eeg extended source localization tensor based vs conventional methods the localization of brain sources based on eeg measurements is a topic that has attracted a lot of attention in the last decades and many different source localization algorithms have been proposed however their performance is limited in the case of several simultaneously active brain regions and low signal to noise ratios to overcome these problems tensor based preprocessing can be applied which consists in constructing a space time frequency stf or space time wave vector stwv tensor and decomposing it using the canonical polyadic cp decomposition in this paper we present a new algorithm for the accurate localization of extended sources based on the results of the tensor decomposition furthermore we conduct a detailed study of the tensor based preprocessing methods including an analysis of their theoretical foundation their computational complexity and their performance for realistic simulated data in comparison to conventional source localization algorithms such as sloreta cortical loreta cloreta and 4 exso music our objective consists on the one hand in demonstrating the gain in performance that can be achieved by tensor based preprocessing and on the other hand in pointing out the limits and drawbacks of this method finally we validate the stf and stwv techniques on real measurements to demonstrate their usefulness for practical applications

meg beamforming using bayesian pca for adaptive data covariance matrix regularization beamformers are a commonly used method for doing source localization from magnetoencephalography meg data a key ingredient in a beamformer is the estimation of the data covariance matrix when the noise levels are high or when there is only a small amount of data available the data covariance matrix is estimated poorly and the signal to noise ratio snr of the beamformer output degrades one solution to this is to use regularization whereby the diagonal of the covariance matrix is amplified by a pre specified amount however this provides improvements at the expense of a loss in spatial resolution and the parameter controlling the amount of regularization must be chosen subjectively in this paper we introduce a method that provides an adaptive solution to this problem by using a bayesian principle component analysis pca this provides an estimate of the data covariance matrix to give a data driven non arbitrary solution to the trade off between the spatial resolution and the snr of the beamformer output this also provides a method for determining when the quality of the data covariance estimate maybe under question we apply the approach to simulated and real meg data and demonstrate the way in which it can automatically adapt the regularization to give good performance over a range of noise and signal levels

bayesian gaussian process classification with the em ep algorithm gaussian process classifiers gpcs are bayesian probabilistic kernel classifiers in gpcs the probability of belonging to a certain class at an input location is monotonically related to the value of some latent function at that location starting from a gaussian process prior over this latent function data are used to infer both the posterior over the latent function and the values of hyperparameters to determine various aspects of the function recently the expectation propagation ep approach has been proposed to infer the posterior over the latent function based on this work we present an approximate em algorithm the em ep algorithm to learn both the latent function and the hyperparameters this algorithm is found to converge in practice and provides an efficient bayesian framework for learning hyperparameters of the kernel a multiclass extension of the em ep algorithm for gpcs is also derived in the experimental results the em ep algorithms are as good or better than other methods for gpcs or support vector machines svms with cross validation

seeg dipole source localization based on an empirical bayesian approach taking into account forward model uncertainties electromagnetic brain source localization consists in the inversion of a forward model based on a limited number of potential measurements a wide range of methods has been developed to regularize this severely ill posed problem and to reduce the solution space imposing spatial smoothness anatomical constraint or sparsity of the activated source map this last criteria based on physiological assumptions stating that in some particular events e g epileptic spikes evoked potential few focal area of the brain are simultaneously actives has gained more and more interest bayesian approaches have the ability to provide sparse solutions under adequate parametrization and bring a convenient framework for the introduction of priors in the form of probabilistic density functions however the quality of the forward model is rarely questioned while this parameter has undoubtedly a great influence on the solution its construction suffers from numerous approximation and uncertainties even when using realistic numerical models in addition it often encodes a coarse sampling of the continuous solution space due to the computational burden its inversion implies in this work we propose an empirical bayesian approach to take into account the uncertainties of the forward model by allowing constrained variations around a prior physical model in the particular context of seeg measurements we demonstrate on simulations that the method enhance the accuracy of the source time course estimation as well as the sparsity of the resulting source map results on real signals prove the applicability of the method in real contexts

