  Neurons in the brain represent external stimuli via neural codes. These codes
often arise from 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. We
find 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 neural code. We consider the algebraic properties of homomorphisms
between neural rings, which naturally relate to maps between neural codes. We
show that maps between two neural codes are in bijection with ring
homomorphisms between the respective neural rings, and define the notion of
neural ring homomorphism, a special restricted class of ring homomorphisms
which preserve neuron structure. 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 canonical form associated to any neural
code, providing the groundwork for inferring stimulus space features from
neural activity alone.
