Using Algebra to Explore Neural Coding

Our ability to perceive, process, and interact with our environment depends on a correspondence between patterns of neural activity and stimuli in the environment. This is known as neural coding. A combinatorial neural code describes a pattern of neural activity in terms of which subsets of a population of neurons fire together and which do not. Our canonical example is that of convex neural codes arising from place cells, neurons which fire whenever an animal is within the place cell's place field, a convex region of the environment. Convex neural codes have been studied from a variety of perspectives, including mathematical coding theory, geometry, and topology. Here, we present a summary of algebraic approaches to the study of neural coding, including the neural ideal and the neural toric ideal.