Biomedical Engineering Reference
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Thus, the conditional probability of this block given a fixed neighborhood is the
exponential of the energy characterizing physical interactions within the block as
well as with the boundaries. Here, spins are replaced by spiking patterns; space is
replaced with time which is mono-dimensional and oriented: there is no dependence
in the future. Boundary conditions are replaced by the dependence in the past.
8.3.2
Determining the “Best” Markov Chain to Describe an
Experimental Raster
We now show how the formalism of the previous section can be used to analyze
spike trains statistics in experimental rasters.
8.3.2.1
Observables
We call
observable
a function which associates to a raster plot a real number. Typical
choices of observables are
ω
k
1
(
n
1
) which is 1 if neuron
k
1
fires at time
n
1
and
which is 0 otherwise;
ω
k
1
(
n
1
)
ω
k
2
(
n
2
) which is 1 if neuron
k
1
fires at time
n
1
and
neuron
k
2
fires at time
n
2
and which is 0 otherwise, and so on. Another example
is
ω
k
1
(
n
1
)(1
−
ω
k
2
(
n
2
)) which is 1 is neuron
k
1
fires at time
n
1
and neuron
k
2
is silent at time
n
2
. This example stresses that observables can consider as well
events where some neurons are silent. One can also consider more general forms of
observables, e.g., non linear functions of spike blocks (see for example Eqs. (
8.38
)
and (
8.42
)below).
It is a general result from Hammersley and Clifford [
25
] that any function of spike
blocks can be uniquely decomposed as a linear combination of what we call
mono-
mials
in this chapter, namely a function of the form
ω
k
1
(
n
1
)
ω
k
2
(
n
2
)
... ω
k
m
(
n
m
)
which is equal to 1 if and only if neuron
k
1
fires at time
n
1
,
...
,neuron
k
m
fires
at time
n
m
in the raster
ω
. Thus monomials attribute the value '1' to characteristic
spike events.
8.3.2.2
Probabilities and Averages
Let
μ
be a probability on the set of rasters (typically the Gibbs distribution
introduced above). Mathematically, the knowledge of
μ
is equivalent to knowing
the probability
μ
[
ω
m
] of any possible spike block. For an observable
f
we denote
μ
[
f
]
=
fdμ
the average of
f
with respect to
μ
.If
f
is only a function of finite
blocks
ω
m
then:
d
ef
μ
[
f
]=
ω
m
f
(
ω
m
)
μ
[
ω
m
]
,
(8.9)
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