Biomedical Engineering Reference
In-Depth Information
b.
A solution of (6) is called convergent if it
approaches asymptotically some equilib-
rium:
totical convergence of the solution
x
(
t
,
x
0
,
t
0
)
means a “good” remembering process. The
attraction region of a stored pattern may be
viewed as the set of all its distorted or noisy
patterns, which still allow recalling it. The
problem of computing regions of asymptotic
stability for nonlinear systems and also for
analogue neural classification networks is
widely discussed in the literature, but is
beyond the scope of this chapter; the reader
is send to the papers of Noldus
et al.
(1994)
and Loccufier
et al.
(1995).
d. Since there exists also other terms for the
above introduced notions some comments
are necessary.
Monostability
has been intro-
duced by Kalman (1957) and sometimes it
is called
strict mutability
(term introduced
by Popov in 1979), while
quasi-monostabil-
ity
is called by the same author
mutability
and by others
dichotomy
(Gelig, Leonov &
Yakubovich, 1978). The
quasi-gradient-like
property is called sometimes
global asymp-
totics
.
e. In the sequel we shall use the following
terms:
lim
x
(
t
)
=
c
∈
S
(7)
t
→
∞
c.
A solution is called quasi-convergent if it
approaches asymptotically the stationary
set
S
:
lim
d
(
x
(
t
),
S
)
=
0
(8)
t
→
∞
where
⋅
d
denotes the distance from a
point to the set
S.
(
,
S
)
Definition 2:
System (6) is called monostable if
every bounded solution is convergent; it is called
quasi-monostable if every bounded solution is
quasi-convergent.
Definition 3:
System (6) is called gradient-
like if every solution is convergent; it is called
quasi-gradient-like if every solution is quasi-
convergent.
•
Dichotomy:
All bounded solutions tend
to the equilibrium set;
•
Global asymptotics:
All solutions tend
to the equilibrium set;
•
Gradient-like behavior:
The set of
equilibria is stable in the sense of Ly-
apunov and any solution tends asymp-
totically to some equilibrium point.
From the above definitions we have:
a. An equilibrium is a constant solution
T
th
. The
set of equilibria may consist of both stable
and unstable equilibria.
b. For RNN, the stable equilibria are used to
retain different memories; unstable equilib-
ria will never be recalled.
of the equation
x
=
[
x
1
x
]
(
,
x
)
=
0
n
f. Dichotomy signifies some genuine nonoscil-
latory behavior in the sense that there may
exist unbounded solutions but no oscillations
are allowed. In general, the gradient-like
behavior represents the desirable behavior
for almost all systems. If the equilibria are
isolated then global asymptotics and gradi-
ent-like behavior are equivalent (any solution
may not approach the stationary set other-
wise than approaching some equilibrium).
c.
An initial condition for the RNN is any
pattern that is presented at the input of the
network. If for an input pattern
x
0
the evolu-
tion
tx
of the network is such that a
stored pattern
x
is attained in an asymptoti-
cal manner, then the initial pattern is within
the
asymptotic stability
(
attraction
)
region
of the equilibrium pattern and the asymp-
(
,
x
,
t
)
0
0
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