Biomedical Engineering Reference
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by the 'balance' condition set by the strong global inhibition, the activity in the small
selective excitatory sub-population becomes essentially uncoupled from the rest of
the network. Consequently, this sub-population behaves essentially as the weakly
coupled excitatory network of Section 15.3.3. However, a direct consequence of
this scenario is that the CV in persistent activity must be lower than in spontaneous
activity, because the mean inputs are larger in the cue population, while the variance
remains unchanged (see Figure 2). Thus, there is no multistability between several
'balanced states'.
It is therefore possible for small sub-populations within a larger balanced network
to be bistable in a robust way, but at the price that the small sub-populations them-
selves do not remain balanced in both steady-states. Is there an alternative? The idea
would be to find a scenario in which the variance in the cue population also increases
in a significant way from spontaneous to persistent activity, so that the increase in
CV induced by the increase in variance counterbalances the decrease induced by the
increase in the mean. One can even imagine a scenario in which the mean does not
change, but the variance does. Such a scenario was introduced in [93, 94] for a net-
15.6A)
in which the interneurons are also subdivided in selective sub-populations.
Such a network is divided functionally in 'columns' or 'micro-columns' composed
both of excitatory and inhibitory populations. Both populations are activated in a
selective way when their preferred stimulus is shown. Consequently, in a persistent
state, both excitatory and inhibitory populations raise their firing rates. A similar
phenomenon was observed in experiments monitoring the activity of neurons in the
prefrontal cortex of primates during working memory tasks. Recordings of nearby
putative pyramidal cells and interneurons showed that the two sub-populations in-
crease their firing rate during the delay period [92]. This has lead to the postulate of
a micro-columnar organization of the pre-frontal cortex [92].
The 'micro-columnar' network has been studied at the mean-field level [93, 94].
In order to do a systematic investigation of the spiking variability resulting from dif-
ferent types of network organizations, the mean-field theory has been extended to be
self-consistent both at the level of rates and CVs. In the previously discussed models,
Poisson spiking statistics was an assumption, so the irregularity in the spiking activ-
ity in the pre-synaptic spike trains was 'fixed'. In [93] this assumption was relaxed
by assuming that the neuronal spike trains can be described as renewal processes
characterized by their mean rate and CV (a renewal point process is characterized
by independent ISI intervals from an arbitrary distribution). When the statistics of
the renewal spike trains are close to the Poisson case, the output rate and CV of the
post-synaptic neuron can be calculated as a function of the rate and CV of its inputs,
leading to steady state solutions in which both the rate and the CV are calculated
self-consistently. Using this framework, multistability in the micro-columnar net-
work described above has been studied using simple heuristic firing rate dynamics
similar to Equation (15.88). The synaptic interactions between neurons depend on
whether they belong to the same or to different micro-columns, and again, selec-
tive micro-columns are characterized by stronger excitatory recurrent interactions.
In
Figure 15.8,
we show the time course of activity of a bistable micro-column in
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