Abstract
Magnetoencephalography and electroencephalography (M/EEG) are non-invasive modalities that measure the weak electromagnetic fields generated by neural activity. Estimating the location and magnitude of the current sources that generated these electromagnetic fields is an inverse problem. Although it can be cast as a linear regression, this problem is severely ill-posed as the number of observations, which equals the number of sensors, is small. When considering a group study, a common approach consists in carrying out the regression tasks independently for each subject using techniques such as MNE or sLORETA. An alternative is to jointly localize sources for all subjects taken together, while enforcing some similarity between them. By pooling S subjects in a single joint regression, the number of observations is S times larger, potentially making the problem better posed and offering the ability to identify more sources with greater precision. Here we show how the coupling of the different regression problems can be done through a multi-task regularization that promotes focal source estimates. To take into account intersubject variabilities, we propose the Minimum Wasserstein Estimates (MWE). Thanks to a new joint regression method based on optimal transport (OT) metrics, MWE does not enforce perfect overlap of activation foci for all subjects but rather promotes spatial proximity on the cortical mantle. Besides, by estimating the noise level of each subject, MWE copes with the subject-specific signal-to-noise ratios with only one regularization parameter. On realistic simulations, MWE decreases the localization error by up to 4 mm per source compared to individual solutions. Experiments on the Cam-CAN dataset show improvements in spatial specificity in population imaging compared to individual models such as dSPM as well as a state-of-the-art Bayesian group level model. Our analysis of a multimodal dataset shows how multi-subject source localization reduces the gap between MEG and fMRI for brain mapping.
Highlights
Magnetoencephalography (MEG) measures the magnetic field surrounding the head, while Electroencephalography (EEG) measures the electric potential at the surface of the scalp
It can be regarded as a linear regression problem in high dimension. This linearity follows directly from Maxwell equations. This inverse problem is inherently “ill-posed”: the number of potential sources is larger than the number of MEG and EEG sensors, which implies that, even in the absence of noise, different neural activity patterns could result in the same electromagnetic field measurements
We evaluate the performance of all models knowing the ground truth by comparing the best estimates on a grid of hyperparameters in terms of three metrics: the mean squared error (MSE) to quantify accuracy in amplitude estimation, AUC and a generalized Earth mover distance (EMD) to assess supports estimation
Summary
Magnetoencephalography (MEG) measures the magnetic field surrounding the head, while Electroencephalography (EEG) measures the electric potential at the surface of the scalp. This linearity follows directly from Maxwell equations This inverse problem is inherently “ill-posed”: the number of potential sources is larger than the number of MEG and EEG sensors, which implies that, even in the absence of noise, different neural activity patterns could result in the same electromagnetic field measurements. In the M/EEG source imaging literature, to our knowledge, the only contribution formulating the problem as a multi-task regression model employs a Group Lasso with an 21 block sparse norm [39]. Following similar ideas that lead to the concomitant Lasso [46, 51, 58] or the multi-task Lasso [43], the Minimum Wasserstein Estimates (MWE) was first proposed in [31] Both MTW and MWE rely on convex 1 norm penalties which tend to promote sparse solutions at the expense of an amplitude bias.
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