Algebra and Combinatorics Seminar

Date: February 16, 2018

Time: 3:00PM - 3:50PM

Location: BLOC 628

Speaker: Alex Kunin, Penn State University

Title: Hyperplane neural codes and the polar complex

Abstract: This talk concerns combinatorial and algebraic questions arising from neuroscience. Combinatorial codes arise in a neuroscience setting as sets of co-firing neurons in a population; abstractly, they record intersection patterns of sets in a cover of a space. Hyperplane codes are a class of combinatorial codes that arise as the output of a one layer feed-forward neural network, such as Perceptron. Here we establish several natural properties of non-degenerate hyperplane codes, in terms of the {\it polar complex} of the code, a simplicial complex associated to any combinatorial code. We prove that the polar complex of a non-degenerate hyperplane code is shellable. Moreover, we show that all currently known properties of hyperplane codes follow from the shellability of the appropriate polar complex. Lastly, we connect this to previous work by examining some algebraic properties of the Stanley-Reisner ideal associated to the polar complex. This is joint work with Vladimir Itskov and Zvi Rosen.