The algorithm for canonical forms of neural ideals

To elucidate the combinatorial architecture of neural codes, the neural ideal $J_C$, an algebraic object, was introduced. Represented in its canonical form, $J_C$ provides a succinct characterization of the inherent receptive field architecture within the code. The polynomials in $J_C$ are also instrumental in determining the relationships among the neurons' receptive fields. Consequently, the computation of the collection of canonical forms is pivotal. In this paper, based on the study of relations between pseudo-monomials, the authors present a computationally efficient iterative algorithm for the canonical forms of the neural ideal. Additionally, we introduce a new relationship among the neurons' receptive fields, which can be characterized by if-and-only-if statements, relating both to $J_C$ and to a larger ideal of a code $I(C)$.