  We study the existence and linear stability of stationary periodic solutions
to a neural field model, an intergo-differential equation of the Hammerstein
type. Under the assumption that the activation function is a discontinuous step
function and the kernel is decaying sufficiently fast, we formulate necessary
and sufficient conditions for the existence of a special class of solutions
that we call 1-bump periodic solutions. We then analyze the stability of these
solutions by studying the spectrum of the Frechet derivative of the
corresponding Hammerstein operator. We prove that the spectrum of this operator
agrees up to zero with the spectrum of a block Laurent operator. We show that
the non-zero spectrum consists of only eigenvalues and obtain an analytical
expression for the eigenvalues and the eigenfunctions. The results are
illustrated by multiple examples.
