Biomedical Engineering Reference
In-Depth Information
gives the following result (Danciu & Răsvan,
2005)
that a time-varying periodic signal repeats in
time with some periodicity, whereas an almost
periodic signal is composed of two or more peri-
odic signals with incommensurable frequencies;
here the term incommensurable means that the
ratio of the frequencies is an irrational number,
which implies that the phase relationship between
different periodic signals changes on every cycle
forever.
The model for Hopfield-type neural networks
with time delays and time-varying stimuli has
the form
Theorem 4.
System (34) is exponentially stable
for sigmoidal functions, in particular for (35),
provided there exist positive definite matrices,
P
0
, P
i
, R
j
, and a positive value
m
, such that
T
P
>
U
(
0
U
(
0
,
R
>
0
,
0
j
m
2
∑
=
2
P
−
2
k
c
I
>
0
j
j
ij
i
1
(39)
The
Linear Matrix Inequalities
(39) may be
computed using the available commercial soft-
ware on this topic. Cleary (39) gives a condition
on the weights
c
ij
and the nonlinearity constants
k
. The symmetry requirement for
c
ji
is no longer
necessary, what gives a sensible relaxation of the
stability restrictions since symmetry is not a result
of the learning process.
Remark that the conditions (39) are valid
regardless the values of the delays hence we ob-
tained what is usually called
delay-independent
stability
. Also stability is
exponential
since both
the Krasovskii-Lyapunov functional and its de-
rivative are quadratic functionals. Stability is also
global
since the functional and the inequality for
the derivative are valid globally.
m
(
)
∑
x
(
t
)
=
−
a
x
(
t
)
−
c
f
x
(
t
−
)
+
d
(
t
)
,
i
=
1
m
.
i
i
i
ij
j
j
ij
i
1
(40)
The forcing stimuli
d
i
(
t
) are periodic or almost
periodic (see for instance the allures in Figure 5
and Figure 6) and the main mathematical problem
is to find conditions on the systems to ensure
existence and exponential stability of a unique
global (i.e. defined on
) solution that has the
features of a limit regime, i.e., it is not defined
by the initial conditions and is of the same type
as the stimulus—periodic or almost periodic,
respectively.
We shall present here for comparison some
results for the two approaches of the absolute
stability theory for the estimates of systems' solu-
tions: Lyapunov function(al) and the frequency
domain inequality of Popov.
The first result (Danciu & Răsvan, 2007) is
based on the application of the Lyapunov func-
tional (26) but restricted to be only quadratic in
the state variables (li
i
=
0, δ
ij
=
0)
Synchronization Problems for Neural
Networks with Time Delays
In the specific literature of the neuroscience do-
main are reported studies on the rhythmic activi-
ties in the nervous system (see for instance Kopell,
2000) and on the synchronization of the oscillatory
responses with the external time-varying inputs
(König & Schillen, 1991). Both rhythmicity and
synchronization suggest some recurrence and this
implies coefficients and stimuli being periodic
or almost periodic in the mathematical model
of the artificial neural networks. Let us mention
0
m
1
m
( )
∑
∑
∫
V
x
(
⋅
)
=
x
2
(
0
)
+
x
2
(
)
d
i
i
ij
j
2
i
=
1
j
=
1
−
ij
(41)
with pi
i
>
0, r
ij
>
0,
. The derivative func-
i
, =
j
1
m
tional corresponding to
in (40) is
d
i
( ≡
t
0
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