Since convolution and correlation are bilinear
operators, that is, linear in each of their arguments, if one
of the arguments is relatively fixed (as it would be, for
example, when a sensory signal is correlated with a learned
pattern), the operator is linear in its other argument:
for fixed 
.
Patterns of neural connectivity are often equivalent to a
convolution or correlation with a fixed field.
For example, the dependence of the activity at 
on the
activity at 
might fall off as some simple function
(e.g. Gaussian) of the distance between u and v, or as
some more complex (e.g. nonsymmetrical) function of the
relation between u and v.
In the former case we have a radial connectivity field
, in the latter a connectivity
kernel 
.
In either case, the contribution of region A to the
activity at 
can be written
.
Therefore, the field 
contributed to B by A is
defined
[ _u(t) = __v-u _v(t) v , ]
which is 
, the convolution of the
(unvarying) connectivity kernel 
with the activity
field 
.
Viewing such connectivity patterns as convolutions may
illuminate their function.
For example, by the ``convolution theorem'' of Fourier
analysis, the convolution 
is equivalent to the multiplication 
,
where 
and 
are the Fourier transforms
(over the space domain) of the activity fields and K is the
Fourier transform of the connectivity kernel.
Thus 
represents the spatial frequency spectrum, at
time t, of activity in region A, and K represents a
(comparatively unvarying) spatial frequency ``window''
applied to this activity by its connectivity to B.
For example, if 
is a Gaussian, then K is also
Gaussian, and the effect of the connections is spatial
low-pass filtering of the activity in A.
Many linear operators on fields can be approximated by
convolutions implemented by neural connectivity. We will
illustrate this with one useful operator, the derivative.
Suppose we have a one dimensional field 
and we want to
compute its derivative 
.
It happens that the derivative can be written as a convolution
with the derivative of the Dirac delta
function
(MacLennan 1990):
.
Like the Dirac delta, its derivative is not physically
realizable, but we can compute an approximation that is
adequate for neural computation.
To see this, suppose that we low-pass filter 
before
computing its derivative; this is reasonable, since the
frequency content of 
is limited by neural
resolution.
In particular, suppose we filter 
by convolving it with
a Gaussian 
; thus we will compute the approximate
derivative
.
But convolution is associative, so this is equivalent to
.
The parenthesized expression is the derivative of the
Gaussian function, so we see that an approximate derivative
of a field can be computed by convolving it with the
derivative of a Gaussian (which is easily implemented through
neural connectivity):
[ ' ' . ]
The derivative is approximate because of the filter applied
to 
, the transfer function of which is the Fourier
transform of 
, which is itself Gaussian.
It should be noted that such an analysis can be applied when
regions A and B are coextensive, and so no real
``projection'' is involved.
For example, A and B might represent two populations of
neurons in the same region, so that the connectivity field
or L reflects how cells of type B depend on
neighboring cells of type A.
Indeed, A and B might be the same cells, if we are
describing how their recurrent activity depends on their own
preceding activity and that of their neighbors. Thus we
might have a linear differential field equation of the form
or, more generally,
.
(See Section 4 for examples.)