  We study the Maslov index as a tool to analyze stability of steady state
solutions to a reaction-diffusion equation in one spatial dimension. We show
that the path of unstable subspaces associated to this equation is governed by
a matrix Riccati equation whose solution $S$ develops singularities when
changes in the Maslov index occur. Our main result proves that at these
singularities the change in Maslov index equals the number of eigenvalues of
$S$ that increase to $+\infty$ minus the number of eigenvalues that decrease to
$-\infty$.
