2.3.2. Phase Slope Index

The phase slope index (PSI) is an effective brain connectivity measure that assesses the direction of coupling between two oscillatory signals of similar frequencies [13]. Given two time series **<sup>x</sup>** = {*xi*}*<sup>l</sup> <sup>t</sup>*=<sup>1</sup> and **<sup>y</sup>** = {*yi*}*<sup>l</sup> <sup>t</sup>*=1, the PSI is defined as the slope of the phase of the cross-spectra between **x** and **y**:

$$PSI(\mathbf{x}\rightarrow\mathbf{y}) = \Diamond(\sum\_{f\in F} \mathbb{C}\_{\mathbf{xy}}^\*(f)\mathbb{C}\_{\mathbf{xy}}(f+df)),\tag{11}$$

where *C***xy**(*f*) = *S***xy**/ *S***xy**, *<sup>S</sup>***xy** is the complex coherence, *<sup>S</sup>***xy** <sup>∈</sup> <sup>C</sup> is the cross-spectrum between **<sup>x</sup>** and **<sup>y</sup>**, *<sup>S</sup>***xx**, *<sup>S</sup>***yy** <sup>∈</sup> <sup>C</sup> are the auto-spectrums of **<sup>x</sup>** and **<sup>y</sup>**, *d f* <sup>∈</sup> <sup>R</sup><sup>+</sup> is the frequency resolution, *F* stands for the set of frequencies over which the slope is summed, and  indicates selecting only the imaginary part of the sum [12]. If the PSI, as defined in Equation (11), is positive, then there is directed interaction from **x** to **y** in *F*. Conversely, if the PSI is negative, the directed interaction goes from **y** to **x**. Note that by definition the PSI is an antisymmetric measure: *PSI*(**x** → **y**) = −*PSI*(**y** → **x**).
