Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 61

SN

2 ∫

N A . (fig. 20.1)

We complete the relation for N 4 by drawing the net (bold line):

( ) , NSNN

A

∫ ◊ (fig. 20.2)

4 2

which is, of course,

( ) .

N ∫ S SN ◊ N

4 2 2

We approach now the expression for N 3 and draw (bold line):

( ) ∫ SN ◊ N N , (fig. 20.3)

A 2 B

and complete the whole net by drawing (bold line):

∫ ( v.

NSNN B (fig. 20.4)

3 1

Although this simple example does not do justice to the profoundness of
the McCulloch-Pitts theorem, it emphasizes not only the important rela-
tionship between formal networks and formal logic but also the minimal
structural necessity to accommodate functional requirements.

4.3. Interaction Networks of Discrete, Linear Elements
We now turn our attention to networks which are composed of “linear ele-
ments”, i.e., of McCulloch elements operating with low thresholds (qª1)
and weak signals ( f i << 1/(D t +D t R ) in an asynchronous network, or of
Sherrington elements (m=0) which perform algebraic summation on their
inputs.
The formalism which handles the situation of an arbitrary number of
interacting elements has completely been worked out by Hartline (1959)
who showed in a series of brilliant experiments the mutual inhibitory action
of proximate fibers in the optic stalk of the horseshoe crab by illuminating
various neighbors of a particular ommatidium in the crab's compound eye.

FIGURE 20. Stepwise development of a McCulloch-Pitts network that computes the
“illusion of heat and cold”. Bold lines represent added network elements (q-2,
everywhere).