Optimally estimating Granger causality
It is a rule, rather than an exception, that physiological oscillations of various frequencies are present in fMRI signals. Here we presented a theoretical framework to show how to more reliably and precisely estimate Granger causality from experimental datasets with time-varying properties caused by the physiological oscillations. Within this framework, the Granger causality was redefined as a global index measuring directed information flow between time series with the time-varying properties. Both theoretical analyses and numerical examples demonstrated that the Granger causality is an increase function of the temporal resolution used in estimation, reflecting the general physics principle of coarse graining, which causes information loss by smoothing out very fine-scale details both in time and space. This was also confirmed by results that the Granger causality on the finer spatial-temporal scales considerably outperforms the traditional approach in terms of the improved consistency between two resting-state scans for the same subject. For practice, the proposed theoretical framework was implemented by combining together several modules, including the optimal time window dividing, parameter estimation in the fine temporal and spatial scales to optimally estimate the Granger causality. Taking together, our approach provided a novel and robust framework to estimate the Granger causality from the fMRI, EEG and related data.