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In this paper, we revisit the problem of Bayesian model selection (BMS) at the group level. We originally addressed this issue in Stephan et al. (2009), where models are treated as random effects that could differ between subjects, with an unknown population distribution. Here, we extend this work, by (i) introducing the Bayesian omnibus risk (BOR) as a measure of the statistical risk incurred when performing group BMS, (ii) highlighting the difference between random effects BMS and classical random effects analyses of parameter estimates, and (iii) addressing the problem of between group or condition model comparisons. We address the first issue by quantifying the chance likelihood of apparent differences in model frequencies. This leads to the notion of protected exceedance probabilities. The second issue arises when people want to ask “whether a model parameter is zero or not” at the group level. Here, we provide guidance as to whether to use a classical second-level analysis of parameter estimates, or random effects BMS. The third issue rests on the evidence for a difference in model labels or frequencies across groups or conditions. Overall, we hope that the material presented in this paper finesses the problems of group-level BMS in the analysis of neuroimaging and behavioural data. •Some conceptual issues with group-level Bayesian Model Selection are still outstanding.•We provide a complete picture of the statistical risk incurred when performing BMS.•We address the problem of between-group and between-condition comparisons.•We examine the difference between BMS and classical random effects analyses.

The Default Mode Network (DMN) is a brain system that mediates internal modes of cognitive activity, showing higher neural activation when one is at rest. Nowadays, there is a lot of interest in assessing functional interactions between its key regions, but in the majority of studies only association of Blood-oxygen-level dependent (BOLD) activation patterns is measured, so it is impossible to identify causal influences. There are some studies of causal interactions (i.e., effective connectivity), however often with inconsistent results. The aim of the current work is to find a stable pattern of connectivity between four DMN key regions: the medial prefrontal cortex (mPFC), the posterior cingulate cortex (PCC), left and right intraparietal cortex (LIPC and RIPC). For this purpose functional magnetic resonance imaging (fMRI) data from 30 healthy subjects (1000 time points from each one) was acquired and spectral dynamic causal modeling (DCM) on a resting-state fMRI data was performed. The endogenous brain fluctuations were explicitly modeled by Discrete Cosine Set at the low frequency band of 0.0078-0.1 Hz. The best model at the group level is the one where connections from both bilateral IPC to mPFC and PCC are significant and symmetrical in strength (p < 0.05). Connections between mPFC and PCC are bidirectional, significant in the group and weaker than connections originating from bilateral IPC. In general, all connections from LIPC/RIPC to other DMN regions are much stronger. One can assume that these regions have a driving role within the DMN. Our results replicate some data from earlier works on effective connectivity within the DMN as well as provide new insights on internal DMN relationships and brain's functioning at resting state.

Abstract Background Occurrence of low blood taurine concentrations (B-TauC) and predisposing factors to taurine deficiency in English Cocker Spaniels (ECS) are incompletely understood. Objectives Investigate the occurrence of low B-TauC in a Swedish population of ECS and evaluate the association between B-TauC and dog characteristics, clinical variables, and diet composition. Animals One-hundred eighty privately owned ECS. Methods Dogs were prospectively recruited and underwent physical examination, blood analyses, and echocardiographic and ophthalmic examinations. Dogs with clinical signs of congestive heart failure (CHF) also underwent thoracic radiography. Taurine concentrations were analyzed in plasma (EDTA and heparin) and whole blood. Diets consumed by the dogs at the time of the examination were analyzed for dietary taurine- (D-TauC), cysteine- (D-CysC), and methionine concentrations (D-MetC). Results Fifty-three of 180 dogs (29%) had low B-TauC, of which 13 (25%) dogs had clinical and radiographic signs of CHF, increased echocardiographic left ventricular (LV) dimensions and volumes, and impaired LV systolic function. Five (9%) dogs with low B-TauC had retinal abnormalities. Dietary MetC, dietary animal protein source (red/white meat), and age were associated with B-TauC in the final multivariable regression model (P < .001, R2adj = .39). Conclusions and Clinical Importance Low B-TauC suggests that taurine deficiency may play a role in the development of myocardial failure and CHF in ECS. Low D-MetC and diets with red meat as the animal protein source were associated with low B-TauC. Dogs with B-TauC below the normal reference range were older than dogs with normal concentrations.

