Order-forcing in Neural Codes

Neural Codes and the Factor Complex

We introduce the factor complex of a neural code, and show how intervals and maximal codewords are captured by the combinatorics of factor complexes. We use these results to obtain algebraic and combinatorial characterizations of max-intersection-complete codes, as well as a new combinatorial characterization of intersection-complete codes.

神经码与神经环自同态

Neural Codes and Neural ring endomorphisms

本文研究神经码的嵌入维数猜想，证明两类特例并给出逆命题反例，同时引入Doublet maximal码和Circulant codes两类新神经码，刻画其开凸性与最大交完备性的等价关系，并计算相关神经环自同态数量。

Universal Gröbner Bases of Toric Ideals of Combinatorial Neural Codes

In the 1970s, O'Keefe and Dostrovsky discovered that certain neurons, called place cells, in an animal's brain are tied to its location within its arena. A combinatorial neural code is a collection of $0/1$-vectors which encode the patterns of co-firing activity among the place cells. Gross, Obatake, and Youngs have recently used techniques from toric algebra to study when a neural code is $0$- $1$-, or $2$-inductively pierced: a property that allows one to reconstruct a Venn diagram-like planar figure that acts as a geometric schematic for the neural co-firing patterns. This article examines …

Minimal Embedding Dimensions of Connected Neural Codes

In the past few years, the study of receptive field codes has been of large interest to mathematicians. Here we give a complete characterization of receptive field codes realizable by connected receptive fields and we give the minimal embedding dimensions of these codes. In particular, we show that all connected codes are realizable in dimension at most 3. To our knowledge, this is the first family of receptive field codes for which the exact characterization and minimal embedding dimension is known.

具有四个最大码字的神经码的凸性

Convexity of Neural Codes with Four Maximal Codewords

本文研究具有四个最大码字的神经码的凸性，发现多数情况下凸性由缺乏局部障碍刻画，部分情况还需排除“轮”障碍，并基于神经复形在20种情形中完全刻画了15种。

Obstructions to convexity in neural codes

How does the brain encode spatial structure? One way is through hippocampal neurons called place cells, which become associated to convex regions of space known as their receptive fields: each place cell fires at a high rate precisely when the animal is in the receptive field. The firing patterns of multiple place cells form what is known as a convex neural code. How can we tell when a neural code is convex? To address this question, Giusti and Itskov identified a local obstruction, defined via the topology of a code's simplicial complex, and proved that convex neural codes have no local obstr…

Non-monotonicity of closed convexity in neural codes

Neural codes are lists of subsets of neurons that fire together. Of particular interest are neurons called place cells, which fire when an animal is in specific, usually convex regions in space. A fundamental question, therefore, is to determine which neural codes arise from the regions of some collection of open convex sets or closed convex sets in Euclidean space. This work focuses on how these two classes of codes -- open convex and closed convex codes -- are related. As a starting point, open convex codes have a desirable monotonicity property, namely, adding non-maximal codewords preserve…

Wheels: A New Criterion for Non-convexity of Neural Codes

We introduce new geometric and combinatorial criteria that preclude a neural code from being convex, and use them to tackle the classification problem for codes on six neurons. Along the way, we give the first example of a code that is non-convex, has no local obstructions, and has simplicial complex of dimension two. We also characterize convexity for neural codes for which the simplicial complex is pure of low or high dimension.

The neural ring: using algebraic geometry to analyze neural codes

Neurons in the brain represent external stimuli via neural codes. These codes often arise from stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can - in principle - be extracted from neural codes. This motivates us to define the neural ring and a related neural …

The neural ring: an algebraic tool for analyzing the intrinsic structure of neural codes

Neurons in the brain represent external stimuli via neural codes. These codes often arise from stereotyped stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can - in principle - be extracted from neural codes. This motivates us to define the neural ring and a rel…