Directed neural interactions in fMRI: a comparison between Granger Causality and Effective Connectivity

Understanding how neural populations interact is crucial to understand brain function. Most common approaches to infer neural interactions are based on Granger causality (GC) analyses and effective connectivity (EC) models of neural time series. However, an in-depth investigation of the similarity and complementarity of these approaches is currently lacking. GC and EC are classically thought to provide complementary information about the interdependence between neural signals. Whereas GC quantifies the amount of predictability between time series and it is interpreted as a measure of information flow, EC quantifies the amount and sign of the interaction, and it is often interpreted as the causal influence that a neural unit exert over another. Here, we show that, in the context of functional magnetic resonance imaging (fMRI) data analysis and first-order autoregressive models, GC and EC share common assumptions and are mathematically related. More precisely, by defining a corrected version of GC accounting for unequal noise variances affecting the source and target node, we show that the two measures are linked by an approximately quadratic relation, where positive or negative values of EC are associated with identical values of GC. While the relation is obtained in limit of infinite sampling time, we use simulations to show that it can be observed in finite data samples as classically observed in neuroimaging studies, provided sufficiently long sampling, multiple sessions or group averaging. Finally, we compare the GC and EC analyses on fMRI data from the Human Connectome Project, and obtain results consistent with simulation outcomes. While GC and EC analyses do not provide reliable estimates at the single subject or single connection level, they become stable at the group level (more than approximately 20 subjects), where the predicted relation between GC and EC can be clearly observed from the data. To conclude, our study provides a common mathematical framework to make grounded methodological choices in the reconstruction and analysis of directed brain networks from neuroimaging time series.