Georgopoulos (1995) surveys research on population
coding in motor cortex of the direction of arm motion.
The population codes are naturally treated as fields, and the
transformations of directions are simple field computations.
We consider a region 
in motor cortex in which activity
is observed in anticipation of reaching motions.
Each cell 
has a preferred direction
in three-dimensional space.
Cell activity 
falls off with the cosine of the angle
between the reaching direction 
and the
preferred direction 
.
Since (for normalized vectors) the cosine is equal to the
inner product of the vectors, 
, we can express the activity:
for some constants a and b.
Thus the motor cortex represents a vector field 
of the preferred directions, and the population coding of an
intended motion 
is a scalar activity field
given by the inner product of the motion
with the preferred-direction field.
There is another way of looking at the population coding
of a motion 
, which is sometimes more
illuminating.
Since all the neurons have the same receptive field profile,
we may rewrite
Eq. 3
in terms of a radial function 
of the difference
between the preferred and intended direction vectors:
[ _u = (_u - r) , ]
where
[
(v)
= a + b - b \| v \|^2/2 .
]
This is because the Euclidean distance is related to the
inner product in a simple way:
(provided 
).
Now let 
be the direction field, defined over
three-dimensional space, that corresponds to 
.
That is, the value of 
at neural location u equals
the value of 
at spatial location 
,
or 
, which we may abbreviate
.
For simplicity we suppose 
is one-to-one, so we
can define 
by 
.
Notice that 
effects a change of coordinates from
neural coordinates to three-dimensional space.
The direction field 
can also be expressed as the result of convolving
the receptive field 
with an idealized direction field
, a Dirac delta,
which has an infinite spike at 
but is zero elsewhere:
[ = _r . ]
This is because convolving 
with 
effectively translates the center of 
to 
;
equivalently, the convolution blurs the idealized direction
field 
by the receptive field profile 
.