prior knowledge on cortex organization in the reconstruction of source current densities from eeg abstract the reconstruction of the generators of electroencephalographic eeg signals is important for understanding brain processes since the inverse problem has no unique solution additional knowledge or assumptions are needed often results from other anatomical or functional measurement modalities are difficult to interpret directly in terms of eeg source strengths but they provide valuable information about the functional similarity between brain regions for example in form of parcellations we propose a novel approach to the incorporation of such parcellations as priors into the reconstruction of distributed source current densities from eeg two algorithms are described based on a surface constrained loreta low resolution electromagnetic tomography approach the first patchloreta1 uses both topological neighborhood and prior information to define smoothness while the second patchloreta2 neglects topological neighborhood computer simulations using a smooth reconstruction surface on the brain envelope reveal important aspects of the algorithms performance in particular the influences of noise and incongruence between measurements and prior information it turns out that patchloreta1 makes efficient use of the provided prior information and at the same time is quite robust towards faulty priors as well as noise the algorithms are also tested on the localization of the sources of event related potentials here both the smooth brain and folded cortical surfaces serve as reconstruction spaces we find that patchloreta1 becomes ineffective on the folded cortex while patchloreta2 yields plausible results we also discuss the extension of the proposed algorithms to other types of priors and propose ways to overcome shortcomings of the current implementation

hierarchical bayesian inference for the eeg inverse problem using realistic fe head models depth localization and source separation for focal primary currents the estimation of the activity related ion currents by measuring the induced electromagnetic fields at the head surface is a challenging and severely ill posed inverse problem this is especially true in the recovery of brain networks involving deep lying sources by means of eeg meg recordings which is still a challenging task for any inverse method recently hierarchical bayesian modeling hbm emerged as a unifying framework for current density reconstruction cdr approaches comprising most established methods as well as offering promising new methods our work examines the performance of fully bayesian inference methods for hbm for source configurations consisting of few focal sources when used with realistic high resolution finite element fe head models the main foci of interest are the correct depth localization a well known source of systematic error of many cdr methods and the separation of single sources in multiple source scenarios both aspects are very important in the analysis of neurophysiological data and in clinical applications for these tasks hbm provides a promising framework and is able to improve upon established cdr methods such as minimum norm estimation mne or sloreta in many aspects for challenging multiple source scenarios where the established methods show crucial errors promising results are attained additionally we introduce wasserstein distances as performance measures for the validation of inverse methods in complex source scenarios

bayesian eeg source localization using a structured sparsity prior this paper deals with eeg source localization the aim is to perform spatially coherent focal localization and recover temporal eeg waveforms which can be useful in certain clinical applications a new hier archical bayesian model is proposed with a multivariate bernoulli laplacian structured sparsity prior for brain activity this distribution approximates a mixed l20 pseudo norm regularization in a bayesian framework a partially collapsed gibbs sampler is proposed to draw samples asymptotically distributed according to the posterior of the proposed bayesian model the generated samples are used to estimate the brain activity and the model hyperparameters jointly in an unsupervised framework two different kinds of metropolis hastings moves are introduced to accelerate the convergence of the gibbs sampler the first move is based on multiple dipole shifts within each mcmc chain whereas the second exploits proposals associated with different mcmc chains experiments with focal synthetic data shows that the proposed algorithm is more robust and has a higher recovery rate than the weighted l21 mixed norm regularization using real data the proposed algorithm finds sources that are spatially coherent with state of the art methods namely a multiple sparse prior approach and the champagne algorithm in addition the method estimates waveforms showing peaks at meaningful timestamps this information can be valuable for activity spread characterization

