Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 33

In general, if we have, for all m:

( ) =

F ij lm l , ,

we have in the connection matrix

c ij = 0.

If we wish to establish whether or not an element e i is actively connected
to itself, we again set up a transition matrix, only replacing m by l.An
example of the hypothetical transition matrix of a self-connected element,
capable of three states:

l
l¢ = F (l, l) 123

1 2--

l 2 -1-

3 --3

This element oscillates between states 1 and 2 but stays calmly in 3 when
in 3.
The state transition matrix that describes the action of, say, ( k - 1) ele-
ments on some other element is, of course, of k dimensions. In this case the
state labels for the i th element may be called l i . An example of such a
matrix for two elements e 2 , e 3 , acting on e 1 is given in fig. 8.
If the states of our elements represent some physical variable which may
undergo continuous changes, for instance, if these states represent the mag-
nitudes of an electrical potential, or of a pressure, or of a pulse frequency,
the symbol S i itself may be taken to represent this magnitude and eq. (12),
which described the state transitions of element e j under the influence of
element e i , assumes now the form of a differential equation

d
d

S
t

j

[

()

]

= F St , (14)

ij i

FIGURE 8. State transition matrix for three elements.