  Neurons in the brain represent external stimuli via neural codes. These codes
often arise from stereotyped stimulus-response maps, associating to each neuron
a convex receptive field. An important problem confronted by the brain is to
infer properties of a represented stimulus space without knowledge of the
receptive fields, using only the intrinsic structure of the neural code. How
does the brain do this? To address this question, it is important to determine
what stimulus space features can - in principle - be extracted from neural
codes. This motivates us to define the neural ring and a related neural ideal,
algebraic objects that encode the full combinatorial data of a neural code. Our
main finding is that these objects can be expressed in a "canonical form" that
directly translates to a minimal description of the receptive field structure
intrinsic to the code. We also find connections to Stanley-Reisner rings, and
use ideas similar to those in the theory of monomial ideals to obtain an
algorithm for computing the primary decomposition of pseudo-monomial ideals.
This allows us to algorithmically extract the canonical form associated to any
neural code, providing the groundwork for inferring stimulus space features
from neural activity alone.
