Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 90

to consider, for it is this property that makes the computations in the
network invariant to stimulus translations (see eq. (85)).
The general solution for response, given explicitly in terms of stimulus
interaction function, is, for the x -component:

()

Fu
Fu

+•

Ú

S

() =

- i xu

r x 12 p ed u , (112)

()

12 - p

-•

k

where F S and F k are the Fourier transforms of stimulus distribution s and
interaction function K respectively:

+•

() =

Ú

()

Fu 12psed x

i xu

x , (113a)

S

-•

+•

Ú

() = ()

Fu 12p Kt ed

i tu

t , (113b)

k

-•

with t representing the x component of the distance and i the imaginary unit

(114)

t = D , i = - 1 .

x

The expressions are valid for the other components, mutatis mutandis .
With respect to these results two comments are in order. First, one should
observe the analogy of eq. (110) with the result obtained in the case of inter-
action of individually disti n guishable elements (eq. (65)).

n

Â

() = () + () rs j j a r i , (65)

ij

i = 1

where summation over the activity of individual elements and the interac-
tion coefficients a ij correspond to integration and interaction function in
(110) respectively. However, the cumbersome matrix inversions as sug-
gested in eqs. (66) to (74) disappear, because the Fourier transforms in eqs.
(112) and (113) perform these inversions—so to say—in one stroke. Thus,
a general study of network structures will have to proceed along the lines
suggested here, otherwise sheer manipulatory efforts may attenuate the
enthusiasm for exploring some worthwhile possibilities.
The other comment refers to our earlier observation of the functional
equivalence of discrete action and interaction nets (see fig. 23). The ques-
tion arises whether or not an action network can be found that has pre-
cisely the same stimulus-response characteristic as a given interaction
network of cell assemblies. It is not insignificant that this question can be
answered in the affirmative. Indeed, it can be shown that a functional equiv-
alent action net with action function A ( t ) can be generated from a given
interaction net with interaction function K ( t ) by the Fourier transform

1
12

+•

() =

Ú

At 12 p ed

i tu

u .

(115)

()

- p Fu

-•

k

These transforms are extensively tabulated (Magnus and Oberhettinger,
1949) and permit one to establish quickly the desired relationships.