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

When separated generators exhibit a stable phase difference of, for
example, 30 degrees then this can not be explained by volume conduction. 1
As will be explained in later sections correlation coefficient methods such as
the Pearson product correlation (e.g., “co-modulation” and “Lexicor
correlation”) do not compute phase and are therefore incapable of
controlling for volume conduction. The use of complex numbers and the
cross-spectrum is essential for studies of brain connectivity not only because
of the ability to control volume conduction but also because of the need to
measure the fine temporal details and temporal history of coupling or
“connectivity” within and between different regions of the brain.
Figure 1 is an illustration of the cross-spectrum of volume conduction
vs. connectivity in which a sine wave is generated inside a sphere with
sensors on the surface. The top shows the zero phase lag recordings of a
sine wave and illustrates volume conduction in which the solid angle from
the source to the surface is equal in all directions. The bottom shows
recordings with significant phase differences which can not be accounted for
by volume conduction and must be due to “connections” in the interior of
the sphere. As discussed in more detail in section 9, the cross-spectrum is
the sum of the in-phase potentials (i.e., cospectrum) and out-of-phase
potentials (i.e., quadspectrum). The in-phase component contains volume
conduction and the synchronous activation of local neural generators. The
out-of-phase component contains the network or connectivity contributions
from locations distant to a given source. In other words, the cospectrum =
volume conduction and the quadspectrum = non-volume conduction which
can be separated and analyzed by independently evaluating the cospectrum
and quadspectrum (see section 9).
1 Theoretically, large phase differences can be produced by volume conduction when there is a deep and
temporally stable tangential dipole that has a positive and negative pole with an inverse electrical field at
opposite ends of the human skull. In this instance, phase difference is maximal at the spatial extremes and
approximates zero half way between the two ends of the standing dipole. However, this is a special
situation that is sometimes present in evoked potential studies but is absent in spontaneous EEG studies.
In the case of spontaneous EEG there is no time locked event by which to synchronize potentials that result
in a standing dipole, instead, there is an instantaneous summation of millions of ongoing rhythmic
pyramidal cell dipoles with different orientations averaged over time.