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electromagnetic coupling (magnetic induction) in the frequencies up to about
1 MHz; thus, the quasi-static approximation of Maxwell equations holds throughout
the spectrum of interest (Nunez and Srinivasan
2006
, p. 535
-
540). Finally, for
source oscillations below 40 Hz, it has been veri
ed experimentally that capacitive
effects are also negligible, implying that potential difference is in phase with the
corresponding generator (Nunez and Srinivasan
2006
, p. 61). These phenomena
strongly support the
superposition principle
, according to which the relation
between neocortical dipolar
fields and scalp potentials may be approximated by a
system of linear equations (Sarvas
1987
). We can therefore employ a
linear BSS
model
. Because of these properties of volume conduction, scalp EEG potentials
describe an
instantaneous mixture
of the
fields emitted by several dipoles extending
over large cortical areas. Whether this is a great simpli
cation, we need to keep in
mind that it does not hold true for all cerebral phenomena. Rather, it does at the
macroscopic spatial scale concerned by EEG.
The goal of EEG blind source separation (BSS) is to
“
isolate
”
in space and time
the generators of the observed EEG as much as possible, counteracting the mixing
caused by volume conduction and maximizing the signal-to-noise ratio (SNR). First
explored in our laboratory during the
rault
and Jutten
1986
), BSS has enjoyed considerable interest worldwide only starting a
decade later, inspired by the seminal papers of Jutten and H
first half of the 1980s (Ans et al.
1985
;H
é
é
rault (
1991
), Comon
(
1994
), and Bell and Sejnowski (
1995
). Thanks to its
flexibility and power, BSS has
today greatly expanded encompassing a wide range of applications such as speech
enhancement, image processing, geophysical data analysis, wireless communica-
tion, and biological signal analysis (Comon and Jutten,
2010
).
8.3
The BSS Problem for EEG, ERS/ERD, and ERP
For
N
scalp sensors and
M
fixed location and orien-
tation in the analyzed time interval, the linear BSS model simply states the
superposition principle discussed above, i.e.,
≤
N
EEG dipolar
fields with
vð
t
Þ
¼
Asð
t
Þþgð
t
Þ
ð
8
:
1
Þ
vð
t
Þ2<
N
A 2<
N
M
is a time-
is the
sensor measurement vector
at sample
t
,
sð
t
Þ2<
M
holds the time course of the
source components, and
gð
t
Þ2<
N
is additive noise, temporally white, possibly
uncorrelated with
invariant full column rank
mixing matrix,
and with spatially uncorrelated components. Equation (
8.1
)
states that each observation
sð
t
Þ
vð
t
Þ
(EEG) is a linear combination (mixing) of sources
sð
t
Þ
, given by the coef
cients in the corresponding column of matrix
A
. Neither
sð
t
Þ
nor
is known, that is why the problem is said to be
blind
. Our source estimation is
given by
A
sð
t
Þ
¼
Bvð
t
Þ
ð
8
:
2
Þ
