Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 30

FIGURE 5. Cascade of action networks (a) and equivalent action network (b).

Translated into network language, this says that an action network with all
receptors connected to the same effectors remains invariant when cascaded
an arbitrary number of times. As an example, consider the action matrix

1
4

1
4

1
4

1
4

0 0

1
4

1
4

1
4

1
4

011101
011101
011101
011101
011101
011101

0 0

1
4

1
4

1
4

1
4

0 0

A 6 = = ¥ 4 ,

1
4

1
4

1
4

1
4

0 0

1
4

1
4

1
4

1
4

0 0

1
4

1
4

1
4

1
4

0 0

for which the equivalent network is represented in fig. 6. Call p the number
of effectors contacted ( p = 4 in fig. 6), and N ( A m ) the normalized matrix
whose elements are

= 1

n

p a

. (10)

ij ij

Clearly,

m

Â

n ij = 1 ,

1

and k cascades give

( ) .

ApNA

k

=

k

(11)

m