a switching multi scale dynamical network model of eeg meg we introduce a new generative model of the encephalography eeg meg data the inversion of which allows for inferring the locations and temporal evolution of the underlying sources as well as their dynamical interactions the proposed switching mesostate space model smsm builds on the multi scale generative model for eeg meg by daunizeau and friston 2007 smsm inherits the assumptions that 1 bioelectromagnetic activity is generated by a set of distributed sources 2 the dynamics of these sources can be modelled as random fluctuations about a small number of mesostates and 3 the number of mesostates engaged by a cognitive task is small additionally four generalising assumptions are now included 4 the mesostates interact according to a full dynamical causal network dcn that can be estimated 5 the dynamics of the mesostates can switch between multiple approximately linear operating regimes 6 each operating regime remains stable over finite periods of time temporal clusters and 7 the total number of times the mesostates dynamics can switch is small the proposed model adds therefore a level of flexibility by accommodating complex brain processes that cannot be characterised by purely linear and stationary gaussian dynamics importantly the smsm furnishes a new interpretation of the eeg meg data in which the source activity may have multiple discrete modes of behaviour each with approximately linear dynamics this is modelled by assuming that the connection strengths of the underlying mesoscopic dcn are time dependent but piecewise constant i e they can undergo discrete changes over time a variational bayes inversion scheme is derived to estimate all the parameters of the model by maximising a negative free energy lower bound on the model evidence this bound is used to select among different model choices that are defined by the number of mesostates as well as by the number of stationary linear regimes the full model is compared to a simplified version that uses no dynamical assumptions as well as to a standard eeg inversion technique the comparison is carried out using an extensive set of simulations and the application of smsm to a real data set is also demonstrated our results show that for experimental situations in which we have some a priori belief that there are multiple approximately linear dynamical regimes the proposed smsm provides a natural modelling tool

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

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

a mesostate space model for eeg and meg abstract we present a multi scale generative model for eeg that entails a minimum number of assumptions about evoked brain responses namely 1 bioelectric activity is generated by a set of distributed sources 2 the dynamics of these sources can be modelled as random fluctuations about a small number of mesostates 3 mesostates evolve in a temporal structured way and are functionally connected i e influence each other and 4 the number of mesostates engaged by a cognitive task is small e g between one and a few a variational bayesian learning scheme is described that furnishes the posterior density on the models parameters and its evidence since the number of meso sources specifies the model the model evidence can be used to compare models and find the optimum number of meso sources in addition to estimating the dynamics at each cortical dipole the mesostate space model and its inversion provide a description of brain activity at the level of the mesostates i e in terms of the dynamics of meso sources that are distributed over dipoles the inclusion of a mesostate level allows one to compute posterior probability maps of each dipole being active i e belonging to an active mesostate critically this model accommodates constraints on the number of meso sources while retaining the flexibility of distributed source models in explaining data in short it bridges the gap between standard distributed and equivalent current dipole models furthermore because it is explicitly spatiotemporal the model can embed any stochastic dynamical causal model e g a neural mass model as a markov process prior on the mesostate dynamics the approach is evaluated and compared to standard inverse eeg techniques using synthetic data and real data the results demonstrate the added value of the mesostate space model and its variational inversion

selecting forward models for meg source reconstruction using model evidence we investigated four key aspects of forward models for distributed solutions to the meg inverse problem 1 the nature of the cortical mesh constraining sources derived from an individual s mri or inverse normalised from a template mesh 2 the use of single sphere overlapping spheres or boundary element model bem head models 3 the density of the cortical mesh 3000 vs 7000 vertices and 4 whether source orientations were constrained to be normal to that mesh these were compared within the context of two types of spatial prior on the sources a single prior corresponding to a standard l2 minimum norm mnm inversion or multiple sparse priors msp the resulting generative models were compared using a free energy approximation to the bayesian model evidence after fitting multiple epochs of responses to faces or scrambled faces statistical tests of the free energy across nine participants showed clear superiority of msp over mnm models with the former reconstructing deeper sources furthermore there was 1 no evidence that an individually defined cortical mesh was superior to an inverse normalised canonical mesh but 2 clear evidence that a bem was superior to spherical head models provided individually defined inner skull and scalp meshes were used finally for msp models there was evidence that the combination of 3 higher density cortical meshes and 4 dipoles constrained to be normal to the mesh was superior to lower density or freely oriented sources in contrast to the mnm models in which free orientation was optimal these results have practical implications for meg source reconstruction particularly in the context of group studies

anatomically informed basis functions for eeg source localization combining functional and anatomical constraints distributed linear solutions have frequently been used to solve the source localization problem in eeg here we introduce an approach based on the weighted minimum norm wmn method that imposes constraints using anatomical and physiological information derived from other imaging modalities the anatomical constraints are used to reduce the solution space a priori by modeling the spatial source distribution with a set of basis functions these spatial basis functions are chosen in a principled way using information theory the reduced problem is then solved with a classical wmn method further functional constraints can be introduced in the weighting of the solution using fmri brain responses to augment spatial priors we used simulated data to explore the behavior of the approach over a range of the model s hyperparameters to assess the construct validity of our method we compared it with two established approaches to the source localization problem a simple weighted minimum norm and a maximum smoothness loreta like solution this involved simulations using single and multiple sources that were analyzed under different levels of confidence in the priors

