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A
mij
=
.
m
Addition and subtraction of two matrices of like order is simply carried
out by adding or subtracting corresponding elements:
(1)
(2)
It is easy to see that matrix addition or subtraction corresponds to super-
position or subposition in our networks. However, we should be prepared
to obtain in some of the new entries numbers that are larger than unity or
less than zero, if by change the network superimposed over an existent one
has between two elements a connection that was there in the first place.
Hence, the new entry will show a number “2” which is not permitted accord-
ing to our old rules of representing connections in matrices (which admit-
ted only “ones” and “zeros” as entries). We may, at our present state of
insight, gracefully ignore this peculiarity by insisting that we cannot do more
than connect, which gives a “one” in the matrix. Reluctantly, we may there-
fore adopt the rule that whenever matrix manipulation produces entries
c
ij
> 1, we shall substitute “1” and for entries
c
ij
< 0, we shall substitute “0”.
Nevertheless, this tour de force leaves us with an unsatisfactory aftertaste
and we may look for another interpretation. Clearly, the numbers
c
ij
that
will appear in the entries of the matrix indicate the numbers of parallel
paths that connect element
e
i
with element
e
j
. In a situation where some
“agent” is passed between these elements, this multiplicity of parallel path-
ways can easily be interpreted by assuming that a proportionate multiple
of this agent is being passed between these elements. The present skeleton
of our description of networks does not yet permit us to cope with this
situation, simply because our elements are presently only symbolic blobs,
indicating the convergence and divergence of lines, but incapable of any
operations. However, it is significant that the mere manipulations of the
concepts of our skeleton compel us to bestow our “elements” with more
vitality than we were willing to grant them originally. We shall return to this
point at the end of the chapter; presently, however, we shall adopt the
pedestrian solution to the problem of multiple entries as suggested above,
namely, by simply chopping all values down to “0” and “1” in accordance
with our previous recommendations.
Having eliminated some of the scruples which otherwise may have
spoiled unrestricted use of matrix calculus in dealing with our networks, we
may now approach a problem that has considerable significance in the phys-
iological case, namely, the treatment of cascades of action networks. By a
cascade of two action networks
A
m
and
B
m
, symbolically represented by
Cas(
AB
)
m
we simply define a network consisting of 3
m
elements, in which
all general effectors of
A
m
are identical with the general receptors of
B
m
.
Fig. 5a gives a simple example. The question arises as to whether or not
such a cascade can be represented by an equivalent single action net.
ABC c
mm m ij
±== ,
m
cab
ij
=±.
ij
ij
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