Granger causality maps for Langevin systems

In this paper, we derive an analytical expression for Granger causality rates in vector Ornstein-Uhlenbeck processes that yields a meaningful solution even where local dynamics are unstable. Applied to Granger causality maps for Langevin systems, this fills in the “holes” left by the earlier approach and gives a computationally efficient way of calculating the maps. We also show that the rate is invariant under rescaling of the overall fluctuation intensity, so that the maps can effectively be calculated for nonlinear dynamical systems.