Multi-spectral phase coherence
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Multi-spectral phase coherence (MSPC) is a generalized cross-frequency coupling metric introduced by Yang and colleagues in 2016. MSPC can be used to quantify
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
phase coupling between a set of base frequencies and their
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
/ 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: : \Psi(f_i,a_i) = \left\langle \exp \left( j\left(\sum_i a_i\varphi(f_i) -\varphi \left(f_\text\right)\right) \right) \right\rangle where \varphi(f_i) is the phase at frequency f_i , a_i is the weight of f_i to a harmonic/intermodulation frequency f_\text = \sum_i a_if_i ), and \langle \cdot \rangle represents the average over realizations. Bi-phase locking value,{{cite journal, last2=Ojemann, first2=JG, last3=Sorensen, first3=LB, date=15 May 2009, title=Bi-phase locking – a tool for probing non-linear interaction in the human brain., journal=NeuroImage, volume=46, issue=1, pages=123–32, pmid=19457390, last1=Darvas, first1=F, doi=10.1016/j.neuroimage.2009.01.034, pmc=2778057 also called bi-phase coherence in the literature, is a special case of MSPC when a_1=a_2=1 , i = 1, 2. The time-delay can be estimated from the phase lag when MSPC is computed between signals.


References

Nonlinear functional analysis Spectral theory