed. The reconstructed source distributions are no longer concentrated. (2) By contrast, the results of the proposed GFTbased methods outperform benchmark methods in most cases after applying the spatial graph filters. They both show good stability for varied neighborhood levels and SNR settings. Particularly, the performance of the proposed GFTbased \(\ell _1\) regularization family (i.e., \(\ell _1\)norm and \(\ell _{2,1}\)norm) exhibits better performance on reconstructing source extents without losing its advantage in sparse focal source reconstruction and outperforms other methods in most instances.
Causal time series reconstruction An order KÂ =Â 1 causal time series was generated at different levels of noise. As the reconstructed time series at the source space is impacted by the strength of regularizations, to rectify this impact, we use the correlation of two sets of time series to quantify the similarity between the reconstructed time series and the ground true time series. A detailed comparison of different methods is given in TableÂ 2, and an example of source reconstruction at 20Â dB SNR is illustrated in Fig.Â 5.
It is worth noting that the proposed GFTbased methods are better than their original counterparts in most cases. Even at low SNR levels, the proposed methods can also perform well. Particularly, the GFTbased \(\ell _1\) regularization family has a significant performance improvement compared to that of their original form which is consistent with the findings in the source localization simulation above, and this phenomenon can be observed in Fig.Â 5, where the reconstruction results of the GFTbased \(\ell _1\) regularization family method are significantly better than the original method, especially the reconstruction of the source signal trend during some time intervals.
3.2 Real data experiments
We further validated the proposed methodology on a real dataset that is publicly accessible through the MNEPython package [38]. In this dataset acquirement, checkerboard patterns were presented into the left and right visual field, interspersed by tones to the left or right ear with a stimuli interval of 750Â ms. The subject was asked to press a key with the right index finger as soon as possible after the appearance of a smiley face was presented at the center of the visual field [40]. Evoked Response Potentials (ERP) were extracted from the MEG measurements, and then we averaged these ERPs for source reconstruction under MNE, MCE, \(\ell _{2,1}\), dSPM, sLORETA with and without the proposed GFT operations. The averaged spikes are shown in Fig.Â 6, and the reconstructed source distributions are shown in Fig.Â 7.
From Fig.Â 7, we can see that the source area estimated by MNE, MCE, \(\ell _{2,1}\), sLORETA, and dSPM is highly broad. By contrast, the proposed GFTbased methods provide more sparse focal source reconstructions. Moreover, the reconstructed focal for the proposed GFTbased methods falls primarily on areas with the strongest source signal, while others would spread to several regions. Obviously, the spatial graph filter in the proposed GFTbased methods promotes a concentrated and accurate estimation of the visual zone.
4 Discussion
We showed that the subspace composed of lowfrequency spatial graph filters can be used to approximate the compact and extended source activation patterns, which is a nonparametric approach, thus empowering the classical ESI methods to have better reconstruction performance for source extents reconstruction. This approach is an easy and effective preprocessing technique compared to predefined spatial gradient based methods or datadriven methods in a deep learning paradigm. While we validated 5 classical methods, it is worth noting that most of the other classical approaches, such as recursive multiple signal classifier (MUSIC), recursively applied and projected MUSIC (RAP MUSIC), lowresolution brain electromagnetic tomography (LORETA), FOCUSS, weighted minimum norm (WMN), shrinking LORETAFOCUSS, hybrid weighted minimum norm method, recursive sLORETAFOCUSS, standardized shrinking LORETAFOCUSS (SSLOFO), etc., can use the proposed GFT in the source space before applying each specific method.
5 Conclusion
In this study, we improved the classical source localization methods by using spatial graph filters to solve the inverse problem of ESI. The proposed methodology enjoys the advantage of reconstructing focal source extents with sparsity and minimizing the impact of the noise by transforming the estimation of the source signal into a projected subspace spanned by spatial frequency graph filters. Numerical experiments demonstrated that the proposed method performs particularly well on source extents, yields excellent robustness when the SNR level is low, and can better reconstruct the source time series, specifically the performance of the \(\ell _1\) family regularization can be greatly improved with the GFT. In the experiment on real data we performed, the proposed methodology provides a satisfactory reconstruction with more concentrated source distribution and more stability to noise than benchmark algorithms.
Availability of data and materials
The public data used can be accessed through: https://mne.tools/stable/index.html. The code will be accessible upon request to the corresponding authors.
