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

32- Auto-Channel Cross-Frequency Phase Difference (ACP) is a
measure of the phase difference between two phase difference time series at
two frequencies recorded from one location. Phase difference between
two time series and two frequencies is defined as a point on the unit circle
and is represented in degrees or radians and is “normalized” with respect to
frequency (i.e., independent of frequency because r = 1). For example, a
phase difference of 45 0 is the same for the standard EEG frequency bands of
delta, theta, alpha, beta, gamma, etc. Because of this fact and because of the
physics of superposition of waves the bi-spectral phase measure is a useful
measure of local generator signals that are coupled at different frequencies
and exhibit bi-frequency phase locking. The first and second derivatives of
bi-frequency phase coupling are similar to the inter-coupling measures and
are useful measures of “transition states” or bifurcation points and stability
measures of homeostatic systems measured from a single location and given
superposition of waves from many different locations.
The equation for use with a hand calculator to compute Auto Bi-Spectral
Phase (f 1 , f 2 ) or ACP is:
Eq. 30
ACP =
Arc tan
∑
N
∑
N
( a(
x'
f ) v(
x'
' f ) − b(
x'
f ) u(
x'
' f ))
1
( a(
x'
f ) u(
x''
f ) + b(
x'
f ) v(
x''
f ))
1
2
2
1
1
2
2
Where x’ = frequency 1 and x’’ = frequency two recorded from the same
channel and N = number of time samples (for cospectrum and quadspectrum
calculation see section 9).
33- Cross-Channel Cross-Frequency Phase Difference (CCP) is a
measure of the phase difference between two real valued phase difference
time series at two frequencies recorded from two different locations. This
is an important measure of network dynamics and communication at
different frequencies across space. Because instantaneous phase is a scalar
and a real number then the commutation properties of algebra hold and the
use of the Fourier transform is valid to compute the arctangent of the
quadspectrum and cospectrum. Phase difference between two locations and
two frequencies is defined as a point on the unit circle and is represented in