Combinatorics, geometry and topology of neural codes

Given a collection of neurons, consider all of the subsets which are observed to fire at the same time. One can view this as a simplicial complex with missing facets (for example, if neurons 1 and 2 fire together and neuron 1 is never observed firing on its own). We review recent work on such codes by Curto, Giusti, Itskov and their collaborators, focusing on aspects of when such codes are consistent with a “place field” model, and the interplay between such codes and the neural network architecture which can realize them.