A distributed spatio-temporal EEG/MEG inverse solver

We propose a novel l(1)l(2)-norm inverse solver for estimating the sources of EEG/MEG signals. Based on the standard l(1)-norm inverse solvers, this sparse distributed inverse solver integrates the l(1)-norm spatial model with a temporal model of the source signals in order to avoid unstable activation patterns and "spiky" reconstructed signals often produced by the currently used sparse solvers. The joint spatio-temporal model leads to a cost function with an l(1)l(2)-norm regularizer whose minimization can be reduced to a convex second-order cone programming (SOCP) problem and efficiently solved using the interior-point method. The efficient computation of the SOCP problem allows us to implement permutation tests for estimating statistical significance of the inverse solution. Validation with simulated and human MEG data shows that the proposed solver yields source time course estimates qualitatively similar to those obtained through dipole fitting, but without the need to specify the number of dipole sources in advance. Furthermore, the l(1)l(2)-norm solver achieves fewer false positives and a better representation of the source locations than the conventional l(2) minimum-norm estimates.     

Time-coherent expansion of MEG/EEG cortical sources

In this study we estimated the spatial extent of cortical areas of time-coherent activity using the inverse problem in magneto/electroencephalography (MEEG). The model discussed here uses classical regularization tools in order to force the inverse solution to be piecewise coherent. First, the cortex was seeded by focal dipolar sources. Then, a time-coherent expansion (TCE) onto the cortical surface was performed in order to obtain surface source models composed of patches with uniform current density. Patches represent extended cortical regions with one single time course per active area. Results obtained from synthetic data show that using the TCE method is relevant even with a low signal-to-noise ratio, although the final estimation is often slightly biased. We applied the TCE method to evoked magnetic fields obtained after electrical stimulation of fingers in order to estimate the somatotopic cortical maps of the primary somatosensory cortex.     

Bayesian model averaging in EEG/MEG imaging

In this paper, the Bayesian Theory is used to formulate the Inverse Problem (IP) of the EEG/MEG. This formulation offers a comparison framework for the wide range of inverse methods available and allows us to address the problem of model uncertainty that arises when dealing with different solutions for a single data. In this case, each model is defined by the set of assumptions of the inverse method used, as well as by the functional dependence between the data and the Primary Current Density (PCD) inside the brain. The key point is that the Bayesian Theory not only provides for posterior estimates of the parameters of interest (the PCD) for a given model, but also gives the possibility of finding posterior expected utilities unconditional on the models assumed. In the present work, this is achieved by considering a third level of inference that has been systematically omitted by previous Bayesian formulations of the IP. This level is known as Bayesian model averaging (BMA). The new approach is illustrated in the case of considering different anatomical constraints for solving the IP of the EEG in the frequency domain. This methodology allows us to address two of the main problems that affect linear inverse solutions (LIS): (a) the existence of ghost sources and (b) the tendency to underestimate deep activity. Both simulated and real experimental data are used to demonstrate the capabilities of the BMA approach, and some of the results are compared with the solutions obtained using the popular low-resolution electromagnetic tomography (LORETA) and its anatomically constraint version (cLORETA).     

Random field-union intersection tests for EEG/MEG imaging

Electrophysiological (EEG/MEG) imaging challenges statistics by providing two views of the same spatiotemporal data: topographic and tomographic. Until now, statistical tests for these two situations have developed separately. This work introduces statistical tests for assessing simultaneously the significance of spatiotemporal event-related potential/event-related field (ERP/ERF) components and that of their sources. The test for detecting a component at a given time instant is provided by a Hotelling's T(2) statistic. This statistic is constructed in such a manner to be invariant to any choice of reference and is based upon a generalized version of the average reference transform of the data. As a consequence, the proposed test is a generalization of the well-known Global Field Power statistic. Consideration of tests at all time instants leads to a multiple comparison problem addressed by the use of Random Field Theory (RFT). The Union-Intersection (UI) principle is the basis for testing hypotheses about the topographic and tomographic distributions of such ERP/ERF components. The performance of the method is illustrated with actual EEG recordings obtained from a visual experiment of pattern reversal stimuli.     

