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Fig. 3. A simple bilinear extension of the linear dynamic system shown in Fig. 1. This is an example for a concrete implementation of the neural state equation of DCM for fMRI. Note the role of the bilinear terms which model context-dependent (additive) changes of the strengths of the connections from the right to the left hemisphere (circled elements in the B matrix)

Fig. 3. A simple bilinear extension of the linear dynamic system shown in Fig. 1. This is an example for a concrete implementation of the neural state equation of DCM for fMRI. Note the role of the bilinear terms which model context-dependent (additive) changes of the strengths of the connections from the right to the left hemisphere (circled elements in the B matrix)

Friston et al. (2000). Briefly, it consists of a set of differential equations that describe the relations between four haemodynamic state variables, using five parameters (6h). More specifically, changes in neural activity elicit a vasodila-tory signal that leads to increases in blood flow and subsequently to changes in blood volume and deoxyhemoglobin content. The predicted BOLD signal is a non-linear function of blood volume and deoxyhemoglobine content. Details of the haemodynamic model can be found in other publications (Friston et al. 2000; Stephan et al. 2004). Figure 4 provides a conceptual overview of DCM for fMRI.

The combined neural and haemodynamic parameter set 0 = {0n,0h} is estimated from the measured BOLD data, using a fully Bayesian approach with empirical priors for the haemodynamic parameters and conservative shrinkage priors for the coupling parameters. Details of the parameter estimation scheme, which rests on a gradient ascent procedure embedded into an expectation maximization (EM) algorithm and uses a Laplace (i.e. Gaussian) approximation to the true posterior, can be found in Friston (2002). Eventually, the posterior distributions of the parameter estimates can be used to test hypotheses about connection strengths. Usually, these hypotheses concern context-dependent changes in coupling. If there is uncertainty about the connectional structure of the modeled system, or if one would like to compare competing hypotheses (represented by different DCMs), a Bayesian model selection procedure can be used to find the DCM that exhibits an optimal balance between model fit and model complexity (Penny et al. 2004b).

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