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

However, when using circular statistics the mean phase angle or phase
difference is relative to an arbitrary reference or starting point which is
difficult to define with spontaneous EEG. Spontaneous EEG is perfectly
useful because subjects are alert and holding themselves still or with no
motion as a reference and the magnitude and direction of a shift in phase
angle is all that is relevant (see section 15).
The phase difference is defined as:
Eq. 19 -
Phase difference (f) = Arctan
( Smoothedquadspectrum(
f ))
( Smoothed cos pectrum(
f ))
In the numerical example in Table II,
Phase difference (or angle at 1.25 Hz) = Arctan 0.031/0.073 = 22.7 o
Two oscillators are frequency locked when the first derivative of the
phase difference has a stable periodic orbit even if there is a difference in
phase between the two oscillators. Two oscillators are entrained when they
are frequency locked in a 1:1 fashion with no phase difference. Two
oscillators are phase locked where there is a stable phase difference that is
not 1:1 (e.g., 2:3). Two oscillators are synchronized when they are phase
locked independent of the absolute value of the phase difference, e.g., when
the 1 st derivative of the time series of phase ≈ 0. Synchronization is in-phase
when the phase difference = 0 and out-of-phase is when the phase difference
≠ 0. Two oscillators are said to be synchronized in anti-phase when the
phase difference = 180 0 . Frequency locking without phase locking is called
phase trapping. The relationship between all of these definitions is depicted
in figure 9.