Mechanisms of evoked and induced responses in MEG/EEG

Cortical responses, recorded by electroencephalography and magnetoencephalography, can be characterized in the time domain, to study event-related potentials/fields, or in the time-frequency domain, to study oscillatory activity. In the literature, there is a common conception that evoked, induced, and on-going oscillations reflect different neuronal processes and mechanisms. In this work, we consider the relationship between the mechanisms generating neuronal transients and how they are expressed in terms of evoked and induced power. This relationship is addressed using a neuronally realistic model of interacting neuronal subpopulations. Neuronal transients were generated by changing neuronal input (a dynamic mechanism) or by perturbing the systems coupling parameters (a structural mechanism) to produce induced responses. By applying conventional time-frequency analyses, we show that, in contradistinction to common conceptions, induced and evoked oscillations are perhaps more related than previously reported. Specifically, structural mechanisms normally associated with induced responses can be expressed in evoked power. Conversely, dynamic mechanisms posited for evoked responses can induce responses, if there is variation in neuronal input. We conclude, it may be better to consider evoked responses as the results of mixed dynamic and structural effects. We introduce adjusted power to complement induced power. Adjusted power is unaffected by trial-to-trial variations in input and can be attributed to structural perturbations without ambiguity.     

Lexical influences on speech perception: a Granger causality analysis of MEG and EEG source estimates

Behavioral and functional imaging studies have demonstrated that lexical knowledge influences the categorization of perceptually ambiguous speech sounds. However, methodological and inferential constraints have so far been unable to resolve the question of whether this interaction takes the form of direct top-down influences on perceptual processing, or feedforward convergence during a decision process. We examined top-down lexical influences on the categorization of segments in a /s/-/integral/ continuum presented in different lexical contexts to produce a robust Ganong effect. Using integrated MEG/EEG and MRI data we found that, within a network identified by 40 Hz gamma phase locking, activation in the supramarginal gyrus associated with wordform representation influences phonetic processing in the posterior superior temporal gyrus during a period of time associated with lexical processing. This result provides direct evidence that lexical processes influence lower level phonetic perception, and demonstrates the potential value of combining Granger causality analyses and high spatiotemporal resolution multimodal imaging data to explore the functional architecture of cognition.     

A unified Bayesian framework for MEG/EEG source imaging

The ill-posed nature of the MEG (or related EEG) source localization problem requires the incorporation of prior assumptions when choosing an appropriate solution out of an infinite set of candidates. Bayesian approaches are useful in this capacity because they allow these assumptions to be explicitly quantified using postulated prior distributions. However, the means by which these priors are chosen, as well as the estimation and inference procedures that are subsequently adopted to affect localization, have led to a daunting array of algorithms with seemingly very different properties and assumptions. From the vantage point of a simple Gaussian scale mixture model with flexible covariance components, this paper analyzes and extends several broad categories of Bayesian inference directly applicable to source localization including empirical Bayesian approaches, standard MAP estimation, and multiple variational Bayesian (VB) approximations. Theoretical properties related to convergence, global and local minima, and localization bias are analyzed and fast algorithms are derived that improve upon existing methods. This perspective leads to explicit connections between many established algorithms and suggests natural extensions for handling unknown dipole orientations, extended source configurations, correlated sources, temporal smoothness, and computational expediency. Specific imaging methods elucidated under this paradigm include the weighted minimum l(2)-norm, FOCUSS, minimum current estimation, VESTAL, sLORETA, restricted maximum likelihood, covariance component estimation, beamforming, variational Bayes, the Laplace approximation, and automatic relevance determination, as well as many others. Perhaps surprisingly, all of these methods can be formulated as particular cases of covariance component estimation using different concave regularization terms and optimization rules, making general theoretical analyses and algorithmic extensions/improvements particularly relevant.     

