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FIGURE 26. Cascading of binomial action function of
m
th function of (
m
+ 1)th
order.
Expanding s around
x
we obtain:
∂s
∂
1
2
∂s
∂
2
1
6
∂s
∂
3
(
)
=
()
±
2
3
s
xx
±
D
s
x
D
x
+
D
x
±
D
x
+
...,
x
x
2
x
3
which, inserted above, gives
2
4
∂s
∂
1
12
∂s
∂
()
=
r
u
D
2
x
+
D
4
x
+
....
x
2
x
4
Neglecting fourth order and higher terms, this lateral inhibition net extracts
everywhere the second derivative of the stimulus distribution. It can easily
be shown that the
m
th binomial action functions will extract the 2
m
th deriv-
ative of the stimulus. In other words, for uniform stimulus, strong or weak,
stationary or oscillating, these nets will not respond. However, this could
have been seen by the structure of their binomial action function, since
+
Â
1
m
2
m
Ê
Ë
ˆ
¯
=
D
()
0
.
m
+
D
-
m
(ii) McCulloch Element; Asynchronism
We change the
modus operandi
of our elements, adopt a McCulloch
element with unit threshold (q=1), and operate the net asynchronously.
We ask for the output frequency of each effector element, given the stim-
ulus distribution. For simplicity, we write our equations in terms of the ON
probability
p
of the elements. With numbers 1, 2, 3, we label the afferent
fibers in the receptor field (see fig. 25c). The truth table is easily established,
giving an output ON for input states (010), (011) and (110) only. The sur-
viving Bernoulli products (eq. (25)) are:
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