Nonlinear DCM (Stephan et al. 2008) introduces a parametric matrix D that allows neuronal activity in one region to change the connectivity between other regions (Fig. 3.1 This is in contrast to bilinear dynamics (Eq. 3.2) in which, perhaps unrealistically, effective connectivity can be changed via ‘modulatory inputs’.

The motivation for this extension is to address ‘neuronal gain control’ between two neuronal states that are gated by other states (Stephan et al. 2008). The approach also models the neuronal origin of modulatory influences such as ‘short-term synaptic plasticity’ (Stephan et al. 2008). Applications based on nonlinear DCM can be found in den Ouden et al. (2010) and Table 3.1 (Desseilles et al. 2011; Neufang et al. 2011).

Neuronal activity is linked to fMRI signals via an extended Balloon model (Buxton et al. 1998, 2004) and BOLD signal model (Stephan et al. 2007b). The haemodynamic model specifies how changes in neuronal activity give rise to changes in blood oxygenation that is measured with fMRI. For each region i, neuronal activities are translated into BOLD signals via the interactions between the neuronal state z _i and haemodynamic state variables: the vasodilatory signal, the flow induced, changes in volume, and changes in deoxyhaemoglobin. The observed BOLD signals are produced by a nonlinear model that integrates over the states, y = g(v, q), where the evolution of v and q over time depends on self-regulatory feedback as well as s and f (cf. to Fig. 3 in Friston et al. 2003). The equations for the haemodynamics are described in detail elsewhere (Grubb et al. 1974; Buxton et al. 1998; Mandeville et al. 1999; Friston et al. 2000).

Two classes of prior densities are used in DCM; they are placed over coupling and haemodynamic parameters θ = {A, B, C, θ ^h}. DCM uses ‘shrinkage priors’ over coupling parameters. These are zero-mean Gaussian priors with a variance that is chosen to reflect realistic ranges in effective connectivity seen in fMRI studies. These shrinkage priors move the posterior estimates towards zero, especially when the likelihood has a less precise distribution. For example, the posterior expectation will ‘shrink’ to its prior expectation given a likelihood with a very large variance. However, a likelihood that has high precision (inverse variance) enforces the posterior to deviate from zero. Prior variances can also be chosen to reflect anatomical knowledge, for example, probabilistic tractography (Stephan et al. 2009). Haemodynamic priors in DCM reflect empirical knowledge about blood flow and oxygenation dynamics in the brain (Buxton et al. 1998, 2004). The prior densities of the five haemodynamic parameters θ ^h = {κ, γ, τ, α, ρ} that mediate the interactions among these states are set based on empirical measures (see Eq. (3.3) and Table 3.1 in Friston et al. 2003). These priors have since been updated in light of more recent data (Penny 2012).

DCMs are fitted to data using the Variational Laplace (VL) algorithm described in (Friston et al. 2007). This is an iterative algorithm which approximates the posterior distribution over parameters with a Gaussian distribution. The parameters of this distribution are updated so as to minimise the distance between the approximate and true posterior, in the sense of Kullback-Leibler divergence – a distance measure between probability densities (MacKay 2003). The VL algorithm provides estimates of two quantities. The first is the posterior density over model parameters p(θ|m, y) that can be used to make inferences about model parameters θ. The second is the probability of the data given the model, otherwise known as the model evidence p(y|m).

Inferences about DCM parameters – based on single models – require strong assumptions about network structure based on well-established neurophysiological, cognitive or anatomical evidence. For instance, Almeida and colleagues demonstrated that the uncinate fasciculus, which provides orbitofronto-amygdala circuitry, mediates emotional regulation and is abnormal in bipolar disorder patients, in terms of fractional anisotropy (Almeida et al. 2009). Both single-model studies reviewed here (Almeida et al. 2009, 2011) used a fully connected model to allow all possible coupling parameters to be estimated. They found that left-sided, top-down orbitomedial prefrontal cortex (OMPFC) to amygdala and right-sided bottom-up amygdala to OMPFC distinguished bipolar from major depression.

In general, model evidence is not straightforward to compute, since this computation involves integrating out the dependence on model parameters:

Therefore, an approximation to the model evidence is required. DCM uses the free-energy approximation to the model evidence provided by the VL algorithm. The model evidence, and the VL approximation to it, naturally embodies the accuracy-complexity trade-off that is the hallmark of a good model (Pitt and Myung 2002

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