Biomedical Engineering Reference
In-Depth Information
be observed. Spike trains statistics is assumed to be summarized by an hidden
probability
μ
characterizing the probability of spiking patterns. One current goal
in experimental analysis of spike trains is to approximate
μ
from data. We describe
here several theoretical tools allowing to handle this question. Our presentation is
based on the notion of transition probabilities. In this context we introduce Gibbs
distributions, which is one of the main theoretical concept of this chapter. Gibbs
distributions are usually considered in the stationary case where they are obtained
from the maximal entropy principle. Their definition via transition probabilities,
adopted in this chapter, affords the consideration of Gibbs distribution in the more
general context of non-stationary dynamics with possibly infinite memory.
8.3.1
Spike Statistics
8.3.1.1
Raster Plots
We consider a network of
N
neurons. We assume that there is a minimal time scale
δ>
0 corresponding to the minimal resolution of the spike time, constrained by
biophysics and by measurements methods (typically
δ
∼
1 ms) [
8
,
9
]. Without loss
of generality (change of time units) we set
δ
=1, so that spikes are recorded
at integer times. One then associates to each neuron
k
and each integer time
n
a
variable
ω
k
(
n
)=1if neuron
k
fires at time
n
and
ω
k
(
n
)=0otherwise. A
spiking
pattern
is a vector
ω
(
n
)
d
ef
=[
ω
k
(
n
)]
N
k
=1
which tells us which neurons are firing at
N
the set of spiking patterns. A
spike block
is a finite
ordered list of spiking patterns, written:
time
n
. We note
A
=
{
0
,
1
}
ω
n
2
n
1
=
{
ω
(
n
)
}
{n
1
≤n≤n
2
}
,
where spike times have been prescribed between the times
n
1
to
n
2
(i.e.,
n
2
−
n
1
+1
time steps). The
range
of a block is
n
2
−
n
1
+1, the number of time steps from
Nn
possible blocks
with
N
neurons and range
n
. For example,
N
=3neurons and
n
=2time steps
the possible blocks are:
⎛
n
2
−
n
1
+1
. Thus, there are 2
n
1
to
n
2
. The set of such blocks is
A
⎞
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
00
00
00
01
00
00
10
00
00
11
00
00
11
11
10
11
11
11
⎝
⎠
;
⎝
⎠
;
⎝
⎠
;
⎝
⎠
;
...
⎝
⎠
;
⎝
⎠
.
We call a
raster plot
a bi-infinite sequence
ω
d
ef
+
∞
n
=
−∞
, of spiking patterns.
This notion corresponds to its biological counterpart (Sect.
8.2.2
) with the obvious
difference that experimental raster plots are finite. The consideration of infinite
sequences is more convenient on the mathematical side but, at several places, we
discuss the effects of having finite experimental rasters on spike statistics estimation.
The set of raster plots is denoted
=
{
ω
(
n
)
}
X
=
A
.
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