Information Technology Reference
In-Depth Information
In fact the Laplace transform of (
7.155
) can, using (
7.159
), be written
1
R
(
u
)
=
,
(7.160)
−
ψ(
1
u
)
from which it is clear that the number of events is determined directly by the event prob-
ability density. Note that the aged event probability density
t
)
ψ(
t
,
expressed by (
7.154
)
requires an independent determination of
, to which we now turn our attention.
To realize the phenomenological condition necessary for
R
ψ(
t
)
→∞
requires that the reduction of response intensity may, but need not, have a lower
limit, depending on whether or not the statistics of the neural network are ergodic
or non-ergodic. To establish this condition, consider the relation between the rate of
event production and the event probability density given in Laplace space by (
7.160
).
Recalling the Laplace transform of the hyperbolic distribution
(
t
)
to decrease as
t
)
μ
−
1
e
uT
1
ψ(
E
uT
μ
−
u
)
=
(μ
−
1
)(
1
−
μ)(
uT
−
,
(7.161)
we can write the asymptotic expansion as
u
→
0 to obtain to lowest order in
u
in the
case of no finite first moment
0
ψ(
)
μ
−
1
lim
u
u
)
=
1
−
(
2
−
μ)(
uT
+··· ;
1
<μ<
2
,
(7.162)
→
and for a finite first moment but a diverging second moment
0
ψ(
)
μ
−
1
lim
u
u
)
=
1
+
u
t
−
(
2
−
μ)(
uT
+··· ;
2
<μ<
3
,
(7.163)
→
in which the lowest-order terms in powers of
u
of the two functions in (
7.161
) have been
retained in both expressions. Inserting (
7.162
) into the expression for the rate of event
production yields
−
μ
1
)
=
(
uT
)
0
R
lim
u
(
u
−
μ)
;
1
<μ<
2
,
(7.164)
(
2
→
and inserting (
7.163
) yields
1
(
2
−
μ)
0
R
lim
u
(
u
)
=
−
−
μ
;
2
<μ<
3
,
(7.165)
2
3
u
t
t
(
uT
)
→
respectively. Using the Tauberian theorem relating the time and Laplace domains,
t
n
−
1
(
1
u
n
⇔
)
,
(7.166)
n
we obtain from (
7.164
) for the rate of event production as
t
→∞
for the non-ergodic
case
T
t
2
−
μ
sin
[
π(μ
−
1
)
]
R
(
t
)
≈
;
1
<μ<
2
,
(7.167)
π
T
and for the ergodic case
1
μ
−
2
; 2
T
t
1
1
R
(
t
)
≈
+
<μ<
3
.
(7.168)
t
3
−
μ
