  Let $K$ be a periodic cell complex endowed with a covering $q:K\to G$ where
$G$ is a finite quotient space of equivalence classes under translations acting
on $K$. We assume $G$ is embedded in a space whose homotopy type is a $d$-torus
for some $d$, which introduces "toroidal cycles" in $G$ which do not lift to
cycles in $K$ by $q$ . We study the behaviour of toroidal and non-toroidal
cycles for the case $K$ is 1-periodic, i.e. $G=K/\mathbb{Z}$ for some free
action of $\mathbb{Z}$ on $K$. We show that toroidal cycles can be entirely
classified by endomorphisms on the homology of unit cells of $K$, and moreover
that toroidal cycles have a sense of unimodality when studying the persistent
homology of $G$.
