Biomedical Engineering Reference
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response to a transient input. Panels A and B show the average rate and CV of
the excitatory and inhibitory sub-populations in the micro-column. In this network,
since the micro-column remains balanced in the elevated rate state, the CV of both
sub-populations remains close to one in the delay period. Both the similar courses
of activity of the excitatory and inhibitory populations, and the high CV during el-
evated persistent activity are consistent with measurements from prefrontal neurons
in working memory tasks [30, 92]. The reason for this behavior is that the mean
current to both sub-populations (see the lower panel for the case of the excitatory
sub-population) remains approximately constant as the network switches between
its two stable states. The increase in firing rate is due to an increase in the fluctu-
ations in the current. Indeed, as a result of this, the CV actually increases in the
elevated firing rate state.
This increase in CV is in contrast to the decrease in CV in models in which the
network is not balanced in the elevated activity state, like the networks described
in the previous two sections. This qualitative difference between relative change in
spiking variability in these scenarios should, in principle, be experimentally testable,
although the small difference in CV observed in
Figure 15.8
would be hard to detect
in experimentally recorded spike trains, due to limited sampling problems. Further
experimental data are needed to resolve this issue.
As suggested at the beginning of this section, this scenario still suffers from a fine
tuning problem. The range of multistability in the network with balanced persistent
state is extremely small for realistic numbers of inputs per cell [93]. In fact, such
multistability vanishes in the large
C
limit, even if the sub-populations are taken
to scale as 1
√
C
, because in that limit the difference in the fluctuations between
spontaneous and persistent activity vanishes.
The fundamental problem which precludes robust bistability in balanced networks
is the different scaling of the first two moments of the input current with the number
of inputs and with the connection strength. While the mean scales as
JC
, the variance
scales as
J
2
C
. It is thus impossible to find a scaling relationship between
J
and
C
that keeps both moments finite when
C
/
•.
It is possible that cross-correlations in the activity of different neurons might pro-
vide a solution to the 'linearity' problem of balanced networks. The different scaling
of the mean and the variance is a direct consequence of the fact that we have as-
sumed the different inputs to the cell to be independent, so that the variance of their
linear sum is the sum of their variances. If the inputs to the cell showed signifi-
cant correlations, the variance would now scale as
→
2
, in which case any scaling
relationship between
J
and
C
would have the same effect on the mean and on the
variance. It would therefore be of great interest to incorporate cross-correlations in a
self-consistent manner into the picture we have been describing in this chapter.
(
JC
)
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