solving the eeg inverse problem based on space time frequency structured sparsity constraints abstract we introduce stout spatio temporal unifying tomography a novel method for the source analysis of electroencephalograpic eeg recordings which is based on a physiologically motivated source representation our method assumes that only a small number of brain sources are active throughout a measurement where each of the sources exhibits focal smooth but localized characteristics in space time and frequency this structure is enforced through an expansion of the source current density into appropriate spatio temporal basis functions in combination with sparsity constraints this approach combines the main strengths of two existing methods namely sparse basis field expansions haufe et al 2011 and time frequency mixed norm estimates gramfort et al 2013 by adjusting the ratio between two regularization terms stout is capable of trading temporal for spatial reconstruction accuracy and vice versa depending on the requirements of specific analyses and the provided data due to allowing for non stationary source activations stout is particularly suited for the localization of event related potentials erp and other evoked brain activity we demonstrate its performance on simulated erp data for varying signal to noise ratios and numbers of active sources our analysis of the generators of visual and auditory evoked n200 potentials reveals that the most active sources originate in the temporal and occipital lobes in line with the literature on sensory processing

hierarchical bayesian estimates of distributed meg sources theoretical aspects and comparison of variational and mcmc methods magnetoencephalography meg provides millisecond scale temporal resolution for noninvasive mapping of human brain functions but the problem of reconstructing the underlying source currents from the extracranial data has no unique solution several distributed source estimation methods based on different prior assumptions have been suggested for the resolution of this inverse problem recently a hierarchical bayesian generalization of the traditional minimum norm estimate mne was proposed in which the variance of distributed current at each cortical location is considered as a random variable and estimated from the data using the variational bayesian vb framework here we introduce an alternative scheme for performing bayesian inference in the context of this hierarchical model by using markov chain monte carlo mcmc strategies in principle the mcmc method is capable of numerically representing the true posterior distribution of the currents whereas the vb approach is inherently approximative we point out some potential problems related to hyperprior selection in the previous work and study some possible solutions a hyperprior sensitivity analysis is then performed and the structure of the posterior distribution as revealed by the mcmc method is investigated we show that the structure of the true posterior is rather complex with multiple modes corresponding to different possible solutions to the source reconstruction problem we compare the results from the vb algorithm to those obtained from the mcmc simulation under different hyperparameter settings the difficulties in using a unimodal variational distribution as a proxy for a truly multimodal distribution are also discussed simulated meg data with realistic sensor and source geometries are used in performing the analyses

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

diffusion based spatial priors for functional magnetic resonance images we recently outlined a bayesian scheme for analyzing fmri data using diffusion based spatial priors harrison l m penny w ashburner j trujillo barreto n friston k j 2007 diffusion based spatial priors for imaging neuroimage 38 677 695 the current paper continues this theme applying it to a single subject functional magnetic resonance imaging fmri study of the auditory system we show that spatial priors on functional activations based on diffusion can be formulated in terms of the eigenmodes of a graph laplacian this allows one to discard eigenmodes with small eigenvalues to provide a computationally efficient scheme furthermore this formulation shows that diffusion based priors are a generalization of conventional laplacian priors penny w d trujillo barreto n j friston k j 2005 bayesian fmri time series analysis with spatial priors neuroimage 24 350 362 finally we show how diffusion based priors are a special case of gaussian process models that can be inverted using classical covariance component estimation techniques like restricted maximum likelihood patterson h d thompson r 1974 maximum likelihood estimation of components of variance paper presented at 8th international biometrics conference constanta romania the convention in spm is to smooth data with a fixed isotropic gaussian kernel before inverting a mass univariate statistical model this entails the strong assumption that data are generated smoothly throughout the brain however there is no way to determine if this assumption is supported by the data because data are smoothed before statistical modeling in contrast if a spatial prior is used smoothness is estimated given non smoothed data explicit spatial priors enable formal model comparison of different prior assumptions e g that data are generated from a stationary i e fixed throughout the brain or non stationary spatial process indeed for the auditory data we provide strong evidence for a non stationary process which concurs with a qualitative comparison of predicted activations at the boundary of functionally selective regions

