In the investigation of biological population codes, investigators frequently seek to identify the ``coordinate system'' in which the underlying variable x is coded. Recent work [Sanger 1994] has shown that it is not possible to define a unique coordinate system for a population code. In the current framework, this result can be summarized by noting that the density mapping theorem shows that different population codes can be mapped into each other using linear transformations that perform shift operations, so that it is relatively easy to change coordinate systems merely by re-labelling the values assigned to cell outputs.
The probability technique proposed here allows us to go beyond the lack of a unique coordinate system to investigate whether there is a best coordinate system about which the population gives maximal information. I propose the following
Definition: An ``optimal'' coordinate system has the property that a uniform distribution of input samples with respect to this coordinate system yields a maximal entropy density over the distributed representation.
For example, suppose that a set of cells have local receptive fields that
are uniformly distributed in polar coordinates with even spacing along the
r and 
dimensions. If sample inputs are drawn from a uniform
distribution in Cartesian 
coordinates, then many of the cells
with receptive fields near the origin will be under-stimulated, and the
resulting output density will not have maximum entropy. But if the input
samples are selected uniformly in 
then the population code
will have maximum entropy.
To find the optimal coordinate system, choose inputs uniformly with respect
to some coordinate system x. Measure the cell outputs 
, and
compute a transformation kernel 
such that 
is
constant. Let f be the function represented by 
. Then the
optimal coordinate system is given by 
.
Note that optimality is determined by the distribution of cell receptive field centers. If this distribution is uniform with respect to a candidate coordinate system, then that coordinate system is optimal. For instance, the distribution of preferred hand movement directions for cells in motor cortex is approximately uniform in Cartesian coordinates [Georgopoulos 1986], but the distribution of preferred hand force directions is not [Kalaska et al. 1989].
An interesting consequence of these results is that we can test whether or not an animal has in fact optimized a distributed representation. If the representation is optimal, then the actual distribution of sensory inputs should be approximately uniform within the optimal coordinate system. If the distribution is significantly different then the representation is not optimal for the actually occurring environment statistics.