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).     

The coupled dipole model: an integrated model for multiple MEG/EEG data sets

Often MEG/EEG is measured in a few slightly different conditions to investigate the functionality of the human brain. This kind of data sets show similarities, though are different for each condition. When solving the inverse problem (IP), performing the source localization, one encounters the problem that this IP is ill-posed: constraints are necessary to solve and stabilize the solution to the IP. Moreover, a substantial amount of data is needed to avoid a signal to noise ratio (SNR) that is too poor for source localizations. In the case of similar conditions, this common information can be exploited by analyzing the data sets simultaneously. The here proposed coupled dipole model (CDM) provides an integrated method in which these similarities between conditions are used to solve and stabilize the inverse problem. The coupled dipole model is applicable when data sets contain common sources or common source time functions. The coupled dipole model uses a set of common sources and a set of common source time functions (STFs) to model all conditions in one single model. The data of each condition are mathematically described as a linear combination of these common spatial and common temporal components. This linear combination is specified in a coupling matrix for each data set. The coupled dipole model was applied in two simulation studies and in one experimental study. The simulations show that the errors in the estimated spatial and temporal parameters decrease compared to the standard separate analyses. A decrease in position error of a factor of 10 was shown for the localization of two nearby sources. In the experimental application, the coupled dipole model was shown to be necessary to obtain a plausible solution in at least 3 of 15 conditions investigated. Moreover, using the CDM, a direct comparison between parameters in different conditions is possible, whereas in separate models, the scaling of the amplitude parameters varies in general from data set to data set.     

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.     

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.     

A novel method for reliable and fast extraction of neuronal EEG/MEG oscillations on the basis of spatio-spectral decomposition

Neuronal oscillations have been shown to underlie various cognitive, perceptual and motor functions in the brain. However, studying these oscillations is notoriously difficult with EEG/MEG recordings due to a massive overlap of activity from multiple sources and also due to the strong background noise. Here we present a novel method for the reliable and fast extraction of neuronal oscillations from multi-channel EEG/MEG/LFP recordings. The method is based on a linear decomposition of recordings: it maximizes the signal power at a peak frequency while simultaneously minimizing it at the neighboring, surrounding frequency bins. Such procedure leads to the optimization of signal-to-noise ratio and allows extraction of components with a characteristic "peaky" spectral profile, which is typical for oscillatory processes. We refer to this method as spatio-spectral decomposition (SSD). Our simulations demonstrate that the method allows extraction of oscillatory signals even with a signal-to-noise ratio as low as 1:10. The SSD also outperformed conventional approaches based on independent component analysis. Using real EEG data we also show that SSD allows extraction of neuronal oscillations (e.g., in alpha frequency range) with high signal-to-noise ratio and with the spatial patterns corresponding to central and occipito-parietal sources. Importantly, running time for SSD is only a few milliseconds, which clearly distinguishes it from other extraction techniques usually requiring minutes or even hours of computational time. Due to the high accuracy and speed, we suggest that SSD can be used as a reliable method for the extraction of neuronal oscillations from multi-channel electrophysiological recordings.     

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.     

Maturation of obligatory auditory responses and their neural sources: evidence from EEG and MEG

Neural auditory responses are known to change from childhood to adulthood. The most prominent components of the event-related potentials (ERPs) found in children are the P1 and N2, while the P1 and N1 are strongest in adults. Previous dipole localizations showed regions of the auditory cortex (AC) underlying these responses. An N1 in children, however, has only been observed in older age or under certain experimental conditions different than commonly applied in adults. The current study aimed to further elucidate on auditory processing and related components in school-aged children. To do this, MEG and EEG was recorded in adults and 9 to 10year old children, while presenting pure tones either repetitively or randomly among tones of different pitch. Furthermore, the current paradigm was explicitly designed to not only investigate the P1 and N2 in children, but moreover to examine N1 modulations based on different refractory states caused by the two conditions. Our results are clear cut. In adults, P1(m) and N1(m) components were localized in AC regions, with the N1(m) largely attenuated for repetitive tones. The P1(m) and N2(m) components observed in children were also localized in AC regions. Most importantly, ERP modulations in the N1 time window (i.e., larger responses for random than repetitive tones) were remarkably similar for adults and children, both in amplitude and latency. This effect indicates that the N1 sub-component reflecting frequency-specific refractoriness is fully developed in 9 to 10year old children. Thus, previous interpretations on the function and maturation of the N1 need reconsideration.     

