  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.
