The current study will last for 12 weeks and will involve people with AD. Participants will be initially examined with an overnight brain wave study to assess for silent epileptic (seizure-like) activity. Presence of epileptic activity on the screening exam is not required to enter the trial. Participants will then be assigned to groups in a randomized manner. One group will receive levetiracetam for 4 weeks, then no drug for 4 weeks, and then placebo for 4 weeks. For another group, the order of treatments will be reversed. The cognitive abilities of participants will be retested every 4 weeks and compared to those at the beginning. The cognitive tests include a virtual-reality navigation test of memory and computerized tests of mental flexibility and problem solving. The participants will be monitored with a magnetoencephalogram (MEG) with simultaneous EEG (M/EEG) at each visit. M/EEG is a highly effective non-invasive method for identifying brain regions of epileptic activity. The investigators will need to recruit 36 randomized participants to test the study hypotheses. This study will take place at the University of California, San Francisco (UCSF) Memory and Aging Center, which is a large referral site for dementias, and the Department of Radiology. The overall goal of the study is to demonstrate that levetiracetam provides cognitive benefit in AD, particularly in patients who have silent epileptic activity.