Consider a scalar variable x that represents some measurable external
quantity. Suppose that sensory processing encodes the variable x over a
population of cells 
such that each cell has a
localized
``receptive field'' for values of x. For example, we could use a
Gaussian shaped receptive field such as
where the standard deviation s determines the specificity and overlap of
the receptive
fields, and the means 
determine the centers. Then any value of x
is represented as a local pattern of
activity over the set 
.
Now, define 
to be a noisy measured value of x with
additive
independent random noise 
. Let the probability distribution of
be equal to the receptive field shape 
. For example, if
is as above, then let 
be Normal with variance 
.
Then we have
In other words, each basis function output can be interpreted as the
probability that the
noisy measurement 
was in fact generated by the uncorrupted value
. The value of 
for which 
is maximized is
the maximum likelihood (ML) estimate of x given 
.
We can compute the maximum a posteriori (MAP) estimate of x given
using Bayes' rule:
Note that this is proportional to 
if 
and
are uniform. In this case, the set
of 
's gives the probability density of
x for each measurement 
.
We can average the basis function outputs over multiple values of x to obtain the expected value:
and this expression gives a smoothed and subsampled version of 
. In
particular, if the receptive fields are all the same shape so that
, then equation 1
describes sampled values from the convolution 
and approximates
. Thus the
expected values 
give a sampled and smoothed approximation
to the input
probability density 
in a manner similar to Parzen Windows density
estimation. (
can also be interpreted as a sampled but not
smoothed version of 
.)
The above analogy provides the theoretical basis for the analysis of this
type of population code. Note that it applies only to population codes
described by local receptive fields. The noise term 
is an artificial
theoretical construction that allows the analogy to be made. A receptive
field in the population code corresponds to the noise distribution in the
probability analogy, and a wider receptive field is equivalent to a higher
noise variance.
