Biomedical Engineering Reference
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where the mean and variance of the current are given respectively by
m
(
n
)=
m
ext
+
CJ
n
(15.69)
s
2
s
ext
+
CJ
2
n
(
n
)=
.
(15.70)
In this 'extended' mean-field theory, not only the mean inputs are included in the
description, but also the fluctuations around the 'mean-field' are relevant. As em-
phasized above, this approach is only applicable to network states in which neurons
fire in an approximately Poissonian way, and when the low connection probability
makes the emission processes of neurons essentially uncorrelated. Moreover, since
J
and
C
are finite, this approach is only approximate. However, simulations show it
gives very accurate results when
C
nt
m
is large (several hundreds) and
J
/
(
C
m
V
th
)
is
small (less than several percent), as seem to be the case in cortex [10, 24, 19]. Equa-
tion (15.68) can again be solved graphically to obtain the self-consistent, steady-state
firing rates in the network (see below).
It is straightforward to extend this description to a two population network of
excitatory and inhibitory neurons. The equations are, for finite
C
E
,
C
I
,(E-to-E)
J
EE
,
(I-to-E)
J
EI
, (E-to-I)
J
IE
, and (I-to-I)
J
II
:
n
E
=
f
(
m
E
,
s
E
)
(15.71)
n
I
=
f
(
m
I
,
s
I
)
(15.72)
m
E
=
m
ext E
+[
C
E
J
EE
n
E
−
C
I
J
EI
n
I
]
m
I
=
m
ext I
+[
C
E
J
IE
n
E
−
C
I
J
II
n
I
]
s
ext E
+
C
E
J
EE
n
E
+
C
I
J
EI
n
I
s
E
=
s
ext I
+
C
E
J
IE
n
E
+
C
I
J
II
n
I
.
s
I
=
(15.73)
The stationary states of these two population networks and their stability properties
have been studied extensively [11, 19]. Since the number of connections per neu-
ron in these networks is large, they behave qualitatively like the balanced networks
discussed in the previous section.
15.3.1.4
Spatial distribution of activity in finite heterogeneous networks
Mean-field equations have been derived for heterogeneous networks of binary neu-
rons [119] and for heterogeneous networks of noisy LIF neurons [10]. Consider a
network of
N
neurons in which the probability that two neurons are connected is
C
/
N
1. Each neuron will receive
C
connections on average, and the cell-to-cell
fluctuations in the number of afferents will be order
√
C
. In principle, when
C
is
large, the fluctuations in the number of connections are small compared to the mean.
However, since networks of excitatory and inhibitory cells settle down in a balanced
state in which excitation and inhibition cancel each other out to within 1
/
√
C
(see
above), the effective average input to the neurons becomes of the same order as its
fluctuations, and this is reflected in wide distributions of firing rates in the steady
states. To calculate this distributions self-consistently one proceeds as follows: The
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