Spatiotemporal EEG/MEG source analysis based on a parametric noise covariance model

Joint Learning of Full-Structure Noise in Hierarchical Bayesian Regression Models

We consider the reconstruction of brain activity from electroencephalography (EEG). This inverse problem can be formulated as a linear regression with independent Gaussian scale mixture priors for both the source and noise components. Crucial factors influencing the accuracy of the source estimation are not only the noise level but also its correlation structure, but existing approaches have not addressed the estimation of noise covariance matrices with full structure. To address this shortcoming, we develop hierarchical Bayesian (type-II maximum likelihood) models for observations with latent variables for source and noise, which are estimated jointly from data. As an extension to classical sparse Bayesian learning (SBL), where across-sensor observations are assumed to be independent and identically distributed, we consider Gaussian noise with full covariance structure. Using the majorization-maximization framework and Riemannian geometry, we derive an efficient algorithm for updating the noise covariance along the manifold of positive definite matrices. We demonstrate that our algorithm has guaranteed and fast convergence and validate it in simulations and with real MEG data. Our results demonstrate that the novel framework significantly improves upon state-of-the-art techniques in the real-world scenario where the noise is indeed non-diagonal and full-structured. Our method has applications in many domains beyond biomagnetic inverse problems.

Bayesian Algorithms for Joint Estimation of Brain Activity and Noise in Electromagnetic Imaging

Simultaneously estimating brain source activity and noise has long been a challenging task in electromagnetic brain imaging using magneto- and electroencephalography. The problem is challenging not only in terms of solving the NP-hard inverse problem of reconstructing unknown brain activity across thousands of voxels from a limited number of sensors, but also for the need to simultaneously estimate the noise and interference. We present a generative model with an augmented leadfield matrix to simultaneously estimate brain source activity and sensor noise statistics in electromagnetic brain imaging (EBI). We then derive three Bayesian inference algorithms for this generative model (expectation-maximization (EBI-EM), convex bounding (EBI-Convex) and fixed-point (EBI-Mackay)) to simultaneously estimate the hyperparameters of the prior distribution for brain source activity and sensor noise. A comprehensive performance evaluation for these three algorithms is performed. Simulations consistently show that the performance of EBI-Convex and EBI-Mackay updates is superior to that of EBI-EM. In contrast to the EBI-EM algorithm, both EBI-Convex and EBI-Mackay updates are quite robust to initialization, and are computationally efficient with fast convergence in the presence of both Gaussian and real brain noise. We also demonstrate that EBI-Convex and EBI-Mackay update algorithms can reconstruct complex brain activity with only a few trials of sensor data, and for resting-state data, achieving significant improvement in source reconstruction and noise learning for electromagnetic brain imaging.

The IAS-MEEG Package: A Flexible Inverse Source Reconstruction Platform for Reconstruction and Visualization of Brain Activity from M/EEG Data

We present a standalone Matlab software platform complete with visualization for the reconstruction of the neural activity in the brain from MEG or EEG data. The underlying inversion combines hierarchical Bayesian models and Krylov subspace iterative least squares solvers. The Bayesian framework of the underlying inversion algorithm allows to account for anatomical information and possible a priori belief about the focality of the reconstruction. The computational efficiency makes the software suitable for the reconstruction of lengthy time series on standard computing equipment. The algorithm requires minimal user provided input parameters, although the user can express the desired focality and accuracy of the solution. The code has been designed so as to favor the parallelization performed automatically by Matlab, according to the resources of the host computer. We demonstrate the flexibility of the platform by reconstructing activity patterns with supports of different sizes from MEG and EEG data. Moreover, we show that the software reconstructs well activity patches located either in the subcortical brain structures or on the cortex. The inverse solver and visualization modules can be used either individually or in combination. We also provide a version of the inverse solver that can be used within Brainstorm toolbox. All the software is available online by Github, including the Brainstorm plugin, with accompanying documentation and test data.

Introducing block-Toeplitz covariance matrices to remaster linear discriminant analysis for event-related potential brain-computer interfaces

Objective.Covariance matrices of noisy multichannel electroencephalogram (EEG) time series data provide essential information for the decoding of brain signals using machine learning methods. However, small datasets and high dimensionality make it hard to estimate these matrices. In brain-computer interfaces (BCI) based on event-related potentials (ERP) and a linear discriminant analysis (LDA) classifier, the state of the art covariance estimation uses shrinkage regularization. As this is a general covariance regularization approach, we aim at improving LDA further by better exploiting the domain-specific characteristics of the EEG to regularize the covariance estimates.Approach.We propose to enforce a block-Toeplitz structure for the covariance matrix of the LDA, which implements an assumption of signal stationarity in short time windows for each channel.Main results.An offline re-analysis of data collected from 213 subjects under 13 different event-related potential BCI protocols showed a significantly increased binary classification performance of this 'ToeplitzLDA' compared to shrinkage regularized LDA (up to 6 AUC points,p < 0.001) and Riemannian classification approaches (up to 2 AUC points,p < 0.001). In an unsupervised visual speller application, this improvement would translate to a relative reduction of spelling errors by 81% on average for 25 subjects. Additionally, aside from lower memory and reduced time complexity for LDA training, ToeplitzLDA proves to be robust against drastic increases of the number of temporal features.Significance.The proposed covariance estimation allows BCI researchers to improve classification rates and reduce calibration times of BCI protocols using event-related potentials and thus support the usability of corresponding applications. Its lower computational and memory needs could make it a valuable algorithm especially for mobile BCIs.

Multiple Source Detection based on Spatial Clustering and Its Applications on Wearable OPM-MEG

OBJECTIVE

Magnetoencephalography (MEG) is a non-invasive technique that measures the magnetic fields of brain activity. In particular, a new type of optically pumped magnetometer (OPM)-based wearable MEG system has been developed in recent years. Source localization in MEG can provide signals and locations of brain activity. However, conventional source localization methods face the difficulty of accurately estimating multiple sources. The present study presented a new parametric method to estimate the number of sources and localize multiple sources. In addition, we applied the proposed method to a constructed wearable OPM-MEG system.