statistical parametric mapping for event related potentials ii a hierarchical temporal model in this paper we describe a temporal model for event related potentials erp in the context of statistical parametric mapping spm in brief we project channel data onto a two dimensional scalp surface or into three dimensional brain space using some appropriate inverse solution we then treat the spatiotemporal data in a mass univariate fashion this implicitly factorises the model into spatial and temporal components the key contribution of this paper is the use of observation models that afford an explicit distinction between observation error and variation in the expression of erps this distinction is created by employing a two level hierarchical model in which the first level models the erp effects within subject and trial type while the second models differences in erp expression among trial types and subjects by bringing the analysis of erp data into a classical hierarchical i e mixed effects framework many apparently disparate approaches e g conventional p300 analyses and time frequency analyses of stimulus locked oscillations can be reconciled within the same estimation and inference procedure inference proceeds in the normal way using t or f statistics to test for effects that are localised in peristimulus time or in some time frequency window the use of f statistics is an important generalisation of classical approaches because it allows one to test for effects that lie in a multidimensional subspace i e of unknown but constrained form we describe the analysis procedures the underlying theory and compare its performance to established techniques

variational free energy and the laplace approximation this note derives the variational free energy under the laplace approximation with a focus on accounting for additional model complexity induced by increasing the number of model parameters this is relevant when using the free energy as an approximation to the log evidence in bayesian model averaging and selection by setting restricted maximum likelihood reml in the larger context of variational learning and expectation maximisation em we show how the reml objective function can be adjusted to provide an approximation to the log evidence for a particular model this means reml can be used for model selection specifically to select or compare models with different covariance components this is useful in the context of hierarchical models because it enables a principled selection of priors that under simple hyperpriors can be used for automatic model selection and relevance determination ard deriving the reml objective function from basic variational principles discloses the simple relationships among variational bayes em and reml furthermore we show that em is formally identical to a full variational treatment when the precisions are linear in the hyperparameters finally we also consider briefly dynamic models and how these inform the regularisation of free energy ascent schemes like em and reml

systematic regularization of linear inverse solutions of the eeg source localization problem distributed linear solutions of the eeg source localization problem are used routinely here we describe an approach based on the weighted minimum norm method that imposes constraints using anatomical and physiological information derived from other imaging modalities to regularize the solution in this approach the hyperparameters controlling the degree of regularization are estimated using restricted maximum likelihood reml eeg data are always contaminated by noise e g exogenous noise and background brain activity the conditional expectation of the source distribution given the data is attained by carefully balancing the minimization of the residuals induced by noise and the improbability of the estimates as determined by their priors this balance is specified by hyperparameters that control the relative importance of fitting and conforming to prior constraints here we introduce a systematic approach to this regularization problem in the context of a linear observation model we have described previously in this model basis functions are extracted to reduce the solution space a priori in the spatial and temporal domains the basis sets are motivated by knowledge of the evoked eeg response and information theory in this paper we focus on an iterative expectation maximization procedure to jointly estimate the conditional expectation of the source distribution and the reml hyperparameters on which this solution rests we used simulated data mixed with real eeg noise to explore the behavior of the approach with various source locations priors and noise levels the results enabled us to conclude i solutions in the space of informed basis functions have a high face and construct validity in relation to conventional analyses ii the hyperparameters controlling the degree of regularization vary largely with source geometry and noise the second conclusion speaks to the usefulness of using adaptative reml hyperparameter estimates

