English: This image is Figure 1 of Causal Modelling and Brain Connectivity in Functional Magnetic Resonance Imaging by Karl Friston.

Figure 1. Models of Effective Connectivity
This schematic shows the underlying equations on which dynamic (DCM) and Granger (GCM) causal models are based.
In DCM for fMRI, bilinear differential equations describe the changes in neuronal activity x(t)_i in terms of linearly separable components that reflect the influence of other regional state variables. Known deterministic inputs u(t) elicit a change in neuronal states directly though c_i or increase the coupling parameters a_ij in proportion to the bilinear coupling parameters b_ij. The neuronal states enter a region-specific haemodynamic model to produce the outputs y(t)_i.
GCM tries to model the ensuing dependencies among the outputs with a time-lagged linear regression of the current response on previous responses (up to an order denoted by p).

In both models, the data contain observation noise ε(t) that is added to regional observations. The DCM is effectively a state-space model formulated in continuous time; whereas the GCM is a vector autoregression model in discrete time. See Figure 2 for a fuller explanation of the haemodynamic part of the model.