Conference Agenda

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Session Overview
Session
MS148, part 2: Algebraic neural coding
Time:
Wednesday, 10/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F-105
53 seats, 70m^2

Presentations
10:00am - 12:00pm

Algebraic Neural Coding

Chair(s): Nora Youngs (Colby College), Zvi Rosen (Florida Atlantic University, United States of America)

Neuroscience aims to decipher how the brain represents information via the firing of neurons. Place cells of the hippocampus have been demonstrated to fire in response to specific regions of Euclidean space. Since this discovery, a wealth of mathematical exploration has described connections between the algebraic and combinatorial features of the firing patterns and the shape of the space of stimuli triggering the response. These methods generalize to other types of neurons with similar response behavior. At the SIAM AG meeting, we hope to bring together a group of mathematicians doing innovative work in this exciting field. This will allow experts in commutative algebra, combinatorics, geometry and topology to connect and collaborate on problems related to neural codes, neural rings, and neural networks.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Sunflowers of Convex Sets and New Obstructions to Convexity

R. Amzi Jeffs
University of Washington

Any collection of convex open sets in R^d gives rise to an associated neural code. The question of which codes can be realized in this way has been an open problem for a number of years, and although recent literature has described rich combinatorial and geometric obstructions to convexity, a full classification (even conjectural) is far out of reach. I will describe some new obstructions based on sunflowers of convex open sets, and show how these obstructions differ fundamentally from those which have been investigated previously.

 

Convex Codes and Oriented Matroids

Caitlin Lienkaemper
The Pennsylvania State University

Convex neural codes describe the intersection patterns of collections of convex open sets. Representable oriented matroids describe the intersection patterns of collections of half spaces—that is, of convex sets with convex complements. It is thus natural to view convex codes as a generalization of oriented matroids. In this talk, we will make this relationship precise. First, using a new notion of neural code morphism, we show that a code has a realization with convex polytopes if and only if it is the image of a representable matroid under such a morphism. This allows us to translate the problem of whether a code has a convex polytope realization into a matroid completion problem. Next, we enumerate all neural codes which are images of small representable matroids, and use the relationship between convex codes and oriented matroids to define new signatures of convexity and non-convexity. This is joint work with Alex Kunin and Zvi Rosen.

 

Sufficient Conditions for 1- and 2- Inductively Pierced Codes

Nida Obatake
Texas A&M University

Neural codes are binary codes in {0, 1}^n ; here we focus on the ones which represent the firing patterns of a type of neurons called place cells. There is much interest in determining which neural codes can be realized by a collection of convex sets. However, drawing representations of these convex sets, particularly as the number of neurons in a code increases, can be very difficult. Nevertheless, for a class of codes that are said to be k-inductively pierced for k = 0, 1, 2 there is an algorithm for drawing Euler diagrams. Here we use the toric ideal of a code to show sufficient conditions for a code to be 1- or 2-inductively pierced, so that we may use the existing algorithm to draw realizations of such codes.

 

Progress Toward a Classification of Inductively Pierced Codes via Polyhedra

Robert Davis
Harvey Mudd College

A difficult problem in the field of combinatorial neural codes is to determine when a given code can be represented in the plane as intersections of convex sets and their complements. If the code is 2-inductively pierced, then there exists a polynomial-time algorithm which constructs such a representation in the plane and which uses closed discs as the convex sets. Recently, Gross, Obatake, and Youngs provided a way to classify 2-inductively pierced codes for up to three neurons by considering a special weight order on ideals of polynomials associated to the codes. In this talk, we present progress toward extending their result for an arbitrary number of neurons. We focus on the use of state polytopes of homogeneous toric ideals, which encode their distinct reduced Gröbner bases. It is the properties of these bases that we aim to connect to being 2-inductively pierced.