Hierarchical Bayesian estimation for MEG inverse problem

Source current estimation from MEG measurement is an ill-posed problem that requires prior assumptions about brain activity and an efficient estimation algorithm. In this article, we propose a new hierarchical Bayesian method introducing a hierarchical prior that can effectively incorporate both structural and functional MRI data. In our method, the variance of the source current at each source location is considered an unknown parameter and estimated from the observed MEG data and prior information by using the Variational Bayesian method. The fMRI information can be imposed as prior information on the variance distribution rather than the variance itself so that it gives a soft constraint on the variance. A spatial smoothness constraint, that the neural activity within a few millimeter radius tends to be similar due to the neural connections, can also be implemented as a hierarchical prior. The proposed method provides a unified theory to deal with the following three situations: (1) MEG with no other data, (2) MEG with structural MRI data on cortical surfaces, and (3) MEG with both structural MRI and fMRI data. We investigated the performance of our method and conventional linear inverse methods under these three conditions. Simulation results indicate that our method has better accuracy and spatial resolution than the conventional linear inverse methods under all three conditions. It is also shown that accuracy of our method improves as MRI and fMRI information becomes available. Simulation results demonstrate that our method appropriately resolves the inverse problem even if fMRI data convey inaccurate information, while the Wiener filter method is seriously deteriorated by inaccurate fMRI information.     

Combining sparsity and rotational invariance in EEG/MEG source reconstruction

We introduce Focal Vector Field Reconstruction (FVR), a novel technique for the inverse imaging of vector fields. The method was designed to simultaneously achieve two goals: a) invariance with respect to the orientation of the coordinate system, and b) a preference for sparsity of the solutions and their spatial derivatives. This was achieved by defining the regulating penalty function, which renders the solutions unique, as a global l(1)-norm of local l(2)-norms. We show that the method can be successfully used for solving the EEG inverse problem. In the joint localization of 2-3 simulated dipoles, FVR always reliably recovers the true sources. The competing methods have limitations in distinguishing close sources because their estimates are either too smooth (LORETA, Minimum l(1)-norm) or too scattered (Minimum l(2)-norm). In both noiseless and noisy simulations, FVR has the smallest localization error according to the Earth Mover's Distance (EMD), which is introduced here as a meaningful measure to compare arbitrary source distributions. We also apply the method to the simultaneous localization of left and right somatosensory N20 generators from real EEG recordings. Compared to its peers FVR was the only method that delivered correct location of the source in the somatosensory area of each hemisphere in accordance with neurophysiological prior knowledge.     

Independent component analysis of short-time Fourier transforms for spontaneous EEG/MEG analysis

Analysis of spontaneous EEG/MEG needs unsupervised learning methods. While independent component analysis (ICA) has been successfully applied on spontaneous fMRI, it seems to be too sensitive to technical artifacts in EEG/MEG. We propose to apply ICA on short-time Fourier transforms of EEG/MEG signals, in order to find more “interesting” sources than with time-domain ICA, and to more meaningfully sort the obtained components. The method is especially useful for finding sources of rhythmic activity. Furthermore, we propose to use a complex mixing matrix to model sources which are spatially extended and have different phases in different EEG/MEG channels. Simulations with artificial data and experiments on resting-state MEG demonstrate the utility of the method.     

Localization of extended brain sources from EEG/MEG: the ExSo-MUSIC approach

We propose a new MUSIC-like method, called 2q-ExSo-MUSIC (q ≥ 1). This method is an extension of the 2q-MUSIC (q ≥ 1) approach for solving the EEG/MEG inverse problem, when spatially-extended neocortical sources ("ExSo") are considered. It introduces a novel ExSo-MUSIC principle. The novelty is two-fold: i) the parameterization of the spatial source distribution that leads to an appropriate metric in the context of distributed brain sources and ii) the introduction of an original, efficient and low-cost way of optimizing this metric. In 2q-ExSo-MUSIC, the possible use of higher order statistics (q ≥ 2) offers a better robustness with respect to Gaussian noise of unknown spatial coherence and modeling errors. As a result we reduced the penalizing effects of both the background cerebral activity that can be seen as a Gaussian and spatially correlated noise, and the modeling errors induced by the non-exact resolution of the forward problem. Computer results on simulated EEG signals obtained with physiologically-relevant models of both the sources and the volume conductor show a highly increased performance of our 2q-ExSo-MUSIC method as compared to the classical 2q-MUSIC algorithms.     

