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(72)
Rr
=
=
S
-1
.
n
ij
n
n
Let |
D
| denote the characteristic determinant |
s
ij
|, and
S
ij
the product of
(-1)
i
+
j
with the determinant obtained from |
s
ij
| by striking out the
i
th row
and
j
th column, then the response matrix elements
r
ij
are given by
S
D
ji
r
=
,
(73)
ij
and we have the solution for the responses:
(74)
r
=
R
s
.
n
n
n
Clearly, a solution for r
n
can be obtained only if the characteristic deter-
minant
D
does not vanish, otherwise all responses approach infinity, which
implies that the system of interacting elements is unstable. It is important
to note that stability is by no means guaranteed if the interactions are
inhibitory, for the inhibition of an inhibition is, of course, facilitation.
The actual calculation of a response matrix, given the numerical inter-
action matrix, is an extremely cumbersome procedure that requires the
calculation of
n
2
matrices, each of which demands the calculation of
n
! prod-
ucts consisting of
n
factors each, that is
n
3
·
n
! operations all together. Under
these circumstances it is clear that manual computation can be carried out
for only the most simple cases, while slightly more sophisticated situations
must be handled by high speed digital computers; even they prove insuffi-
cient if the number of elements goes beyond about, say, 50. However, the
horseshoe crab performs these operations in a couple of milliseconds by
simultaneous parallel computation in the fibers of the optic tract. The
Limulus
' eye is—so to say—made for matrix inversion.
In order to clarify procedures we give a simple example of four elements
e
1
,
e
2
,
e
3
,
e
4
which are thought to be placed at the four corners of a square
labeled clockwise with
e
1
in the NW corner. We assume mutual interaction
to take place between neighbors only, and all coefficients to be alike at
a
.
The connection matrix for this configuration is
e
j
1234
1
0101
2
1010
e
i
3
0101
4
1010
and the numerical interaction matrix
0
a
0
a
a
0
a
0
A
=
.
4
0
a
0
a
a
0
a
0
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