Biomedical Engineering Reference
In-Depth Information
It corresponds to impose constraints only on firing rates of neurons. We have
ζ
β
=
k
=1
(1 +
e
β
k
) and the corresponding Gibbs distribution is easy to compute:
n
N
e
β
k
ω
k
(
l
)
1+
e
β
k
μ
[
ω
m
]=
.
(8.25)
l
=
m
k
=1
Thus, the corresponding statistical model is such that spikes are independent. We
callita
Bernoulli model
. The parameter
β
k
is directly related to the firing rate
r
k
since
r
k
=
μ
(
ω
k
(0) = 1 ) =
e
β
k
1+
e
β
k
, so that we may rewrite (
8.25
)as:
n
N
r
ω
k
(
l
)
k
μ
[
ω
m
]=
1
−ω
k
(
l
)
,
(1
−
r
k
)
l
=
m
k
=1
the classical probability of coin tossing with independent probabilities.
Another prominent example of range-1 potential is inspired from statistical
physics of magnetic systems and has been used by Schneidman and collaborators in
[
60
] for the analysis of retina data (Sect.
8.4
). It is called
Ising potential
and reads,
with our notations:
N
φ
(
ω
(0)) =
β
k
ω
k
(0) +
β
kj
ω
k
(0)
ω
j
(0)
−
log
ζ
β
.
(8.26)
k
=1
1
≤j<k≤N
The corresponding Gibbs distribution provides a statistical model where syn-
chronous pairwise synchronizations
ω
k
(0)
ω
j
(0) between neurons are taken into
account, but neither higher order spatial correlations nor other time correlations are
considered. The function
ζ
β
is the classical partition function (
8.23
).
The Ising model is well known in statistical physics and the analysis of spike
statistics with this type of potential benefits from a diversity of methods leading to
really efficient algorithms to obtain the parameters
β
from data [
11
,
51
,
55
,
71
].
8.3.2.10
Markovian Potentials
Let us now consider potentials of the form (
8.7
) allowing to consider spatial
dependence as well as time dependence upon a past of depth
D
.
Consider first a stationary Markov chain with memory depth 1. The potential has
the form:
N−
1
N−
1
k−
1
0
φ
(
ω
0
β
kjτ
ω
k
(0)
ω
j
(
τ
)
−
log
ζ
β
(
ω
0
β
k
ω
k
(0)+
−
1
)
.
(8.27)
−
1
)=
k
=0
k
=0
j
=0
τ
=
−
1
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