How does Buddhism explain consciousness if there is no self?

One major differing point between Hinduism and Buddhism is the understanding of the self. Buddhist reject the Brahminical notion of an eternal, non-material soul substance that is distinct from the mind, body and sense organs. How then, is consciousness explained? Acts of cognition must occur from a *cogniser*, a subject that knows, because consciousness does not belong to the body or the mind. Could it not be therefore inferred that this *cogniser* exists as the self - a non-material soul? How can Buddhism explain consciousness?