  Olfactory systems use a small number of broadly sensitive receptors to
combinatorially encode a vast number of odors. We propose a method of decoding
such distributed representations by exploiting a statistical fact: receptors
that do not respond to an odor carry more information than receptors that do
because they signal the absence of all odorants that bind to them. Thus, it is
easier to identify what the odor is not, rather than what the odor is. For
realistic numbers of receptors, response functions, and odor complexity, this
method of elimination turns an underconstrained decoding problem into a
solvable one, allowing accurate determination of odorants in a mixture and
their concentrations. We construct a neural network realization of our
algorithm based on the structure of the olfactory pathway.