METHODS

We used spatial clustering of the dipole spatial distribution to detect sources. The spatial distribution of dipoles was obtained by segmenting the MEG data temporally into slices and then estimating the parameters of the dipoles on each data slice using the particle swarm optimization algorithm. Spatial clustering was performed using the spatial-temporal density-based spatial clustering of applications with a noise algorithm. The performance of our approach for detecting multiple sources was compared with that of four typical benchmark algorithms using the OPM-MEG sensor configuration.

RESULTS

The simulation results showed that the proposed method had the best performance for detecting multiple sources. Moreover, the effectiveness of the method was verified by a multimodel sensory stimuli experiment on a real constructed 31-channel OPM-MEG.

CONCLUSION

Our study provides an effective method for the detection of multiple sources.

SIGNIFICANCE

With the improvement of the source localization methods, MEG may have a wider range of applications in neuroscience and clinical research.

Classification of Event-Related Potentials with Regularized Spatiotemporal LCMV Beamforming

The usability of EEG-based visual brain–computer interfaces (BCIs) based on event-related potentials (ERPs) benefits from reducing the calibration time before BCI operation. Linear decoding models, such as the spatiotemporal beamformer model, yield state-of-the-art accuracy. Although the training time of this model is generally low, it can require a substantial amount of training data to reach functional performance. Hence, BCI calibration sessions should be sufficiently long to provide enough training data. This work introduces two regularized estimators for the beamformer weights. The first estimator uses cross-validated L2-regularization. The second estimator exploits prior information about the structure of the EEG by assuming Kronecker–Toeplitz-structured covariance. The performances of these estimators are validated and compared with the original spatiotemporal beamformer and a Riemannian-geometry-based decoder using a BCI dataset with P300-paradigm recordings for 21 subjects. Our results show that the introduced estimators are well-conditioned in the presence of limited training data and improve ERP classification accuracy for unseen data. Additionally, we show that structured regularization results in lower training times and memory usage, and a more interpretable classification model.

Equivalent current dipole sources of neurofeedback training-induced alpha activity through temporal/spectral analytic techniques

Much of the work in alpha NFT has focused on evaluating changes in alpha amplitude. However, the generation mechanism of training-induced alpha activity has not yet been clarified. The present study aimed to identify sources of training-induced alpha activity through four temporal/spectral analytic techniques, i.e., the max peak average (MPA), positive average (PA), negative average (NA) and event-related spectral perturbation average (ERSPA) methods. Thirty-five healthy participants were recruited into an alpha group receiving feedback of 8–12-Hz amplitudes, and twenty-eight healthy participants were recruited into a control group receiving feedback of random 4-Hz amplitudes from the range of 7 to 20 Hz. Twelve sessions were performed within 4 weeks (3 sessions per week). The control group had no change in the amplitude spectrum. In contrast, twenty-nine participants in the alpha group showed significant alpha amplitude increases exclusively and were identified as “responders”. A whole-head EEG was recorded for the “responders” after NFT. The epochs of training-induced alpha activity from whole-head EEG were averaged by four different methods for equivalent current dipole source analysis. High agreement and Cohen’s kappa coefficients on dipole source localization between each method were observed, showing that the dipole clusters of training-induced alpha activity were consistently located in the precuneus, posterior cingulate cortex (PCC) and middle temporal gyrus. The residual variance (goodness of fit) for dipole estimation of the MPA was significantly smaller than that of the others. Our findings indicate that the precuneus, PCC and middle temporal gyrus play important roles in enhancing training-induced alpha activity. The four averaging methods (especially the MPA method) were suitable for investigating sources of brainwaves. Additionally, three dipoles can be used for dipole source analysis of training-induced alpha activity in future research, especially the training sites are around the central regions.

Unification of sparse Bayesian learning algorithms for electromagnetic brain imaging with the majorization minimization framework

Methods for electro- or magnetoencephalography (EEG/MEG) based brain source imaging (BSI) using sparse Bayesian learning (SBL) have been demonstrated to achieve excellent performance in situations with low numbers of distinct active sources, such as event-related designs. This paper extends the theory and practice of SBL in three important ways. First, we reformulate three existing SBL algorithms under the majorization-minimization (MM) framework. This unification perspective not only provides a useful theoretical framework for comparing different algorithms in terms of their convergence behavior, but also provides a principled recipe for constructing novel algorithms with specific properties by designing appropriate bounds of the Bayesian marginal likelihood function. Second, building on the MM principle, we propose a novel method called LowSNR-BSI that achieves favorable source reconstruction performance in low signal-to-noise-ratio (SNR) settings. Third, precise knowledge of the noise level is a crucial requirement for accurate source reconstruction. Here we present a novel principled technique to accurately learn the noise variance from the data either jointly within the source reconstruction procedure or using one of two proposed cross-validation strategies. Empirically, we could show that the monotonous convergence behavior predicted from MM theory is confirmed in numerical experiments. Using simulations, we further demonstrate the advantage of LowSNR-BSI over conventional SBL in low-SNR regimes, and the advantage of learned noise levels over estimates derived from baseline data. To demonstrate the usefulness of our novel approach, we show neurophysiologically plausible source reconstructions on averaged auditory evoked potential data.

Neural silences can be localized rapidly using noninvasive scalp EEG

A rapid and cost-effective noninvasive tool to detect and characterize neural silences can be of important benefit in diagnosing and treating many disorders. We propose an algorithm, SilenceMap, for uncovering the absence of electrophysiological signals, or neural silences, using noninvasive scalp electroencephalography (EEG) signals. By accounting for the contributions of different sources to the power of the recorded signals, and using a hemispheric baseline approach and a convex spectral clustering framework, SilenceMap permits rapid detection and localization of regions of silence in the brain using a relatively small amount of EEG data. SilenceMap substantially outperformed existing source localization algorithms in estimating the center-of-mass of the silence for three pediatric cortical resection patients, using fewer than 3 minutes of EEG recordings (13, 2, and 11mm vs. 25, 62, and 53 mm), as well for 100 different simulated regions of silence based on a real human head model (12 ± 0.7 mm vs. 54 ± 2.2 mm). SilenceMap paves the way towards accessible early diagnosis and continuous monitoring of altered physiological properties of human cortical function.

