by Paus in this volume for alternative approaches in humans). The values of A represent the influences of system elements over each other and thus correspond to the effective connectivity within the system. Finally, the values of the matrix C represent the magnitude of the direct effects that external (e.g. sensory) inputs have on system elements. By setting a particular parameter cij to be zero, we disallow for a direct effect of the external input Uj on xi (see Fig. 1 for an example). A and C represent the system parameters (0) that one needs to estimate when fitting this model to measured data. Simple linear models of this kind have found widespread application in various scientific disciplines (von Bertalanffy 1969). In Sect. 6, we will see that Dynamic Causal Modelling (DCM, Friston et al. 2003) extends the above formulation by bilinear terms that model context-dependencies of intrinsic connection strengths.

It should be noted that the framework outlined here is concerned with dynamic systems in continuous time and thus uses differential equations. The same basic ideas, however, can also be applied to dynamic systems in discrete time (using difference equations), as well as to "static" systems where the system is at equilibrium at each point of observation. The latter perspective, which is useful for regression-like equations, is used by classic system models for functional neuroimaging data, e.g. psycho-physiological interactions (PPI; Friston et al. 1997), structural equation modeling (SEM; Mcintosh et al. 1994; B├╝chel & Friston 1997) or multivariate autoregressive models (MAR; Harrison et al. 2003; Gobel et al. 2003). These will be described in the following sections.

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