Biomedical Engineering Reference
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of f
is much larger than one at the self-consistent rate n
ss
and this steady
state is, thus, unstable. The conclusion here is that low firing rates are expected to be
hard to achieve in purely excitatory networks unless they are weakly coupled.
On the other hand, when the network is inhibitory (
Figure 15.4B
;
m
ext
V
(
m
V
(
n
)
,
s
V
)
4mV),
a supra-threshold external input is required to obtain an active network. The function
f now decreases as a function of n (due to the fact that the mean decreases with
n). Equation (15.87) is now trivially satisfied when the coupling is predominantly
inhibitory,
=
20
.
J
0. Hence, a network state at this rate is stable. In the balanced
network of Section 15.3.1.2, inhibition strongly dominates recurrence because it has
to compensate for the external inputs. In this limit, the slope becomes infinitely
negative. Note, however, that this strong stability of the purely inhibitory network is
peculiar to synaptic couplings without latency. In presence of a latency, oscillatory
instabilities appear even in strongly noisy networks [19, 24, 26].
Another simplified rate dynamics which has been frequently used is
<
t
d
n
)
dt
=
−
(
t
C Js
m
ext
V
n
(
t
)+
f
(
+
(
t
)
,
s
V
)
s
(
t
)=
t
syn
n
(
t
)
,
(15.88)
where t remains unspecified. Although the fixed points of the systems described
by Equations (15.80) and (15.88) are the same, this latter scheme neglects the dy-
namics for the synaptic variable, and instead uses an arbitrary time constant for the
process by which the firing rate attains its steady state value. In conditions of high
noise, Equations (15.80) seem, therefore, better suited to describe the time course of
network activity than Equations (15.88).
We can extend Equations (15.80) to allow the description of a network with two
(excitatory and inhibitory) neural populations, with firing rates n
E
and n
I
, and with
synaptic latencies. If synaptic activation has a latency of t
lE
for excitation and t
lI
for
inhibition, then we have
ds
E
)
dt
=
−
(
t
s
E
(
t
)
/
t
syn
,
E
+
n
E
(
t
−
t
lE
)
m
ext E
V
J
EE
s
E
(
J
EI
C
I
s
I
(
s
V
)
n
E
(
t
)=
f
E
(
+
C
E
t
)
−
t
)
,
ds
I
(
)
dt
=
−
t
s
I
(
t
)
/
t
syn
,
I
+
n
I
(
t
−
t
lI
)
m
ext I
J
IE
s
E
(
J
II
C
I
s
I
(
s
V
)
.
n
I
(
t
)=
f
I
(
+
C
E
t
)
−
t
)
,
(15.89)
V
Equations (15.89) are Wilson-Cowan type dynamical mean-field equations which
are 'derived' from an underlying biophysical description of neurons and synapses.
In principle, the behavior of this model can be compared quantitatively (albeit ap-
proximately) with that of the original large-scale network of irregularly spiking LIF
neurons. One should bear in mind, however, that when time delays are included, the
dynamics become significantly richer, and the analysis more complicated. In fact,
rigurously speaking, in the presence of temporal delays the system becomes infinite-
dimensional, even if one deals with a single population (a function evaluated at
t
+
t,
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