Robust estimation of noise for electromagnetic brain imaging with the champagne algorithm

Robust estimation of the number, location, and activity of multiple correlated brain sources has long been a challenging task in electromagnetic brain imaging from M/EEG data, one that is significantly impacted by interference from spontaneous brain activity, sensor noise, and other sources of artifacts. Recently, we introduced the Champagne algorithm, a novel Bayesian inference algorithm that has shown tremendous success in M/EEG source reconstruction. Inherent to Champagne and most other related Bayesian reconstruction algorithms is the assumption that the noise covariance in sensor data can be estimated from “baseline” or “control” measurements. However, in many scenarios, such baseline data is not available, or is unreliable, and it is unclear how best to estimate the noise covariance. In this technical note, we propose several robust methods to estimate the contributions to sensors from noise arising from outside the brain without the need for additional baseline measurements. The incorporation of these methods for diagonal noise covariance estimation improves the robust reconstruction of complex brain source activity under high levels of noise and interference, while maintaining the performance features of Champagne. Specifically, we show that the resulting algorithm, Champagne with noise learning, is quite robust to initialization and is computationally efficient. In simulations, performance of the proposed noise learning algorithm is consistently superior to Champagne without noise learning. We also demonstrate that, even without the use of any baseline data, Champagne with noise learning is able to reconstruct complex brain activity with just a few trials or even a single trial, demonstrating significant improvements in source reconstruction for electromagnetic brain imaging.

Brain Activity Mapping from MEG Data via a Hierarchical Bayesian Algorithm with Automatic Depth Weighting

A recently proposed iterated alternating sequential (IAS) MEG inverse solver algorithm, based on the coupling of a hierarchical Bayesian model with computationally efficient Krylov subspace linear solver, has been shown to perform well for both superficial and deep brain sources. However, a systematic study of its ability to correctly identify active brain regions is still missing. We propose novel statistical protocols to quantify the performance of MEG inverse solvers, focusing in particular on how their accuracy and precision at identifying active brain regions. We use these protocols for a systematic study of the performance of the IAS MEG inverse solver, comparing it with three standard inversion methods, wMNE, dSPM, and sLORETA. To avoid the bias of anecdotal tests towards a particular algorithm, the proposed protocols are Monte Carlo sampling based, generating an ensemble of activity patches in each brain region identified in a given atlas. The performance in correctly identifying the active areas is measured by how much, on average, the reconstructed activity is concentrated in the brain region of the simulated active patch. The analysis is based on Bayes factors, interpreting the estimated current activity as data for testing the hypothesis that the active brain region is correctly identified, versus the hypothesis of any erroneous attribution. The methodology allows the presence of a single or several simultaneous activity regions, without assuming that the number of active regions is known. The testing protocols suggest that the IAS solver performs well with both with cortical and subcortical activity estimation.

Optimal design for parameter estimation in EEG problems in a 3D multilayered domain

The fundamental problem of collecting data in the ``best way'' in order to assure statistically efficient estimation of parameters is known as Optimal Experimental Design. Many inverse problems consist in selecting best parameter values of a given mathematical model based on fits to measured data. These are usually formulated as optimization problems and the accuracy of their solutions depends not only on the chosen optimization scheme but also on the given data. We consider an electromagnetic interrogation problem, specifically one arising in an electroencephalography (EEG) problem, of finding optimal number and locations for sensors for source identification in a 3D unit sphere from data on its boundary. In this effort we compare the use of the classical D-optimal criterion for observation points as opposed to that for a uniform observation mesh. We consider location and best number of sensors and report results based on statistical uncertainty analysis of the resulting estimated parameters.

Separability tests for high-dimensional, low sample size multivariate repeated measures data

Longitudinal imaging studies have moved to the forefront of medical research due to their ability to characterize spatio-temporal features of biological structures across the lifespan. Valid inference in longitudinal imaging requires enough flexibility of the covariance model to allow reasonable fidelity to the true pattern. On the other hand, the existence of computable estimates demands a parsimonious parameterization of the covariance structure. Separable (Kronecker product) covariance models provide one such parameterization in which the spatial and temporal covariances are modeled separately. However, evaluating the validity of this parameterization in high-dimensions remains a challenge. Here we provide a scientifically informed approach to assessing the adequacy of separable (Kronecker product) covariance models when the number of observations is large relative to the number of independent sampling units (sample size). We address both the general case, in which unstructured matrices are considered for each covariance model, and the structured case, which assumes a particular structure for each model. For the structured case, we focus on the situation where the within subject correlation is believed to decrease exponentially in time and space as is common in longitudinal imaging studies. However, the provided framework equally applies to all covariance patterns used within the more general multivariate repeated measures context. Our approach provides useful guidance for high dimension, low sample size data that preclude using standard likelihood based tests. Longitudinal medical imaging data of caudate morphology in schizophrenia illustrates the approaches appeal.

Kronecker product linear exponent AR(1) correlation structures for multivariate repeated measures

Longitudinal imaging studies have moved to the forefront of medical research due to their ability to characterize spatio-temporal features of biological structures across the lifespan. Credible models of the correlations in longitudinal imaging require two or more pattern components. Valid inference requires enough flexibility of the correlation model to allow reasonable fidelity to the true pattern. On the other hand, the existence of computable estimates demands a parsimonious parameterization of the correlation structure. For many one-dimensional spatial or temporal arrays, the linear exponent autoregressive (LEAR) correlation structure meets these two opposing goals in one model. The LEAR structure is a flexible two-parameter correlation model that applies to situations in which the within-subject correlation decreases exponentially in time or space. It allows for an attenuation or acceleration of the exponential decay rate imposed by the commonly used continuous-time AR(1) structure. We propose the Kronecker product LEAR correlation structure for multivariate repeated measures data in which the correlation between measurements for a given subject is induced by two factors (e.g., spatial and temporal dependence). Excellent analytic and numerical properties make the Kronecker product LEAR model a valuable addition to the suite of parsimonious correlation structures for multivariate repeated measures data. Longitudinal medical imaging data of caudate morphology in schizophrenia illustrates the appeal of the Kronecker product LEAR correlation structure.

