This final grade essay presented below will consist of the analysis of a two different groups of subjects, to determine the existent differences between them in the EEG results on the sleep stages. With this work we will try to identify the most remarkable differences in the stage II of the sleep, where the spindles show up and where’s the information to take into account. The spindle is short oscillatory burst that appear during sleep. They have been recently related with some cognitive task, like learning and memory consolidation. Through the Hilbert-Huang transformation, we implemented a spindle detector and the purpose is to apply on both different groups to establish the differences. The diseased group is affected with a neuro-cognitive disease called Down Syndrome. This work aims to be the first step in this long way ahead that is studying what is affecting a minority population affected with Down Syndrome.