Biomedical Engineering Reference
In-Depth Information
able Taylor-series expansion about the origin (it
is analytic), then the function
2
m
m
m
E
:
→
with
∑
∑
W
(
x
)
=
−
a
(
x
)
d
′
(
x
)
b
(
x
)
−
c
d
(
x
)
i
i
i
i
i
i
ij
j
j
the form
i
=
1
j
=
1
(15)
1
T
E
( =
z
z
Cz
(12)
The condition that the Lyapunov function (14)
to be nonincreasing in time along the systems
trajectories implies
2
where the matrix
C
is symmetric, has the sig-
nificance of the potential energy of the system.
The negative of this function may be a Lyapunov
function candidate for the system (Kosko, 1992).
In this case the Lyapunov function was obtained
in a natural way and is of the quadratic form.
xW
. This inequality
holds provided
a
i
(l) > 0 and
d
i
(l) are monotone
non-decreasing. If additionally
d
i
(
x
i
) are strictly
increasing, then the set where
W =
0 consists
of equilibria only; according to
Lemma 2
in
this case the system has
global asymptotics
(is
quasi-gradient like
) i.e. every solution approaches
asymptotically the stationary set. As already said,
the above conditions on the system's parameters
are only sufficiently for this behavior.
Let us remark that if functions
d
i
(
x
i
) are strictly
increasing, then (13) may be written as
( ≤
0
Qualitative behaviors for General
Competitive Cohen-Grossberg
Neural Networks
Consider the general case of the Cohen-Grossberg
(1983) competitive neural network
x
−
=
A
(
x
)
gradV
(
x
)
(16)
m
∑
=
x
=
a
(
x
)
b
(
x
)
−
c
d
(
x
)
,
i
=
1
m
,
i
i
i
i
i
ij
j
j
which makes system (16) a
pseudo-gradient
system
- compare to (9); here
A
(
x
) is a diagonal
matrix with the entries (
op. cit.
)
j
1
(13)
with
c
ij
= c
ji
(the symmetry condition). Worth
mentioning that equations (13) account not only for
the model of the neural networks just mentioned
but they also include: Volterra-Lotka equations
of population biology, Gilpin and Ayala system
of growth and competition, Hartline-Ratliff equa-
tions related to the Limulus retina, Eigen and
Schuster equation for the evolutionary selection
of macromolecular quasi-species (
op. cit.
).
To system (13) it is associated the Lyapunov
function (Cohen-Grossberg, 1983)
a
(
x
)
A
(
x
)
=
i
i
ij
ij
(17)
′
d
(
x
)
i
i
For a
gradient-like behavior
of system (13)
one has to find the conditions ensuring that the
stationary set consists of equilibrium points only
and more, that these equilibria are isolated. Since
the model of system (13) is more general and the
stationary set consists of a large number of equi-
libria, the study of these qualitative properties is
done on particular cases.
m
V
:
→
x
1
m
m
m
i
Qualitative behaviors for a General
Network with Several
Sector-Restricted Nonlinearities
∑∑
∑
∫
V
(
x
)
=
c
d
(
x
)
d
(
x
)
−
b
(
)
d
′
(
)
d
ij
i
i
j
j
i
i
2
1
1
1
0
(14)
Its derivative along the solutions of the system
(13) is
Consider the general model of a RNN (Noldus
et al.
, 1994, 1995)
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