Summary: We give intrinsic characterizations of neural rings and homomorphisms between them. Also we introduce the notion of a basic monomial code map and characterize monomial code maps as compositions of basic monomial code maps. Finally, we characterize monomial isomorphisms between neural codes. Our work is based on the paper [Abel Symp. 15, 163–180 (2020; Zbl 1448.62214)] by C. P. Curto and N. Youngs about neural ring homomorphisms and maps between neural codes and on the paper [SIAM J. Appl. Algebra Geom. 4, No. 1, 99–122 (2020; Zbl 1453.94155)] by R. A. Jeffs about morphisms of neural rings.