  Betti curves of symmetric matrices were introduced in (Giusti et. al., 2015)
as a new class of matrix invariants that depend only on the relative ordering
of matrix entries. These invariants are computed using persistent homology, and
can be used to detect underlying structure in biological data that may
otherwise be obscured by monotone nonlinearities. Here we prove three theorems
that fully characterize the Betti curves of rank 1 symmetric matrices. We then
illustrate how these Betti curve signatures arise in natural data obtained from
calcium imaging of neural activity in zebrafish.
