EEG and Brain Connectivity: A Tutorial - Bio-Medical Instruments, Inc.

30- Cross-Channel Cross-Frequency Coherence (CCC) is a measure of
the phase consistency between two different frequencies recorded from two
different locations. For example, the phase consistency between theta (4-7
Hz) and High Beta (20 – 40 Hz) EEG signals in two spatially separated
channels F3 and F4 of the 10/20 system of EEG electrode location.
Mathematically, the Cross-Channel Cross-Frequency Coherence (CCC)
is defined as the ratio of the auto-spectra and cross-spectra for two
channels, X and Y and two frequencies f 1 and f 2 . We again refer to the
definitions of the cospectrum and the quadspectrum (see section 9) and then
we define the cross bi-spectral coherence:
Eq. = 28
(
CCC =
∑
N
( a(
x'
f ) u(
y'
f ) + b(
x'
f ) v(
y'
f )))
1
2
∑
N
( a(
x'
f )
1
1
2
2
+ b(
x'
f )
2
2
2
+ (
)
∑
N
∑
N
( a(
x'
f ) v(
y'
f ) − b(
x'
f ) u(
y'
f )))
u(
y'
f )
2
1
2
2
+ v(
y'
f )
2
2
)
1
2
2
Where x’ = channel 1 and y’ = channel 2, f 1 = frequency 1 in channel 1 and
f 2 = frequency 2 in channel 2. N and the summation sign represents
averaging over frequencies in the raw spectrogram or averaging replications
of a given frequency or both. The numerator and denominator of bicoherence
always refers to smoothed or averaged values, and, when there are
N replications or N frequencies then each bi-coherence value has 2N degrees
of freedom.
31- Bi-Spectral Phase
Bi-spectral phase difference is generically defined as:
Eq. 29 –
Phase difference (f 1 ,f 2 ) = Arctan
( Smoothedquadspectrum(
f1,
f
2
))
( Smoothed cos pectrum(
f , f ))
1
2
Like bi-coherence there are two subdivisions of bi-spectral phase: 1- Auto
Bi-spectral phase and 2- Cross Bi-spectral phase.