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(i) Ideal One-to-One Mapping
Assume two widely extending, but closely spaced, parallel surfaces,
representing layers L p and L q . Perpendicular to L p emerge parallel fibers
which synapse with their corresponding elements in L q without error and
without deviation. In this case the mapping function k is simply Dirac's
Delta Function*
(
) =
(
)
2
kr r
,
d
r
-
r
,
12
1
2
π
•=
0
for
for
xx
xx
Ó
o
(
) =
d xx
-
o
o
and
+•
Ú
(
)
d xx x
-
d .
=
o
-•
For simplicity, let us assume that the transfer function k is a constant a .
Hence
(
) =
2
(
)
Kr r
,
a
d
r
-
r
,
12
1
2
and, after eq. (96):
Ú
() =
2
(
) ( )
()
r
r
a r
d
-
r
s
r
d
Aar
=
s
.
2
1
2
1
1
1
L p
As was to be expected, in this simple case the response is a precise replica
of the stimulus, multiplied by some proportionality constant.
(ii) Ideal Mapping with Perturbation
Assume we have the same layers as before, with the same growth program
for fiber descending upon L q , but this time the layers are thought to be much
further apart. Consequently, we may expect the fibers to be affected by
random perturbations, and a fiber bundle leaving at r 1 and destined for
r 2 = r 1 will be scattered according to a normal (Gaussian) distribution.
Hence, we have for the mapping function
(
) =
(
)
kr r
,
12
p
h
2
exp
-
D
2
2
h
2
12
with
2
= rr
D 2
1
2
and h representing the variance of the distribution.
* The exponent in d indicates the dimensionality of the manifold considered. Here
it is two-dimensional, hence d 2 .
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