Mathematical Models in Cognitive Science: Neural Fields and Exemplar Dynamics

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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.