Bayesian Meets Wavelets in spacetime – BMWs

A preferred approach to solving the inverse M/EEG problem, known as the variational approach, seeks solutions that minimize a certain cost function. This function measures the adequacy of the solution to both the experimental data and an a priori model, which should be realistic and expressive (capable of capturing a wide variety of situations). The compromise between these two objectives is determined by parameters, which are all the more numerous as the model is capable of reproducing a wide variety of situations, and which are often difficult to determine. Furthermore, the computational cost of numerical resolution in large dimensions can be high.

The BMWs project develops spatio-temporal models of brain activity, with prior assumptions of sparsity, whose parameters are estimated in a hierarchical Bayesian framework. This framework includes Gaussian models and Bernoulli-Gaussian models. The latter allow for an explicit representation of the source support and its estimation, which constitutes a first approach to the problem of quantifying the uncertainty associated with the solution obtained, while remaining compatible with efficient algorithms adapted to large dimensions.

Estimating the spatial extent of brain activity is even more difficult than determining its location. The problem is addressed by combining the aforementioned hierarchical Bayesian approaches with sparse decompositions of the sources sought on multiscale dictionaries defined on the surface of the cortex: brain activity is modeled as a function on the surface of the cortex, equal to the sum of a small number of elementary solutions called wavelets.

In the variational framework, the formulation of the problem in this transformed domain effectively leads to extended solutions, whose spatial extent can be estimated and validated by well-chosen metrics, but the problem of parameter tuning remains open. Integration into a hierarchical Bayesian framework solves this latter problem. This approach allows simultaneous EEG, MEG, and intracerebral data to be processed and used for validation.

The project complements a reference software suite called MNE-python, commonly used in the field of MEG/EEG signal processing and source localization. The software developments are designed to be complementary to MNE-Python, but an independent tool is also being developed to be made available to the community without requiring prior knowledge of MNE-python.

A key strength of the project is the combination of EEG/MEG with intracerebral measurements (SEEG). These data are used to evaluate solutions to the inverse M/EEG problem and compare the approaches developed within the project with state-of-the-art algorithms.

Variational and Bayesian approaches have been developed, which make it possible to estimate brain sources and identify their spatial support. For example, a method based on a Bernoulli–Gaussian model (named LEMUR) estimates the support of the brain sources and associates with each point of it an uncertainty measure derived from its activation probability. The SBL (Sparse Bayesian Learning) approach, combined with modeling sources as superpositions of cortical wavelets (i.e. functions defined on the surface of the cortex), produces solutions whose support is controlled (and therefore provides a measure of spatial extent), and which are realistic from a quantitative point of view, which is not the case for the vast majority of existing approaches. The calculation cost of the corresponding algorithms remains very reasonable, and compatible with common applications.

Another approach developed is based on a formulation of the inverse problem firstly involving a convolutional decomposition: a decomposition carried out at the level of the sensors can be transferred to the source space, which makes it possible to separate the temporal analysis from the spatial estimation and to reduce the cost of the inversion steps.

The evaluation of brain source localization and reconstruction results requires the use of appropriate validation metrics. In this project, new metrics were introduced and used which allow, better than existing metrics, to quantify the precision of the solution in terms of location, spatial extent and depth. Used on simulated data, but also real data in combination with annotations, these metrics confirmed the increased precision of the source estimates provided by w-SBL.

The project led to the development of a software base in Python bringing together the main methods from BMWs, made available as open source (https://gitlab.inria.fr/mkowalsk/bmws). The implementations were structured in a modular way to facilitate experiments, comparisons between approaches and subsequent developments. Particular attention was paid to compatibility with the MNE-Python ecosystem, allowing the direct use of the tools developed in standard M/EEG processing pipelines.

The source reconstruction methods developed in the project were tested on synthetic data and annotated real-world data (from the database associated with MNE-python), allowing for an evaluation of their quality. They were also tested and validated on real-world data acquired by the project's DYNAMAP team, for which SEEG data were also available. Comparison of this SEEG data with the MEG reconstructions proved very promising.

