Biomedical Engineering Reference
In-Depth Information
This case has been investigated in [
34
] for spike train analysis of spike trains in the
parietal cat cortex, assuming stationarity.
More general form of time-translation invariant potentials of the form:
0
φ
n
ω
0
D
=
β
i
1
,n
1
,...,i
l
,n
l
ω
i
1
(
n
1
)
...ω
i
l
(
n
l
)
,
(8.28)
−
l
=
−D
P
(
N,D
)
,
where the sum
P
(
N,D
)
holds on the set of non repeated pairs of integers (
i,n
)
with
i
, have been considered in [
75
]
All examples treated above concerns stationary situations. In the non-stationary
case, the entropy rate is not defined and the Gibbs distributions cannot be defined
via the maximal entropy principle, while it is still possible to define them as done
in Sect.
8.3.1.5
. Now, the most general form for non-stationary Markov chain with
memory depth
D
corresponds to potentials of the form:
∈{
1
,...,N
}
and
n
∈{−
D,...,
0
}
0
φ
n
ω
n−D
=
β
i
1
,n
1
,...,i
l
,n
l
(
n
)
ω
i
1
(
n
+
n
1
)
...ω
i
l
(
n
+
n
l
)
,
(8.29)
l
=
−D
P
(
N,D
)
,
In the non-stationary case the coefficients
β
i
1
,n
1
,...,i
l
,n
l
(
n
) depend explicitly on
n
.
They can be chosen so that (
8.29
) is normalized.
8.3.2.11
Non-Markovian Potentials
One can conceptually extend the definition of Markovian potentials to the case
when
D
. This corresponds to a process with an infinite memory called
“chain with complete connections” which is widely studied in the mathematical and
mathematical physics literature (see [
33
] for a review). In this case the potential
→−∞
is a “function”
φ
n
ω
n
depending on a infinite past
ω
n−
1
−∞
. Although this case
seems rather abstract, it turns out that the only known examples where spike train
statistics can be exactly characterized in neural networks models are of this form.
An example is given below. Moreover, this potential form allows to recover all the
examples above.
−∞
8.3.2.12
How Good Is the Estimation?
Once we have chosen a set of constraints and once we have found the parameters
β
to match (
8.13
), how can we check the goodness of the model? Additionally,
changing the set of constraints provides another model. How can we compare two
statistical models?
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