Betweenness, brain models, random number generators

介数、大脑模型、随机数生成器

• 批准号: RGPIN-2014-05599
• 负责人: Chvatal, Vasek
• 金额: $ 2.84万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611726
• 关键词: Betweenness; brain; models; random; number

I propose to work in two mutually unrelated areas: On the one hand, I propose investigating a conjectured generalization of a classical theorem concerning points and lines in the plane and on the other hand, I propose investigating the extent to which classical models of the brain can exhibit an erratic, disorderly, apparently unpredictable behaviour characteristic of an epileptic brain before onset of a seizure. Any number of points in the plane determine a number of lines, each of which passes through at least two of these points. A theorem in euclidean geometry, dating back to the 1930s, asserts that the number of such lines is at least the number of the points unless all of the points lie on a single line (in which case this is the only line they determine). A conjecture announced in 2008 by my former student Xiaomin Chen and myself aspires to generalize this theorem to the setting of so-called metric spaces, introduced in 1906 by Maurice Frechet. A metric space is a set where a nonnegative real number is associated with every unordered pair of points; this number is referred to as the distance between the two points; it equals zero if and only if the two points are identical; it satisfies the 'triangle inequality', meaning that the distance from A to B plus the distance from B to C is at least the distance from A to C. In 1924, Karl Menger proposed to say that a point B in a metric space lies between points A and C to mean that the distance from A to B plus the distance from B to C equals the distance from A to C. Xiaomin and I defined the line XY in a metric space (where X,Y are two distinct points) as the set of all points Z such that one of X,Y,Z lies between the other two (in particular, the line XY includes both points X and Y) and we conjectured that in every metric space with n points there are at least n lines unless some line consists of all n points (in which case there may be additional lines as well). A proof of this conjecture would reveal an iceberg, of which the original euclidean geometry theorem is just a tip. The motto 'be wise, generalize!' is the stimulus for much progress in mathematics and generalizations may have unexpected applications: one of the well-known examples is the use of non-euclidean geometry in special relativity. Even if the conjecture should turn out to be false in its full generality, it is already known to be true in four special cases. Demarcating the metric spaces where it holds true would be most interesting. Epilepsy is a group of neurologic conditions, the common and fundamental characteristic of which is recurrent, unprovoked epileptic seizures. This affliction is widespread: there are over 50 million epilepsy sufferers in the world today. In attempts to study epilepsy in selected patients, firing patterns of neurons that are located predominantly in their cerebral cortex are recorded as time series called electroencephalograms (EEG). Even though different types of seizures have different EEG manifestations, one frequent occurrence is a transition from an irregular, disorderly EEG before the seizure (the pre-ictal state) to more organized sustained rhythm of spikes or sharp waves during the seizure (the ictal state). In an effort to better understand the development of seizures, I intend to engineer existing models of the brain, so that they display seizure-like behaviour; my first short-term objective is to simulate the pre-ictal flutter of apparently unpredictable firing patterns. I propose to begin with the simplest, and chronologically first, model of the brain, the McCulloch-Pitts networks, before moving on to the biologically more plausible networks of spiking neurons.

我建议在两个互不相关的领域开展工作：一方面，我建议研究有关平面上的点和线的经典定理的推测概括，另一方面，我建议研究大脑的经典模型在癫痫发作之前可以在多大程度上表现出癫痫大脑的不稳定、无序、明显不可预测的行为特征。 平面上任意数量的点确定多条线，每条线至少经过这些点中的两个。欧几里得几何中的一个定理可以追溯到 20 世纪 30 年代，它断言此类线的数量至少等于点的数量，除非所有点都位于一条线上（在这种情况下，这是它们确定的唯一线）。我以前的学生陈晓敏和我自己在 2008 年宣布的一个猜想，旨在将这个定理推广到 Maurice Frechet 于 1906 年提出的所谓度量空间的设置。度量空间是一个集合，其中非负实数与每个无序点对相关联；该数字称为两点之间的距离；当且仅当两点相同时它才等于零；它满足“三角不等式”，即从 A 到 B 的距离加上从 B 到 C 的距离至少等于从 A 到 C 的距离。1924 年，卡尔·门格尔提出，度量空间中的点 B 位于 A 和 C 点之间，这意味着从 A 到 B 的距离加上 B 到 C 的距离等于 A 到 C 的距离。小敏和我在度量空间中定义了直线 XY（其中 X、Y 是两个不同的点） 作为所有点 Z 的集合，其中 X、Y、Z 之一位于其他两个点之间（特别是，线 XY 包括点 X 和 Y），并且我们推测在每个具有 n 个点的度量空间中，至少有 n 条线，除非某条线由所有 n 个点组成（在这种情况下也可能有其他线）。 这个猜想的证明将揭示一座冰山，而最初的欧几里得几何定理只是冰山一角。座右铭是“明智，概括！”是数学进步的刺激因素，概括可能会产生意想不到的应用：众所周知的例子之一是在狭义相对论中使用非欧几里得几何。即使这个猜想在其全部普遍性上被证明是错误的，但在四种特殊情况下已经知道它是正确的。划定它适用的度量空间将是最有趣的。 癫痫是一组神经系统疾病，其共同和基本特征是反复无端癫痫发作。这种痛苦很普遍：当今世界有超过 5000 万癫痫患者。在尝试研究选定患者的癫痫症时，主要位于大脑皮层的神经元的放电模式被记录为时间序列，称为脑电图（EEG）。尽管不同类型的癫痫发作有不同的脑电图表现，但一种常见的情况是从癫痫发作前的不规则、紊乱的脑电图（发作前状态）转变为癫痫发作期间更有组织的持续的尖峰或尖波节律（发作状态）。 为了更好地了解癫痫发作的发展过程，我打算改造现有的大脑模型，使它们表现出类似癫痫发作的行为；我的第一个短期目标是模拟明显不可预测的射击模式的发作前颤动。我建议从最简单且按时间顺序排列的大脑模型开始，即麦卡洛克-皮茨网络，然后再转向生物学上更合理的尖峰神经元网络。