comparison of the spatial resolution of source imaging techniques in high density eeg and meg abstract background the present study aims at evaluating and comparing electrical and magnetic distributed source imaging methods applied to high density electroencephalography hdeeg and magnetoencephalography meg data we used resolution matrices to characterize spatial resolution properties of minimum norm estimate mne dynamic statistical parametric mapping dspm standardized low resolution electromagnetic tomography sloreta and coherent maximum entropy on the mean cmem an entropy based technique the resolution matrix provides information of the point spread functions psf and of the crosstalk functions ct this latter being also called source leakage as it reflects the influence of a source on its neighbors methods the spatial resolution of the inverse operators was first evaluated theoretically and then with real data acquired using electrical median nerve stimulation on five healthy participants we evaluated the dipole localization error dle and the spatial dispersion sd of each psf and ct map results cmem showed the smallest spatial spread sd for both psf and ct maps whereas localization errors dle were similar for all methods whereas cmem sd values were lower in meg compared to hdeeg the other methods slightly favored hdeeg over meg in real data cmem provided similar localization error and significantly less spatial spread than other methods for both meg and hdeeg whereas both meg and hdeeg provided very accurate localizations all the source imaging methods actually performed better in meg compared to hdeeg according to all evaluation metrics probably due to the higher signal to noise ratio of the data in meg conclusion our overall results show that all investigated methods provide similar localization errors suggesting very accurate localization for both meg and hdeeg when similar number of sensors are considered for both modalities intrinsic properties of source imaging methods as well as their behavior for well controlled tasks suggest an overall better performance of cmem in regards to spatial resolution and spatial leakage for both hdeeg and meg this indicates that cmem would be a good candidate for studying source localization of focal and extended generators as well as functional connectivity studies

source localization with meg data a beamforming approach based on covariance thresholding summary reconstructing neural activities using non invasive sensor arrays outside the brain is an ill posed inverse problem since the observed sensor measurements could result from an infinite number of possible neuronal sources the sensor covariance based beamformer mapping represents a popular and simple solution to the above problem in this article we propose a family of beamformers by using covariance thresholding a general theory is developed on how their spatial and temporal dimensions determine their performance conditions are provided for the convergence rate of the associated beamformer estimation the implications of the theory are illustrated by simulations and a real data analysis

meg and eeg data fusion simultaneous localisation of face evoked responses we present an empirical bayesian scheme for distributed multimodal inversion of electromagnetic forward models of eeg and meg signals we used a generative model with common source activity and separate error components for each modality under this scheme the weightings of error for each modality relative to source components are estimated automatically from the data by optimising the model evidence this obviates the need for arbitrary user defined weightings to evaluate the scheme we acquired three types of data simultaneously from twelve participants total magnetic flux as recorded by 102 magnetometers orthogonal in plane gradients of the magnetic field as recorded by 204 planar gradiometers and voltage differences in the electrical field recorded by 70 electrodes we assessed the relative precision of each sensor type in terms of signal to noise ratio snr using empirical sample variances and optimised estimators from the generative model we then compared the localisation of face evoked responses using each modality separately with that obtained by their fusion under the common generative model finally we quantified the conditional precisions of the source estimates using their posterior covariance confirming that eeg can improve meg based source reconstructions

the choquet integral for the aggregation of interval scales in multicriteria decision making this paper addresses the question of which models fit with information concerning the preferences of the decision maker over each attribute and his preferences about aggregation of criteria interacting criteria we show that the conditions induced by these information plus some intuitive conditions lead to a unique possible aggregation operator the choquet integral

distributed non fragile filtering in sensor networks with energy constraints this paper investigates the distributed filtering problem for sensor networks with energy constraint and filter gain variation firstly two strategies are proposed to reduce energy consumption i e the measurement size and the transmission rate reduction strategies to reduce the measurement size a useful measurement selection policy is introduced to choose which element of local measurements is transmitted the selected measurement which has a smaller size than the original one is then stochastically transmitted in order to reflect the imprecision in filter implementation the additive filter gain variation problem is considered based on the switched system approach and some stochastic system analysis methods a sufficient condition is obtained such that the filtering error system is exponentially stable in the mean square sense and a prescribed h performance level is guaranteed filter gains are shown to be determined by solving a set of linear matrix inequalities lmis finally a simulation example is given to demonstrate the effectiveness of the proposed design

