Dynamic causal modeling
It uses nonlinear state-space models in continuous time, specified using stochastic or ordinary differential equations.
[1] 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.
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.
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.
[2] 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.
are of key interest, which for example represent connection strengths that may change under different experimental conditions.
The neural model in DCM for fMRI is a Taylor approximation that captures the gross causal influences between brain regions and their change due to experimental inputs (see picture).
A more efficient scheme for resting state data was subsequently introduced which operates in the frequency domain, called DCM for Cross-Spectral Density (CSD).
Regression DCM operates in the frequency domain, but linearizes the model under certain simplifications, such as having a fixed (canonical) haemodynamic response function.
DCM for EEG and MEG data use more biologically detailed neural models than fMRI, due to the higher temporal resolution of these measurement techniques.
Conductance-based models derive from the equivalent circuit representation of the cell membrane developed by Hodgkin and Huxley in the 1950s.
[29] This provides two useful quantities: the log marginal likelihood or model evidence
Generally, this cannot be calculated explicitly and is approximated by a quantity called the negative variational free energy
Neuroimaging studies typically investigate effects that are conserved at the group level, or which differ between subjects.
There are two predominant approaches for group-level analysis: random effects Bayesian Model Selection (BMS)[30] and Parametric Empirical Bayes (PEB).
The analysis pipeline for the BMS approach procedure follows a series of steps: Alternatively, Parametric Empirical Bayes (PEB) [31] can be used, which specifies a hierarchical model over parameters (e.g., connection strengths).
The variational Bayesian methods used for model estimation in DCM are based on the Laplace assumption, which treats the posterior over parameters as Gaussian.
Sampling approaches provide the gold standard; however, they are time-consuming and have typically been used to validate the variational approximations in DCM.
