Recent clinical data showed that patients with Alzheimer Disease (AD) might present epilepsy at early stages of the disease (Cretin et al., 2016, Vossel et al., 2016). In mice models of Alzheimer disease, preclinical researchers observed an increase of epileptic events during Rapid Eye Movement (REM) sleep, which is very unusual. This study aims at testing if patients with AD present an exacerbation of epileptic events during REM sleep, which could constitute an early biomarker of the disease. Investigators will evaluate the incidence of epilepsy during each sleep stage in 40 patients with early or moderate forms of AD and in 40 healthy subjects. Investigators will also look for a link between epilepsy during sleep in AD participants and memory performances, brain damage (by using MRI scans) and in the case of patients, the phenotype of the Apolipoprotein E(ApoE) gene.