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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.

Stimulus repetition normally causes reduced neural activity in brain regions that process that stimulus. Some theories claim that this “repetition suppression” reflects local mechanisms such as neuronal fatigue or sharpening within a region, whereas other theories claim that it results from changed connectivity between regions, following changes in synchrony or top-down predictions. In this study, we applied dynamic causal modeling (DCM) on a public fMRI dataset involving repeated presentations of faces and scrambled faces to test whether repetition affected local (self-connections) and/or between-region connectivity in left and right early visual cortex (EVC), occipital face area (OFA) and fusiform face area (FFA). Face “perception” (faces versus scrambled faces) modulated nearly all connections, within and between regions, including direct connections from EVC to FFA, supporting a non-hierarchical view of face processing. Face “recognition” (familiar versus unfamiliar faces) modulated connections between EVC and OFA/FFA, particularly in the left hemisphere. Most importantly, immediate and delayed repetition of stimuli were also best captured by modulations of connections between EVC and OFA/FFA, but not self-connections of OFA/FFA, consistent with synchronization or predictive coding theories, though also possibly reflecting local mechanisms like synaptic depression.

Advances in social neuroscience have made neural signatures of social exchange measurable simultaneously across people. This has identified brain regions differentially active during social interaction between human dyads, but the underlying systems-level mechanisms are incompletely understood. This paper introduces dynamic causal modeling and Bayesian model comparison to assess the causal and directed connectivity between two brains in the context of hyperscanning (h-DCM). In this setting, correlated neuronal responses become the data features that have to be explained by models with and without between-brain (effective) connections. Connections between brains can be understood in the context of generalized synchrony, which explains how dynamical systems become synchronized when they are coupled to each another. Under generalized synchrony, each brain state can be predicted by the other brain or a mixture of both. Our results show that effective connectivity between brains is not a feature within dyads per se but emerges selectively during social exchange. We demonstrate a causal impact of the sender's brain activity on the receiver of information, which explains previous reports of two-brain synchrony. We discuss the implications of this work; in particular, how characterizing generalized synchrony enables the discovery of between-brain connections in any social contact, and the advantage of h-DCM in studying brain function on the subject level, dyadic level, and group level within a directed model of (between) brain function.

This technical note introduces adiabatic dynamic causal modelling, a method for inferring slow changes in biophysical parameters that control fluctuations of fast neuronal states. The application domain we have in mind is inferring slow changes in variables (e.g., extracellular ion concentrations or synaptic efficacy) that underlie phase transitions in brain activity (e.g., paroxysmal seizure activity). The scheme is efficient and yet retains a biophysical interpretation, in virtue of being based on established neural mass models that are equipped with a slow dynamic on the parameters (such as synaptic rate constants or effective connectivity). In brief, we use an adiabatic approximation to summarise fast fluctuations in hidden neuronal states (and their expression in sensors) in terms of their second order statistics; namely, their complex cross spectra. This allows one to specify and compare models of slowly changing parameters (using Bayesian model reduction) that generate a sequence of empirical cross spectra of electrophysiological recordings. Crucially, we use the slow fluctuations in the spectral power of neuronal activity as empirical priors on changes in synaptic parameters. This introduces a circular causality, in which synaptic parameters underwrite fast neuronal activity that, in turn, induces activity-dependent plasticity in synaptic parameters. In this foundational paper, we describe the underlying model, establish its face validity using simulations and provide an illustrative application to a chemoconvulsant animal model of seizure activity.

This technical note presents a dynamic causal modelling (DCM) procedure for evaluating different models of neurovascular coupling in the human brain – using combined electromagnetic (M/EEG) and functional magnetic resonance imaging (fMRI) data. This procedure compares the evidence for biologically informed models of neurovascular coupling using Bayesian model comparison. First, fMRI data are used to localise regionally specific neuronal responses. The coordinates of these responses are then used as the location priors in a DCM of electrophysiological responses elicited by the same paradigm. The ensuing estimates of model parameters are then used to generate neuronal drive functions, which model pre- or post-synaptic activity for each experimental condition. These functions form the input to a model of neurovascular coupling, whose parameters are estimated from the fMRI data. Crucially, this enables one to evaluate different models of neurovascular coupling, using Bayesian model comparison – asking, for example, whether instantaneous or delayed, pre- or post-synaptic signals mediate haemodynamic responses. We provide an illustrative application of the procedure using a single-subject auditory fMRI and MEG dataset. The code and exemplar data accompanying this technical note are available through the statistical parametric mapping (SPM) software.

This paper asks whether integrating multimodal EEG and fMRI data offers a better characterisation of functional brain architectures than either modality alone. This evaluation rests upon a dynamic causal model that generates both EEG and fMRI data from the same neuronal dynamics. We introduce the use of Bayesian fusion to provide informative (empirical) neuronal priors – derived from dynamic causal modelling (DCM) of EEG data – for subsequent DCM of fMRI data. To illustrate this procedure, we generated synthetic EEG and fMRI timeseries for a mismatch negativity (or auditory oddball) paradigm, using biologically plausible model parameters (i.e., posterior expectations from a DCM of empirical, open access, EEG data). Using model inversion, we found that Bayesian fusion provided a substantial improvement in marginal likelihood or model evidence, indicating a more efficient estimation of model parameters, in relation to inverting fMRI data alone. We quantified the benefits of multimodal fusion with the information gain pertaining to neuronal and haemodynamic parameters – as measured by the Kullback-Leibler divergence between their prior and posterior densities. Remarkably, this analysis suggested that EEG data can improve estimates of haemodynamic parameters; thereby furnishing proof-of-principle that Bayesian fusion of EEG and fMRI is necessary to resolve conditional dependencies between neuronal and haemodynamic estimators. These results suggest that Bayesian fusion may offer a useful approach that exploits the complementary temporal (EEG) and spatial (fMRI) precision of different data modalities. We envisage the procedure could be applied to any multimodal dataset that can be explained by a DCM with a common neuronal parameterisation.