  Sensory processing is hard because the variables of interest are encoded in
spike trains in a relatively complex way. A major goal in sensory processing is
to understand how the brain extracts those variables. Here we revisit a common
encoding model in which variables are encoded linearly. Although there are
typically more variables than neurons, this problem is still solvable because
only a small number of variables appear at any one time (sparse prior).
However, previous solutions usually require all-to-all connectivity,
inconsistent with the sparse connectivity seen in the brain. Here we propose a
principled algorithm that provably reaches the MAP inference solution but using
sparse connectivity. Our algorithm is inspired by the mouse olfactory bulb, but
our approach is general enough to apply to other modalities; in addition, it
should be possible to extend it to nonlinear encoding models.