Variational Bayesian inversion of the equivalent current dipole model in EEG/MEG

In magneto- and electroencephalography (M/EEG), spatial modelling of sensor data is necessary to make inferences about underlying brain activity. Most source reconstruction techniques belong to one of two approaches: point source models, which explain the data with a small number of equivalent current dipoles and distributed source or imaging models, which use thousands of dipoles. Much methodological research has been devoted to developing sophisticated Bayesian source imaging inversion schemes, while dipoles have received less such attention. Dipole models have their advantages; they are often appropriate summaries of evoked responses or helpful first approximations. Here, we propose a variational Bayesian algorithm that enables the fast Bayesian inversion of dipole models. The approach allows for specification of priors on all the model parameters. The posterior distributions can be used to form Bayesian confidence intervals for interesting parameters, like dipole locations. Furthermore, competing models (e.g., models with different numbers of dipoles) can be compared using their evidence or marginal likelihood. Using synthetic data, we found the scheme provides accurate dipole localizations. We illustrate the advantage of our Bayesian scheme, using a multi-subject EEG auditory study, where we compare competing models for the generation of the N100 component.     

A verifiable solution to the MEG inverse problem

Magnetoencephalography (MEG) is a non-invasive brain imaging technique with the potential for very high temporal and spatial resolution of neuronal activity. The main stumbling block for the technique has been that the estimation of a neuronal current distribution, based on sensor data outside the head, is an inverse problem with an infinity of possible solutions. Many inversion techniques exist, all using different a-priori assumptions in order to reduce the number of possible solutions. Although all techniques can be thoroughly tested in simulation, implicit in the simulations are the experimenter's own assumptions about realistic brain function. To date, the only way to test the validity of inversions based on real MEG data has been through direct surgical validation, or through comparison with invasive primate data. In this work, we constructed a null hypothesis that the reconstruction of neuronal activity contains no information on the distribution of the cortical grey matter. To test this, we repeatedly compared rotated sections of grey matter with a beamformer estimate of neuronal activity to generate a distribution of mutual information values. The significance of the comparison between the un-rotated anatomical information and the electrical estimate was subsequently assessed against this distribution. We found that there was significant (P < 0.05) anatomical information contained in the beamformer images across a number of frequency bands. Based on the limited data presented here, we can say that the assumptions behind the beamformer algorithm are not unreasonable for the visual-motor task investigated.     

A neural mass model for MEG/EEG: coupling and neuronal dynamics

Although MEG/EEG signals are highly variable, systematic changes in distinct frequency bands are commonly encountered. These frequency-specific changes represent robust neural correlates of cognitive or perceptual processes (for example, alpha rhythms emerge on closing the eyes). However, their functional significance remains a matter of debate. Some of the mechanisms that generate these signals are known at the cellular level and rest on a balance of excitatory and inhibitory interactions within and between populations of neurons. The kinetics of the ensuing population dynamics determine the frequency of oscillations. In this work we extended the classical nonlinear lumped-parameter model of alpha rhythms, initially developed by Lopes da Silva and colleagues [Kybernetik 15 (1974) 27], to generate more complex dynamics. We show that the whole spectrum of MEG/EEG signals can be reproduced within the oscillatory regime of this model by simply changing the population kinetics. We used the model to examine the influence of coupling strength and propagation delay on the rhythms generated by coupled cortical areas. The main findings were that (1) coupling induces phase-locked activity, with a phase shift of 0 or pi when the coupling is bidirectional, and (2) both coupling and propagation delay are critical determinants of the MEG/EEG spectrum. In forthcoming articles, we will use this model to (1) estimate how neuronal interactions are expressed in MEG/EEG oscillations and establish the construct validity of various indices of nonlinear coupling, and (2) generate event-related transients to derive physiologically informed basis functions for statistical modelling of average evoked responses.     

Multiple sparse priors for the M/EEG inverse problem

This paper describes an application of hierarchical or empirical Bayes to the distributed source reconstruction problem in electro- and magnetoencephalography (EEG and MEG). The key contribution is the automatic selection of multiple cortical sources with compact spatial support that are specified in terms of empirical priors. This obviates the need to use priors with a specific form (e.g., smoothness or minimum norm) or with spatial structure (e.g., priors based on depth constraints or functional magnetic resonance imaging results). Furthermore, the inversion scheme allows for a sparse solution for distributed sources, of the sort enforced by equivalent current dipole (ECD) models. This means the approach automatically selects either a sparse or a distributed model, depending on the data. The scheme is compared with conventional applications of Bayesian solutions to quantify the improvement in performance.     

