Application of Bayesian inference to fMRI data analysis

Multilevel testing of constraints induced by structural equation modeling in fMRI effective connectivity analysis: A proof of concept

In functional MRI (fMRI), effective connectivity analysis aims at inferring the causal influences that brain regions exert on one another. A common method for this type of analysis is structural equation modeling (SEM). We here propose a novel method to test the validity of a given model of structural equation. Given a structural model in the form of a directed graph, the method extracts the set of all constraints of conditional independence induced by the absence of links between pairs of regions in the model and tests for their validity in a Bayesian framework, either individually (constraint by constraint), jointly (e.g., by gathering all constraints associated with a given missing link), or globally (i.e., all constraints associated with the structural model). This approach has two main advantages. First, it only tests what is testable from observational data and does allow for false causal interpretation. Second, it makes it possible to test each constraint (or group of constraints) separately and, therefore, quantify in what measure each constraint (or, e..g., missing link) is respected in the data. We validate our approach using a simulation study and illustrate its potential benefits through the reanalysis of published data.

Bayesian Inference for Functional Dynamics Exploring in fMRI Data

This paper aims to review state-of-the-art Bayesian-inference-based methods applied to functional magnetic resonance imaging (fMRI) data. Particularly, we focus on one specific long-standing challenge in the computational modeling of fMRI datasets: how to effectively explore typical functional interactions from fMRI time series and the corresponding boundaries of temporal segments. Bayesian inference is a method of statistical inference which has been shown to be a powerful tool to encode dependence relationships among the variables with uncertainty. Here we provide an introduction to a group of Bayesian-inference-based methods for fMRI data analysis, which were designed to detect magnitude or functional connectivity change points and to infer their functional interaction patterns based on corresponding temporal boundaries. We also provide a comparison of three popular Bayesian models, that is, Bayesian Magnitude Change Point Model (BMCPM), Bayesian Connectivity Change Point Model (BCCPM), and Dynamic Bayesian Variable Partition Model (DBVPM), and give a summary of their applications. We envision that more delicate Bayesian inference models will be emerging and play increasingly important roles in modeling brain functions in the years to come.

Fast joint detection-estimation of evoked brain activity in event-related FMRI using a variational approach

In standard within-subject analyses of event-related functional magnetic resonance imaging (fMRI) data, two steps are usually performed separately: detection of brain activity and estimation of the hemodynamic response. Because these two steps are inherently linked, we adopt the so-called region-based joint detection-estimation (JDE) framework that addresses this joint issue using a multivariate inference for detection and estimation. JDE is built by making use of a regional bilinear generative model of the BOLD response and constraining the parameter estimation by physiological priors using temporal and spatial information in a Markovian model. In contrast to previous works that use Markov Chain Monte Carlo (MCMC) techniques to sample the resulting intractable posterior distribution, we recast the JDE into a missing data framework and derive a variational expectation-maximization (VEM) algorithm for its inference. A variational approximation is used to approximate the Markovian model in the unsupervised spatially adaptive JDE inference, which allows automatic fine-tuning of spatial regularization parameters. It provides a new algorithm that exhibits interesting properties in terms of estimation error and computational cost compared to the previously used MCMC-based approach. Experiments on artificial and real data show that VEM-JDE is robust to model misspecification and provides computational gain while maintaining good performance in terms of activation detection and hemodynamic shape recovery.

Hemodynamic-informed parcellation of fMRI data in a joint detection estimation framework

Identifying brain hemodynamics in event-related functional MRI (fMRI) data is a crucial issue to disentangle the vascular response from the neuronal activity in the BOLD signal. This question is usually addressed by estimating the so-called hemodynamic response function (HRF). Voxelwise or region-/parcelwise inference schemes have been proposed to achieve this goal but so far all known contributions commit to pre-specified spatial supports for the hemodynamic territories by defining these supports either as individual voxels or a priori fixed brain parcels. In this paper, we introduce a joint parcellation-detection-estimation (JPDE) procedure that incorporates an adaptive parcel identification step based upon local hemodynamic properties. Efficient inference of both evoked activity, HRF shapes and supports is then achieved using variational approximations. Validation on synthetic and real fMRI data demonstrate the JPDE performance over standard detection estimation schemes and suggest it as a new brain exploration tool.