Review of the BCI Competition IV

The BCI competition IV stands in the tradition of prior BCI competitions that aim to provide high quality neuroscientific data for open access to the scientific community. As experienced already in prior competitions not only scientists from the narrow field of BCI compete, but scholars with a broad variety of backgrounds and nationalities. They include high specialists as well as students. The goals of all BCI competitions have always been to challenge with respect to novel paradigms and complex data. We report on the following challenges: (1) asynchronous data, (2) synthetic, (3) multi-class continuous data, (4) session-to-session transfer, (5) directionally modulated MEG, (6) finger movements recorded by ECoG. As after past competitions, our hope is that winning entries may enhance the analysis methods of future BCIs.

A time-varying source connectivity approach to reveal human somatosensory information processing

Exploration of neural sources and their effective connectivity based on transient changes in electrophysiological activities to external stimuli is important for understanding brain mechanisms of sensory information processing. However, such cortical mechanisms have not yet been well characterized in electrophysiological studies since (1) it is difficult to estimate the stimulus-activated neural sources and their activities and (2) it is difficult to identify transient effective connectivity between neural sources in the order of milliseconds. To address these issues, we developed a time-varying source connectivity approach to effectively capture fast-changing information flows between neural sources from high-density Electroencephalography (EEG) recordings. This time-varying source connectivity approach was applied to somatosensory evoked potentials (SEPs), which were elicited by electrical stimulation of right hand and recorded using 64 channels from 16 subjects, to reveal human somatosensory information processing. First, SEP sources and their activities were estimated, both at single-subject and group level, using equivalent current dipolar source modeling. Then, the functional integration among SEP sources was explored using a Kalman smoother based time-varying effective connectivity inference method. The results showed that SEPs were mainly generated from the contralateral primary somatosensory cortex (SI), bilateral secondary somatosensory cortex (SII), and cingulate cortex (CC). Importantly, we observed a serial processing of somatosensory information in human somatosensory cortices (from SI to SII) at earlier latencies (<150 ms) and a reciprocal processing between SII and CC at later latencies (>200 ms).

Model Selection and Estimation in the Matrix Normal Graphical Model

Motivated by analysis of gene expression data measured over different tissues or over time, we consider matrix-valued random variable and matrix-normal distribution, where the precision matrices have a graphical interpretation for genes and tissues, respectively. We present a l(1) penalized likelihood method and an efficient coordinate descent-based computational algorithm for model selection and estimation in such matrix normal graphical models (MNGMs). We provide theoretical results on the asymptotic distributions, the rates of convergence of the estimates and the sparsistency, allowing both the numbers of genes and tissues to diverge as the sample size goes to infinity. Simulation results demonstrate that the MNGMs can lead to better estimate of the precision matrices and better identifications of the graph structures than the standard Gaussian graphical models. We illustrate the methods with an analysis of mouse gene expression data measured over ten different tissues.

The primary somatosensory cortex largely contributes to the early part of the cortical response elicited by nociceptive stimuli

Research on the cortical sources of nociceptive laser-evoked brain potentials (LEPs) began almost two decades ago (Tarkka and Treede, 1993). Whereas there is a large consensus on the sources of the late part of the LEP waveform (N2 and P2 waves), the relative contribution of the primary somatosensory cortex (S1) to the early part of the LEP waveform (N1 wave) is still debated. To address this issue we recorded LEPs elicited by the stimulation of four limbs in a large population (n=35). Early LEP generators were estimated both at single-subject and group level, using three different approaches: distributed source analysis, dipolar source modeling, and probabilistic independent component analysis (ICA). We show that the scalp distribution of the earliest LEP response to hand stimulation was maximal over the central-parietal electrodes contralateral to the stimulated side, while that of the earliest LEP response to foot stimulation was maximal over the central-parietal midline electrodes. Crucially, all three approaches indicated hand and foot S1 areas as generators of the earliest LEP response. Altogether, these findings indicate that the earliest part of the scalp response elicited by a selective nociceptive stimulus is largely explained by activity in the contralateral S1, with negligible contribution from the secondary somatosensory cortex (S2).

General bounds for electrode mislocation on the EEG inverse problem

We analyze the effect of electrode mislocation on the electroencephalography (EEG) inverse problem using the Cramér-Rao bound (CRB) for single dipolar source parameters. We adopt a realistic head shape model, and solve the forward problem using the Boundary Element Method; the use of the CRB allows us to obtain general results which do not depend on the algorithm used for solving the inverse problem. We consider two possible causes for the electrode mislocation, errors in the measurement of the electrode positions and an imperfect registration between the electrodes and the scalp surfaces. For 120 electrodes placed in the scalp according to the 10-20 standard, and errors on the electrode location with a standard deviation of 5mm, the lower bound on the standard deviation in the source depth estimation is approximately 1mm in the worst case. Therefore, we conclude that errors in the electrode location may be tolerated since their effect on the EEG inverse problem are negligible from a practical point of view.

Shrinkage approach for EEG covariance matrix estimation

We present a shrinkage estimator for the EEG spatial covariance matrix of the background activity. We show that such an estimator has some advantages over the maximum likelihood and sample covariance estimators when the number of available data to carry out the estimation is low. We find sufficient conditions for the consistency of the shrinkage estimators and results concerning their numerical stability. We compare several shrinkage schemes and show how to improve the estimator by incorporating known structure of the covariance matrix.

Bayesian analysis of matrix normal graphical models

We present Bayesian analyses of matrix-variate normal data with conditional independencies induced by graphical model structuring of the characterizing covariance matrix parameters. This framework of matrix normal graphical models includes prior specifications, posterior computation using Markov chain Monte Carlo methods, evaluation of graphical model uncertainty and model structure search. Extensions to matrix-variate time series embed matrix normal graphs in dynamic models. Examples highlight questions of graphical model uncertainty, search and comparison in matrix data contexts. These models may be applied in a number of areas of multivariate analysis, time series and also spatial modelling.