Functional neuroimaging research on depression has traditionally targeted neural networks associated with the psychological aspects of depression. In this study, instead, we focus on alterations of sensorimotor function in depression. We used resting-state functional MRI data and dynamic causal modeling (DCM) to assess the hypothesis that depression is associated with aberrant effective connectivity within and between key regions in the sensorimotor hierarchy. Using hierarchical modeling of between-subject effects in DCM with parametric empirical Bayes we first established the architecture of effective connectivity in sensorimotor cortices. We found that in (interoceptive and exteroceptive) sensory cortices across participants, the backward connections are predominantly inhibitory, whereas the forward connections are mainly excitatory in nature. In motor cortices these parities were reversed. With increasing depression severity, these patterns are depreciated in exteroceptive and motor cortices and augmented in the interoceptive cortex, an observation that speaks to depressive symptomatology. We established the robustness of these results in a leave-one-out cross-validation analysis and by reproducing the main results in a follow-up dataset. Interestingly, with (nonpharmacological) treatment, depression-associated changes in backward and forward effective connectivity partially reverted to group mean levels. Overall, altered effective connectivity in sensorimotor cortices emerges as a promising and quantifiable candidate marker of depression severity and treatment response.

•Variational Laplace (VL) is a scheme for Bayesian modelling.•VL is widely used in neuroimaging, in particular DCM.•This paper provides a tutorial explanation of the math and algorithms.•New standalone code is provided to enable re-implementation.•The supplementary materials provide worked derivations. This article details a scheme for approximate Bayesian inference, which has underpinned thousands of neuroimaging studies since its introduction 15 years ago. Variational Laplace (VL) provides a generic approach to fitting linear or non-linear models, which may be static or dynamic, returning a posterior probability density over the model parameters and an approximation of log model evidence, which enables Bayesian model comparison. VL applies variational Bayesian inference in conjunction with quadratic or Laplace approximations of the evidence lower bound (free energy). Importantly, update equations do not need to be derived for each model under consideration, providing a general method for fitting a broad class of models. This primer is intended for experimenters and modellers who may wish to fit models to data using variational Bayesian methods, without assuming previous experience of variational Bayes or machine learning. Accompanying code demonstrates how to fit different kinds of model using the reference implementation of the VL scheme in the open-source Statistical Parametric Mapping (SPM) software package. In addition, we provide a standalone software function that does not require SPM, in order to ease translation to other fields, together with detailed pseudocode. Finally, the supplementary materials provide worked derivations of the key equations.

Bayesian model selection (BMS) is a powerful method for determining the most likely among a set of competing hypotheses about the mechanisms that generated observed data. BMS has recently found widespread application in neuroimaging, particularly in the context of dynamic causal modelling (DCM). However, so far, combining BMS results from several subjects has relied on simple (fixed effects) metrics, e.g. the group Bayes factor (GBF), that do not account for group heterogeneity or outliers. In this paper, we compare the GBF with two random effects methods for BMS at the between-subject or group level. These methods provide inference on model-space using a classical and Bayesian perspective respectively. First, a classical (frequentist) approach uses the log model evidence as a subject-specific summary statistic. This enables one to use analysis of variance to test for differences in log-evidences over models, relative to inter-subject differences. We then consider the same problem in Bayesian terms and describe a novel hierarchical model, which is optimised to furnish a probability density on the models themselves. This new variational Bayes method rests on treating the model as a random variable and estimating the parameters of a Dirichlet distribution which describes the probabilities for all models considered. These probabilities then define a multinomial distribution over model space, allowing one to compute how likely it is that a specific model generated the data of a randomly chosen subject as well as the exceedance probability of one model being more likely than any other model. Using empirical and synthetic data, we show that optimising a conditional density of the model probabilities, given the log-evidences for each model over subjects, is more informative and appropriate than both the GBF and frequentist tests of the log-evidences. In particular, we found that the hierarchical Bayesian approach is considerably more robust than either of the other approaches in the presence of outliers. We expect that this new random effects method will prove useful for a wide range of group studies, not only in the context of DCM, but also for other modelling endeavours, e.g. comparing different source reconstruction methods for EEG/MEG or selecting among competing computational models of learning and decision-making.

