Variational Bayesian Parameter Estimation Techniques for the General Linear Model

Variational Bayes (VB), variational maximum likelihood (VML), restricted maximum likelihood (ReML), and maximum likelihood (ML) are cornerstone parametric statistical estimation techniques in the analysis of functional neuroimaging data. However, the theoretical underpinnings of these model parameter estimation techniques are rarely covered in introductory statistical texts. Because of the widespread practical use of VB, VML, ReML, and ML in the neuroimaging community, we reasoned that a theoretical treatment of their relationships and their application in a basic modeling scenario may be helpful for both neuroimaging novices and practitioners alike. In this technical study, we thus revisit the conceptual and formal underpinnings of VB, VML, ReML, and ML and provide a detailed account of their mathematical relationships and implementational details. We further apply VB, VML, ReML, and ML to the general linear model (GLM) with non-spherical error covariance as commonly encountered in the first-level analysis of fMRI data. To this end, we explicitly derive the corresponding free energy objective functions and ensuing iterative algorithms. Finally, in the applied part of our study, we evaluate the parameter and model recovery properties of VB, VML, ReML, and ML, first in an exemplary setting and then in the analysis of experimental fMRI data acquired from a single participant under visual stimulation

Neural surprise in somatosensory Bayesian learning

Tracking statistical regularities of the environment is important for shaping human behavior and perception. Evidence suggests that the brain learns environmental dependencies using Bayesian principles. However, much remains unknown about the employed algorithms, for somesthesis in particular. Here, we describe the cortical dynamics of the somatosensory learning system to investigate both the form of the generative model as well as its neural surprise signatures. Specifically, we recorded EEG data from 40 participants subjected to a somatosensory roving-stimulus paradigm and performed single-trial modeling across peri-stimulus time in both sensor and source space. Our Bayesian model selection procedure indicates that evoked potentials are best described by a non-hierarchical learning model that tracks transitions between observations using leaky integration. From around 70ms post-stimulus onset, secondary somatosensory cortices are found to represent confidence-corrected surprise as a measure of model inadequacy. Indications of Bayesian surprise encoding, reflecting model updating, are found in primary somatosensory cortex from around 140ms. This dissociation is compatible with the idea that early surprise signals may control subsequent model update rates. In sum, our findings support the hypothesis that early somatosensory processing reflects Bayesian perceptual learning and contribute to an understanding of its underlying mechanisms

Early haemodynamic changes observed in patients with epilepsy, in a visual experiment and in simulations

Objective: The objective of this study was to investigate whether previously reported early blood oxygen level dependent (BOLD) changes in epilepsy could occur as a result of the modelling techniques rather than physiological changes. Methods: EEG-fMRI data were analysed from seven patients with focal epilepsy, six control subjects undergoing a visual experiment, in addition to simulations. In six separate analyses the event timing was shifted by either -9,-6,-3,+3,+6 or +9 s relative to the onset of the interictal epileptiform discharge (IED) or stimulus. Results: The visual dataset and simulations demonstrated an overlap between modelled haemodynamic response function (HRF) at event onset and at \ub13 s relative to onset, which diminished at \ub16 s. Pre-spike analysis at -6 s improved concordance with the assumed IED generating lobe relative to the standard HRF in 43% of patients. Conclusion: The visual and simulated dataset findings indicate a form of "temporal bleeding", an overlap between the modelled HRF at time 0 and at \ub13 s which attenuated at \ub16 s. Pre-spike analysis at -6 s may improve concordance. Significance: This form of analysis should be performed at 6 s prior to onset of IED to minimise temporal bleeding effect. The results support the presence of relevant BOLD responses occurring prior to IEDs

EEG-fMRI Based Information Theoretic Characterization of the Human Perceptual Decision System

The modern metaphor of the brain is that of a dynamic information processing device. In the current study we investigate how a core cognitive network of the human brain, the perceptual decision system, can be characterized regarding its spatiotemporal representation of task-relevant information. We capitalize on a recently developed information theoretic framework for the analysis of simultaneously acquired electroencephalography (EEG) and functional magnetic resonance imaging data (fMRI) (Ostwald et al. (2010), NeuroImage 49: 498–516). We show how this framework naturally extends from previous validations in the sensory to the cognitive domain and how it enables the economic description of neural spatiotemporal information encoding. Specifically, based on simultaneous EEG-fMRI data features from n = 13 observers performing a visual perceptual decision task, we demonstrate how the information theoretic framework is able to reproduce earlier findings on the neurobiological underpinnings of perceptual decisions from the response signal features' marginal distributions. Furthermore, using the joint EEG-fMRI feature distribution, we provide novel evidence for a highly distributed and dynamic encoding of task-relevant information in the human brain

Multimodal Functional Network Connectivity: An EEG-fMRI Fusion in Network Space

EEG and fMRI recordings measure the functional activity of multiple coherent networks distributed in the cerebral cortex. Identifying network interaction from the complementary neuroelectric and hemodynamic signals may help to explain the complex relationships between different brain regions. In this paper, multimodal functional network connectivity (mFNC) is proposed for the fusion of EEG and fMRI in network space. First, functional networks (FNs) are extracted using spatial independent component analysis (ICA) in each modality separately. Then the interactions among FNs in each modality are explored by Granger causality analysis (GCA). Finally, fMRI FNs are matched to EEG FNs in the spatial domain using network-based source imaging (NESOI). Investigations of both synthetic and real data demonstrate that mFNC has the potential to reveal the underlying neural networks of each modality separately and in their combination. With mFNC, comprehensive relationships among FNs might be unveiled for the deep exploration of neural activities and metabolic responses in a specific task or neurological state

An information theoretic approach to EEG-fMRI integration

The integration of signals from electro-encephalography (EEG) and functional magnetic resonance imaging (fMRI), acquired simultaneously from the same observer, holds great potential for the elucidation of the neurobiological underpinnings of human brain function. However, the most appropriate way in which to combine the data in order to achieve this goal is not clear. In this thesis, a symmetric and data-driven route to the integration of multimodal functional brain imaging data based on information theory is proposed. As a proof of principle, the framework, which was originally developed in the study of neuronal population codes, is applied in the experimental context of visually evoked responses and the neural underpinnings of visual perceptual decisions. The implications, benefits, and limitations of this theoretical framework for the analysis of simultaneously acquired EEG and fMRI data are discussed

Variational Bayesian Parameter Estimation Techniques for the General Linear Model

Variational Bayes (VB), variational maximum likelihood (VML), restricted maximum likelihood (ReML), and maximum likelihood (ML) are cornerstone parametric statistical estimation techniques in the analysis of functional neuroimaging data. However, the theoretical underpinnings of these model parameter estimation techniques are rarely covered in introductory statistical texts. Because of the widespread practical use of VB, VML, ReML, and ML in the neuroimaging community, we reasoned that a theoretical treatment of their relationships and their application in a basic modeling scenario may be helpful for both neuroimaging novices and practitioners alike. In this technical study, we thus revisit the conceptual and formal underpinnings of VB, VML, ReML, and ML and provide a detailed account of their mathematical relationships and implementational details. We further apply VB, VML, ReML, and ML to the general linear model (GLM) with non-spherical error covariance as commonly encountered in the first-level analysis of fMRI data. To this end, we explicitly derive the corresponding free energy objective functions and ensuing iterative algorithms. Finally, in the applied part of our study, we evaluate the parameter and model recovery properties of VB, VML, ReML, and ML, first in an exemplary setting and then in the analysis of experimental fMRI data acquired from a single participant under visual stimulation