A space-frequency analysis of MEG background processes

In MEG source localization the estimated source parameters will be more reliable when the spatiotemporal covariance of the noise and background activity is taken into account. Since this covariance is in general too large to estimate based on the data and to invert efficiently, different parametrizations have been proposed in the literature. These models can be seen as special cases of the general decomposition of the covariance into a sum of Kronecker products of spatial matrices X(n) and temporal matrices T(n) (Van Loan, 2000). In this study we investigate the assumption of the matrices T(n) being Toeplitz. If so, the covariance matrix in the space-frequency domain will have an approximate block-diagonal structure, facilitating inversion, which is a prerequisite for source localization. In this study we address the question whether the Toeplitz approximation is valid for data sets obtained in visual evoked field, auditory evoked field, somatosensory evoked field experiments and data sets containing spontaneous activity. It turns out that on average 87% is in the block-diagonal of the sample covariance, which is close to the values obtained for real Toeplitz matrices T(n). This implies that the space-frequency domain is very interesting for source localization since the major part of the entire covariance can be incorporated in that domain straightforwardly. Finally, the two major processes in the background activity are characterized in terms of their spatial and frequency patterns, yielding a focal and a non-focal pattern in 8 of 10 data sets analyzed in this study. The focal pattern represents the alpha frequency at parieto-occipital areas, whereas the non-focal pattern is more widespread both in space and in frequency.

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.

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.

A probabilistic algorithm integrating source localization and noise suppression for MEG and EEG data

We have developed a novel probabilistic model that estimates neural source activity measured by MEG and EEG data while suppressing the effect of interference and noise sources. The model estimates contributions to sensor data from evoked sources, interference sources and sensor noise using Bayesian methods and by exploiting knowledge about their timing and spatial covariance properties. Full posterior distributions are computed rather than just the MAP estimates. In simulation, the algorithm can accurately localize and estimate the time courses of several simultaneously active dipoles, with rotating or fixed orientation, at noise levels typical for averaged MEG data. The algorithm even performs reasonably at noise levels typical of an average of just a few trials. The algorithm is superior to beamforming techniques, which we show to be an approximation to our graphical model, in estimation of temporally correlated sources. Success of this algorithm using MEG data for localizing bilateral auditory cortex, low-SNR somatosensory activations, and for localizing an epileptic spike source are also demonstrated.

Modeling spatiotemporal covariance for magnetoencephalography or electroencephalography source analysis

We propose a new model to approximate spatiotemporal noise covariance for use in neural electromagnetic source analysis, which better captures temporal variability in background activity. As with other existing formalisms, our model employs a Kronecker product of matrices representing temporal and spatial covariance. In our model, spatial components are allowed to have differing temporal covariances. Variability is represented as a series of Kronecker products of spatial component covariances and corresponding temporal covariances. Unlike previous attempts to model covariance through a sum of Kronecker products, our model is designed to have a computationally manageable inverse. Despite increased descriptive power, inversion of the model is fast, making it useful in source analysis. We have explored two versions of the model. One is estimated based on the assumption that spatial components of background noise have uncorrelated time courses. Another version, which gives closer approximation, is based on the assumption that time courses are statistically independent. The accuracy of the structural approximation is compared to an existing model, based on a single Kronecker product, using both Frobenius norm of the difference between spatiotemporal sample covariance and a model, and scatter plots. Performance of ours and previous models is compared in source analysis of a large number of single dipole problems with simulated time courses and with background from authentic magnetoencephalography data.

Surface-source modeling and estimation using biomagnetic measurements

We propose a number of electric source models that are spatially distributed on an unknown surface for biomagnetism. These can be useful to model, e.g., patches of electrical activity on the cortex. We use a realistic head (or another organ) model and discuss the special case of a spherical head model with radial sensors resulting in more efficient computations of the estimates for magnetoencephalography. We derive forward solutions, maximum likelihood (ML) estimates, and Cramér-Rao bound (CRB) expressions for the unknown source parameters. A model selection method is applied to decide on the most appropriate model. We also present numerical examples to compare the performances and computational costs of the different models and illustrate when it is possible to distinguish between surface and focal sources or line sources. Finally, we apply our methods to real biomagnetic data of phantom human torso and demonstrate the applicability of them.

Bayesian estimation of evoked and induced responses

We describe an extension of our empirical Bayes approach to magnetoencephalography/electroencephalography (MEG/EEG) source reconstruction that covers both evoked and induced responses. The estimation scheme is based on classical covariance component estimation using restricted maximum likelihood (ReML). We have focused previously on the estimation of spatial covariance components under simple assumptions about the temporal correlations. Here we extend the scheme, using temporal basis functions to place constraints on the temporal form of the responses. We show how the same scheme can estimate evoked responses that are phase-locked to the stimulus and induced responses that are not. For a single trial the model is exactly the same. In the context of multiple trials, however, the inherent distinction between evoked and induced responses calls for different treatments of the underlying hierarchical multitrial model. We derive the respective models and show how they can be estimated efficiently using ReML. This enables the Bayesian estimation of evoked and induced changes in power or, more generally, the energy of wavelet coefficients.

