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r
=
a
.
ij
ij
m
m
This result is of significance insofar as it shows that two entirely different
structures have precisely the same stimulus-response characteristic. For
example, Hartline's observation of inhibitory interaction amongst the fibers
in the optic stalk of the horseshoe crab can be explained equally well by an
appropriate post-ommatidial action net. It is only the anatomical evidence
of the absence of such nets which forces us to assume that interaction
processes are responsible for the observed phenomena.
The converse of the above theorem “for each action net there exists a
functionally equivalent interaction net” is true only if the characteristic
determinant of the inverse of A m does not vanish.
We consider k + 1 cascaded layers L i ( i = 0,1,..., k ) with the transmitting
layer L o the locus of receptors proper, and with all elements in layer L i -1
acting upon all elements in layer L i , their actions defined by an action
matrix A mi . The action performed by the receptors e o j on the ultimate effec-
tors e k 1 is again (see eq. (3)) defined by the matrix product of all A mi
k
) = '
(
Cas
m AA
,
,...,
A
A
.
(78)
m
2
mk
mi
1
Hence, we have the following theorem:
Any cascaded network of a finite number of layers, each acting upon its
follower with an arbitrary action matrix can be replaced by a functionally
equivalent single action net with an action matrix
k
= '
()
()
k
l
a
a
.
(79)
ij
ij
m
m
l
=
1
Again, gross structural differences may lead to indistinguishable
performances.
We generalize our observation in Chapter 1, eq. (8) concerning the
invariance of certain action nets to cascading. An action matrix with all rows
alike
*
aa
ij
=
,
kj
and for which
m
Â
* =
a ij
1
j
=
1
is invariant to being cascaded. Let
A mij
*
=
,
m
we have
[
AA
m
*]
k
.
(80)
m
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