Biomedical Engineering Reference
In-Depth Information
where
M
inv
is the set of all possible stationary probabilities
ν
on the set of rasters
with
N
neurons;
h
[
ν
] is the entropy of
ν
and
ν
[
φ
] is the average value of
φ
with respect to the probability
ν
. Looking at the second equality, the variational
principle (
8.17
) selects, among all possible probability
ν
,
one
probability which
realizes the supremum, the Gibbs distribution
μ
.
The quantity
P
(
φ
) is called the
topological pressure
. For a normalized potential
it is equal to 0. However, the variational principle (
8.17
) holds for non-normalized
potentials as well i.e., functions which are not the logarithm of a probability
[
10
,
28
,
57
].
In particular, consider a function of the form:
K
H
β
(
ω
0
−D
)=
β
k
O
k
(
ω
)
,
(8.18)
k
=1
where
O
k
are observables,
β
k
real numbers and
β
denotes the vector of
β
k
's,
k
=1
,...,
K
. We assume that each observable depends on spikes in a time interval
{−
.
To the non-normalized potential
D,...,
0
}
H
β
(
ω
0
−D
) one can associate a normalized
potential
φ
of the form:
φ
(
ω
0
−
D
)=
H
β
(
ω
0
D
)
−
log
ζ
β
(
ω
0
D
)
,
(8.19)
−
−
where
ζ
(
ω
0
−D
) is a function that can explicitly computed. In short, one can associate
to the potential
H
β
(
ω
0
−D
) a matrix with positive coefficient;
ζ
β
(
ω
0
−D
) is is a
function of the (real positive) largest eigenvalue of this matrix as well as of the
corresponding right eigenvector (see [
75
] for details). This function depends on
the model-parameters
β
. The topological pressure is the logarithm of the largest
eigenvalue.
In this way,
H
β
defines a stationary Markov chain with memory depth
D
, with
transition probabilities:
P
ω
(0)
ω
−
1
e
H
β
(
ω
−
D
)
ζ
β
(
ω
0
−D
=
.
(8.20)
−D
)
Denote
μ
β
the Gibbs distribution of this Markov chain. The topological pressure
P
(
φ
β
) obeys:
∂
P
(
φ
β
)
∂β
k
=
μ
β
[
O
k
]
,
(8.21)
while its second derivative controls the covariance of the Gaussian matrix character-
izing the fluctuations of empirical averages of observables about their mean. Note
that those fluctuations are Gaussian if the second derivative of
P
is defined. This
holds if all transitions probabilities are positive.
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