Biomedical Engineering Reference
In-Depth Information
P
∑
µ
=
1
(
η
J
0
J
(
a
,
a
)
ij
ai
aj
=
−
f
)(
η
−
f
)
i
=
j
;
a
=
IT
,
PF
(16.11)
(
−
)
f
1
f
N
t
P
∑
µ
=
1
(
η
g
J
(
a
,
b
)
ij
ai
bj
=
−
f
)(
η
−
f
)
∀
i
,
j
;
a
=
b
.
(16.12)
f
(
1
−
f
)
N
t
The intra-modular connections are such that a number
P
of sparse independent
configurations of neural activity are dynamically stable, constituting the possible
sustained activity states in each module. This is expressed by saying that each mod-
ule has learned
P
binary patterns
ai
=
, each of them signalling
which neurons are active in each of the sustained activity configurations. Each vari-
able
{
η
0
,
1
,
µ
=
1
,...,
P
}
ai
is allowed to take the values 1 and 0 with probabilities
f
and
respec-
tively, independently across neurons and across patterns. The inter-modular connec-
tions reflect the temporal associations between the sustained activity states of each
module. In this way, every stored pattern
(
1
−
f
)
in the IT module has an associated pattern
in the PF module which is labelled by the same index. The normalization constant
N
t
=
µ
was chosen so that the sum of the magnitudes of the inter- and the
intra-modular connections remains constant and equal to 1 while their relative values
are varied. When this constraint is imposed the strength of the connections can be ex-
pressed in terms of a single independent parameter
g
measuring the relative intensity
of the inter- vs. the intra-modular connections (
J
0
can be set equal to 1 everywhere).
Both modules implicitly include an inhibitory population of neurons receiving and
sending signals to the excitatory neurons through uniform synapses. In this case the
inhibitory population can be treated as a single inhibitory neuron with an activity
dependent only on the mean activity of the excitatory population.
N
(
J
0
+
g
)
We chose the
transduction function of the inhibitory neuron to be linear with slope
.
Since the number of neurons in a typical network one may be interested in is very
large, e.g.,
10
6
, the analytical treatment of the set of coupled differential
equations (16.10) becomes untractable. On the other hand, when the number of neu-
rons is large, a reliable description of the asymptotic solutions of these equations
can be found using the techniques of statistical mechanics [40]. In this framework,
instead of characterizing the states of the system by the state of every neuron, this
characterization is performed in terms of
macroscopic
quantities called
order pa-
rameters
which measure and quantify some global properties of the network as a
whole. The relevant order parameters appearing in the description of the system are
the overlap of the state of each module with each of the stored patterns
m
a
and the
average activity of each module
x
a
, defined respectively as:
10
5
∼
−
1
1
N
∑
i
(
η
ai
−
N
∑
i
ν
m
a
=
f
)
ν
η
;
x
a
=
η
,
(16.13)
ai
ai
where the symbol
...
η
stands for an average over the stored patterns.
Using the free energy per neuron of the system at zero temperature
(which
is not written explicitly to reduce the technicalities to a minimum), [70] and [67]
F
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