Syllabus for MA861 A1: Partial Differential Equations in Biology

Instructor: Prof. Samuel A. Isaacson
Office: MCS 231
Email:
Time and Location: T/TH 2-3:15PM, MCS 144
Office Hours: T/TH 3:30-4:30pm in MCS 231.

Course Webpage (all course material will be posted here):

http://math.bu.edu/people/isaacson/ma861-fall-2017/ma861.html

Description:

This course provides an introduction to methods of analyzing and simulating a variety of PDEs arising from biological applications. PDEs that will be studied include those describing the motion of individual molecules (the linear diffusion equation, Fokker-Planck equation, and Kramers equation); those describing electrophysiology (the nonlinear Hodgkin-Huxley equation for action potential propagation in nerves, Biodomain equations for propagation of action potentials in cardiac tissue, and general PDEs for electrodiffusion); and those describing coupled fluid-structure interactions (the immersed boundary model coupling the Navier Stokes equations to elasticity models). The course will introduce both analytical methods (matched asymptotic analysis, singular perturbation methods, traveling wave analysis, homogenization, and more rigorous existence and uniqueness methods), along with numerical methods for the simulation of these equations (First Passage Monte Carlo Method, finite difference methods for Fokker-Planck equations, finite difference methods for nonlinear reaction-diffusion equations on trees, finite difference methods for coupled nonlinear parabolic-elliptic systems, and the Immersed Boundary Method).

Course Work and Grading:

Students will be responsible for giving two classes towards the end of the semester based on reading material provided by the instructor.

Tentative Course Outline:

  1. PDEs for Single Particle Motion

    1. Diffusion Equation

      1. First passage times to reach a small target (matched asymptotic expansions).

      2. Simulating exact first passage time statistics with the Walk on Spheres method.

      3. Inhibition and diffusion gradients.

      4. Asymptotic limit of fluctuating boundary conditions; the Robin boundary condition.

    2. Fokker-Planck Equation for Drift-Diffusion

      1. First passage times to reach a small target (matched asymptotic expansions).

      2. The Robin boundary condition as the limit of a steep potential barrier.

    3. Kramers Equation

      1. Diffusion and Fokker-Planck equation as overdamped limits.

      2. Boundary Conditions for the Kramers equation and overdamped limits.

  2. PDEs for Electrophysiology

    1. Cable and Hodgkin-Huxley Equations

      1. Physical derivation.

      2. Phase plane analysis.

      3. Singular perturbation traveling wave analysis.

      4. Numerical solution of the Hodgkin-Huxley PDE model on branching trees.

    2. Bidomain Equations

      1. Cardiac excitation models.

      2. Formulation of the Bidomain equations.

      3. Numerical Methods for solving the Bidomain equations.

      4. Asymptotic homogenization methods for deriving the bidomain PDEs from single-cell models.

      5. Methods for incorporating the effect of gap junctions and fiber architecture of the heart.

    3. General Electrodiffusion Models Based on the Poisson-Nernst-Planck Equation.

      1. Well-posedness, existence, and uniqueness.

      2. Electroneutrality.

      3. Numerical methods for solving PNP-based models.

  3. PDEs for Fluid-Structure Interaction (Time-Permitting)

    1. The Immersed Boundary Equations for viscous fluids (Navier-Stokes equations) coupled to immersed elastic structures.

    2. The Immersed Boundary Method for solving the Immersed Boundary equations.

Prerequisites:

Undergraduate level linear algebra, ODEs and PDEs. Basic graduate real analysis and PDEs could be helpful for some topics but is not required.

Requirements and Grading:

50% - Class Attendance

50% - Two Presented Lectures