Changes in response contingencies require adjusting ones assumptions about outcomes of behaviors. Such adaptation processes are driven by reward prediction error (RPE) signals which reflect the inadequacy of expectations. Signals resembling RPEs are known to be encoded by mesencephalic dopamine neurons projecting to the striatum and frontal regions. Although regions that process RPEs, such as the dorsal anterior cingulate cortex (dACC), have been identified, only indirect evidence links timing and network organization of RPE processing in humans. In electroencephalography (EEG), which is well known for its high temporal resolution, the feedback-related negativity (FRN) has been suggested to reflect RPE processing. Recent studies, however, suggested that the FRN might reflect surprise, which would correspond to the absolute, rather than the signed RPE signals. Furthermore, the localization of the FRN remains a matter of debate. In this simultaneous EEG–functional magnetic resonance imaging (fMRI) study, we localized the FRN directly using the superior spatial resolution of fMRI without relying on any spatial constraint or other assumption. Using two different single-trial approaches, we consistently found a cluster within the dACC. One analysis revealed additional activations of the salience network. Furthermore, we evaluated the effect of signed RPEs and surprise signals on the FRN amplitude. We considered that both signals are usually correlated and found that only surprise signals modulate the FRN amplitude. Last, we explored the pathway of RPE signals using dynamic causal modeling (DCM). We found that the surprise signals are directly projected to the source region of the FRN. This finding contradicts earlier theories about the network organization of the FRN, but is in line with a recent theory stating that dopamine neurons also encode surprise-like saliency signals. Our findings crucially advance the understanding of the FRN. We found compelling evidence that the FRN originates from the dACC. Furthermore, we clarified the functional role of the FRN, and determined the role of the dACC within the RPE network. These findings should enable us to study the processing of surprise and adjustment signals in the dACC in healthy and also in psychiatric patients. •The feedback-related negativity (FRN) is associated with surprise signals.•The FRN is rather associated with absolute than signed reward prediction errors.•EEG-informed fMRI consistently locates the FRN in the dorsal anterior cingulum.•Surprise signals are directly projected to the dorsal anterior cingulum.

Metastability is a key source of itinerant dynamics in the brain; namely, spontaneous spatiotemporal reorganization of neuronal activity. This itinerancy has been the focus of numerous dynamic functional connectivity (DFC) analyses – developed to characterize the formation and dissolution of distributed functional patterns over time, using resting state fMRI. However, aside from technical and practical controversies, these approaches cannot recover the neuronal mechanisms that underwrite itinerant (e.g., metastable) dynamics—due to their descriptive, model-free nature. We argue that effective connectivity (EC) analyses are more apt for investigating the neuronal basis of metastability. To this end, we appeal to biologically-grounded models (i.e., dynamic causal modelling, DCM) and dynamical systems theory (i.e., heteroclinic sequential dynamics) to create a probabilistic, generative model of haemodynamic fluctuations. This model generates trajectories in the parametric space of EC modes (i.e., states of connectivity) that characterize functional brain architectures. In brief, it extends an established spectral DCM, to generate functional connectivity data features that change over time. This foundational paper tries to establish the model’s face validity by simulating non-stationary fMRI time series and recovering key model parameters (i.e., transition probabilities among connectivity states and the parametric nature of these states) using variational Bayes. These data are further characterized using Bayesian model comparison (within and between subjects). Finally, we consider practical issues that attend applications and extensions of this scheme. Importantly, the scheme operates within a generic Bayesian framework – that can be adapted to study metastability and itinerant dynamics in any non-stationary time series.

Dynamic causal modeling (DCM) is a generic Bayesian framework for inferring hidden neuronal states from measurements of brain activity. It provides posterior estimates of neurobiologically interpretable quantities such as the effective strength of synaptic connections among neuronal populations and their context-dependent modulation. DCM is increasingly used in the analysis of a wide range of neuroimaging and electrophysiological data. Given the relative complexity of DCM, compared to conventional analysis techniques, a good knowledge of its theoretical foundations is needed to avoid pitfalls in its application and interpretation of results. By providing good practice recommendations for DCM, in the form of ten simple rules, we hope that this article serves as a helpful tutorial for the growing community of DCM users.