Spatio-temporal Granger causality: a new framework
  That physiological oscillations of various frequencies are present in fMRI signals is the rule, not the exception. Herein, we propose a novel theoretical framework, spatio-temporal Granger causality, which allows us to more reliably and precisely estimate the Granger causality from experimental datasets possessing time-varying properties caused by physiological oscillations. Within this framework, Granger causality is redefined as a global index measuring the directed information flow between two time series with time-varying properties. Both theoretical analyses and numerical examples demonstrate that Granger causality is a monotonically increasing function of the temporal resolution used in the estimation. This is consistent with the general principle of coarse graining, which causes information loss by smoothing out very fine-scale details in time and space. Our results confirm that the Granger causality at the finer spatio-temporal scales considerably outperforms the traditional approach in terms of an improved consistency between two resting-state scans of the same subject. To optimally estimate the Granger causality, the proposed theoretical framework is implemented through a combination of several approaches, such as dividing the optimal time window and estimating the parameters at the fine temporal and spatial scales. Taken together, our approach provides a novel and robust framework for estimating the Granger causality from fMRI, EEG, and other related data. 
Qiang Luo, Wenlian Lu, Wei Cheng, Pedro A. Valdes-Sosa, Xiaotong Wen,   Mingzhou Ding and Jianfeng Feng
stat.AP