Neural Ideals in SageMath

Strengthening Relationships between Neural Ideals and Receptive Fields

Neural codes are collections of binary vectors that represent the firing patterns of neurons. The information given by a neural code $C$ can be represented by its neural ideal $J_C$. In turn, the polynomials in $J_C$ can be used to determine the relationships among the receptive fields of the neurons. In a paper by Curto et al., three such relationships, known as the Type 1-3 relations, were linked to the neural ideal by three if-and-only-if statements. Later, Garcia et al. discovered the Type 4-6 relations. These new relations differed from the first three in that they were related to $J_C$ b…

Neural ideals and stimulus space visualization

A neural code $\mathcal{C}$ is a collection of binary vectors of a given length n that record the co-firing patterns of a set of neurons. Our focus is on neural codes arising from place cells, neurons that respond to geographic stimulus. In this setting, the stimulus space can be visualized as subset of $\mathbb{R}^2$ covered by a collection $\mathcal{U}$ of convex sets such that the arrangement $\mathcal{U}$ forms an Euler diagram for $\mathcal{C}$. There are some methods to determine whether such a convex realization $\mathcal{U}$ exists; however, these methods do not describe how to draw a …

Gröbner Bases of Neural Ideals

The brain processes information about the environment via neural codes. The neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code. On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Gröbner basis with respect to that monomial order. How are these two types of generat…

利用神经回路中非理想因素的脑学习原理

Brain Learning Principles Utilizing Non-Ideal Factors in Neural Circuits

本文探讨了非理想因素如何作为进化设计原则，提升大脑的鲁棒性、适应性和创造力。

神经环：分析神经代码内在结构的代数工具

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

大脑中的神经元通过神经代码代表外部刺激。这些代码通常来自于刻板的刺激-反应图，将每个神经元与凸形感受野相关联。大脑面临的一个重要问题是在不知道感受野的情况下，仅使用神经代码的内在结构来推断所表示的刺激空间的属性。大脑是如何做到这一点的？为了解决这个问题，重要的是要确定什么样的刺激空间特征可以-原则上-从神经代码中提取。这促使我们定义神经环和相关的神经理想，代数对象编码的神经代码的完整组合数据。我们的主要发现是，这些对象可以表示在一个“规范形式”，直接转化为一个最小的描述感受野结构的内在代码。我们还发现连接到斯坦利-Reisner环，并使用类似的思想，在理论中的单项式理想，以获得一个算法，用于计算准单项式理想的主要分解。这使我们能够通过算法提取与任何神经代码相关的规范形式，为仅从神经活动推断刺激空间特征提供基础。

On the identification of $k$-inductively pierced codes using toric ideals

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 condition…