All models in DCM have the following basic form: \begin{align} \dot{z}&=f(z,u,\theta^{(n)}) \\ y&=g(z,\theta^{(h)})+\epsilon \end{align} The first equality describes the change in neural activity z with respect to time (i.e. \dot{z}), which cannot be directly observed using non-invasive functional imaging modalities. The evolution of neural activity over time is controlled by a neural function f with parameters \theta^{(n)} and experimental inputs u. The neural activity in turn causes the timeseries y (second equality), which are generated via an observation function g with parameters \theta^{(h)}. Additive observation noise \epsilon completes the observation model. Usually, the neural parameters \theta^{(n)} 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 f and observation model g and setting appropriate
priors over the parameters; e.g. selecting which connections should be switched on or off.
Functional MRI 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). This is coupled with a detailed biophysical model of the generation of the blood oxygen level dependent (BOLD) response and the MRI signal, which was supplemented with a model of neurovascular coupling. Additions to the neural model have included interactions between excitatory and inhibitory neural populations and non-linear influences of neural populations on the coupling between other populations. DCM for resting state studies was first introduced in Stochastic DCM, which estimates both neural fluctuations and connectivity parameters in the time domain, using
Generalized Filtering. A more efficient scheme for resting state data was subsequently introduced which operates in the frequency domain, called DCM for Cross-Spectral Density (CSD). Both of these can be applied to large-scale brain networks by constraining the connectivity parameters based on the functional connectivity. Another recent development for resting state analysis is Regression DCM implemented in the Tapas software collection (see
Software implementations). Regression DCM operates in the frequency domain, but linearizes the model under certain simplifications, such as having a fixed (canonical) haemodynamic response function. The enables rapid estimation of large-scale brain networks.
EEG / MEG DCM for EEG and MEG data use more biologically detailed neural models than fMRI, due to the higher temporal resolution of these measurement techniques. These can be classed into physiological models, which recapitulate neural circuitry, and phenomenological models, which focus on reproducing particular data features. The physiological models can be further subdivided into two classes. Conductance-based models derive from the equivalent circuit representation of the cell membrane developed by Hodgkin and Huxley in the 1950s. Convolution models were introduced by
Wilson & Cowan and Freeman in the 1970s and involve a convolution of pre-synaptic input by a synaptic kernel function. Some of the specific models used in DCM are as follows: • Physiological models: • Convolution models: • DCM for evoked responses (DCM for ERP). This is a biologically plausible neural mass model, extending earlier work by Jansen and Rit. It emulates the activity of a cortical area using three neuronal sub-populations (see picture), each of which rests on two operators. The first operator transforms the pre-synaptic firing rate into a Post-Synaptic Potential (PSP), by
convolving pre-synaptic input with a synaptic response function (kernel). The second operator, a
sigmoid function, transforms the
membrane potential into a firing rate of action potentials. • DCM for LFP (Local Field Potentials). Extends DCM for ERP by adding the effects of specific ion channels on spike generation. • Canonical Microcircuit (CMC). Used to address hypotheses about laminar-specific ascending and descending connections in the brain, which underpin the
predictive coding account of functional brain architectures. The single pyramidal cell population from DCM for ERP is split into deep and superficial populations (see picture). A version of the CMC has been applied to model multi-modal MEG and fMRI data. • Neural Field Model (NFM). Extends the models above into the spatial domain, modelling continuous changes in current across the cortical sheet. • Conductance models: • Neural Mass Model (NMM) and Mean-field model (MFM). These have the same arrangement of neural populations as DCM for ERP, above, but are based on the
Morris-Lecar model of the barnacle muscle fibre, which in turn derives from the
Hodgin and Huxley model of the giant squid axon. • • Phenomenological models: • DCM for phase coupling. Models the interaction of brain regions as Weakly Coupled Oscillators (WCOs), in which the rate of change of phase of one oscillator is related to the phase differences between itself and other oscillators. == Model estimation ==