Information Technology Reference
In-Depth Information
whose solution for
q
as a function of time
t
is:
qS
o
(
)
q
=
1
-
e
-
l
t
,
(35)
l
if at time
t
= 0 we also have
q
= 0.
Let D
t
* be the time required to accumulate the amount
q
* at the element,
and let the activity
S
change during many such time intervals. Clearly, the
“internal frequency” of this element is
f
= 1 D *.
t
(36)
Inserting these into eq. (35):
qS
(
)
o
q
*
=
1
-
e
-
l
f
,
(37)
l
we have an expression that relates the frequency
f
with the stimulus activ-
ity
S
. For convenience we introduce new variables
x
and
y
which represent
normalized input and output activity respectively and which are defined by:
xSq q
=
o
l
*,
yf
=
l
.
(38)
With these, eq. (37) can be rewritten:
1
=-
-
e,
1
1
y
(39)
x
or, solved for
y
:
[
(
)
]
y
=
1
ln
x
-
ln
x
-
1
.
(40)
This relation shows two interesting features. First, it establishes a threshold
for excitation:
x
o
= 1
or
Sqq
q
=
l
*.
o
Because for the logarithmic function to be real its argument must be posi-
tive, or
x
1. It may be noted that the threshold frequency
S
q
is given only
in terms of the element's intrinsic properties l,
q
o
and
q
*.
The second feature of the transfer function of this element is that for
large values of
x
it becomes a linear element. This is easily seen if we use
the approximations
1
ª+
1
e
1
-
e
and
(
)
ee
ln 1 +
Search WWH ::
Custom Search