Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 67

4.4.1. Linear Elements

Consider a set of n = 2 m linear elements, half of which are general recep-
tors and the other half general effectors. A weak geometrical constraint,
which does not affect the generality of some of the following theorems but
facilitates description, is to assume spatial separation of receptors and effec-
tors. The locus of all general receptors, e 1 i , we shall call “transmitting layer”
L 1 , and the locus of all general effectors, e 2 j ,( i , j = 1,2,3,..., m ) the “col-
lecting layer” L 2 , regardless of the dimensionality of these loci, i.e., whether
these elements are arranged in a one-dimensional array, on a two-
dimensional surface, or in a specifiable volume.
We consider the fraction of activity in element e 1 i that is passed on to an
element e 2 i as its partial stimulus a ij ( i ). The action coefficients for the m 2
pairs define the numerical action matrix

(75)

A mij = .

m

With linear elements in the collector layer their responses are the algebraic
sum of their partial stimuli:

m

Â

() = () r j a s i , (76)

ij

i

or in a matrix notation

r s

mmm

= A . (77)

Since this result is in complete analogy to interaction nets (eq. (74))
where the response matrix R n establishes the stimulus-response relation-
ship, we have the following theorem:
Any stable interaction network composed of m elements can be repre-
sented by a functionally equivalent action network composed of 2 m ele-
ments (see fig. 23):

FIGURE 23. Equivalence of action network with interaction network.