Reconstructing the direction of information flow between brain regions ("causality") is crucial when studying evidence-based network models of the brain. In functional MRI (fMRI), the relatively low signal to noise ratio (SNR) and short acquisitions duration pose significant challenges when analyzing brain region which interact with high redundancy. In this context, classical bivariate Granger Causality (GC) results in massive overestimation of artefactual causal links, hampering realistic brain network representation. Using Multivariate Vector Autoregressive Models (MVAR), we aimed to extend bivariate Granger causality in fMRI to measure the true additional directed influence between two signals in the presence of a large number of additional interdependent signals. We call this approach Globally Conditioned Granger Causality (GCGC).

Estimating GCGC from variable Y_t to the variable X_t (Y→ X) amounts to testing the null-hypothesis that knowledge of the past of Y_t does not improve the prediction of the future of X_t. To this end, in the GCGC approach we employ two models: first, the "restricted" AR model for X­_t, which includes the past values of X_t itself and Z_t, which accounts for all other variables except Y_t. Second, the "unrestricted" AR model, which includes all variables X_t, Y_t, and Z_t [1]

\begin{equation}X_t={\mathbf{A}}\,(X_{t-1}^{(m)}\oplus{Z}_{t-1}^{(m)})+{\varepsilon_t}'\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(restricted model)}\end{equation}

\begin{equation}X_t={\mathbf{A'}}\,(X_{t-1}^{(m)}\oplus{Y}_{t-1}^{(m)}\oplus{Z}_{t-1}^{(m)})+{\varepsilon_t}'\,\,\,\,\,\,\text{(unrestricted model)}\end{equation}

Classical bivariate GC does not explicitly account for Z_t. The determinants of the estimated variance matrices are then used to define the so called causality "strength" $$$\mathcal{F}$$$ as the natural logarithm of the ratio of the two determinants obtained from the restricted and unrestricted model [2, 3]:

$${\mathcal{F}_{Y\rightarrow{X}|Z}}=\ln{\left(\frac{\det{\Sigma}}{\det{\Sigma'}}\right)}$$

Where $$$\Sigma$$$ and $$$\Sigma'$$$ are the covariance matrices of $$${\varepsilon_t}$$$ and $$${\varepsilon_t}'$$$ respectively. $$${\mathcal{F}_{Y\rightarrow{X}|Z}}$$$ follows a $$$\chi^2$$$-distribution; here, a non-parametric bootstrap routine is employed for significance thresholding [4, 5].

First, we characterize CGCG as a function of network density through massively parallel simulation of networks of interacting noisy duffing oscillators (Figure 1). The ability of detecting true causal links while rejecting false causal links is quantified as the area under the Receiver Operating Characteristic (ROC) curve (AUC) constructed as a function of the threshold in causality strength employed in accepting the existence of a causal connection (Figure 2). Successively, we explore the structure of GCGC-based networks in the human brain in functional MRI (fMRI) data from 100 unrelated healthy subjects scanned at rest at 3T (4 sessions of 1200 volumes per subject, TR=0.72s, SMS multiband) within the HCP database. To this end, we locally aggregate the average BOLD signal 116 regions of interest (ROIs) using the Automated Anatomical Labeling (AAL) atlas [6]. Additionally, we generalize the definition of GCGC to the multivariate case and apply this framework to the first-time study of true directed interactions between whole-brain sub-networks (Sensory-Motor, Default-Mode, Left/Right executive, Salience, Primary-Visual, Secondary-Visual) which are pre-defined on the basis of anatomical localization within the AAL.Figure 3 shows the results of comparing bivariate GC to the GCGC approach in synthetic networks as a function of autoregressive order and network density. Across all networks and all orders, the GCGC approach resulted in a notably higher AUC than the bivariate GC approach. Figure 4 shows median network strength matrices across all 100 subjects as well as an index of matrix asymmetry (see Figure caption). Figure 5 shows the results of computing between-network multivariate GC in the 100 unrelated HCP subjects.The GCGC framework consistently provides higher sensitivity and specificity in detecting causal links between noisy sources, with increasing benefit (with respect to the bivariate case) as network density increases. The HCP GCGC matrices reveal a clear-cut underlying sparse networks structure with extremely high consistency across all 100 HCP subjects which is not discernible using the classical GC approach. Also, multivariate GCGC analysis demonstrated first-time evidence of a concerted directional interaction between different subnetworks of the brain Interestingly, we found that the salience network, a group of regions which play a critical role in integrating environmental sensory data with internal “visceral, autonomic, and hedonic markers” [7] sends “top-down” inputs to the brain circuits involved in sensory-motor and cognitive processes as well as in the “default-mode” neural functioning. Therefore, it may be that the process of salience attribution re-orients or re-directs the other networks (and ultimately the subject’s behavior) towards stimuli identified as fundamental for survival. While this interpretation remains speculative pending task-based validation studies, we have shown that the GCGC framework provides a superior tool to study whole-brain effective connectivity, and multivariate GCGC analysis provides an additional layer of information with respect to classical bivariate approaches.