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the other hand, is large enough to give negligible fluctuations in the total
activity, so we are justified in translating our previous concepts, which apply
to individual elements
e
i
,
e
j
as, e.g., stimulus s(
i
), response s(
j
), into a for-
malism that permits us to deal with assemblies of elements rather than with
individuals. Moreover, as long as these elements connect with other ele-
ments over a distance appreciably larger than the uncertainty of its deter-
mination, and there is a considerable fraction of cortical neurons fulfilling
this condition, we are still able to utilize the geometrical concepts as before.
To this end we drop the cellular individuality and refer only to the activity
of cell assemblies localizable within a certain volume.
In analogy to the concept of “number density” of neurons, i.e., the
number of neurons per unit volume at a certain point (
xyz
) in the brain,
we define “stimulus density” s(
xyz
) in terms of activity per unit volume as
the total activity
S
measured in a certain volume, when this volume shrinks
around the point (
xyz
) to “arbitrary” small dimensions:
S
V
d
d
S
V
(
)
=
s
xyz
lim
=
.
(93)
V
Æ
0
Similarly, we have for the response density at (
uvw
):
R
V
d
d
R
V
(
)
=
r
uvw
lim
=
,
(94)
V
Æ
0
if
R
stands for the total response activity in a macroscopic region.
We wish to express the action exerted by the stimulus activity around
some point in a transmitting “layer”
L
1
on to a point in a receiving layer
L
2
. In analogy to our previous considerations we may formally introduce a
“distributed action function”
K
(
xyz
,
uvw
) which defines the incremental
contribution to the response density dr(
uvw
) from the stimulus activity that
prevails in an incremental volume d
V
1
around a point (
xyz
) in the trans-
mitting layer. This activity is, with our definition of stimulus density s(
xyz
)
d
V
1
. Consequently
(
)
=
(
) (
)
(95)
d
r
uvw
K xyz uvw
,
s
xyz
d
V
.
1
In other words,
K
expresses the fraction per unit volume of the activity
around point (
xyz
) that contributes to the response at (
uvw
). The total
response elicited at point (
uvw
) from all regions in layer
L
1
is clearly the
summation of all incremental contributions, if we assume that all cells
around (
uvw
) are linear elements. Hence, we have
(
)
=
Ú
(
) (
)
r
uvw
K xyz uvw
,
s
xyz
d
V
,
(96)
1
V
1
where
V
1
, the subscript to the integral sign, indicates that the integration
has to be carried out over the whole volume
V
1
representing the extension
of layer
L
1
.
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