Biomedical Engineering Reference
In-Depth Information
RECURRENT NEURAL NETWORkS
AFFECTED by TIME-DELAyS
−
1
1
n
(
)
∑
=
v
(
t
)
=
v
+
(
v
)
−
v
R
+
I
,
k
=
1
n
k
k
j
j
k
kj
k
R
C
C
k
k
k
j
1
(22)
A more realistic model for RNN has to take into
account the dynamics affected by time delays
due to the signal propagation at the synapses
level or to the reacting delays in the case of the
artificial neural network. Since these delays
may introduce oscillations or even may lead to
the instability of the network, the dynamical
behaviors of the delayed neural networks have
been investigating starting with the years of '90
(Marcus & Westervelt, 1989; Gopalsamy & He,
1994; Driessche & Zou, 1998; Danciu & Răsvan,
2001, 2005, 2007).
Investigating the dynamics of time delayed
neural networks require some mathematical
preliminaries. Let us recall that for systems
modeled by ordinary differential equations (6) a
fundamental presumption is that the future evo-
lution of the system is completely determined by
the current value of the state variables, i.e., one
can obtain the value of the state
x
(
t
) for
where,
C
k
models the membrane capacitance,
R
k
- the transmembrane resistance,
I
k
- the external in-
put current,
v
k
- the instantaneous transmembrane
potential, 1/
R
kj
,
, =
represent the intercon-
nection weight conductances between different
neurons. The identically sigmoidal functions
k
k
j
1
n
,
j
k
1=
satisfying (19) are introduced in the electri-
cal circuit by nonlinear amplifiers with bounded,
monotone increased characteristics.
It can be verified that equations (22) are of the
type (18) when
n
n
1
1
n
1
∑
A
=
diag
−
+
,
C
R
R
1
k
k
kj
k
=
1
(
)
,
(
)
,
1
n
k
n
k
f
(
v
)
=
col
(
v
)
h
=
−
col
I
C
k
k
k
k
=
1
=
( )
,
t
∈
t
[
0
∞
,
)
n
k
C* = I,
B
=
−
ΓΛ
,
Γ
=
diag
1
C
k
=
1
once the initial condition
tx
=
is known. For
systems with time delays the future evolution
of the state variables
x
(
t
) not only depends on
their current value
x
(
t
0
), but also on their past
values
x
(x), where
(
0
)
x
0
( )
n
Λ
=
1
R
.
kj
k
,
j
=
1
Chosing
=
and
t
−∈
, x > 0. In this
case the initial condition is not a simple point
[
,
t
]
k
k
0
0
n
∑
=
q
=
1
/
R
+
(
/
R
)
,
∈
0
in the state space, but a real valued vector
function defined on the interval
m
x
k
k
kj
[
t
−
,
t
]
j
1
, i.e.,
0
0
the frequency domain inequality (20) holds
provided the matrix of synaptic weights
Λ
is
symmetric.
One obtained that the system (22) is in the case
of
Theorem 2,
thus gradient-like if the symmetry
condition
R
ij
= R
ji
is verified. This condition,
mentioned also in (Noldus
et al.
, 1994, 1995) is
quite known in the stability studies for neural
networks and it is a standard design condition
since the choice of the synaptic parameters is
controlled by network adjustment in the process
of “learning”.
. For this reason one choice
for the state space of the time delay systems is
the space
m
:
[
t
−
,
t
]
→
0
0
of continuous
m
(
−
,
)
-valued
mappings defined on the interval
[−
. Time de-
lay systems are modeled by
functional diferential
equations
with the general form
,
x
=
f
(
t
,
x
)
(23)
t
m
,
where
x
∈
m
and generally, one
f
:
×
→
denotes by
t
a segment of the function ψ defined
as
[−∈
. Equation
(23) indicates that in order to determine the future
(
)
=
(
t
+
)
with
,
t
t
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