There is considerable evidence that humans and monkeys are able to continuously transform images for various purposes. Aside from introspection, such evidence comes from the behavioral experiments pioneered by Shepard (e.g. Shepard & Cooper 1982) and, more recently, from direct neuronal measurement of motor cortex (surveyed in Georgopoulos 1995).
Droulez & Berthoz (1991b) give an algorithm for the
continuous transformation of direction fields, specifically, for the
updating, when the eye moves, of the remembered location, relative
to the retina, of an ocular saccade.
Suppose the field 
is a population code in retinal coordinates
for the destination of the saccade.
If in time 
the eye moves by a vector 
in retinal
coordinates, then the field encoding the destination of the saccade
must be updated according to the equation
[ (+r, t+t) = (r,t) . ]
Eye motion is assumed to be encoded by a two-dimensional
rate-encoded velocity vector 
, which gives the eye velocity in retinal
coordinates.
It is easy to show that
(The gradient 
points in the direction of the peak,
provided there is only one peak; if there are multiple targets, it
points to the nearest target.)
This equation, which gives a discrete update after a time 
,
can be converted into a equation for the continuous updating of
by taking the limit as 
:
[ . = v . ]
This can be understood as follows: Since 
represents the
motion of the eye relative to the retinal field, 
represents the
direction in which the field peak should move.
In front the peak (that is, in its direction of required movement), the
gradient, which points toward the peak, points in the opposite
direction to 
. Therefore 
at that point
will be negative, which means that 
, and the field intensity in the front of the peak increases.
Conversely, behind the peak the gradient points in the same
direction as the required movement, so 
,
which means 
, and the field
intensity on the back of the peak decreases.
Therefore, the peak moves in the required direction.
Equation 4 must be recast for neural computation,
since the vector field 
has to be represented by two
neural populations (for the two dimensions of retinal coordinates).
Thus we write
[ v=
v_x x +
v_y y
.]
Since the neural population is discrete and the neurons have
receptive fields with some diameter, the neural representation imposes a
low-pass filter on the direction field. Writing 
for a
two-dimensional Gaussian, the filtered field can be written
and substituted into
Eq. 4:
As we've seen, the derivatives of the filtered field can be written as
convolutions with derivatives of Gaussians, so
,
where 
is a derivative of a Gaussian along the x-axis
and constant along the y-axis. Thus,
[ (t+t) =
_xy + t(
v_x '_x
+ v_y '_y ). ]
Significantly, when Droulez & Berthoz (1991b) started with a
one-dimensional network of the form
[
+ t v]
and trained it, by a modified Hebbian rule, to compute the updated
population code, they found that after training 
was approximately
Gaussian, and 
was an approximate derivative of a Gaussian.
Droulez & Berthoz (1991a) suggest biologically plausible
neural circuits that can update the direction field 
, which can
be expressed in field computational terms as follows.
A field of interneurons S (sum) forms the sum of the activities of
nearby neurons, 
, while interneuron fields
and 
estimate the partial derivatives by a
means of excitatory and inhibitory synapses,
,
.
Next, a field of interneurons P (product) computes the inner product
of the velocity vector and the field gradient by means of conjunctive
synapses:
.
The neurons in the direction field compute the sum of the S and
P interneurons, which then becomes the new value of the direction
field, 
.
Thus Droulez & Berthoz's (1991a) proposed neuronal
architecture corresponds to the following field equations, all implemented
through local connections:
