Biomedical Engineering Reference
In-Depth Information
and, by induction, the probability of a block
ω
m
,
∀
n,n
−
m
≥
D
is given by:
n
P
ω
(
l
)
ω
l−
1
l−D
P
ω
m
+
D−
1
.
P
[
ω
m
]=
(8.2)
m
l
=
m
+
D
Thus, knowing the probability of occurrence of the block
ω
m
+
D−
m
one can infer
the probability of forthcoming blocks by the mere multiplication of transition
probabilities.
Given the (joint) probability
P
[
ω
m
] the (marginal) probability of sub-blocks
can be easily obtained, since for
m
≤
n
1
≤
n
2
≤
n
,
∗
(
n
1
,n
2
)
P
ω
n
2
n
1
=
P
[
ω
m
]
,
(8.3)
m,n
where
∗
(
n
1
,n
2
)
means that we sum up over all possible spiking patterns in the interval
m,n
{
(i.e., we sum up over all possible values of
ω
(
m
)
,...,ω
(
n
1
−
1)
,ω
(
n
2
+1)
,...,ω
(
n
)).
As a consequence, from (
8.2
)to(
8.3
), the probability of the spike block
ω
n−D
,
of range
D
,is:
m, n
}
excluding the interval
{
n
1
,n
2
}
∗
(
n
−
D,n
)
n
P
ω
(
l
)
ω
l−
1
P
ω
n−D
=
l−D
P
ω
m
+
D−
1
.
(8.4)
m
m,n
l
=
m
+
D
Knowing the probability of an initial block of range
D
(here
ω
m
+
D−
m
) one infers
from this equation the probability of subsequent blocks of range
D
. Equation (
8.4
)
can also be expressed in terms of vector-matrices multiplication, and the main
properties of the Markov chain can be deduced from linear algebra and matrices
spectra theorems [
64
]. For compactness we shall not use this possibility here though,
(see [
74
] for further details).
However, this equation shows that the “future” of the Markov chain (the proba-
bility of occurrence of blocks) depends on an initial condition (here
P
ω
m
+
D−
m
),
which is a priori undetermined. Moreover, there are a priori infinitely many possible
choices for the initial probability.
8.3.1.4
Asymptotic of the Markov Chain
Assume now that
n
.
Practically, this limit corresponds to considering that the system began to exist in
a distant past (defined by the initial condition of the Markov chain) and that it has
evolved long enough, i.e., over a time larger than relaxation times in the system, so
−
m
→
+
∞
in Eq. (
8.4
) and more precisely that
m
→−∞
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