Mathematical Biology Projects

Book: Nonlinear Dynamics and Chaos, by Steven Strogatz

Prerequisites: Linear algebra (e.g. MAT 211). Exposure to differential equations in some sense.

Biological systems are, more often than not, complex, interconnected, and non-linear. In addition, limited experimental data make the complete understanding of a given system difficult-to-impossible. With this in mind, many scientists turn to modeling of these systems as a way to understand them more fully. A prominent class of these models fall under dynamical systems, taking the form of a differential equation.

The purpose of this project will be to explore the field of continuous-time dynamical systems, using biological models as a guide throughout the process. We will look at phase-plane analysis, bifurcations, and introduce the concept of chaos. Supplementary topics that may be discussed, depending on the interests of the student (common modeling assumptions, integration techniques, interpretations of system characteristics in an intuitive way, etc).

Book: Mathematical Foundations of Neuroscience, by Ermentrout and Terman

Prerequisites: Calculus I-IV (e.g. MAT 203, 303).

In the 1950's, Hodgkin and Huxley experimentally derived a system of non-linear ordinary differential equations that, to this day, people use to understand the behaviors of neurons. The goals of this DRP will be to understand how these equations come about and the ways in which people understand the solutions to this equation. The former will take us through a tour of how differential equations are used in chemistry and physics. The latter will take us on a tour of some concepts in dynamical systems. (These are covered in Chapters 1 and 3 in the book). Students will then simulate a solution of the Hodgkin and Huxley equations by implementing a numerical ODE solver in MATLAB/Octave. Time permitting, based on student interest, additional chapters from the book can be explored.