Multi-spectral phase coherence

Multi-spectral phase coherence (MSPC) is a generalized cross-frequency coupling metric introduced by Yang and colleagues in 2016.[1] MSPC can be used to quantify nonlinear phase coupling between a set of base frequencies and their harmonic/intermodulation frequencies. MSPC is a model-free method, which can provide a system description, including (i) the order of the nonlinearity, (ii) the direction of interaction, (iii) the time delay in the system, and both (iv) harmonic and (v) intermodulation coupling.

The MSPC is defined as:

${\displaystyle \Psi (f_{i},a_{i})=\left\langle \exp \left(j\left(\sum _{i}a_{i}\varphi (f_{i})-\varphi \left(f_{\text{sum}}\right)\right)\right)\right\rangle }$

where ${\displaystyle \varphi (f_{i})}$ is the phase at frequency ${\displaystyle f_{i}}$, ${\displaystyle a_{i}}$ is the weight of ${\displaystyle f_{i}}$ to a harmonic/intermodulation frequency ${\displaystyle f_{\text{sum}}=\sum _{i}a_{i}f_{i}}$), and ${\displaystyle \langle \cdot \rangle }$ represents the average over realizations.

Bi-phase locking value,[2] also called bi-phase coherence in the literature, is a special case of MSPC when ${\displaystyle a_{1}=a_{2}=1}$, ${\displaystyle i=1,2.}$

The time-delay can be estimated from the phase lag when MSPC is computed between signals.