The theoretical aspects of Bayesian inference for the M/EEG source reconstruction problem deserve to be further explored. For example, in the case of the Bernoulli-Gauss model, the question arises of the links between marginal estimation of support and joint estimation of Maximum a Posteriori type, and their respective implications in terms of bias and robustness, with implications for the quantification of the uncertainty on the solution of the inverse problem. Similar questions arise in the context of the SBL and MEM approaches. In the latter case, the modeling of temporal dynamics remains to be integrated into a priori models.

Furthermore, the results establishing links between convolutional decomposition, time-frequency factorizations and inverse problem invite us to explore more closely the relationships between convolutional decompositions and source separation methods such as ICA, with the aim of better exploiting the temporal structure for source localization.

The approach developed for locating sources and estimating their spatial extent, although freed from the difficult problem of estimating hyper-parameters, still relies on structural parameters linked to the construction of multi-scale dictionaries, which have a certain impact on the solution. Work in progress systematically studies this impact on the basis of validation metrics developed in the project, and will provide the user with more precise and well-argued recommendations. In the same spirit, the use of other types of multi-scale dictionaries, for example bases rather than overcomplete dictionaries, will certainly be useful for dealing with volumetric (rather than superficial) modeling of the cortex, for which the computational cost increases significantly.

We have shown that the source reconstruction results obtained can, to some extent, be validated on intracerebral data when such data are available. This, however, requires taking into account the highly irregular sampling of SEEG. The approach taken is to project the sources reconstructed from MEG onto the SEEG sensor space and to use carefully chosen comparison metrics. The initial results of the project, while promising, require further refinement, as does the methodology used. Once this is done, a next step will be to use such information directly in the inverse problem to improve its resolution.

Finally, the data that are beginning to be acquired on the new OPM sensors also represent a new challenge for these approaches.

Magneto and Electroencephalography (M/EEG) are brain imaging modalities used to measure the electrical activity of the brain. Thanks to their very high temporal resolutions, these are the only techniques that allow direct monitoring of neural activity over time. M/EEG brain imaging is used clinically for patients with epileptic seizures.

The EEG (electro-encephalography) and MEG (magneto-encephalography) source reconstruction problems are strongly ill-conditioned. Current approaches suffer from severe limitations, among which a poor spatial resolution, the absence of modeling of temporal dynamics, and the difficulty to fuse modalities. In addition, these only provide source estimates (or statistical maps from which activity is inferred), while a more complete probabilistic description of source activation would be desirable.
The main objective of BMWs is to develop and combine for the first time several state-of-the art techniques to estimate the fine spatio-temporal dynamics of brain networks with unprecedented precision and with quantification of uncertainties, and to implement the developed approach within MNE-Python, a common, open source, software platform.

Within BMWs, we will develop Bayesian frameworks for joint processing of EEG/MEG data within a common space-time formulation, and obtain not only quantitative spacetime source estimates but also activation probabilities and uncertainty measures for brain regions of interest. For that we will rely on state of the art models (Bernoulli-Gauss and variants) that have proven their efficiency in signal and image processing applications.
We will also combine wavelet expansions in time and space with sparse and group sparse priors to obtain better localized source estimates. While temporal wavelets are routinely used in EEG/MEG data analysis, the use of spatial wavelets, namely wavelets defined on the cortical surface (designed specifically for that purpose) is original and can be expected to improve the relevance of models.
Last, we will exploit combined surface (EEG and MEG) and intracerebral (SEEG) measurements. The latter will provide a ground truth of unprecedented precision, that will be used to compare prior models and if possible to constrain the EEG/MEG based reconstruction and therefore obtain improved source estimates. For validation we will benefit of unique recordings of MEG, EEG and SEEG performed in patients in presurgical evaluation of epilepsy.

The outcomes of the project will be made available to the neuroimaging community thanks to their integration in the open source software platform MNE-Python, and a challenge on space-time source reconstruction based upon joint EEG/MEG/SEEG data obtained within the project will be set up.