A more thorough explanation:

The Durham rule is a test used in criminal law to determine if a defendant is not criminally responsible for an act due to mental disease or defect. It was established in the case of Durham v. United States in 1954. However, it is no longer accepted in any American jurisdiction as it has been criticized for being too broad.

For example, if a person with a mental illness commits a crime, they may use the Durham rule as a defense to argue that their mental illness caused them to commit the crime.

The insanity defense is an affirmative defense used in criminal law to allege that a mental disorder caused the accused to commit the crime. Unlike other defenses, a successful insanity defense may not result in acquittal but instead in a special verdict ("not guilty by reason of insanity") that usually leads to the defendant's commitment to a mental institution.

For example, if a person with a mental disorder commits a crime, they may use the insanity defense to argue that their mental disorder caused them to commit the crime and they should not be held criminally responsible.

The black-rage insanity defense is a defense based on an African-American's violent eruption of anger induced at least partly by racial tensions. This defense was first used in the mid-1990s.

For example, if an African-American person commits a violent crime due to the anger and frustration caused by racial tensions, they may use the black-rage insanity defense to argue that their mental state was affected by the racial tensions and they should not be held criminally responsible.