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is connected to the input of the i-th neuron through a resistance R
ij
. The nonlinear
function that describes connection between neurons g
i
.
u
/ is a sigmoidal one. By
defining the parameters
T
ij
C
i
c
i
D
1
I
i
C
i
b
i
D
C
i
R
i
˛
ij
D
(8.2)
Thus, Eq. (
8.1
) can be rewritten as
x
i
.t/ Db
i
x
i
.t/ C
P
jD1
˛
ij
g
j
.x
j
.t// C c
i
1in
(8.3)
which can be also written in matrix form as
x.t/ D
Bx
.t/ C
Ag
.x.t// C C
(8.4)
where
x.t/ D .x
1
.t/; ;x
n
.t//
T
B D diag.b
1
; ;b
n
/
AD .a
ij
/
nn
(8.5)
C D .c
1
; ;c
n
/
T
g.x/ D .g
1
.x
1
/; ;g
n
.x
n
//
T
One can consider the case
b
i
D
P
jD1
j˛
ij
j >0;c
i
01in
(8.6)
Moreover, by assuming a symmetric network structure it holds
(8.7)
˛
ij
D ˛
ji
1ijn
which means that A is a symmetric matrix. It is also known that neural networks are
subjected to noise. For example, if every external input I
i
is perturbed in the way
I
i
!I
i
C
i
w
i
.t/, where
w
i
.t/ is white noise, then the stochastically perturbed neural
network is described by a set of stochastic differential equations (Wiener processes)
of the form
(8.8)
dx
.t/ D Œ
Bx
.t/ C
Ag
.x.t// C C
dt
C
1
d
w
1
.t/
where
1
D .
1
=C
1
; ;
n
=C
n
/
T
. Moreover, if the connection matrix element T
ij
is perturbed in the way T
ij
!T
ij
C
ij
w
2
.t/, where
w
2
.t/ is another white noise
independent of
w
1
.t/, then the stochastically perturbed neural network can be
described as
(8.9)
dx
.t/ D Œ
Bx
.t/ C
Ag
.x.t// C C
dt
C
1
d
w
1
.t/ C
2
g.x.t//
dw
2
.t/
where
2
D .
ij
=C
i
/
nn
. In general one can describe the stochastic neural network
by a set of stochastic differential equations of the form