A Bayesian framework for simultaneously modeling neural and behavioral data

Scientists who study cognition infer underlying processes either by observing behavior (e.g., response times, percentage correct) or by observing neural activity (e.g., the BOLD response). These two types of observations have traditionally supported two separate lines of study. The first is led by cognitive modelers, who rely on behavior alone to support their computational theories. The second is led by cognitive neuroimagers, who rely on statistical models to link patterns of neural activity to experimental manipulations, often without any attempt to make a direct connection to an explicit computational theory. Here we present a flexible Bayesian framework for combining neural and cognitive models. Joining neuroimaging and computational modeling in a single hierarchical framework allows the neural data to influence the parameters of the cognitive model and allows behavioral data, even in the absence of neural data, to constrain the neural model. Critically, our Bayesian approach can reveal interactions between behavioral and neural parameters, and hence between neural activity and cognitive mechanisms. We demonstrate the utility of our approach with applications to simulated fMRI data with a recognition model and to diffusion-weighted imaging data with a response time model of perceptual choice.

Variational solution to the joint detection estimation of brain activity in fMRI

We address the issue of jointly detecting brain activity and estimating underlying brain hemodynamics from functional MRI data. We adopt the so-called Joint Detection Estimation (JDE) framework that takes spatial dependencies between voxels into account. We recast the JDE into a missing data framework and derive a Variational Expectation-Maximization (VEM) algorithm for its inference. It follows a new algorithm that has interesting advantages over the previously used intensive simulation methods (Markov Chain Monte Carlo, MCMC): tests on artificial data show that the VEM-JDE is more robust to model mis-specification while additional tests on real data confirm that it achieves similar performance in much less computation time.

Characterization of the hemodynamic modes associated with interictal epileptic activity using a deformable model-based analysis of combined EEG and functional MRI recordings

Simultaneous electroencephalography and functional magnetic resonance imaging (EEG/fMRI) have been proposed to contribute to the definition of the epileptic seizure onset zone. Following interictal epileptiform discharges, one usually assumes a canonical hemodynamic response function (HRF), which has been derived from fMRI studies in healthy subjects. However, recent findings suggest that the hemodynamic properties of the epileptic brain are likely to differ significantly from physiological responses. Here, we propose a simple and robust approach that provides HRFs, defined as a limited set of gamma functions, optimized so as to elicit strong activations after standard model-driven statistical analysis at the single subject level. The method is first validated on healthy subjects using experimental data acquired during motor, visual and memory encoding tasks. Second, interictal EEG/fMRI data measured in 10 patients suffering from epilepsy are analyzed. Results show dramatic changes of activation patterns, depending on whether physiological or pathological assumptions are made on the hemodynamics of the epileptic brain. Our study suggests that one cannot assume a priori that HRFs in epilepsy are similar to the canonical model. This may explain why a significant fraction of EEG/fMRI exams in epileptic patients are inconclusive after standard data processing. The heterogeneous perfusion in epileptic regions indicates that the properties of brain vasculature in epilepsy deserve careful attention.

A bayesian spatio - temporal approach for the analysis of FMRI data with non - stationary noise

In this work, the bayesian framework is used for the analysis of fMRI data. The novelty of the proposed approach is the introduction of a spatio - temporal model used to estimate the variance of the noise across the images and the voxels. The proposed approach is based on a spatio - temporal version of Generalized Linear Model (GLM). To estimate the regression parameters of the GLM as well as the variance components of the noise, the Variational Bayesian (VB) Methodology is employed. The use of VB methodology results in an iterative algorithm, where the estimation of the regression coefficients and the estimation of variance components of the noise, across images and across voxels, are alternated in an elegant and fully automated way. The proposed approach is compared with the Weighted Least Squares (WLS) approach and both methods are evaluated on a real fMRI experiment.

Conjunction analysis and propositional logic in fMRI data analysis using Bayesian statistics

PURPOSE

To evaluate logical expressions over different effects in data analyses using the general linear model (GLM) and to evaluate logical expressions over different posterior probability maps (PPMs).

MATERIALS AND METHODS

In functional magnetic resonance imaging (fMRI) data analysis, the GLM was applied to estimate unknown regression parameters. Based on the GLM, Bayesian statistics can be used to determine the probability of conjunction, disjunction, implication, or any other arbitrary logical expression over different effects or contrast. For second-level inferences, PPMs from individual sessions or subjects are utilized. These PPMs can be combined to a logical expression and its probability can be computed. The methods proposed in this article are applied to data from a STROOP experiment and the methods are compared to conjunction analysis approaches for test-statistics.

RESULTS

The combination of Bayesian statistics with propositional logic provides a new approach for data analyses in fMRI. Two different methods are introduced for propositional logic: the first for analyses using the GLM and the second for common inferences about different probability maps.

CONCLUSION

The methods introduced extend the idea of conjunction analysis to a full propositional logic and adapt it from test-statistics to Bayesian statistics. The new approaches allow inferences that are not possible with known standard methods in fMRI.