approximate bayesian inference in conditionally independent hierarchical models parametric empirical bayes models abstract we consider two stage models of the kind used in parametric empirical bayes peb methodology calling them conditionally independent hierarchical models we suppose that there are k units which may be experimental subjects cities study centers etcetera at the first stage the observation vectors yi for units i 1 k are independently distributed with densities p yi i or more generally p yi i at the second stage the unit specific parameter vectors i are iid with densities p i the peb approach proceeds by regarding the second stage distribution as a prior and noting that if were known inference about could be based on its posterior since is not known the simplest peb methods estimate the parameter by maximum likelihood or some variant and then treat as if it were known to be equal to this estimate although this procedure is sometimes satisfactory a well known defect is that it neglects the uncertainty due to the estimation of in this article w

time frequency mixed norm estimates sparse m eeg imaging with non stationary source activations magnetoencephalography meg and electroencephalography eeg allow functional brain imaging with high temporal resolution while solving the inverse problem independently at every time point can give an image of the active brain at every millisecond such a procedure does not capitalize on the temporal dynamics of the signal linear inverse methods minimum norm dspm sloreta beamformers typically assume that the signal is stationary regularization parameter and data covariance are independent of time and the time varying signal to noise ratio snr other recently proposed non linear inverse solvers promoting focal activations estimate the sources in both space and time while also assuming stationary sources during a time interval however such an hypothesis only holds for short time intervals to overcome this limitation we propose time frequency mixed norm estimates tf mxne which use time frequency analysis to regularize the ill posed inverse problem this method makes use of structured sparse priors defined in the time frequency domain offering more accurate estimates by capturing the non stationary and transient nature of brain signals state of the art convex optimization procedures based on proximal operators are employed allowing the derivation of a fast estimation algorithm the accuracy of the tf mxne is compared to recently proposed inverse solvers with help of simulations and by analyzing publicly available meg datasets

population level inferences for distributed meg source localization under multiple constraints application to face evoked fields we address some key issues entailed by population inference about responses evoked in distributed brain systems using magnetoencephalography meg in particular we look at model selection issues at the within subject level and feature selection issues at the between subject level using responses evoked by intact and scrambled faces around 170 ms m170 we compared the face validity of subject specific forward models and their summary statistics in terms of how estimated responses reproduced over subjects at the within subject level we focused on the use of multiple constraints or priors for inverting distributed source models we used restricted maximum likelihood reml estimates of prior covariance components in both sensor and source space and show that their relative importance is conserved over subjects at the between subject level we used standard anatomical normalization methods to create posterior probability maps that furnish inference about regionally specific population responses we used these to compare different summary statistics namely i whether to test for differences between condition specific source estimates or whether to test the source estimate of differences between conditions and ii whether to accommodate differences in source orientation by using signed or unsigned absolute estimates of source activity

meg source localization under multiple constraints an extended bayesian framework reconstructing the sources of magnetoencephalographic meg measurements is an ill posed inverse problem which does not admit a unique solution forward modeling of the observed signal and prior information or constraints on the solution are needed many types of constraints have been considered however a general framework is required that can accommodate for multiple priors and enables the evaluation of their contribution with this in mind we developed an empirical bayesian framework based on hierarchical linear models our approach entails the estimation of both the expected source distribution and its conditional variance the latter being constrained by an empirically determined mixture of prior variance components corresponding to the constraints one wants to introduce finally a second level of inference based on bayesian model selection enables one to compare different combinations of priors we describe here the methodological framework the estimation scheme and illustrate its application on a single subject meg data set

controlling false positive rates in mass multivariate tests for electromagnetic responses we address the problem of controlling false positive rates in mass multivariate tests for electromagnetic responses in compact regions of source space we show that mass univariate thresholds based on sensor level multivariate thresholds approximated using roy s union intersection principle are unduly conservative we then consider a bonferroni correction for source level tests based on the number of unique lead field extrema for a given source space the sensor indices corresponding to the maxima and minima for each dipolar lead field are listed and the number of unique extrema is given by the number of unique pairs in this list using a multivariate beamformer formulation we validate this heuristic against empirical permutation thresholds for mass univariate and mass multivariate tests of induced and evoked responses for a variety of source spaces using simulated and real data we also show that the same approximations hold when dealing with a cortical manifold rather than a volume and for mass multivariate minimum norm solutions we demonstrate that the mass multivariate framework is not restricted to tests on a single contrast of effects cf roy s maximum root but also accommodates multivariate effects cf wilk s lambda

