Stochastic Modelling and Simulation in the Natural and Cognitive Sciences

Paul Tupper

Abstract

My research concerns the modeling of various phenomena in the natural sciences and in linguistics and psychology, as well as the subsequent analysis and simulation of those models. I focus on problems where I can use my knowledge and experience with dynamical systems and stochastic processes, especially stochastic differential equations.There are several subprojects that my collaborators and I work on.Computational Modeling in PhonologyPhonology is the study of the sound patterns of languages. There are two aspects of phonology (and of other areas of linguistics) where computational methods can play a big role. One is in modeling the actual production of speech: everything from the neurology to the acoustics of uttering words. The other is the historical change of languages: how languages came to be what they are today. I pursue both these areas with my colleague John Alderete (Linguistics SFU) and with linguist Andrew Wedel (Arizona).Stochastic Differential EquationsRoughly, stochastic differential equations (SDEs) are differential equation driven by noise. In many situations they are the natural way to model systems that evolve in time under the influence of random fluctuations. Besides employing SDEs as models in various circumstances, we are active in working out the theoretical details of how these models should be used. Often, this involves playing a proscriptive role, that is, explaining how SDEs should not be used. For example, with Nilima Nigam (Mathematics, SFU) and Marc Ryser (PhD student, McGill), we have demonstrated that a common approach to modeling with stochastic partial differential equations is unsound, as the models are not well posed, and fully resolved numerical solutions give nonsensical results. Another example is my work with MSc student Xin Yang. We are demonstrating that a certain class of SDE models does not actually arise naturally as a renormalized limit of microscopic dynamics, and by using it researchers are introducing unwarranted assumptions.Dynamic Field Theory in Cognitive ScienceSome of the most interesting and successful models of mental processing that are emerging now are based on ideas from dynamical systems. Often the models consist of coupled systems of nonlinear ordinary differential equations. Recently though there has been great success in using dynamic fields (coupled systems of nonlinear partial differential equations) to model cognitive phenomena that have a spatial aspect, such as vision and motion-planning. I am working with Mark Blair (SFU Cognitive Science) to develop mathematical models of a human experimental behaviour that Blair observes in his lab.