Biomedical Engineering Reference
In-Depth Information
•
The brain response to the consecutive stimuli is of the same morphology
•
The ongoing activity is a stationary, zero mean noise
In commonly used analysis a simple additive model is assumed where the mea-
sured stochastic signal
x
(
t
)
is a sum of two terms: the deterministic EP signal
s
(
t
)
. According to this model the
i
th
realization
and the uncorrelated zero mean noise
n
(
t
)
of the signal is:
(
)=
(
)+
(
)
x
i
t
s
t
n
i
t
(4.7)
Averaging across
N
repetitions gives:
Ns
N
i
=
1
x
i
(
t
)=
N
i
=
1
n
i
(
t
)
1
N
1
N
x
(
t
)=
(
t
)+
(4.8)
The expected value of the averaged signal is:
E
[
x
(
t
)] =
s
(
t
)
(4.9)
N
∑
i
=
1
n
i
1
since for zero mean noise we have
E
[
(
t
)] =
0. The variance of
x
(
t
)
is:
⎡
2
⎤
⎦
=
1
N
N
i
=
1
n
i
(
t
)
1
N
σ
n
(
t
)
σ
x
(
t
)
=
⎣
E
(4.10)
since
s
(
t
)
is deterministic. Therefore the signal to noise ratio in terms of amplitude
improves proportionaly to
√
N
.
For the potentials appearing in the first tenths of a second after the sensory stim-
ulation, such as BAEP, the simple additive model (equation 4.7) can be regarded as
valid [John et al., 1978]. The validity of the model for longer latency potentials is
doubtful. Nevertheless, due to its simplicity, its applications in the practice of EP
analysis is overwhelming. A model closer to reality can be formulated assuming that
the response is a stochastic process
s
i
(
)
t
. Then the model (equation 4.7) becomes
x
i
(
t
)=
s
i
(
t
)+
n
i
(
t
)
(4.11)
This model implies that the response may not be completely described by the mean
value
x
but also by higher statistical moments (Sect. 1.1) as, e.g., variance.
Indeed, it should be stressed that the averaging procedure is not as simple as it
may appear at first glance. The simple additive model in which an EP is a sum of
deterministic signal and uncorrelated background noise holds only in a few ideal
cases. One can expect that there are interferences between the neural activity evoked
by the stimulus and the ongoing activity resulting in the reorganization of the latter
[Sayers et al., 1974]. In spite of this fact, one may still accept the simple additive EP
model, but he must be aware of the violation of the assumptions. It should be realized
that the signal in reality is not deterministic, but stochastic. Thus it is recommended
to evaluate the variance of the EP along with its mean value. The assumption of
(
t
)
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