Dynamic causal modeling (DCM) is a framework for specifying models, fitting them to data and comparing their evidence using Bayesian model comparison. It uses nonlinear state-space models in continuous time, specified using stochastic or ordinary differential equations. DCM was initially developed for testing hypotheses about neural dynamics. In this setting, differential equations describe the interaction of neural populations, which directly or indirectly give rise to functional neuroimaging data e.g., functional magnetic resonance imaging (fMRI), magnetoencephalography (MEG) or electroencephalography (EEG). Parameters in these models quantify the directed influences or effective connectivity among neuronal populations, which are estimated from the data using Bayesian statistical methods.
Procedure
DCM is typically used to estimate the coupling among brain regions and the changes in coupling due to experimental changes (e.g., time or context). A model of interacting neural populations is specified, with a level of biological detail dependent on the hypotheses and available data. This is coupled with a forward model describing how neural activity gives rise to measured responses. Estimating the generative model identifies the parameters (e.g. connection strengths) from the observed data. Bayesian model comparison is used to compare models based on their evidence, which can then be characterised in terms of parameters.
DCM studies typically involve the following stages:
Experimental design. Specific hypotheses are formulated and an experiment is conducted. Data preparation. The acquired data are pre-processed (e.g., to select relevant data features and remove confounds). Model specification. One or more forward models (DCMs) are specified for each dataset. Model estimation. The model(s) are fitted to the data to determine their evidence and parameters. Model comparison. The evidence for each model is used for Bayesian Model Comparison (at the single-subject level or at the group level) to select the best model(s). Bayesian model averaging (BMA) is used to compute a weighted average of parameter estimates over different models.
The key stages are briefly reviewed below.
Experimental design
Functional neuroimaging experiments are typically either task-based or examine brain activity at rest (resting state). In task-based experiments, brain responses are evoked by known deterministic inputs (experimentally controlled stimuli). These experimental variables can change neural activity through direct influences on specific brain regions, such as evoked potentials in the early visual cortex, or via a modulation of coupling among neural populations; for example, the influence of attention. These two types of input - driving and modulatory - are parameterized separately in DCM. To enable efficient estimation of driving and modulatory effects, a 2x2 factorial experimental design is often used - with one factor serving as the driving input and the other as the modulatory input.
Resting state experiments have no experimental manipulations within the period of the neuroimaging recording. Instead, hypotheses are tested about the coupling of endogenous fluctuations in neuronal activity, or in the differences in connectivity between sessions or subjects. The DCM framework includes models and procedures for analysing resting state data, described in the next section.
Model specification
All models in DCM have the following basic form:
The first equality describes the change in neural activity with respect to time (i.e. ), which cannot be directly observed using non-invasive functional imaging modalities. The evolution of neural activity over time is controlled by a neural function with parameters and experimental inputs . The neural activity in turn causes the timeseries (second equality), which are generated via an observation function with parameters . Additive observation noise completes the observation model. Usually, the neural parameters are of key interest, which for example represent connection strengths that may change under different experimental conditions.
Specifying a DCM requires selecting a neural model and observation model and setting appropriate priors over the parameters; e.g. selecting which connections should be switched on or off.