Biomedical Engineering Reference
In-Depth Information
The relationship between observed potentials
V
(
r
i
,
r
j
,
t
) and brain sources
P
(
r
,
t
)
depends on the anatomy and physiology of brain tissue (especially the cerebral cor-
tex and its white matter connections) and the physics of volume conduction through
the human head. This topic is concerned with
quantitative electroencephalography
,
consisting of mathematical transformations of recorded potential to new dependent
variables
X
and independent variables
ξ
1
,
ξ
2
,
ξ
3
, …; that is,
(
)
(
)
Vt
r
,,
→
X
ξξ ξ
12 3
, , ,
(1.1)
i
j
The transformations of (1.1) provide important estimates of source dynamics
P
(
r
,
t
) that supplement the unprocessed data
V
(
r
i
,
r
j
,
t
). In the case of transformed elec-
trode references, the new dependent variable
X
retains its identity as an electric poten-
tial. With surface Laplacian and dura imaging transformations (high-resolution
EEGs),
X
is proportional to estimated brain surface potential. Other transformations
include Fourier transforms, principal/independent components analysis, constrained
inverse solutions (source localization), correlation dimension/Lyapunov exponents,
and measures of phase locking, including coherence and Granger causality.
Some EEG transformations have clear physical and physiological motivations;
others are more purely mathematical. Fourier transforms, for example, are clearly
useful across many applications because specific EEG frequency bands are associated
with specific brain states. Other transformations have more limited appeal, in some
cases appearing to be no more than mathematics in search of application. How does
one distinguish mathematical methods that truly benefit EEG from methods that
merely demonstrate fancy mathematics? Our evaluation of the accuracy and efficacy
of quantitative EEG cannot be limited to mathematical issues; close consideration of
EEG physics and physiology is also required. One obvious approach, which unfortu-
nately is substantially underemployed in EEG, is to adopt physiologically based
dynamic and volume conduction models to evaluate the proposed transforms
X
(
ξ
1
,
ξ
2
,
ξ
3
, …). If transformed variables reveal important dynamic properties of the known
sources modeled in such simulations, they may be useful with genuine EEG data; if
not, there is no apparent justification for the transform. Several examples of appro-
priate and inappropriate transforms are discussed in [2].
1.2
A Window on the Mind
Since the first human recordings in the early 1920s and their widespread acceptance
10 years later, it has been known that the amplitude and frequency content of EEGs
reveals substantial information about brain state. For example, the voltage record
during deep sleep has dominant frequencies near 1 Hz, whereas the eyes-closed
waking alpha state is associated with near-sinusoidal oscillations near 10 Hz. More
quantitative analyses allow for identification of distinct sleep stages, depth of anes-
thesia, seizures, and other neurological disorders. Such methods may also reveal
robust EEG correlations with cognitive processes: mental calculations, working
memory, and selective attention. Modern methods of EEG are concerned with both
temporal and spatial properties given by the experimental scalp potential function
(
)
()
( )
Vt
rr
,,
=
Φ
r
,
t
−
Φ
r
,
t
(1.2)
i
j
i
j
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