References
Wendel K, VĂ¤isĂ¤nen O, Malmivuo J, Gencer NG, Vanrumste B, Durka P, MagjareviÄ‡ R, Supek S, Pascu ML, Fontenelle H et al (2009) EEG/MEG source imaging: methods, challenges, and open issues. Comput Intell Neurosci
Michel CM, Brunet D (2019) EEG source imaging: a practical review of the analysis steps. Front Neurol 10:325
Huang G, Liu K, Liang J, Cai C, Gu ZH, Qi F, Li Y, Yu ZL, Wu W (2022) Electromagnetic source imaging via a datasynthesisbased convolutional encoderâ€“decoder network. IEEE Trans Neural Netw Learn Syst
He B, Sohrabpour A, Brown E, Liu Z (2018) Electrophysiological source imaging: a noninvasive window to brain dynamics. Annu Rev Biomed Eng 20:171â€“196
HĂ¤mĂ¤lĂ¤inen MS, Ilmoniemi RJ (1994) Interpreting magnetic fields of the brain: minimum norm estimates. Med Biol Eng Comput 32(1):35â€“42
Dale AM, Liu AK, Fischl BR, Buckner RL, Belliveau JW, Lewine JD, Halgren E (2000) Dynamic statistical parametric mapping: combining fMRI and MEG for highresolution imaging of cortical activity. Neuron 26(1):55â€“67
PascualMarqui RD et al (2002) Standardized lowresolution brain electromagnetic tomography (sLORETA): technical details. Methods Find Exp Clin Pharmacol 24(Suppl D):5â€“12
Uutela K, HĂ¤mĂ¤lĂ¤inen M, Somersalo E (1999) Visualization of magnetoencephalographic data using minimum current estimates. Neuroimage 10(2):173â€“180
Rao BD, KreutzDelgado K (1999) An affine scaling methodology for best basis selection. IEEE Trans Signal Process 47(1):187â€“200
Gorodnitsky IF, George JS, Rao BD (1995) Neuromagnetic source imaging with FOCUSS: a recursive weighted minimum norm algorithm. Electroencephalogr Clin Neurophysiol 95(4):231â€“251
Bore JC, Yi C, Li P, Li F, Harmah DJ, Si Y, Guo D, Yao D, Wan F, Xu P (2018) Sparse EEG source localization using LAPPS: least absolute lP (0 < p < 1) penalized solution. IEEE Trans Biomed Eng 66:1927â€“1939
Babadi B, ObregonHenao G, Lamus C, HĂ¤mĂ¤lĂ¤inen MS, Brown EN, Purdon PL (2014) A subspace pursuitbased iterative greedy hierarchical solution to the neuromagnetic inverse problem. Neuroimage 87:427â€“443
Wipf D, Nagarajan S (2009) A unified Bayesian framework for MEG/EEG source imaging. Neuroimage 44(3):947â€“966
Hashemi A, Cai C, Gao Y, Ghosh S, MĂĽller KR, Nagarajan SS, Haufe S (2022) Joint learning of fullstructure noise in hierarchical Bayesian regression models. IEEE Trans Med Imaging 14:610â€“624
Ghosh S, Cai C, Gao Y, Hashemi A, Haufe S, Sekihara K, Raj A, Nagarajan SS (2023) Bayesian inference for brain source imaging with joint estimation of structured lowrank noise. In: 2023 IEEE 20th international symposium on biomedical imaging (ISBI). IEEE, pp 1â€“5
Wan G, Jiao M, Ju X, Zhang Y, Schweitzer H, Liu F (2023) Electrophysiological brain source imaging via combinatorial search with provable optimality. In: Proceedings of the AAAI conference on artificial intelligence, vol 37, pp 12491â€“12499
Hecker L, Rupprecht R, Elst L, Kornmeier J (2021) Convdip: a convolutional neural network for better EEG source imaging. Front Neurosci 15:569918
Jiao M, Wan G, Guo YN, Wang D, Liu H, Xiang J, Liu F (2022) A graph Fourier transform based bidirectional LSTM neural network for EEG source imaging. Front Neurosci 447
Liu X, Sajda P (2023) Fusing simultaneously acquired EEG and fMRI via hierarchical deep transcoding. In: International conference on brain informatics. Springer, pp 57â€“67
Liu X, Sajda P (2023) Latent neural source recovery via transcoding of simultaneous EEGfMRI. In: International conference on brain informatics. Springer, pp 318â€“330
Jiao M, Yang S, Wang B, Xian X, Semenov YR, Wan G, Liu F (2023) Mmdfesi: multimodal deep fusion of EEG and MEG for brain source imaging. In: International conference on brain informatics. Springer, pp 273â€“285
Baillet S, Mosher JC, Leahy RM (2001) Electromagnetic brain mapping. IEEE Signal Process Mag 18(6):14â€“30
Sohrabpour A, Lu Y, Worrell G, He B (2016) Imaging brain source extent from EEG/MEG by means of an iteratively reweighted edge sparsity minimization (IRES) strategy. Neuroimage 142:27â€“42
Cai C, Diwakar M, Chen D, Sekihara K, Nagarajan SS (2019) Robust empirical Bayesian reconstruction of distributed sources for electromagnetic brain imaging. IEEE Trans Med Imaging 39:567â€“577
