The upper bound of periods realized in a system of 1-dimensional uniform neural networks with refractory period is analyzed for each transition function. As for one function whose refractory period is 2, the period is not to be bounded. For the other functions with the refractory period R the upper bound equals 1 or R1. They contrast with the fact that the period is bounded by 4 for every case without refractory period. The results are obtained by constructing directly the periodic states of every cell from a boundary cell to the other one after another.