Betweenness, brain models, random number generators

我建议在两个相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推测推广；另一方面，我建议研究大脑的经典模型在癫痫发作前表现出不稳定、无序、显然不可预测的行为特征的程度。平面上任意数量的点决定了若干条直线，每条直线至少经过这些点中的两点。欧几里得几何中的一个定理可以追溯到20世纪30年代，该定理断言，除非所有的点都在一条直线上（在这种情况下，这是它们确定的唯一直线），否则这样的直线的数量至少是点的数量。我以前的学生陈晓敏和我在2008年提出了一个猜想，希望将这个定理推广到1906年由莫里斯·弗雷切特引入的所谓度量空间的集合中。度量空间是一个集合，其中一个非负实数与每一个无序点对相关联；这个数字被称为两点之间的距离；当且仅当两点相等时它等于零；它满足三角形不等式，即A到B的距离加上B到C的距离至少等于A到C的距离。1924年，卡尔·门格尔提出，度量空间中的点B位于点a和点C之间意味着从a到B的距离加上从B到C的距离等于从a到C的距离。晓敏和我定义了度量空间中的直线XY（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中的一个位于另外两个点之间(特别是，直线XY包括点X和点Y)，我们推测在每一个有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，也可能有额外的线）。对这一猜想的证明将揭示一个冰山，而最初的欧几里得几何定理只是冰山的一角。座右铭是“明智，概括！”是数学中许多进步的动力，推广可能有意想不到的应用：一个著名的例子是在狭义相对论中使用非欧几里得几何。即使这个猜想在其全部的一般情况下被证明是错误的，它也已经在四个特殊情况下被证明是正确的。在它成立的度量空间中划定界限将是最有趣的。癫痫是一组神经系统疾病，其共同和基本特征是复发性、无因性癫痫发作。这种疾病很普遍：当今世界有5000多万癫痫患者。在对选定的癫痫患者进行研究时，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、无序脑电图（前兆状态）过渡到癫痫发作期间更有组织的持续的尖峰或尖波节律（前兆状态）。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发射模式的临界前颤振。我建议先从最简单的、按时间顺序排列的大脑模型——麦卡洛克-皮茨网络（McCulloch-Pitts networks）开始，然后再转向生物学上更可信的尖峰神经元网络。

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.