Mathematical Models in Cognitive Science: Neural Fields and Exemplar Dynamics

认知科学中的数学模型：神经场和范例动力学

• 批准号: RGPIN-2014-03713
• 负责人: Tupper, Paul
• 金额: $ 1.68万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587376
• 关键词: Mathematical; Models; Cognitive; Science; Neural

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.

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的接口的主题。第三个是关于距离空间理论的一个扩展--距离空间理论。所有这三个项目都为本科生和研究生的培训提供了许多机会。 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对西蒙弗雷泽大学认知科学实验室进行的实验中的眼动跟踪数据进行建模。 这些实验探索了人类受试者如何执行一项任务，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。所收集的数据包括受试者的猜测、猜测的定时和眼睛跟踪数据，即受试者在每个时间点的注视在视频屏幕上的位置。我们的目标是创建一个数学模型的主题的行为，而执行这些任务。一旦我们有了这个模型的计算机实现，我们就可以用它来预测新环境下的行为，然后设计实验来测试这些预测。长期目标是发展对分类背后的认知机制的理解。学生有很多机会参与这个项目，从模型的开发，现有模型的分析，计算实现的问题，最后模型拟合实验数据。 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何在大脑中储存的理论，如元音“a”和“i”。范例动态是范例理论的一个应用，它解释了这些语言范畴是如何形成的，并在几代人的使用过程中发生变化。我们希望对范例动力学模型的工作原理有一个更精确的数学理解。这些模型本身很难直接处理，因为从数学上讲，它们是出生率非线性依赖于现有点的时空点过程。我们将通过用偏微分方程模型近似这些模型来研究这些模型，其中方程描述了语音空间不同区域中样本的密度。我们从这些范例密度方程的研究中学到的东西将使我们能够对范例模型做出精确的陈述，并更仔细地描述语言学中当前范例模型的局限性。这个项目为学生提供了一个极好的机会，在数学和语言学的一个令人兴奋的新接口工作。 第三个项目是关于训练的理论。Different扩展度量空间的概念，而不是考虑只定义在空间中的点对上的度量函数，我们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，因此发展度量空间的类似理论是很有价值的。方面的理论，我将重点放在家庭的结果最小失真嵌入度量到赋范空间，其应用组合优化问题，如Sparling-Cut。与学生和合作者一起，我将发展相应的理论。学生可能要做的重要的第一步是确定L1中多样性的最坏情况最小失真嵌入。最终的目标是开发新的近似算法的NP难问题。