Spatiotemporal noise covariance estimation from limited empirical magnetoencephalographic data

The performance of parametric magnetoencephalography (MEG) and electroencephalography (EEG) source localization approaches can be degraded by the use of poor background noise covariance estimates. In general, estimation of the noise covariance for spatiotemporal analysis is difficult mainly due to the limited noise information available. Furthermore, its estimation requires a large amount of storage and a one-time but very large (and sometimes intractable) calculation or its inverse. To overcome these difficulties, noise covariance models consisting of one pair or a sum of multi-pairs of Kronecker products of spatial covariance and temporal covariance have been proposed. However, these approaches cannot be applied when the noise information is very limited, i.e., the amount of noise information is less than the degrees of freedom of the noise covariance models. A common example of this is when only averaged noise data are available for a limited prestimulus region (typically at most a few hundred milliseconds duration). For such cases, a diagonal spatiotemporal noise covariance model consisting of sensor variances with no spatial or temporal correlation has been the common choice for spatiotemporal analysis. In this work, we propose a different noise covariance model which consists of diagonal spatial noise covariance and Toeplitz temporal noise covariance. It can easily be estimated from limited noise information, and no time-consuming optimization and data-processing are required. Thus, it can be used as an alternative choice when one-pair or multi-pair noise covariance models cannot be estimated due to lack of noise information. To verify its capability we used Bayesian inference dipole analysis and a number of simulated and empirical datasets. We compared this covariance model with other existing covariance models such as conventional diagonal covariance, one-pair and multi-pair noise covariance models, when noise information is sufficient to estimate them. We found that our proposed noise covariance model yields better localization performance than a diagonal noise covariance, while it performs slightly worse than one-pair or multi-pair noise covariance models - although these require much more noise information. Finally, we present some localization results on median nerve stimulus empirical MEG data for our proposed noise covariance model.

Effects of geometric head model perturbations on the EEG forward and inverse problems

We study the effect of geometric head model perturbations on the electroencephalography (EEG) forward and inverse problems. Small magnitude perturbations of the shape of the head could represent uncertainties in the head model due to errors on images or techniques used to construct the model. They could also represent small scale details of the shape of the surfaces not described in a deterministic model, such as the sulci and fissures of the cortical layer. We perform a first-order perturbation analysis, using a meshless method for computing the sensitivity of the solution of the forward problem to the geometry of the head model. The effect on the forward problem solution is treated as noise in the EEG measurements and the Cramér-Rao bound is computed to quantify the effect on the inverse problem performance. Our results show that, for a dipolar source, the effect of the perturbations on the inverse problem performance is under the level of the uncertainties due to the spontaneous brain activity. Thus, the results suggest that an extremely detailed model of the head may be unnecessary when solving the EEG inverse problem.

A generalized spatiotemporal covariance model for stationary background in analysis of MEG data

Using a noise covariance model based on a single Kronecker product of spatial and temporal covariance in the spatiotemporal analysis of MEG data was demonstrated to provide improvement in the results over that of the commonly used diagonal noise covariance model. In this paper we present a model that is a generalization of all of the above models. It describes models based on a single Kronecker product of spatial and temporal covariance as well as more complicated multi-pair models together with any intermediate form expressed as a sum of Kronecker products of spatial component matrices of reduced rank and their corresponding temporal covariance matrices. The model provides a framework for controlling the tradeoff between the described complexity of the background and computational demand for the analysis using this model. Ways to estimate the value of the parameter controlling this tradeoff are also discussed.

The spatiotemporal MEG covariance matrix modeled as a sum of Kronecker products

The single Kronecker product (KP) model for the spatiotemporal covariance of MEG residuals is extended to a sum of Kronecker products. This sum of KP is estimated such that it approximates the spatiotemporal sample covariance best in matrix norm. Contrary to the single KP, this extension allows for describing multiple, independent phenomena in the ongoing background activity. Whereas the single KP model can be interpreted by assuming that background activity is generated by randomly distributed dipoles with certain spatial and temporal characteristics, the sum model can be physiologically interpreted by assuming a composite of such processes. Taking enough terms into account, the spatiotemporal sample covariance matrix can be described exactly by this extended model. In the estimation of the sum of KP model, it appears that the sum of the first 2 KP describes between 67% and 93%. Moreover, these first two terms describe two physiological processes in the background activity: focal, frequency-specific alpha activity, and more widespread non-frequency-specific activity. Furthermore, temporal nonstationarities due to trial-to-trial variations are not clearly visible in the first two terms, and, hence, play only a minor role in the sample covariance matrix in terms of matrix power. Considering the dipole localization, the single KP model appears to describe around 80% of the noise and seems therefore adequate. The emphasis of further improvement of localization accuracy should be on improving the source model rather than the covariance model.

Spatiotemporal Bayesian inference dipole analysis for MEG neuroimaging data

Recently, we described a Bayesian inference approach to the MEG/EEG inverse problem that used numerical techniques to estimate the full posterior probability distributions of likely solutions upon which all inferences were based [Schmidt, D.M., George, J.S., Wood, C.C., 1999. Bayesian inference applied to the electromagnetic inverse problem. Human Brain Mapping 7, 195; Schmidt, D.M., George, J.S., Ranken, D.M., Wood, C.C., 2001. Spatial-temporal bayesian inference for MEG/EEG. In: Nenonen, J., Ilmoniemi, R. J., Katila, T. (Eds.), Biomag 2000: 12th International Conference on Biomagnetism. Espoo, Norway, p. 671]. Schmidt et al. (1999) focused on the analysis of data at a single point in time employing an extended region source model. They subsequently extended their work to a spatiotemporal Bayesian inference analysis of the full spatiotemporal MEG/EEG data set. Here, we formulate spatiotemporal Bayesian inference analysis using a multi-dipole model of neural activity. This approach is faster than the extended region model, does not require use of the subject's anatomical information, does not require prior determination of the number of dipoles, and yields quantitative probabilistic inferences. In addition, we have incorporated the ability to handle much more complex and realistic estimates of the background noise, which may be represented as a sum of Kronecker products of temporal and spatial noise covariance components. This reduces the effects of undermodeling noise. In order to reduce the rigidity of the multi-dipole formulation which commonly causes problems due to multiple local minima, we treat the given covariance of the background as uncertain and marginalize over it in the analysis. Markov Chain Monte Carlo (MCMC) was used to sample the many possible likely solutions. The spatiotemporal Bayesian dipole analysis is demonstrated using simulated and empirical whole-head MEG data.

