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Fig. 8.1
The model of a Hopfield associative memory with feedback weights
Function g
r
./ can be associated with a repulsion potential, i.e.
r
x
i
V
r
.jjx
i
x
j
jj/ D .x
i
x
j
/g
r
.jjx
i
x
j
jj/
(8.13)
A suitable function g./ that describes the interaction between the weights is given
by [
150
]
x
j
/.a be
jj
x
i
x
j
jj
2
g.x
i
x
j
/ D.x
i
/
(8.14)
2
where the parameters a, b, and c may also be subjected to stochastic variations and
noise terms. It holds that g
a
.x
i
x
j
/ Da, i.e. attraction has a linear behavior
(spring-mass system) jjx
i
x
j
jjg
a
.x
i
x
j
/. Moreover,
x
j
/ D be
jj
x
i
x
j
jj
2
(8.15)
g
r
.x
i
2
which means that g
r
.x
i
x
j
/jjx
i
x
j
jj b is bounded. For the i-th particle with
unit mass m
i
D 1 holds
x
i
D
v
i
;m
i
v
i
D U
i
(8.16)
where the aggregate force is U
i
DK
v
v
i
denotes friction,
while the term F
i
is the propulsion. Assuming zero acceleration
D f
i
CF
i
.Thetermf
i
v
i
D 0 one gets
F
i
D K
v
v
i
, which for K
v
D 1 and m
i
D 1 gives F
i
D
v
i
. Thus an approximate
kinematic model is
x
i
D F
i
(8.17)
