We derived a stability criterion for the totally synchronized state of a periodic network of coupled functional units. The periodicity constraint was the key assumption in deriving our circulant matrix-based stability criterion. The functional units were nonlinear discrete dynamical systems. We assumed exponentially decaying coupling strength versus distance in order to reduce the number of control parameters that would have been introduced due to the functional units' coupling. As a concrete example, we determined the stability domain of the synchronous state for logistic type functional units.