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()
=
(
) ( )
12
d
r
r
Kr
,
r
r
r
d
V
,
(108)
2
1
where
K
, the distributed interaction function, defines again the fraction
per unit volume of the activity prevailing around point
r
1
that is transmit-
ted to point
r
2
, and r(
r
1
)d
V
is clearly the activity of elements in the vicin-
ity of point
r
1
.
Of course, the same physiological interpretation that has been given to
the action function is applicable to the interaction function, except that
properties of symmetry refer to symmetry of exchanges of stimulation
between two points. In other words,
K
(
r
1
,
r
2
) and
K
(
r
2
,
r
1
) describe the pro-
portions that are transmitted from
r
1
to
r
2
, and back from
r
2
to
r
1
respec-
tively. These proportions may not necessarily be the same.
In addition to the stimuli contributed by elements of its own network,
each element may, or may not, receive stimulation from fibers descending
upon this network from other systems that do not receive fibers from the
network under consideration. We denote the elementary stimulation so
contributed to
r
2
by ds(
r
2
). It is, of course, no restriction to assume that
these fibers stem from another network, say
V
o
, that functions as action
network on to our system. In this case ds(
r
2
) may be directly replaced by
dr(
r
2
) or eq. (95), noting, however, that the action function in this expres-
sion has to be changed into, say,
A
(
r
o
,
r
2
), where
r
o
indicates positions of ele-
ments in this donor system. With ds(
r
2
) representing a stimulus from
external sources to elements around
r
2
of our network we have for the total
elementary stimulus at
r
2
:
()
=
()
+
(
) ( )
d
r
r
d
s
r
K r
,
r
r
r
d
V
,
(109)
2
2
1
2
1
which summed over the entire volume
V
gives the desired stimulus-
response relationship for any point in this volume:
()
=
()
+
Ú
(
) ( )
r
r
s
r
Kr
,
r
r
r
d
V
.
(110)
2
2
1
2
1
V
This equation cannot be readily solved by integration, unlike the case for
action networks, because here the unknown quantity r appears not only
explicitly on the left-hand side of this equation, but also implicitly within
the integral. Expressions of this type are called integral equations and (110)
above belongs to the class of integral equations of the second kind. The
function
K
(
r
1
,
r
2
) is usually referred to as the “kernel”, and methods of solu-
tion are known, if the kernel possesses certain properties.
It is fortunate that a general solution for eq. (110) can be obtained (Insel-
berg Von Foerster, 1962, p. 32) if the kernel
K
is a function of only the dis-
tance between points
r
1
and
r
2
:
(
)
=
(
)
=
()
Kr r
,
Kr
-
r
K
D
,
(111)
12
1
2
where D stands again (see eq. (86)) for the vector expressing this distance.
These kernels represent precisely the kind of interaction function we wish
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