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.     

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.     

A novel integrated MEG and EEG analysis method for dipolar sources

The ability of magnetoencephalography (MEG) to accurately localize neuronal currents and obtain tangential components of the source is largely due to MEG's insensitivity to the conductivity profile of the head tissues. However, MEG cannot reliably detect the radial component of the neuronal current. In contrast, the localization accuracy of electroencephalography (EEG) is not as good as MEG, but EEG can detect both the tangential and radial components of the source. In the present study, we investigated the conductivity dependence in a new approach that combines MEG and EEG to accurately obtain, not only the location and tangential components, but also the radial component of the source. In this approach, the source location and tangential components are obtained from MEG alone, and optimal conductivity values of the EEG model are estimated by best-fitting EEG signal, while precisely matching the tangential components of the source in EEG and MEG. Then, the radial components are obtained from EEG using the previously estimated optimal conductivity values. Computer simulations testing this integrated approach demonstrated two main findings. First, there are well-organized optimal combinations of the conductivity values that provide an accurate fit to the combined MEG and EEG data. Second, the radial component, in addition to the location and tangential components, can be obtained with high accuracy without needing to know the precise conductivity profile of the head. We then demonstrated that this new approach performed reliably in an analysis of the 20-ms component from human somatosensory responses elicited by electric median-nerve stimulation.     

A mouse model for studying large-scale neuronal networks using EEG mapping techniques

Human functional imaging studies are increasingly focusing on the identification of large-scale neuronal networks, their temporal properties, their development, and their plasticity and recovery after brain lesions. A method targeting large-scale networks in rodents would open the possibility to investigate their neuronal and molecular basis in detail. We here present a method to study such networks in mice with minimal invasiveness, based on the simultaneous recording of epicranial EEG from 32 electrodes regularly distributed over the head surface. Spatiotemporal analysis of the electrical potential maps similar to human EEG imaging studies allows quantifying the dynamics of the global neuronal activation with sub-millisecond resolution. We tested the feasibility, stability and reproducibility of the method by recording the electrical activity evoked by mechanical stimulation of the mystacial vibrissae. We found a series of potential maps with different spatial configurations that suggested the activation of a large-scale network with generators in several somatosensory and motor areas of both hemispheres. The spatiotemporal activation pattern was stable both across mice and in the same mouse across time. We also performed 16-channel intracortical recordings of the local field potential across cortical layers in different brain areas and found tight spatiotemporal concordance with the generators estimated from the epicranial maps. Epicranial EEG mapping thus allows assessing sensory processing by large-scale neuronal networks in living mice with minimal invasiveness, complementing existing approaches to study the neurophysiological mechanisms of interaction within the network in detail and to characterize their developmental, experience-dependent and lesion-induced plasticity in normal and transgenic animals.     

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.     

Geometrical interpretation of fMRI-guided MEG/EEG inverse estimates

Magneto- and electroencephalography (MEG/EEG) and functional magnetic resonance imaging (fMRI) provide complementary information about the functional organization of the human brain. An important advantage of MEG/EEG is the millisecond time resolution in detecting electrical activity in the cerebral cortex. The interpretation of MEG/EEG signals, however, is limited by the difficulty of determining the spatial distribution of the neural activity. Functional MRI can help in the MEG/EEG source analysis by suggesting likely locations of activity. We present a geometric interpretation of fMRI-guided inverse solutions in which the MEG/EEG source estimate minimizes a distance to a subspace defined by the fMRI data. In this subspace regularization (SSR) approach, the fMRI bias does not assume preferred amplitudes for MEG/EEG sources, only locations. Characteristic dependence of the source estimates on the regularization parameters is illustrated with simulations. When the fMRI locations match the true MEG/EEG source locations, they serve to bias the underdetermined MEG/EEG inverse solution toward the fMRI loci. Importantly, when the fMRI loci do not match the true MEG/EEG loci, the solution is insensitive to those fMRI loci.     

