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cortical patches. For EEG data, this is an effective way to counteract the effect of
volume conduction. In fact, we have seen that the brain tissue behaves
approximately as a linear conductor; thus, observed potentials (mixtures) must
be more correlated than the generating dipolar
fields.
Sources are colored and/or their energy varies over time: Observed potentials
are the summation of postsynaptic potentials over large cortical areas caused by
trains of action potentials carried by afferent
fibers. The action potentials come in
trains/rest periods, resulting in sinusoidal oscillations of the scalp potentials, with
negative shifts during the train discharges and positive shifts during rest. The
periodicity of trains/rest periods is deemed responsible for high-amplitude EEG
rhythms (oscillations) up to about 12 Hz, whereas higher-frequency (>12 Hz) low-
amplitude rhythms may result from sustained (tonic) afferent discharges
(Speckmann and Elegr,
2005
). There is no doubt that an important portion of
spontaneous EEG activity is rhythmic, whence strongly colored (Niedermeyer
2005a
; Steriade
2005
; Buzs
á
ki
2006
, Chap. 6, 7). Some rhythmic waves come in
more or less short bursts. Typical examples are sleep spindles (7
-
14 Hz) (Nie-
dermeyer
2005b
; Steriade
2005
) and frontal theta (4
35 Hz)
waves (Niedermeyer
2005a
). Others are more sustained, as it is the case for slow
delta (1
-
Hz) and beta (13
-
Hz) waves during deep sleep stages III and IV (Niedermeyer
2005b
), the
Rolandic mu rhythms (around 10 Hz and 20 Hz), and posterior alpha rhythms
(8
-
12 Hz) (Niedermeyer
2005a
). In all cases, brain electric oscillations are not
everlasting and one can always de
-
ne time intervals when rhythmic activity is
present and others when it is absent or substantially reduced. Such intervals may
be precisely de
ned based on known reactivity properties of the rhythms. For
example, in event-related synchronization/desynchronization (ERD/ERS:
Pfurtscheller and Lopes da Silva
1999
), which are time-locked, but not phase-
locked, increases and decreases of the oscillating energy (Steriade
2005
) intervals
may be de
ned before and after event onset. On the other hand, event-related
potentials (ERP: Lopes Da Silva
2005
), which are both time-locked and phase-
locked, can be further partitioned in several successive intervals comprising the
different peaks. Such source energy variation signatures can be modeled precisely
by SOS, as we will show with the ensuing ErrP study.
8.6
Approximate Joint Diagonalization of Covariance
Matrices (AJDC)
The SOS BSS method we are considering is consistently solved by
approximate
joint diagonalization
algorithms (Cardoso and Souloumiac
1993
2
; Tichavsky and
Yeredor
2009
). Given a set of covariance matrices {
C
1,
C
2
,
…
}, the AJD seeks a
2
This paper does not consider SOS but fourth-order statistics; however, the algorithms are based
on approximate joint diagonalization of matrices which are the slices of the tensor of fourth-order
cumulants and thus can be used for SOS matrices as well.
