Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 73

FIGURE 25. One dimensional periodic action net. Network (a); action function (b);
receptor function (c); equivalent two-input element network (d); symbolic repre-
sentation (e).

The equivalent net with McCulloch elements having only two inputs is
given in fig. 25d. In fig. 25e its equivalent is symbolized in purely logical
terms. The index 1 in G 1 and K 1 is adopted to discriminate this receptor
and action function from a general class of such functions, K m , the so-called
m th order binomial action function:

2 m

Ê
Ë

ˆ
¯

D

= () K

m *

1 , D =±± 012 , , ,..., ± m . (92)

m + D

These arise from cascading binomial action nets m times, as suggested in
fig. 26:

+=-
-=-
+=
-=+
+=+

D
D
D
D
D

1 , 2 ,

Ï

Ô
Ô

4 , 1 ,

4

Ê
Ë

ˆ
¯ =

D

= ()

*

K 2 1

Ì 6 , 0 ,

2 + D

Ô
Ô

4 , 1 ,

Ó

1 , 2 .

(i) Sherrington Element

In order to obtain stimulus response relationship in the network of fig. 25
we have to specify the operational modality of the elements. First we
assume a strictly linear model (Sherrington element). Let s( x ) and r( u ) be
stimulus and response at points x and u respectively. We have

() = () -- ( ) -+ ( )

r u 2 s x s x D x s x D . x