This article deals with cellular automata (CA) working over periodic configurations, as opposed to standard CA, where the initial configuration is bounded by persistent symbols. We study the capabilities of language recognition and computation of functions over such automata, as well as the complexity classes they define over languages and functions. We show that these new complexity classes coincide with the standard ones starting from polynomial time. As a by-product, we present a CA that solves a somehow relaxed version of the density classification problem.