Figure 7
Three time series \(X_1\), \(X_2\), \(X_3\) are recorded simultaneously (see text for more details). Estimation of the effect of quadratic phase coupling between \(X_1, X_2\) for interacting bifrequencies (\(F_1=8.5\) Hz, \(F_2=10.5\) Hz) on \(X_3\). The panels on the left side refer to the evaluation of the cross-bicoherence phase coupling and the panels on the right side refer to the evaluation of the bivariate Granger causality in the frequency-domain. (A) All cross-bicoherence values are computed in the sector \(Q_{\textrm{I}}\). (B) The cross-bicoherence significance map, showing all bifrequencies (black bins) characterized by significant cross-bicoherence values. (C) The cross-bicoherence plot, showing all significant cross-bicoherence values whose sum is equal to 10.906 in this example. (D) The cross-bicoherence plot for the bifrequencies neighborhood of interest (\(f_1 \in [8,9]\), \(f_2 \in [10,11]\)) (enlargement centered at a green cross mark), shows that the sum of these significant values is equal to 3.604, which yields \(R_\mathrm{BQPC}= 0.330\) (i.e., 3.604/10.906). (E) For the same data, the median unconditional bivariate GC spectrum measures the strength of the causal dependency of the bivariate process \(X_{1,2}\) on \(X_3\) computed with the optimal VAR’s lag model, that is a VAR(7) model in this example. (F) A threshold value \(\theta =0.0301\) is calculated to determine the significant values of Granger causality in the frequency domain. (G) The integration over the curve shows that the sum of all significant values is equal to 0.552. (H) For the spectral interval of interest (\(f \in [18,20]\)), the sum of all significant values is equal to 0.128, which yields \(R_\mathrm{GQPC}= 0.233\) (i.e., 0.128/0.552).