Evaluation and validity of a LORETA normative EEG database

To evaluate the reliability and validity of a Z-score normative EEG database for Low Resolution Electromagnetic Tomography (LORETA), EEG digital samples (2 second intervals sampled 128 Hz, 1 to 2 minutes eyes closed) were acquired from 106 normal subjects, and the cross-spectrum was computed and multiplied by the Key Institute's LORETA 2,394 gray matter pixel T Matrix. After a log10 transform or a Box-Cox transform the mean and standard deviation of the *.lor files were computed for each of the 2394 gray matter pixels, from 1 to 30 Hz, for each of the subjects. Tests of Gaussianity were computed in order to best approximate a normal distribution for each frequency and gray matter pixel. The relative sensitivity of a Z-score database was computed by measuring the approximation to a Gaussian distribution. The validity of the LORETA normative database was evaluated by the degree to which confirmed brain pathologies were localized using the LORETA normative database. Log10 and Box-Cox transforms approximated Gaussian distribution in the range of 95.64% to 99.75% accuracy. The percentage of normative Z-score values at 2 standard deviations ranged from 1.21% to 3.54%, and the percentage of Z-scores at 3 standard deviations ranged from 0% to 0.83%. Left temporal lobe epilepsy, right sensory motor hematoma and a right hemisphere stroke exhibited maximum Z-score deviations in the same locations as the pathologies. We conclude: (1) Adequate approximation to a Gaussian distribution can be achieved using LORETA by using a log10 transform or a Box-Cox transform and parametric statistics, (2) a Z-Score normative database is valid with adequate sensitivity when using LORETA, and (3) the Z-score LORETA normative database also consistently localized known pathologies to the expected Brodmann areas as an hypothesis test based on the surface EEG before computing LORETA.

Model Selection in Spatio-Temporal Electromagnetic Source Analysis

Several methods [model selection procedures (MSPs)] to determine the number of sources in electroencephalogram (EEG) and magnetoencphalogram (MEG) data have previously been investigated in an instantaneous analysis. In this paper, these MSPs are extended to a spatio-temporal analysis if possible. It is seen that the residual variance (RV) tends to overestimate the number of sources. The Akaike information criterion (AIC) and the Wald test on amplitudes (WA) and the Wald test on locations (WL) have the highest probabilities of selecting the correct number of sources. The WA has the advantage that it offers the opportunity to test which source is active at which time sample.

Distinguishing Between Moving and Stationary Sources Using EEG/MEG Measurements With an Application to Epilepsy

Performances of electroencephalography (EEG) and magnetoencephalography (MEG) source estimation methods depend on the validity of the assumed model. In many cases, the model structure is related to physical information. We discuss a number of statistical selection methods to distinguish between two possible models using least-squares estimation and assuming a spherical head model. The first model has a single moving source whereas the second has two stationary sources; these may result in similar EEG/MEG measurements. The need to decide between such models occurs for example in Jacksonian seizures (e.g., epilepsy) or in intralobular activities, where a model with either two stationary dipole sources or a single moving dipole source may be possible. We also show that all of the selection methods discussed choose the correct model with probability one when the number of trials goes to infinity. Finally we present numerical examples and compare the performances of the methods by varying parameters such as the signal-to-noise ratio, source depth, and separation of sources, and also apply the methods to real MEG data for epilepsy.

Statistical Maps for EEG Dipolar Source Localization

We present a method that estimates three-dimensional statistical maps for electroencephalogram (EEG) source localization. The maps assess the likelihood that a point in the brain contains a dipolar source, under the hypothesis of one, two or three activated sources. This is achieved by examining all combinations of one to three dipoles on a coarse grid and attributing to each combination a score based on an F statistic. The probability density function of the statistic under the null hypothesis is estimated nonparametrically, using bootstrap resampling. A theoretical F distribution is then fitted to the empirical distribution in order to allow correction for multiple comparisons. The maps allow for the systematic exploration of the solution space for dipolar sources. They permit to test whether the data support a given solution. They do not rely on the assumption of uncorrelated source time courses. They can be compared to other statistical parametric maps such as those used in functional magnetic resonance imaging (fMRI). Results are presented for both simulated and real data. The maps were compared with LORETA and MUSIC results. For the real data consisting of an average of epileptic spikes, we observed good agreement between the EEG statistical maps, intracranial EEG recordings, and fMRI activations.

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.

Parametric vs. non-parametric statistics of low resolution electromagnetic tomography (LORETA)

This study compared the relative statistical sensitivity of non-parametric and parametric statistics of 3-dimensional current sources as estimated by the EEG inverse solution Low Resolution Electromagnetic Tomography (LORETA). One would expect approximately 5% false positives (classification of a normal as abnormal) at the P < .025 level of probability (two tailed test) and approximately 1% false positives at the P < .005 level. EEG digital samples (2 second intervals sampled 128 Hz, 1 to 2 minutes eyes closed) from 43 normal adult subjects were imported into the Key Institute's LORETA program. We then used the Key Institute's cross-spectrum and the Key Institute's LORETA output files (*.lor) as the 2,394 gray matter pixel representation of 3-dimensional currents at different frequencies. The mean and standard deviation *.lor files were computed for each of the 2,394 gray matter pixels for each of the 43 subjects. Tests of Gaussianity and different transforms were computed in order to best approximate a normal distribution for each frequency and gray matter pixel. The relative sensitivity of parametric vs. non-parametric statistics were compared using a "leave-one-out" cross validation method in which individual normal subjects were withdrawn and then statistically classified as being either normal or abnormal based on the remaining subjects. Log10 transforms approximated Gaussian distribution in the range of 95% to 99% accuracy. Parametric Z score tests at P < .05 cross-validation demonstrated an average misclassification rate of approximately 4.25%, and range over the 2,394 gray matter pixels was 27.66% to 0.11%. At P < .01 parametric Z score cross-validation false positives were 0.26% and ranged from 6.65% to 0% false positives. The non-parametric Key Institute's t-max statistic at P < .05 had an average misclassification error rate of 7.64% and ranged from 43.37% to 0.04% false positives. The nonparametric t-max at P < .01 had an average misclassification rate of 6.67% and ranged from 41.34% to 0% false positives of the 2,394 gray matter pixels for any cross-validated normal subject. In conclusion, adequate approximation to Gaussian distribution and high cross-validation can be achieved by the Key Institute's LORETA programs by using a log10 transform and parametric statistics, and parametric normative comparisons had lower false positive rates than the non-parametric tests.