Using nonlinear models in fMRI data analysis: model selection and activation detection

There is an increasing interest in using physiologically plausible models in fMRI analysis. These models do raise new mathematical problems in terms of parameter estimation and interpretation of the measured data. In this paper, we show how to use physiological models to map and analyze brain activity from fMRI data. We describe a maximum likelihood parameter estimation algorithm and a statistical test that allow the following two actions: selecting the most statistically significant hemodynamic model for the measured data and deriving activation maps based on such model. Furthermore, as parameter estimation may leave much incertitude on the exact values of parameters, model identifiability characterization is a particular focus of our work. We applied these methods to different variations of the Balloon Model (Buxton, R.B., Wang, E.C., and Frank, L.R. 1998. Dynamics of blood flow and oxygenation changes during brain activation: the balloon model. Magn. Reson. Med. 39: 855-864; Buxton, R.B., Uludağ, K., Dubowitz, D.J., and Liu, T.T. 2004. Modelling the hemodynamic response to brain activation. NeuroImage 23: 220-233; Friston, K. J., Mechelli, A., Turner, R., and Price, C. J. 2000. Nonlinear responses in fMRI: the balloon model, volterra kernels, and other hemodynamics. NeuroImage 12: 466-477) in a visual perception checkerboard experiment. Our model selection proved that hemodynamic models better explain the BOLD response than linear convolution, in particular because they are able to capture some features like poststimulus undershoot or nonlinear effects. On the other hand, nonlinear and linear models are comparable when signals get noisier, which explains that activation maps obtained in both frameworks are comparable. The tools we have developed prove that statistical inference methods used in the framework of the General Linear Model might be generalized to nonlinear models.

A wavelet-based statistical analysis of FMRI data: I. motivation and data distribution modeling

We propose a new method for statistical analysis of functional magnetic resonance imaging (fMRI) data. The discrete wavelet transformation is employed as a tool for efficient and robust signal representation. We use structural magnetic resonance imaging (MRI) and fMRI to empirically estimate the distribution of the wavelet coefficients of the data both across individuals and spatial locations. An anatomical subvolume probabilistic atlas is used to tessellate the structural and functional signals into smaller regions each of which is processed separately. A frequency-adaptive wavelet shrinkage scheme is employed to obtain essentially optimal estimations of the signals in the wavelet space. The empirical distributions of the signals on all the regions are computed in a compressed wavelet space. These are modeled by heavy-tail distributions because their histograms exhibit slower tail decay than the Gaussian. We discovered that the Cauchy, Bessel K Forms, and Pareto distributions provide the most accurate asymptotic models for the distribution of the wavelet coefficients of the data. Finally, we propose a new model for statistical analysis of functional MRI data using this atlas-based wavelet space representation. In the second part of our investigation, we will apply this technique to analyze a large fMRI dataset involving repeated presentation of sensory-motor response stimuli in young, elderly, and demented subjects.

Unsupervised learning and mapping of active brain functional MRI signals based on hidden semi-Markov event sequence models

In this paper, a novel functional magnetic resonance imaging (fMRI) brain mapping method is presented within the statistical modeling framework of hidden semi-Markov event sequence models (HSMESMs). Neural activation detection is formulated at the voxel level in terms of time coupling between the sequence of hemodynamic response onsets (HROs) observed in the fMRI signal, and an HSMESM of the hidden sequence of task-induced neural activations. The sequence of HRO events is derived from a continuous wavelet transform (CWT) of the fMRI signal. The brain activation HSMESM is built from the timing information of the input stimulation protocol. The rich mathematical framework of HSMESMs makes these models an effective and versatile approach for fMRI data analysis. Solving for the HSMESM Evaluation and Learning problems enables the model to automatically detect neural activation embedded in a given set of fMRI signals, without requiring any template basis function or prior shape assumption for the fMRI response. Solving for the HSMESM Decoding problem allows to enrich brain mapping with activation lag mapping, activation mode visualizing, and hemodynamic response function analysis. Activation detection results obtained on synthetic and real epoch-related fMRI data demonstrate the superiority of the HSMESM mapping method with respect to a real application case of the statistical parametric mapping (SPM) approach. In addition, the HSMESM mapping method appears clearly insensitive to timing variations of the hemodynamic response, and exhibits low sensitivity to fluctuations of its shape.

