Advancing Research for Neurological Diseases Act of 2015

Amends the Public Health Service Act to require the Centers for Disease Control and Prevention to: (1) enhance and expand infrastructure and activities to track the epidemiology of neurological diseases, including multiple sclerosis and Parkinson's disease; and (2) incorporate information obtained through those activities into a National Neurological Diseases Surveillance System. Requires the Department of Health and Human Services (HHS) to ensure that the System facilitates further research on neurological diseases.

Requires HHS to provide for the collection and storage of information on neurological diseases, including the incidence, prevalence, and other information, to the extent practicable.

Requires HHS to: (1) make information and analysis in the System available to federal agencies and to the public, including researchers; and (2) ensure that privacy and security protections applicable to the System are at least as stringent as the protections under the Health Insurance Portability and Accountability Act.