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

Auto-Channel Cross-Frequency Phase (ACFP) and Cross-Channel
Cross-Frequency Phase (CCFP) are computed in the same manner as in
previous sections by computing the arctangent of the ratio of the
quadspectrum to the cospectrum at each moment of time for the two
transformed phase difference time series.
In summary, there are four categories of the bi-spectrum for the
purposes of relating different frequencies: 1- Auto-Channel Auto-Frequency
(AA), 2- Cross-Channel Auto-Frequency (CA), 3- Auto-Channel Cross-
Frequency (AC) and 4- Cross-Channel Cross-Frequency (CC).
29- Auto Channel Cross-Frequency Coherence (ACC) is defined as the
square of the ratio of the cross-spectra within a single channel at two
different frequencies divided by the product of the auto-spectra. For
example, the auto bi-spectrum between the EEG theta frequency (4 - 7 Hz)
and the beta frequency band (25 – 30 Hz) as recorded from electrode
location F3. To compute auto channel cross-frequency coherence one first
transforms each time point to the complex domain using complex
demodulation and then one computes the Fourier transform of the complex
domain time series.
Eq. 27:
Auto Cross-Frequency Coherence (ACC) (f 1 ,f 2 ) after complex demodulation
(x’,y’) is defined as
(
ACC =
∑
N
( a(
x'
f ) u(
x'
' f ) + b(
x'
f ) v(
x''
f )))
1
2
∑
N
( a(
x'
f )
1
1
2
2
+ b(
x'
f )
2
2
2
+ (
)
∑
N
∑
N
( a(
x'
f ) v(
x''
f ) − b(
x'
f ) u(
x'
' f )))
u(
x''
f )
2
1
2
2
+ v(
x'
' f )
2
2
)
1
2
2
Where x = frequency activity recorded from a single channel and x’ =
frequency 1 and x’’ = frequency two recorded from the same channel. 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 bi-coherence 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.