  In the olfactory system, odor percepts retain their identity despite
substantial variations in concentration, timing, and background. We propose a
novel strategy for encoding intensity-invariant stimuli identity that is based
on representing relative rather than absolute values of the stimulus features.
Because, in this scheme, stimulus identity depends on relative amplitudes of
stimulus features, identity becomes invariant with respect to variations in
intensity and monotonous non-linearities of neuronal responses. In the
olfactory system, stimulus identity can be represented by the identities of the
p strongest responding odorant receptor types out of a species dependent
complement. We show that this information is sufficient to recover sparse
stimuli (odorants) via elastic net loss minimization. Such a minimization has
to be performed under constraints imposed by the relationships between stimulus
features. We map this problem onto the dual problem of minimizing a functional
of Lagrange multipliers. The dual problem, in turn, can be solved by a neural
network whose Lyapunov function represents the dual Lagrangian. We thus propose
that networks in the piriform cortex compute odorant identity and implement
dual computations with the sparse activities of individual neurons representing
the Lagrange multipliers.
