Refers to the medical specialty or hospital department dealing with neurological disorders. In Japanese medical contexts, 神経科 is often used for neurology, while 精神科 is for psychiatry.
He is a neurologist.
Because my headaches persisted, I visited the neurology department.
Compound of 神経 (nerve) and 科 (department, field of study). The term is a direct adaptation of the Western medical concept of neurology.