Suppose a small animal is running around an enclosure (stimulus space). In each location in the enclosure, some subset of the animal’s neurons will fire. One way to record the possible combinations of firing neurons is via a neural code, a subset of {0,1}^n where n is the total number of neurons. An element of the neural code is an n-tuple, with a 1 in the i^th spot corresponding to when the i^th neuron is firing, and a zero corresponding to when the i^th neuron is not firing.

A receptive field is a map f_i: X → ℝ≥0 from the stimulus space X to the average firing rate of a single neuron, i, in response to each stimulus. We also use the term receptive field to mean the subset U_i of X where f_i is strictly positive. A receptive field contains more information than a neural code, and hence one may ask the question: if we are only given a neural code, how much information can we reconstruct about a corresponding receptive field?

In this talk we introduce neural codes, receptive fields, neural ideals and their canonical forms, and how we can use the canonical form of a neural ideal to move between neural codes and receptive fields.