Figure 2. Theoretical profiles of spectral and causality measures computed for the two-node system of Stokes and Purdon 12 (Example 2, where unidirectional causality is imposed from node 1 to node 2).
The system is studied setting a resonance frequency of 50 Hz for the transmitter and of 10 Hz (top row panels), 30 Hz (mid row panels) and 50 Hz (bottom row panels) for the receiver. The fact that the power spectral density (PSD) of the transmitter (S 11(f), a) is the same for the three cases induces, together with the unaltered coefficients determining the causal effects, the same profile for the directed coherence DC 1→2(f) ( c) and the spectral GGC measure GG 1→2(f) ( d). However, the different causal contribution of the transmitter on the receiver is revealed by the partial spectrum S 2|1(f)=S 22(f)·DC 1→2(f) (orange line in ( b)), which quantifies the portion of the overall PSD of the receiver (S 22(f), purple line in ( b)) that is causally explained by the transmitter; the non-explained part (S 2|2(f), green line in ( b)) reflects the autonomous dynamics of the receiver.