This is a first-in-human clinical trial to test whether a protein administered into the brain continuously by gene therapy, Brain-Derived Neurotrophic Factor (BDNF), will slow or prevent cell loss in the brains of people affected by Alzheimer's disease and Mild Cognitive Impairment. The protein may also activate cells in the brain that have not yet deteriorated. Gene therapy refers to the use of a harmless virus to have brain cells make the potentially protective protein, BDNF.