Self-organised transients in a neural mass model of epileptogenic tissue dynamics

Stimulation of human epileptic tissue can induce rhythmic, self-terminating responses on the EEG or ECoG. These responses play a potentially important role in localising tissue involved in the generation of seizure activity, yet the underlying mechanisms are unknown. However, in vitro evidence suggests that self-terminating oscillations in nervous tissue are underpinned by non-trivial spatio-temporal dynamics in an excitable medium. In this study, we investigate this hypothesis in spatial extensions to a neural mass model for epileptiform dynamics.We demonstrate that spatial extensions to this model in one and two dimensions display propagating travelling waves but also more complex transient dynamics in response to local perturbations. The neural mass formulation with local excitatory and inhibitory circuits, allows the direct incorporation of spatially distributed, functional heterogeneities into the model. We show that such heterogeneities can lead to prolonged reverberating responses to a single pulse perturbation, depending upon the location at which the stimulus is delivered.This leads to the hypothesis that prolonged rhythmic responses to local stimulation in epileptogenic tissue result from repeated self-excitation of regions of tissue with diminished inhibitory capabilities. Combined with previous models of the dynamics of focal seizures this macroscopic framework is a first step towards an explicit spatial formulation of the concept of the epileptogenic zone. Ultimately, an improved understanding of the pathophysiologic mechanisms of the epileptogenic zone will help to improve diagnostic and therapeutic measures for treating epilepsy.     

Spatiotemporal dynamics of neural language processing: an MEG study using minimum-norm current estimates

The inferior frontal and superior temporal areas in the left hemisphere are well-known to be crucial for language processing in most right-handed individuals. This has been established by classical neurological investigations and neuropsychological studies along with metabolic brain imaging have recently revealed converging evidence. Here, we use fast neurophysiological brain imaging, magnetoencephalography (MEG), and L1 Minimum-Norm Current Estimates to investigate the time course of cortical activation underlying the magnetic Mismatch Negativity elicited by a spoken word. Left superior-temporal areas became active 136 ms after the information in the acoustic input was sufficient for identifying the word, and activation of the left inferior-frontal cortex followed after an additional delay of 22 ms. By providing answers to the where- and when-questions of cortical activation, MEG recordings paired with current estimates of the underlying cortical sources may advance our understanding of the spatiotemporal dynamics of distributed neuronal networks involved in cognitive processing in the human brain.     

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.     

Intermittent spike-wave dynamics in a heterogeneous, spatially extended neural mass model

Generalised epileptic seizures are frequently accompanied by sudden, reversible transitions from low amplitude, irregular background activity to high amplitude, regular spike-wave discharges (SWD) in the EEG. The underlying mechanisms responsible for SWD generation and for the apparently spontaneous transitions to SWD and back again are still not fully understood. Specifically, the role of spatial cortico-cortical interactions in ictogenesis is not well studied. We present a macroscopic, neural mass model of a cortical column which includes two distinct time scales of inhibition. This model can produce both an oscillatory background and a pathological SWD rhythm. We demonstrate that coupling two of these cortical columns can lead to a bistability between out-of-phase, low amplitude background dynamics and in-phase, high amplitude SWD activity. Stimuli can cause state-dependent transitions from background into SWD. In an extended local area of cortex, spatial heterogeneities in a model parameter can lead to spontaneous reversible transitions from a desynchronised background to synchronous SWD due to intermittency. The deterministic model is therefore capable of producing absence seizure-like events without any time dependent adjustment of model parameters. The emergence of such mechanisms due to spatial coupling demonstrates the importance of spatial interactions in modelling ictal dynamics, and in the study of ictogenesis.     

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.