The Relationship of Discrete DCM and Directed Information in fMRI-Based Causality Analysis

This paper explores the conditional equivalence between discrete dynamic causal modeling (DDCM) and directed information (DI) for fMRI-based causality analysis. First, we briefly revisit the DDCM and its relationship with the conventional continuous time dynamic causal modeling. Second, we prove the conditional equivalence between DDCM and DI in characterizing the causal relationship and identifying the causal side of two brain regions based on the fMRI data. Third, we show that the DI-based causal relationship between the neurostates of two brain regions is equivalent to that between the observed BOLD signals. More specifically, in the noise free case, the DI inference is invariant under hemodynamic convolution as long as the system is invertible. The theoretical results are demonstrated using fMRI data obtained under both resting state and stimulus-based state. Our numerical analysis is consistent with that reported in previous study. Our results further confirm the convergence or conditional equivalences among existing causality analysis tools.