Biomedical Engineering Reference
In-Depth Information
Qualitative behaviors for
Time Delay Neural Networks via
Popov-Like Results Using
Comparison
where * denotes convolution with the diagonal
matrix kernel k.
In order to obtain the time delay system of
the form considered by Popov (31) (
op. cit.
), it is
necessary to introduce another assumption about
(25), namely t
ij
= t
j
,
i
1=
for each
j
. The neural
network with delay will thus be described by
Consider the time delay Hopfield network (24) and
its system in deviations (25). From the point of
view of the qualitative behavior of neural networks
as systems with multiple equilibria, the results of
V.M. Popov (1979) of interest here, are concerned
with
dichotomy
,
global asymptotics
and
gradient
behavior
for two systems that form the so-called
comparison system (SM):
m
m
(
)
∑
x
(
t
)
=
−
x
(
t
)
−
c
h
x
(
t
−
)
,
i
=
1
m
i
i
ij
j
j
j
1
(32)
Note that the equations of the neural network
with delay (32) may be reduced to (31) with k
defined by
(M) - the “model” - which corresponds to a
delayless neural network, e.g., described by (10)
with
a
i
=
1,
∀
i
−
(
t
−
)
e
,
t
≥
j
j
(
t
)
=
m
∑
=
j
u
=
−
u
−
h
(
u
)
,
i
=
1
m
0
,
e
lsewhere
.
i
i
ij
j
j
(29)
j
1
(33)
where p
ij
= p
ji
define a non-singular symmetric ma-
trix P and the functions
j
h
are strictly
increasing and uniformly Lipschitz verifying
(⋅
)
:
In this case it is easy to prove (see Răsvan &
Danciu, 2004; Danciu, 2006) that the time-de-
layed neural network described by (32) inherits
the global properties of (29): dichotomy, global
asymptotics or gradient behavior (the best property
for the neural networks).
h
(
)
−
h
(
)
j
1
j
2
0
<
≤
<
L
,
h
0( =
0
.
j
j
−
1
2
(30)
In this case every bounded solution of (29)
approaches the set of equilibria (dichotomy).
Let
Stability Results for Cellular
Neural Networks with Time-Delays
via “Exact” Lyapunov-Krasovskii
Functional
−
Pm
,
i.e., some small gain condition holds. Then
u
(
t
)
is bounded and if the set of equilibria is finite
(like in the case of neural networks) then system
of (29) is gradient-like (for the proof see V.M.
Popov, 1979).
and assume that
1
|
|
>
0
L
=
max >
L
1
i
i
Cellular Neural Networks
(CNN) introduced
in 1988 by Chua & Yang, are artificial RNN
displaying multidimensional arrays of identical
nonlinear dynamical systems (the so-called cells)
and only local interconnections between the cells
(allowing VLSI implementation). CNN have been
successfully applied to complex image process-
ing, shape extraction and edge detection, but
also for nonlinear partial differential equations
(S) - the “system” - which is described by the
following equation
x
+
+
P
∗
h
( =
x
0
`
(31)
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