Multimodal integration of EEG and MEG data: a simulation study with variable signal-to-noise ratio and number of sensors

Previous simulation studies have stressed the importance of the multimodal integration of electroencephalography (EEG) and magnetoencephalography (MEG) data in the estimation of cortical current density. In such studies, no systematic variations of the signal-to-noise ratio (SNR) and of the number of sensors were explicitly taken into account in the estimation process. We investigated effects of variable SNR and number of sensors on the accuracy of current density estimate by using multimodal EEG and MEG data. This was done by using as the dependent variable both the correlation coefficient (CC) and the relative error (RE) between imposed and estimated waveforms at the level of cortical region of interests (ROI). A realistic head and cortical surface model was used. Factors used in the simulations were: (1). the SNR of the simulated scalp data (with seven levels: infinite, 30, 20, 10, 5, 3, 1); (2). the particular inverse operator used to estimate the cortical source activity from the simulated scalp data (INVERSE, with two levels, including minimum norm and weighted minimum norm); and (3). the number of EEG or MEG sensors employed in the analysis (SENSORS, with three levels: 128, 61, 29 for EEG and 153, 61, or 38 in MEG). Analysis of variance demonstrated that all the considered factors significantly affect the CC and the RE indexes. Combined EEG-MEG data produced statistically significant lower RE and higher CC in source current density reconstructions compared to that estimated by the EEG and MEG data considered separately. These observations hold for the range of SNR values presented by the analyzed data. The superiority of current density estimation by multimodal integration of EEG and MEG was not due to differences in number of sensors between unimodal (EEG, MEG) and combined (EEG-MEG) inverse estimates. In fact, the current density estimate relative to the EEG-MEG multimodal integration involved 61 EEG plus 63 MEG sensors, whereas estimations carried out with the single modalities alone involved 128 sensors for EEG and 153 sensors for MEG. The results of the simulations also suggest that the use of simultaneous 29 EEG sensors during the MEG measurements carried out with full sensor arrangements (153 sensors) returned an accuracy of the cortical source estimate statistically similar to that obtained by combining 64 EEG and 153 MEG sensors.

The temporal prior in bioelectromagnetic source imaging problems

The multiplicity of temporal priors proposed for regularization of the bioelectromagnetic source imaging problems [e.g., the inverse electrocardiogram (ECG) and inverse electroencephalogram (EEG) problems], is discordant with the fact that fundamental statistical principles sharply limit the choice. Thus, our objective is to derive the form of the prior consistent with the general unavailability of temporal constraints. Writing linear formulations of the inverse ECG and inverse EEG problems as H = FG + N (where the ith columns of matrices H, G, and N, are data, signal, and noise vectors at time step i, and F is the transfer matrix), and using the noninformative principle that features of the spatiotemporal prior not supplied a posteriori should be invariant under temporal transformations, we show that the implied spatiotemporal signal autocovariance matrix (of the vector formed by the entries of G) is given in block matrix form [equation in text] where Cg is a matrix of unit trace proportional to the autocovariance matrix of any column of G (representing supplied information regarding the spatial prior), epsilon[.] denotes expectation, superscript ' indicates transpose, [symbol in text] is the Kronecker product, [symbol in text] is Frobenius norm, and the "matrix scalar product" [symbol in text] indicates the inner product of the two vectors formed by the entries of the two adjacent matrices (i.e., A [symbol in text] B [triple bond] trace[A'B]). This result eliminates some uncertainties and ambiguities that have characterized spatiotemporal regularization methods--including eight methods previously introduced in this transactions. Ultimately, the result derives from an implied symmetry principle under which the form of a nontrivial noninformative temporal component of the prior can be identified. Among other things, separability of the spatiotemporal prior in terms of the above Kronecker product can be thought of as the expression of the lack of "entanglement" of the spatial and temporal contributions (a consequence of noninformativity). The approach is generalized to the important cases of non-Gaussian spatial priors, and signal and noise that are not independent (transfer matrix noise). We also demonstrate a means for computational complexity reduction, related to the application of a particular orthogonal transformation, having features dependent on whether or not the transfer matrix represents a surjective mapping.

A mathematical approach to the temporal stationarity of background noise in MEG/EEG measurements

The general spatiotemporal covariance matrix of the background noise in MEG/EEG signals is huge. To reduce the dimensionality of this matrix it is modeled as a Kronecker product of a spatial and a temporal covariance matrix. When the number of time samples is larger than, say, J = 500, the iterative Maximum Likelihood estimation of these two matrices is still too time-consuming to be useful on a routine basis. In this study we looked for methods to circumvent this computationally expensive procedure by using a parametric model with subject-dependent parameters. Such a model would additionally help with interpreting MEG/EEG signals. For the spatial covariance, models have been derived already and it has been shown that measured MEG/EEG signals can be understood spatially as random processes, generated by random dipoles. The temporal covariance, however, has not been modeled yet, therefore we studied the temporal covariance matrix in several subjects. For all subjects the temporal covariance shows an alpha oscillation and vanishes for large time lag. This gives rise to a temporal noise model consisting of two components: alpha activity and additional random noise. The alpha activity is modeled as randomly occurring waves with random phase and the covariance of the additional noise decreases exponentially with lag. This model requires only six parameters instead of 12 J(J + 1). Theoretically, this model is stationary but in practice the stationarity of the matrix is highly influenced by the baseline correction. It appears that very good agreement between the data and the parametric model can be obtained when the baseline correction window is taken into account properly. This finding implies that the background noise is in principle a stationary process and that nonstationarities are mainly caused by the nature of the preprocessing method. When analyzing events at a fixed sample after the stimulus (e.g., the SEF N20 response) one can take advantage of this nonstationarity by optimizing the baseline window to obtain a low noise variance at this particular sample.