Classification of Anatomical Structures in MR Brain Images Using Fuzzy Parameters

We present an algorithm that automatically segments and classifies the brain structures in a set of magnetic resonance (MR) brain images using expert information contained in a small subset of the image set. The algorithm is intended to do the segmentation and classification tasks mimicking the way a human expert would reason. The algorithm uses a knowledge base taken from a small subset of semiautomatically classified images that is combined with a set of fuzzy indexes that capture the experience and expectation a human expert uses during recognition tasks. The fuzzy indexes are tissue specific and spatial specific, in order to consider the biological variations in the tissues and the acquisition inhomogeneities through the image set. The brain structures are segmented and classified one at a time. For each brain structure the algorithm needs one semiautomatically classified image and makes one pass through the image set. The algorithm uses low-level image processing techniques on a pixel basis for the segmentations, then validates or corrects the segmentations, and makes the final classification decision using higher level criteria measured by the set of fuzzy indexes. We use single-echo MR images because of their high volumetric resolution; but even though we are working with only one image per brain slice, we have multiple sources of information on each pixel: absolute and relative positions in the image, gray level value, statistics of the pixel and its three-dimensional neighborhood and relation to its counterpart pixels in adjacent images. We have validated our algorithm for ease of use and precision both with clinical experts and with measurable error indexes over a Brainweb simulated MR set.

Bayesian second-level analysis of functional magnetic resonance images

We propose a new method for the second-level analysis of functional MRI data based on Bayesian statistics. Our method does not require a computationally costly Bayesian model on the first level of analysis. Rather, modeling for single subjects is realized by means of the commonly applied General Linear Model. On the basis of the resulting parameter estimates for single subjects we calculate posterior probability maps and maps of the effect size for effects of interest in groups of subjects. A comparison of this method with the conventional analysis based on t statistics shows that the new approach is more robust against outliers. Moreover, our method overcomes some of the severe problems of null hypothesis significance tests such as the need to correct for multiple comparisons and facilitates inferences which are hard to formulate in terms of classical inferences.

Functional MRI activity characterization using response time shift estimates from curve evolution

Characterizing the response of the brain to a stimulus based on functional magnetic resonance imaging data is a major challenge due to the fact that the response time delay of the brain may be different from one stimulus phase to the next and from pixel to pixel. To enhance detectability, this work introduces the use of a curve evolution approach that provides separate estimates of the response time shifts at each phase of the stimulus on a pixel-by-pixel basis. The approach relies on a parsimonious but simple model that is nonlinear in the time shifts of the response relative to the stimulus and linear in the gains. To effectively use the response time shift estimates in a subspace detection framework, we implement a robust hypothesis test based on a Laplacian noise model. The algorithm provides a pixel-by-pixel functional characterization of the brain's response. The results based on experimental data show that response time shift estimates, when properly implemented, enhance detectability without sacrificing robustness.

Bayesian technique for investigating linearity in event-related BOLD fMRI

Event-related BOLD fMRI data is modeled as a linear time-invariant system. Together with Bayesian inference techniques, a statistical test is developed for rigorously detecting linearity/nonlinearity in the BOLD response system. The test is applied to data collected from eight subjects using an event-related paradigm with a switching checkerboard as the visual stimulus. Analyzed as a group, the results clearly find the response to be nonlinear. When each subject is analyzed individually, however, the results are predominantly nonlinear, but there is some evidence to suggest that there may be a crossover from a linear to a nonlinear regime and vice versa. This could be important when estimating physiological parameters for individuals. Additionally, estimates of the hemodynamic response function and corresponding response were obtained, but there was no consistent appearance of a poststimulus undershoot in the event-related BOLD response.

Anisotropic 2-D and 3-D averaging of fMRI signals

A novel method for denoising functional magnetic resonance imaging temporal signals is presented in this note. The method is based on progressively enhancing the temporal signal by means of adaptive anisotropic spatial averaging. This average is based on a new metric for comparing temporal signals corresponding to active fMRI regions. Examples are presented both for simulated and real two and three-dimensional data. The software implementing the proposed technique is publicly available for the research community.

Modeling the haemodynamic response in fMRI using smooth FIR filters

Modeling the haemodynamic response in functional magnetic resonance (fMRI) experiments is an important aspect of the analysis of functional neuroimages. This has been done in the past using parametric response function, from a limited family. In this contribution, we adopt a semi-parametric approach based on finite impulse response (FIR) filters. In order to cope with the increase in the number of degrees of freedom, we introduce a Gaussian process prior on the filter parameters. We show how to carry on the analysis by incorporating prior knowledge on the filters, optimizing hyper-parameters using the evidence framework, or sampling using a Markov Chain Monte Carlo (MCMC) approach. We present a comparison of our model with standard haemodynamic response kernels on simulated data, and perform a full analysis of data acquired during an experiment involving visual stimulation.