(A) In this case, a neural influence exists from area 1 to area 2, but not from area 2 to area 1. Hemodynamic response functions are identical (overlapping curves labeled 1 and 2). As the coupling strength (see inset) increases, the coherence increases. As the latency (see x-axis) increases, the phase delay increases. GCS, as for coherence, varies primarily with coupling strength. GCD increases with both parameters, as do the individual components of GCD (F1→2 and F2→1). Note that even without any connection from area 2 to area 1, artifactual but statistically-significant Granger influence can be noted as the latency of coupling increases.

(B) In this case, the asymmetric neural influence from area 1 to area 2 remains unchanged, but the hemodynamic response function for area 2 peaks earlier than that for area 1 (taken from Handwerker et.al., 2007). While the coherence value – which does not depend on HRF shape – remains unchanged, both the phase delay and Granger causality values reflect the shape of the HRF, not the underlying neural coupling. Moreover, as the coupling strength increases, the Granger causality value becomes progressively larger in the direction opposite the neural influence.

(C) In the final case, there is a reciprocal influence between areas 1 and 2. The black arrow pointing from 1 to 2, as for (A) and (B), represents the coupling strength that is modified; the gray arrow from 2 to 1 represents a fixed coupling strength of 0.125 and a latency of 100ms. As both the strength and the latency of the connection from 1 to 2 increase, the phase delay and GCD reflect a shift from negative to positive values.

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