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If we pass over each fiber an average amount
¯
of activity, we obtain the
stimulus which is funneled from
x
to
u
by multiplying eq. (101) with this
amount:
()
=
()
=
(
) ( )
(102)
snu
d
d
s
u kxu
,
s
x x
d
,
because
()
=
()
()
=
()
(103)
sn x
s
x
,
sn u
s
u
.
If the fractional stimulus density ds(
u
) at the target were translated directly
into response density, we would have
()
=
()
.
s
uu
However, this is not true, for the arriving fibers will synapse with the target
neurons in a variety of ways (fig. 30). Consequently, the resulting response
will depend upon the kind and strength of these synaptic junctions which
again may be a function of source and target points. To accommodate this
observation we introduce a local transfer function k(
x
,
u
), that relates arriv-
ing stimulus with local response
()
=
(
)
( )
(104)
d
rk s
u
x u
,
d
u
.
With the aid of eq. (102) we are now in a position to relate stimulus density
in the source area to response density in the target area:
()
=
(
) (
) ( )
(105)
d
rk
u
x u k x u
,
,
s
x
d
x
.
Clearly, the product of the two functions k and
k
can be combined to define
one “action function”
(
)
=
(
) (
)
(106)
Kxu
,
k
xu kxu
,
,
,
and eq. (105) reduces simply to
()
=
(
) ( )
(107)
d
r
uKxu xx
,
s
d
.
Comparison of this equation with our earlier expression for the stimulus-
response relationship (eq. (95)) shows an exact correspondence, eq. (107)
representing the
x
-portion of the volume representation in eq. (95).
With this analysis we have gained the important insight that action func-
tions—and clearly also interaction functions—are composed of two parts.
A structural part
k
(
r
1
,
r
2
) defines the geometry of connecting pathways, and
a functional part k(
r
1
,
r
2
) defines the operational modalities of the elements
involved. The possibility of subdividing the action function into two clearly
separable parts introduces a welcome constraint into an otherwise unman-
ageable number of possibilities.
We shall demonstrate the workings of the mathematical concep-
tual machinery so far developed on three simple, but perhaps not trivial,
examples.
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