Biomedical Engineering Reference
In-Depth Information
In practice, however, the neuronal system consists of multiple subsystems called
lobes in neurophysiologic terminology, and works in a way of interaction and mem-
ory. For such a neuronal system with long-range correlation, memory, and interac-
tions, a more generalized entropy formalism was proposed by Tsallis [105]:
iN
=
∑
q
1
−
P
i
TE
=
i
=
1
(3.64)
q
−
1
Tsallis entropy (
TE
) degrades to conventional Shannon entropy (
SE
) when the
entropic index
q
converges to 1. Under the nonextensive entropy framework, for
two interactive systems
A
and
B
, the nonextensive entropy of the combined system
A
B
will follow the quasi-additivity:
(
)
() ()
(
() ()
TE A
∪
B
TE A
k
TE B
k
TE A
k
TE B
k
)
=
+
+
1
−
q
(3.65)
k
where
k
is the Boltzmann constant. When
q
1, (3.65) becomes (3.63). For
q
<
1,
q
=
1, and
q
>
1, we can induce
TE
(
A
∪
B
)
TE
(
A
)
+
TE
(
B
),
TE
(
A
∪
B
)
=
TE
(
A
)
+
TE
(
B
) from (3.65) corresponding to super-
extensive, extensive, and subextensive systems, respectively.
Although Tsallis entropy has been frequently recommended as the generalized
statistical measure in past years [105-108], it is not unique. As the literature shows,
we can use other generalized forms of entropy [109]. One of them is the well-known
Renyi entropy [110], which is defined as follows:
TE
(
B
), and
TE
(
A
∪
B
)
TE
(
A
)
+
⎛
⎜
⎞
⎟
1
M
−
∑
P
i
q
RE
=
log
(3.66)
1
−
q
1
i
when
q
1, it also recovers to the usual Shannon entropy. This expression of
entropy adopts power law-like distribution
x
−β
. The exponent
β
is expressed as a
(
q
) of the Renyi parameter
q
[111]. Renyi entropy of scalp EEG signals has
been proven to be sensitive to the rate of recovery of neurological injury following
global ischemia [98].
In the remaining part of this section, we introduce the methods of using
time-dependent entropy to describe the different rhythmic activities in EEG, and
how to use entropy to quantify the nonstationary level in neurological signals.
function
β
3.3.2 Estimating the Entropy of EEG Signals
EEG signals have been conventionally considered to be random processes, or sto-
chastic signals obeying an autoregressive (and moving averaging) model, also
known as AR and ARMA models. Although the parametric methods, such as the AR
model, have obtained some success in describing EEG signals, the model selection
has always been a critical and time-intensive procedure in these conventional analy-
ses. On the other hand, the amplitude or frequency distribution of EEG signals is
strongly physiologically state dependent, for example, in epilepsy seizure and burst-
ing activities following hypoxic-ischemic brain injury. Figure 3.28 shows some typi-
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