Where Do DCM and GCM Come From?
Granger causality was developed in the social sciences and economics, where the complexities of real-time biophysical processes are not an issue. It should be noted that the general concept of Granger-Schweder causality  is based on martingale theory and is not tied to the specific linear autoregressive models in Figure 1. Martingales are stochastic or random processes that, in essence, have no memory (where their differentials are known as innovations ). Unfortunately, most random fluctuations in biological systems do have memory, because they are generated by dynamical systems. This confounds the application of GCM to biological time-series. The application of GCM to fMRI time-series coincided almost exactly with the inception of DCM. DCM is a newer approach to time-series data that was developed to overcome the limitations of conventional techniques (like structural equation modelling and GCM) and is based on systems theory. Both have enjoyed a fairly rapid growth in develop