Biomedical Engineering Reference
In-Depth Information
cesses. In particular this approach may be useful for spike train analysis and for the
investigation of the relations between point processes and continuous signals.
4.1.7.3.7 Statistical assessment of time-varying connectivity
Functional con-
nectivity is expected to undergo rapid changes in the living brain. This fact should be
taken into account when constructing the statistical tool to test the significance of the
changes in the connectivity related to an event. Especially, it is difficult to keep the
assumption that during the longer baseline epoch the connectivity is stationary. ERC
involves statistical methods for comparing estimates of causal interactions during
pre-stimulus baseline epochs and during post-stimulus activated epochs that do not
require the assumption of stationarity of the signal in either of the epochs. Formally
the method relies on the bivariate smoothing in time and frequency defined as:
Y
f
,
t
=
g
(
f
,
t
)+
ε
f
,
t
(4.34)
(
,
)
where
g
is modeled as penalized thin-plate spline [Ruppert et al., 2003] rep-
resenting the actual SdDTF function and ε
f
,
t
f
t
are independent normal random
σ
2
ε
)
(
,
(
N
) variables. The thin-plate spline function can be viewed as spatial plates
joined at a number of knots. The number of knots minus the number of spline pa-
rameters gives the number of the degrees of freedom.
To introduce the testing framework proposed in [Korzeniewska et al., 2008], we
first introduce some notations. Denote by
f
1
0
f
m
the frequencies where
f
m
is the
number of analyzed frequencies. Also, denote by
t
,...,
t
n
the time index cor-
responding to one window in the baseline, where
t
n
is the total number of baseline
windows. Similarly, denote by
T
=
t
1
,...,
T
n
the time index corresponding to one
window in the post-stimulus period, where
T
n
is the total number of post-stimulus
windows.
The goal is to test for every frequency
f
, and for every baseline/stimulus pair
of time windows
=
T
1
,...,
. More precisely, the implicit null
hypothesis for a given post-stimulus time window
T
at frequency
f
is that:
(
t
,
T
)
,whether
g
(
f
,
t
)=
g
(
f
,
T
)
(
,
)=
(
,
)
(
,
)=
(
,
)
,...,
(
,
)=
(
,
)
H
0
,
f
,
T
:
g
f
t
1
g
f
T
or
g
f
t
2
g
f
T
or
g
f
t
n
g
f
T
(4.35)
with the corresponding alternative
H
1
,
f
,
T
:
g
(
f
,
t
1
)
=
g
(
f
,
T
)
and
g
(
f
,
t
2
)
=
g
(
f
,
T
)
and
,...,
g
(
f
,
t
n
)
=
g
(
f
,
T
)
(4.36)
To test these hypotheses a joint 95% confidence interval for the differences
ˆσ
g
(
g
(
f
,
t
)
−
g
(
f
,
T
)
for
t
=
t
1
,...,
t
n
is constructed. Let
g
(
f
,
t
)
,
f
,
t
)
be the penal-
ized spline estimator of
g
and its associated estimated standard error in each
baseline time window. Similarly, let
g
(
f
,
t
)
ˆσ
g
(
(
f
,
T
)
,
f
,
T
)
be the penalized spline esti-
mator of
g
and its associated estimated standard error in each post-stimulus
time window. Since residuals are independent at points well separated in time, the
central limit theorem applies and we can assume that for every baseline/stimulus pair
of time windows
(
f
,
T
)
(
t
,
T
)
(
g
(
f
,
t
)
−
g
(
f
,
T
))
−
(
g
(
f
,
t
)
−
g
(
f
,
T
))
ˆσ
g
(
∼
N
(
0
,
1
)
(4.37)
f
,
t
)+
ˆσ
g
(
f
,
T
)
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