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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=634966
• 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，我建议研究一个关于平面上的点和线的经典定理的广义化，另一方面，我建议研究大脑的经典模型在多大程度上可以表现出一种不稳定的，无序的，癫痫发作前癫痫大脑的一种明显不可预测的行为特征。平面上任意多个点决定多条线，每一个都通过这些点中的至少两个。欧几里德几何中的一个定理，可以追溯到20世纪30年代，声称这样的线的数量至少是点的数量，除非所有的点都在一条线上（在这种情况下，这是他们确定的唯一一条线）。我和我以前的学生陈晓敏在2008年宣布了一个猜想，希望将这个定理推广到所谓的度量空间的背景下，度量空间是由莫里斯·弗雷歇在1906年提出的。度量空间是一个集合，其中一个非负的真实的数与每个无序的点对相关联;这个数被称为两点之间的距离;当且仅当两点相同时它等于零;它满足“三角不等式”，这意味着从A到B的距离加上从B到C的距离至少是从A到C的距离。1924年，卡尔·门格尔（Karl Menger）提出，度量空间中的点B位于点A和点C之间，表示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XY线（其中X，Y是两个不同的点）作为所有点Z的集合，使得X，Y，Z中的一个位于其他两个之间（特别是，线XY包括点X和Y），并且我们证明了在每个有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，也可以有附加的线）。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。格言是“智慧，概括！是数学进步的动力，而推广可能会有意想不到的应用：一个著名的例子是在狭义相对论中使用非欧几里德几何。即使这个猜想在它的全部一般性中被证明是假的，它在四种特殊情况下也是真的。癫痫是一组神经系统疾病，其共同的和基本的特征是反复发作，无诱因的癫痫发作。这种痛苦是普遍的：今天世界上有超过5000万癫痫患者。在研究选定患者癫痫的尝试中，主要位于大脑皮层的神经元的放电模式被记录为称为脑电图（EEG）的时间序列。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、无序的脑电图过渡到癫痫发作后的脑电图（发作前状态）到癫痫发作期间更有组织的持续的尖峰或尖波节律为了更好地理解癫痫发作的发展，我打算设计现有的大脑模型，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放电模式的发作前颤动。我建议开始先从最简单的、按时间顺序排列的大脑模型--麦卡洛克-皮茨网络开始，然后再讨论生物学上更合理的尖峰神经元网络。