One of the most significant contributions Semmelweis made to neurology was his work on cerebral vasculature and the role of inflammation in neurological disease. His research on the pathology of puerperal fever led him to investigate the cerebral vasculature and its relationship to inflammation. Semmelweis proposed that inflammation of the brain's blood vessels, or vasculitis, played a critical role in the development of various neurological disorders, including meningitis and encephalitis.