Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 52

A graphical representation of the input-output relationship of this
element according to eq. (47) is given in fig. 16 for three different values of
m (0; 0.2; 0.5). For small m (m<0.05) the element is linear with threshold
x o = 1. For large m (ª 0.5) the element is an almost perfect “All or Nothing”
device with threshold

1
1 m .

x o = (48)

-

For values of m in the range 0.1 Æ 0.5 the element displays a logarithmic
transfer function. Indeed, it can easily be shown that in the “vicinity” of

1
1mm .

x ª (49)

( )

-

Eq. (47) can be approximated by

yKx

o =

ln , (50)

1
4

m

K = . (51)

2

[ ] ( ) ( )

1 - m ln 1 - m

It may be noted that this “vicinity” extends over an appreciable range as
can be seen by the approximation according to (50) which in fig. 16 is super-
imposed in thin line over the exact curves given in bold line.
With the choice of the parameter m we are in a position to change the
transfer function of this element considerably. We suggest the following
nomenclature for the elements that arise for different values of m:

FIGURE 16. Transfer function of an
integrating element under various
operating conditions (m) (bold line),
and approximation by a logarithmic
function (thin line).