Betweenness, brain models, random number generators

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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.