The brain processes information about the environment via neural codes. The neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code. On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Grbner basis with respect to that monomial order. How are these two types of generating sets canonical forms and Grbner bases related? Our main result states that if the canonical form of a neural ideal is a Grbner basis, then it is the universal Grbner basis (that is, the union of all reduced Grbner bases). Furthermore, we prove that this situation when the canonical form is a Grbner basis occurs precisely when the universal Grbner basis contains only pseudo-monomials (certain generalizations of monomials). Our results motivate two questions: (1) When is the canonical form a Grbner basis? (2) When the universal Grbner basis of a neural ideal is not a canonical form, what can the non-pseudo-monomial elements in the basis tell us about the receptive fields of the code? We give partial answers to both questions. Along the way, we develop a representation of pseudo-monomials as hypercubes in a Boolean lattice.
