  Johnson-Lindenstrauss (JL) matrices implemented by sparse random synaptic
connections are thought to be a prime candidate for how convergent pathways in
the brain compress information. However, to date, there is no complete
mathematical support for such implementations given the constraints of real
neural tissue. The fact that neurons are either excitatory or inhibitory
implies that every so implementable JL matrix must be sign-consistent (i.e.,
all entries in a single column must be either all non-negative or all
non-positive), and the fact that any given neuron connects to a relatively
small subset of other neurons implies that the JL matrix had better be sparse.
  We construct sparse JL matrices that are sign-consistent, and prove that our
construction is essentially optimal. Our work answers a mathematical question
that was triggered by earlier work and is necessary to justify the existence of
JL compression in the brain, and emphasizes that inhibition is crucial if
neurons are to perform efficient, correlation-preserving compression.