Sohrabpour A, Ye S et al (2016) Noninvasive electromagnetic source imaging and granger causality analysis: an electrophysiological connectome (econnectome) approach. IEEE Trans Biomed Eng 63(12):2474â€“2487
Becker H, Albera L, Comon P, Gribonval R, Merlet I (2014) Fast, variationbased methods for the analysis of extended brain sources. In: 2014 22nd European signal processing conference (EUSIPCO). IEEE, pp 41â€“45
Liu F, Wan G, Semenov YR, Purdon PL (2022) Extended electrophysiological source imaging with spatial graph filters. In: International conference on medical image computing and computerassisted intervention. Springer, pp 99â€“109
Sandryhaila A, Moura JM (2013) Discrete signal processing on graphs: graph Fourier transform. In: 2013 IEEE International conference on acoustics, speech and signal processing. IEEE, pp 6167â€“6170
Defferrard M, Bresson X, Vandergheynst P (2016) Convolutional neural networks on graphs with fast localized spectral filtering. In: Advances in neural information processing systems 29
Wang J, Calhoun VD, Stephen JM, Wilson TW, Wang Yp (2018) Integration of network topological features and graph Fourier transform for fMRI data analysis. In: 2018 IEEE 15th international symposium on biomedical imaging (ISBI 2018). IEEE, pp 92â€“96
Brahim A, Farrugia N (2020) Graph Fourier transform of fMRI temporal signals based on an averaged structural connectome for the classification of neuroimaging. Artif Intell Med 106:101870
Yang S, Jiao M, Xiang J, Kalkanis D, Sun H, Liu F (2023) Rejuvenating classical source localization methods with spatial graph filters. In: International conference on brain informatics. Springer, pp 286â€“296
Hamalainen MS (1984) Interpreting measured magnetic fields of the brain: estimates of current distributions. Report, Helsinki University of Technology
Gramfort A, Kowalski M, HĂ¤mĂ¤lĂ¤inen M (2012) Mixednorm estimates for the M/EEG inverse problem using accelerated gradient methods. Phys Med Biol 57(7):1937
PascualMarqui RD, Michel CM, Lehmann D (1994) Low resolution electromagnetic tomography: a new method for localizing electrical activity in the brain. Int J Psychophysiol 18(1):49â€“65
Fischl B (2012) Freesurfer. Neuroimage 62(2):774â€“781
Tadel F, Baillet S, Mosher JC, Pantazis D, Leahy RM (2011) Brainstorm: a userfriendly application for MEG/EEG analysis. Comput Intell Neurosci 2011:1â€“13
Gramfort A, Luessi M, Larson E, Engemann DA, Strohmeier D, Brodbeck C, Parkkonen L, HĂ¤mĂ¤lĂ¤inen MS (2014) MNE software for processing MEG and EEG data. Neuroimage 86:446â€“460
Haufe S, Ewald A (2019) A simulation framework for benchmarking EEGbased brain connectivity estimation methodologies. Brain Topogr 32(4):625â€“642
Gramfort A, Strohmeier D, Haueisen J, HĂ¤mĂ¤lĂ¤inen MS, Kowalski M (2013) Timefrequency mixednorm estimates: sparse M/EEG imaging with nonstationary source activations. Neuroimage 70:410â€“422
Funding
Research reported in this publication was partially supported by the National Institute of Biomedical Imaging And Bioengineering of the National Institutes of Health under Award Number R21EB033455. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
Author information
Authors and Affiliations
Contributions
SY: formal analysis, software, data curation, and first draft writing. MJ: data curation, software, and reviewediting. JX: reviewediting and supervision. NF: experiment design, and reviewediting. HS: experiment design, and reviewediting. FL: grant acquisition, supervision, conception, and reviewediting. All authors contributed to the article and approved the submitted version.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
IRB was waived for the secondary data analysis.
Competing interests
All the authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the articleâ€™s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articleâ€™s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Yang, S., Jiao, M., Xiang, J. et al. Rejuvenating classical brain electrophysiology source localization methods with spatial graph Fourier filters for source extents estimation. Brain Inf. 11, 8 (2024). https://doi.org/10.1186/s40708024002212
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s40708024002212