Mixed-norm estimates for the M/EEG inverse problem using accelerated gradient methods

Abstract

Magneto- and electroencephalography (M/EEG) measure the electromagnetic fields produced by the neural electrical currents. Given a conductor model for the head, and the distribution of source currents in the brain, Maxwell’s equations allow one to compute the ensuing M/EEG signals. Given the actual M/EEG measurements and the solution of this forward problem, one can localize, in space and in time, the brain regions than have produced the recorded data. However, due to the physics of the problem, the limited number of sensors compared to the number of possible source locations, and measurement noise, this inverse problem is ill-posed. Consequently, additional constraints are needed. Classical inverse solvers, often called Minimum Norm Estimates (MNE), promote source estimates with a small ℓ2 norm. Here, we consider a more general class of priors based on mixed-norms. Such norms have the ability to structure the prior in order to incorporate some additional assumptions about the sources. We refer to such solvers as Mixed-Norm Estimates (MxNE). In the context of M/EEG, MxNE can promote spatially focal sources with smooth temporal estimates with a two-level ℓ1/ℓ2 mixed-norm, while a three-level mixed-norm can be used to promote spatially non-overlapping sources between different experimental conditions. In order to efficiently solve the optimization problems of MxNE, we introduce fast first-order iterative schemes that for the ℓ1/ℓ2 norm give solutions in a few seconds making such a prior as convenient as the simple MNE. Furhermore, thanks to the convexity of the optimization problem, we can provide optimality conditions that guarantee global convergence. The utility of the methods is demonstrated both with simulations and experimental MEG data.

Contrasting functional imaging parametric maps: The mislocation problem and alternative solutions

“…Even larger mislocation errors would be foreseeably obtained for maps reconstructed from electroencephalographic (EEG) signals since they spread more than MEG signals and their propagation is harder to model (H€ am€ al€ ainen et al, 1993). One way to avoid the extreme smoothness of MEG maps is to use spatially sparse imaging methods such as, e.g., minimum current estimation (Uutela et al, 1999), minimum mixed-norm estimation (Gramfort et al, 2013(Gramfort et al, , 2012, or the multiple sparse priors approach (Friston et al, 2008). In this case the MEG maps are very focal, but this opposite extreme poses other problems to contrast two different conditions at the group level (unless maps are smoothed afterwards).…”

Bayesian EEG source localization using a structured sparsity prior

“…The first data set corresponds to the auditory evoked responses to left ear pure tone stimulus while the second one consists of the evoked responses to facial stimulus. The results of the proposed method are compared with the weighted ℓ 21 mixed norm (Gramfort et al, 2012), the Champagne model (Wipf et al, 2010) and the method investigated in Friston et al (2008) based on multiple sparse priors.…”

“…In a previously reported work, we proposed to combine them in a Bayesian framework , using the ℓ 0 pseudo norm to locate the non-zero positions and the ℓ 1 norm to estimate their amplitudes. However the methods studied in Candes (2008), Uutela et al (1999), and consider each time sample independently leading in some cases to unrealistic solutions (Gramfort et al, 2012).…”

“…Structured sparsity has also been applied to M/EEG source localization by Gramfort et al by using the ℓ 21 mixed norm (Gramfort et al, 2012). This approach promotes sparsity among different dipoles (via the ℓ 1 portion of the norm) and groups all the time samples of the same dipole together, forcing them to be either jointly active or inactive (with the ℓ 2 norm portion).…”

Spatio-temporal reconstruction of brain dynamics from EEG with a Markov prior

“…a stimulus after which the activity returns to a baseline level (Gramfort et al, 2013). Both methods employ a mixed-norm scheme to recover what is hypothesized to be a structured sparsity pattern across time, see also (Haufe et al, 2008;Gramfort et al, 2012). These types of convex relaxation schemes are 60 very interesting and are frequently applied to solve inverse problems in general (VegaHernández et al, 2008).…”