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Clearly, a network with such a periodic action function produces outputs in
L q that are invariant to any stimulus distribution which is translated with
same periodicity x o , y o , z o .
In order to have response invariance to stimulus translation everywhere
alone the receptor set L q , we must have:
(85)
x
=
D
x
,
y
=
D
y
,
z
=
D
z
.
o
o
o
From this we may draw several interesting conclusions. First, the action
functions so generated are independent of position, for the smallest inter-
val over which they can be shifted is precisely the order of their period.
Second, under the conditions of translatory invariance the action functions
reduce to sole functions of the difference of the coordinates which localize
the two connected elements in their respective layers:
(
) = ()
(86)
Kpq
,
k
D
,
where D is a vector with components
[
(
)
(
)
(
)
]
(87)
D=
x
-
x
,
y
-
h
,
z
-
z
.
We introduce symmetric, anti-symmetric and spherically symmetric action
functions which have the following properties respectively:
- () =
()
(88)
K
D
K
D ,
S
S
- () =- ()
(89)
K
D
K
D ,
a
a
() =
( .
(90)
KK
r
D
D
r
It is easy to imagine the kind of abstractions these action functions
perform on the set of all stimuli which are presented to the receptor set in
L p , if we assume for a moment that both layers, L p and L q , are planes.
Clearly, in all cases the responses in the effect of set are invariant to all
translations of any stimulus distribution (“pattern”) in the receptor set.
Moreover, K s gives invariance to reversals of stimuli symmetric to axes y =
0 and x = 0 (e.g., 3 into Œ, or M into W ), while K a gives invariance to rever-
sals of stimuli symmetric to y x (e.g., ~ into S ; and > into V ). Finally,
action function K r gives invariance to all stimulus rotations as well as trans-
lations (i.e., some reversals as above plus, e.g., N into Z ). The planes of sym-
metry in three dimensions which correspond to the lines in two dimensions
are clearly the three planes defined by the axes xy , yz , zx , in the first
case, and, in the second case, the three planes defined by the six origin-
centered diagonals that cut through the three pairs of opposite squares in
the unit cube.
Although for analytic purposes the action function has desirable prop-
erties, from an experimental point of view it is by no means convenient. In
order to establish in an actual case the action function of, say, element e p in
the receptor set, it is necessary to keep just this element stimulated while
searching with a microprobe though all fibers of a higher nucleus to pick
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