Biomedical Engineering Reference
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first row) represented by these 4 input spike trains varies around the intended fir-
ing rate
r
(
t
)
.
r
measured
(
t
)
is calculated as the average firing frequency in the interval
[
t
−
30 ms
,
t
]
. Third row of
Figure 18.9
shows that the autocorrelation of both
r
(
t
)
and
r
measured
(
)
vanishes after 30 ms.
Various readout neurons, that all received the same input from the microcircuit
model, had been trained by linear regression to output at various times
t
(more
precisely: at all multiples of 30 ms) the value of
r
measured
(
t
t
)
,
r
measured
(
t
−
30
ms
)
,
r
measured
(
, etc.
Figure 18.10a
shows (on test data not
used for training) the correlation coefficients achieved between the target value and
actual output value for 8 such readouts, for the case of two generic microcircuit
models consisting of 135 and 900 neurons (both with the same distance-dependent
connection probability with m
t
−
60
ms
)
,
r
measured
(
t
−
90
ms
)
2 discussed in Section 18.3). Figure 18.10b shows
that dynamic synapses are essential for this analog memory capability of the circuit,
since the
memory curve
drops significantly faster if one uses instead static (
linear
)
synapses for connections within the microcircuit model. Figure 18.10c shows that
the intermediate
hidden
neurons in the microcircuit model are also essential for this
task, since without them the memory performance also drops significantly.
It should be noted that these memory curves not only depend on the microcircuit
model, but also on the diversity of input spike patterns that may have occurred in
the input before, at, and after that time segment in the past from which one recalls
information. Hence the recall of firing rates is particularly difficult, since there exists
a huge number of diverse spike patterns that all represent the same firing rate. If one
restricts the diversity of input patterns that may occur, substantially longer memory
recall becomes possible, even with a fairly small circuit. In order to demonstrate
this point 8 randomly generated Poisson spike trains over 250 ms, or equivalently
2 Poisson spike trains over 1000 ms partitioned into 4 segments each (see top of
Figure 18.11)
,
were chosen as template patterns. Then spike trains over 1000 ms
were generated by choosing for each 250 ms segment one of the two templates for
this segment, and by jittering each spike in the templates (more precisely: each spike
was moved by an amount drawn from a Gaussian distribution with mean 0 and a
SD that we refer to as
jitter
, see bottom of Figure 18.11). A typical spike train
generated in this way is shown in the middle of Figure 18.11. Because of the noisy
dislocation of spikes it was impossible to recognize a specific template from a single
interspike interval (and there were no spatial cues contained in this single channel
input). Instead, a pattern formed by several interspike intervals had to be recognized
and classified retrospectively. The performance of 4 readout neurons trained by
linear regression to recall the number of the template from which the corresponding
input segment had been generated is plotted in
Figure 18.12
(thin line).
For comparison the memory curve for the recall of firing rates for the same tempo-
ral segments (i.e., for inputs generated as for
Figure 18.10,
but with each randomly
chosen target firing rate
r
=
held constant for 250 instead of 30 ms) is plotted as
thin line in Figure 18.12, both for the same generic microcircuit model consisting of
135 neurons. Figure 18.12 shows that information about spike patterns of past inputs
decays in a generic neural microcircuit model slower than information about firing
rates of past inputs, even if just two possible firing rates may occur. One possible ex-
(
t
)
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