an empirical bayesian solution to the source reconstruction problem in eeg distributed linear solutions of the eeg source localisation problem are used routinely in contrast to discrete dipole equivalent models distributed linear solutions do not assume a fixed number of active sources and rest on a discretised fully 3d representation of the electrical activity of the brain the ensuing inverse problem is underdetermined and constraints or priors are required to ensure the uniqueness of the solution in a bayesian framework the conditional expectation of the source distribution given the data is attained by carefully balancing the minimisation of the residuals induced by noise and the improbability of the estimates as determined by their priors this balance is specified by hyperparameters that control the relative importance of fitting and conforming to various constraints here we formulate the conventional weighted minimum norm wmn solution in terms of hierarchical linear models an expectation maximisation em algorithm is used to obtain a restricted maximum likelihood reml estimate of the hyperparameters before estimating the maximum a posteriori solution itself this procedure can be considered a generalisation of previous work that encompasses multiple constraints our approach was compared with the classic wmn and maximum smoothness solutions using a simplified 2d source model with synthetic noisy data the reml solution was assessed with four types of source location priors no priors accurate priors inaccurate priors and both accurate and inaccurate priors the reml approach proved useful as 1 the regularisation or influence of the a priori source covariance increased as the noise level increased 2 the localisation error le was negligible when accurate location priors were used 3 when accurate and inaccurate location priors were used simultaneously the solution was not influenced by the inaccurate priors the reml solution was then applied to real somatosensory evoked responses to illustrate the application in an empirical setting c 2004 elsevier inc all rights reserved

diffusion based spatial priors for imaging we describe a bayesian scheme to analyze images which uses spatial priors encoded by a diffusion kernel based on a weighted graph laplacian this provides a general framework to formulate a spatial model whose parameters can be optimized the application we have in mind is a spatiotemporal model for imaging data we illustrate the method on a random effects analysis of fmri contrast images from multiple subjects this simplifies exposition of the model and enables a clear description of its salient features typically imaging data are smoothed using a fixed gaussian kernel as a pre processing step before applying a mass univariate statistical model e g a general linear model to provide images of parameter estimates an alternative is to include smoothness in a multivariate statistical model penny w d trujillo barreto n j friston k j 2005 bayesian fmri time series analysis with spatial priors neuroimage 24 350 362 the advantage of the latter is that each parameter field is smoothed automatically according to a measure of uncertainty given the data in this work we investigate the use of diffusion kernels to encode spatial correlations among parameter estimates nonlinear diffusion has a long history in image processing in particular flows that depend on local image geometry romeny b m t 1994 geometry driven diffusion in computer vision kluwer academic publishers can be used as adaptive filters this can furnish a non stationary smoothing process that preserves features which would otherwise be lost with a fixed gaussian kernel we describe a bayesian framework that incorporates non stationary adaptive smoothing into a generative model to extract spatial features in parameter estimates critically this means adaptive smoothing becomes an integral part of estimation and inference we illustrate the method using synthetic and real fmri data

multiple sparse priors for the m eeg inverse problem this paper describes an application of hierarchical or empirical bayes to the distributed source reconstruction problem in electro and magnetoencephalography eeg and meg the key contribution is the automatic selection of multiple cortical sources with compact spatial support that are specified in terms of empirical priors this obviates the need to use priors with a specific form e g smoothness or minimum norm or with spatial structure e g priors based on depth constraints or functional magnetic resonance imaging results furthermore the inversion scheme allows for a sparse solution for distributed sources of the sort enforced by equivalent current dipole ecd models this means the approach automatically selects either a sparse or a distributed model depending on the data the scheme is compared with conventional applications of bayesian solutions to quantify the improvement in performance

reconstructing anatomy from electro physiological data here we show how it is possible to make estimates of brain structure based on meg data we do this by reconstructing functional estimates onto distorted cortical manifolds parameterised in terms of their spherical harmonics we demonstrate that both empirical and simulated meg data give rise to consistent and plausible anatomical estimates importantly the estimation of structure from meg data can be quantified in terms of millimetres from the true brain structure we show for simulated data that the functional assumptions which are closer to the functional ground truth give rise to anatomical estimates that are closer to the true anatomy