Dynamic causal modelling of evoked responses in EEG/MEG with lead field parameterization

Dynamical causal modeling (DCM) of evoked responses is a new approach to making inferences about connectivity changes in hierarchical networks measured with electro- and magnetoencephalography (EEG and MEG). In a previous paper, we illustrated this concept using a lead field that was specified with infinite prior precision. With this prior, the spatial expression of each source area, in the sensors, is fixed. In this paper, we show that using lead field parameters with finite precision enables the data to inform the network's spatial configuration and its expression at the sensors. This means that lead field and coupling parameters can be estimated simultaneously. Alternatively, one can also view DCM for evoked responses as a source reconstruction approach with temporal, physiologically informed constraints. We will illustrate this idea using, for each area, a 4-shell equivalent current dipole (ECD) model with three location and three orientation parameters. Using synthetic and real data, we show that this approach furnishes accurate and robust conditional estimates of coupling among sources and their orientations.     

Grouping of MEG gamma oscillations by EEG sleep spindles

Studies have revealed an association between EEG sleep spindles and processing of memories during sleep. Here we investigated whether there is a temporal relation between sleep spindles and MEG oscillatory activity in the gamma frequency band (> 30 Hz) which is considered to reflect local cortical processing of memory representations. MEG and simultaneous EEG (at Cz) were obtained in subjects during sleep together with standard polysomnography. As expected EEG spindles were correlated with power increases in MEG spindle (12.5-15.5 Hz) power mainly over prefrontal and occipital cortical areas. During EEG spindles we revealed both transient significant increases and decreases in MEG power, with decreases occurring significantly more often than increases. The modulations in gamma power occurred mainly at sites of increased MEG spindle power, and more often during peaks than troughs within the EEG spindle cycle. Cross-frequency coherence analyses confirmed a strong phase-coupling of gamma band activity with the spindle rhythm. The findings are consistent with the idea that spindles provide a fine-tuned temporal frame for integrated cortical memory processing during sleep.     

Simultaneous MEG and intracranial EEG recordings during attentive reading

The relationship between neural oscillations recorded at various spatial scales remains poorly understood partly due to an overall dearth of studies utilizing simultaneous measurements. In an effort to study quantitative markers of attention during reading, we performed simultaneous magnetoencephalography (MEG) and intracranial electroencephalography (iEEG) recordings in four epileptic patients. Patients were asked to attend to a specific color when presented with an intermixed series of red words and green words, with words of a given color forming a cohesive story. We analyzed alpha, beta, and gamma band oscillatory responses to the word presentation and compared the strength and spatial organization of those responses in both electrophysiological recordings. Time-frequency analysis of iEEG revealed a network of clear attention-modulated high gamma band (50–150 Hz) power increases and alpha/beta (9–25 Hz) suppressions in response to the words. In addition to analyses at the sensor level, MEG time-frequency analysis was performed at the source level using a sliding window beamformer technique. Strong alpha/beta suppressions were observed in MEG reconstructions, in tandem with iEEG effects. While the MEG counterpart of high gamma band enhancement was difficult to interpret at the sensor level in two patients, MEG time-frequency source reconstruction revealed additional activation patterns in accordance with iEEG results. Importantly, iEEG allowed us to confirm that several sources of gamma band modulation observed with MEG were indeed of cortical origin rather than EMG muscular or ocular artifact.     

Bayesian brain source imaging based on combined MEG/EEG and fMRI using MCMC

A number of brain imaging techniques have been developed in order to investigate brain function and to develop diagnostic tools for various brain disorders. Each modality has strengths as well as weaknesses compared to the others. Recent work has explored how multiple modalities can be integrated effectively so that they complement one another while maintaining their individual strengths. Bayesian inference employing Markov Chain Monte Carlo (MCMC) techniques provides a straightforward way to combine disparate forms of information while dealing with the uncertainty in each. In this paper we introduce methods of Bayesian inference as a way to integrate different forms of brain imaging data in a probabilistic framework. We formulate Bayesian integration of magnetoencephalography (MEG) data and functional magnetic resonance imaging (fMRI) data by incorporating fMRI data into a spatial prior. The usefulness and feasibility of the method are verified through testing with both simulated and empirical data.