## Mathematical Models of Neurons

## Project Mentor

Professor Bo Deng (Department of Mathematics)

## Prerequisites

Three-semester sequence in college Calculus, plus one-semester in Differential Equations. Familiarity with linear electrical circuits from any first course on ordinary differential equations is a plus but not required. Having some experience with Matlab is also a preferable option.

## Project Description

Neurons are the fundamental building blocks of animal nervous systems. They are the basic units as information encoders and decoders in the communication system that we think the central nervous system is. In this project, students will learn how to construct a mathematical model for neurons, learn how to use phase plane analysis to understand some spiking mechanisms of the model, and learn to use numerical simulation to explore some known and unknown features of the model.

## Matrix and Integral Models for Predicting Population Dynamics

## Project Mentors

Professor Richard Rebarber (Department of Mathematics), Professor Brigitte Tenhumberg (School of Biological Sciences)

## Prerequisites

Introductory course in matrix theory. Some introductory mathematical analysis and familiarity with Matlab or Maple would also be helpful, but are not required.

## Project Description

Matrix and Integral Models: Many topics in ecology involve predicting the behavior of populations. A typical approach to modeling a population is to use Population Projection Matrices. In such models the population is grouped into a small number of stages, for instance categorized by age or by size. This defines a population vector, which is a function of time, where time is measured discretely. The life history parameters describing rates of survival, growth and reproduction are incorporated into the projection matrix. If these rates are mainly determined by a continuous variable such as size, the decomposition of the population into a small number of classes can be inadequate. In these cases it would be better to classify members of the population along a continuum of stages. The population vector is then replaced by a function, and the matrix system is replaced by an Integral Projection Model. The eigenvalues of this matrix (or integral operator) provide predictions for the long-term steady-state behavior of the population. Other properties of the matrix provide predictions for the transient behavior, which is the short-term deviation from this steady state.

## Research Activities

You will learn how to derive matrix and integral model using basic theoretical principles. We will consider how to choose a model which best matches experimentally determined data sets; we are especially interested in how the models match the transient dynamics. We will perform numerical experiments to address the question of whether a matrix model is close to an integral model when the matrix model uses a large number of stages, and work to mathematically prove our conjectures. An important task in modern ecology is to predict what happens to the population dynamics when the transition rates are changed; this could happen due to uncertainty in estimating model parameters, a disturbance of the model (e.g. climate change), or population management strategies. We will address this question numerically and mathematically.

The direction this project takes will be partly determined by current research developments. This field is inherently interdisciplinary, and will be approached from both a mathematical and a biological perspective.

## Differential/Difference Equations

## Project Mentors

Professor Allan Peterson (Department of Mathematics) and Chris Ahrendt (Graduate Student, Department of Mathematics)

## Prerequisites

Calculus sequence and one semester course in Differential Equations.

## Project Description

We will be concerned with problems that come up in differential equations and difference equations. One of our interests will be to see how these two theories can be unified and extended. For example calculus is very useful in differential equations and when one studies difference equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general time scales calculus (which for example also includes quantum calculus). Consequently we get a generalization of difference equations and differential equations to so-called dynamic equations on time scales. A simple application of dynamic equations on time scales is a population model which is discrete in season, dies out in winter, while their eggs are incubating or dormant, and, in season again, when hatching gives rise to a overlapping population. Other potential areas of applications include engineering, biology, economics and finance, and mathematics education. Currently there are about 250 researchers worldwide who have published about 400 research articles in the area of dynamic equations on time scales.

Earlier REU teams have studied the so-called nabla exponential function which is a generalization of the usual exponential function, oscillation of a Euler--Cauchy dynamic equation, oscillation of factored dynamic equations, first order dynamic equations, the Henstock--Kurzweil delta integral, stability theory, and existence and nonexistence of periodic points.

Future projects similar to the ones mentioned above will be considered. Some of our time will be devoted to studying difference equations and no prior knowledge of difference equations is required.