Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 81

or, after cancellation of N x on both sides:

 kxu i
i

( )

1 = , .

all

If we consider now infinitesimal targets of length d u , the above summa-
tion takes on the form of an integral

+•

Ú

( )

kxu , d1 u = .

(98)

-•

This suggests that k ( x , u )d u may be interpreted as the probability for a
fiber which originates at x will terminate within an interval of length d u in
the vicinity of u . Consequently, one interpretation of k ( x , u ) is a “probabil-
ity density function”.
With this observation we may derive an expression for the contribution
of region x to the fiber number density of region u . The corresponding fiber
number densities are clearly defined by

d
d

N
x

d
d

N
u

() = () =

nx and nu .

Since n ( x )d x fibers all together emerge from an interval of length d x in
the vicinity of x , their contribution to the number of fibers in the interval
d u at u is:

() = (

[

)

] () [ ]

2 Nu

(99)

d kxu , d u nx d x

with d 2 indicating that this is an infinitesimal expression of second order
(an infinitesimal amount of an infinitesimal amount. Compare with eq.
(97), its finite counterpart). Dividing in eq. (99) both sides by d u we note
that

() =

d

2 Nu
u

d
d

N
u

Ê
Ë

ˆ
¯

x

() d = d nu (100)

x

d

x

is the contribution of x to the number density of fibers at point u . Using
eqs. (99), (100), we can now express the desired relation between source
and target densities by

() = ( )()

d nu kxunx x , d . (101)

From this point of view, k represents a mapping function that defines the
amount of convergence or divergence of fiber bundles leaving the vicinity
of point x and destined to arrive in the vicinity of point u . Clearly, k rep-
resents an important structural property of the network.
For the present discussion it is irrelevant whether certain neurons around
x are the donors for elements around u , or whether we assume that after
axonal bifurcation some branches are destined to contact elements around
u . In both cases we obtain the same expression for the fractional contribu-
tion from x and u (compare figs. 29a and b).