Mathematical Models in Cognitive Science: Neural Fields and Exemplar Dynamics

我的研究计划包括三个不同的项目。前两个项目是相关的，因为它们涉及数学和认知科学相结合的主题。第三个是关于多样性理论，它是度量空间理论的扩展。这三个项目都为培养本科生和研究生提供了许多机会。第一个认知科学项目涉及认知心理学中的数学建模，特别是使用微分方程式对西蒙·弗雷泽大学认知科学实验室进行的实验中的眼球跟踪数据进行建模。这些实验探索了人类受试者如何完成一项任务，在这项任务中，他们必须学习如何对一组抽象图像进行分类。收集的数据包括受试者的猜测、猜测的时间和眼球跟踪数据，即受试者在每个时间点的视线在视频屏幕上的位置。我们的目标是建立受试者在执行这些任务时的行为的数学模型。一旦我们有了该模型的计算机实现，我们就可以使用它来预测在新环境下的行为，然后设计实验来测试这些预测。长期目标是发展对范畴化背后的认知机制的理解。学生有很多机会从事这个项目，从模型的开发，现有模型的分析，计算实施的问题，最后是模型与实验数据的拟合。第二个认知科学项目涉及语言学中的数学建模，特别是一类被称为范例动力学模型的模型。范例理论是一种关于语言范畴的信息，如元音“a”和“i”是如何存储在大脑中的理论。范例动力学是范例理论的一种应用，它解释了这些语言范畴是如何形成的，并随着几代语言使用者的使用而发生变化。我们想要对样本动力学模型如何工作有一个更精确的数学理解。这些模型本身很难直接使用，因为从数学上讲，它们是时空点过程，出生率非线性地依赖于现有的点。我们将通过用偏微分方程模型来逼近这些模型来研究这些模型，其中偏微分方程模型描述了语音空间不同区域中样本的密度。我们从这些样本密度方程的研究中所学到的将使我们能够对样本模型做出准确的陈述，并更仔细地描述当前样本模型在语言学中的局限性。这个项目为学生提供了一个极好的机会，让他们在一个令人兴奋的数学和语言学的新界面上工作。第三个项目是关于多样性的理论。多样性通过扩展度量空间的概念，而不是考虑仅定义在空间中的点对上的度量函数，而是考虑定义在任意有限点子集上的函数。度量空间理论在数学和计算机科学中有许多重要的应用，因此有必要发展多样性的类比理论。我将关注的理论方面是关于赋范空间中度量的最小失真嵌入的一族结果，以及它在最稀疏割等组合优化问题中的应用。我将与学生和合作者一起发展相应的多样性理论。学生可能要做的一个重要的第一步是确定L1中多样性的最坏情况下最小失真嵌入。最终的目标是为NP-Hard问题开发新的近似算法。

My research program consists of three distinct projects. The first two projects are related in that they concern topics at the interface of mathematics and cognitive science. The third is on the theory of diversities, an extension of the theory of metric spaces. All three projects offer many opportunities for the training of students, both undergraduate and graduate. The first cognitive science project concerns mathematical modelling in cognitive psychology, specifically the use of differential equations to model eye-tracking data from experiments conducted in the Cognitive Science Lab at Simon Fraser University. The experiments explore how human subjects perform a task in which they must learn how to categorize a set of abstract images. The data collected includes the guesses of the subjects, the timing of the guesses, and eye-tracking data, i.e. the location on a video screen of the subject’s gaze at every point in time. Our goal is to create a mathematical model of the subjects' behaviour while performing these tasks. Once we have a computer implementation of the model, we can use it to make predictions about behaviour in novel circumstances, and then design experiments to test these predictions. The long-term goal is to develop an understanding of the cognitive mechanisms behind categorization. There are many opportunities for students to work on this project, from the development of models, the analysis of existing models, issues of computational implementation, and finally the fitting of models to experimental data. The second cognitive science project concerns mathematical modelling in linguistics, in particular, a class of models known as exemplar dynamics models. Exemplar theory is a theory of how information about linguistic categories, such as the vowel sound "a" and "i", is stored in the mind. Exemplar dynamics is an application of exemplar theory to explain how these linguistic categories are formed and changed over generations of use by language speakers. We want to develop a more precise mathematical understanding of how exemplar dynamics models work. These models themselves are difficult to work with directly as they are, mathematically speaking, space-time point processes with birth-rate depending nonlinearly on the existing points. We will study these models by approximating them with partial differential equation models where the equations describe the density of exemplars in different regions of phonetic space. What we learn from the study of these exemplar density equations will allow us to make precise statements about exemplar models, and more carefully describe the limitations of current exemplar models in linguistics. This project provides an excellent opportunity for students to work at an exciting new interface of mathematics and linguistics. The third project is on the theory of diversities. Diversities extend the concept of a metric space by instead of considering a metric function defined only on pairs of points in a space, we consider a function defined on arbitrary finite subsets of points. The theory of metric spaces has many important applications in mathematics and computer science, so it is worthwhile to develop the analogous theory for diversities. The aspect of the theory I will focus on is the family of results on minimal distortion embeddings of metrics into normed spaces, and its application to combinatorial optimization problems such as Sparsest Cut. Together with students and collaborators I will develop the corresponding theory for diversities. An important first step that a student might work on is determining the worst-case minimal distortion embedding of a diversity in L1. An eventual goal is to develop new approximation algorithms for NP-hard problems.