Neural ring homomorphisms and maps between neural codes

Understanding how the brain stores and processes information is central to mathematical neuroscience. Neural data is often represented as a neural code: a set of binary firing patterns $\mathcal{C}\subset\{0,1\}^n$. We have previously introduced the neural ring, an algebraic object which encodes combinatorial information, in order to analyze the structure of neural codes. We now relate maps between neural codes to notions of homomorphism between the corresponding neural rings. Using three natural operations on neural codes (permutation, inclusion, deletion) as motivation, we search for a restricted class of homomorphisms which correspond to these natural operations. We choose the framework of linear-monomial module homomorphisms, and find that the class of associated code maps neatly captures these three operations, and necessarily includes two others - repetition and adding trivial neurons - which are also meaningful in a neural coding context.