MATH 592: TOPICS IN APPLIED MATHEMATICS II
Mathematical Physiology 2007
Grade: The course grade will be determined
from assignment and project grades in the following proportion:
| Assignments |
60% |
| Project
(written part) |
25% |
| Project
(presentation) |
15% |
Project: Part of the course grade will be
determined from a semester project consisting of both written and oral
parts.
- Subject
Material:
You have a choice of two styles for a project:
- A survey of some area
of Mathematical Biology. The topic need not relate to physiology. For
example, mathematical modelling in epidemiology, population dynamics,
cellular
locomotion, evolution, genetics, biomechanics, molecular motors (to
name a few) are not addressed in this course
but each would be a suitable area. For this choice, the list of
references
should be more substantial.
- A detailed analysis
of
a recent model describing some biological system. One emphasis in such
a project might be to address the correlations between
experiments and the
model. A different emphasis would be to detail the mathematics of the
analysis of the model describing the techniques used. Creative
initiative in this type of
project will be encouraged.
All topics must be
approved by me. About half way through the
semester I would
suggest that you meet individually with me to discuss your interests
so that I might give you some guidance about how to proceed.
- Written Part: Should
include
an Abstract, Introduction, Main Body,
Discussion and Bibliography. The Discussion should include a critique
of the mathematical methods and/or models reviewed in the main body.
The paper should be no less than 10 (typed) pages. I would prefer that
LaTeX or TeX be used. Due Date: TBA
- Oral Part:
At the end of the
semester after you have submitted a final
draft of the written part of the project, you will give a short
(15-30min TBA) presentation of your project to the class. The oral
presentations will be scheduled during the Final exam week.
References:
Most of the content of the lectures will be drawn from the first
reference
listed below though it is not "required". I have listed some of my
favorite books below. These books cover many different areas of
mathematical biology and are at different mathematical and biological
levels.
- "Mathematical Physiology", Keener, Sneyd (1998) -
related to this course
- "Mathematical Biology", Murray (1989) - more about
population dynamics and pattern formation
- "Computational Cell Biology", Fall, Marland, Wagner, Tyson
(2000) - generally very micro level biophysical modelling
- "Mathematical Models in Biology", Edelstein-Keshet (2005) -
very good introductory (undergrad even) book
- "Spiking Neuron Models", Gerstner, Kistler (2002) - pretty
good comprehensive book on neuroscience modelling
Computer Simulation Software
xppaut: Later in the semester we will be using a dynamical
systems software
xppaut developed by Bard Ermentrout in the
Department of Mathematics at the University of Pittsburg. An excellent
book for this software is:
- "Simulating, Analyzing and Animating Dynamical Systems",
Ermentrout (2002)
Notes: Posted supplementary notes will be listed
here. Some from a previous version of this course are listed:
- Law
of Mass Action
Overview of the Law of Mass Action in Statistical Mechanics
and Chemical Kinetics.
- Hopf
Bifurcations
A summary of practical theory about Hopf bifurcations
in planar systems.
- Cell
Calcium
A summary of some terminology related to
intracellular cell calcium
- Fast-Slow
Dynamics
A PDF file summarizing Fast and Slow subsystems for ODE
models of physiological systems (very brief introduction)
XPPAUT LIBRARIES OF MODELS:
|
Code |
Ref |
Description |
| Hodgkin Huxley Model |
HH.ode |
|
Hodgkin -Huxley model of
squid axon electrical activity |
| FitzHugh-Nagumo Model |
FHN.ode |
|
FitzHugh Nagumo model
with
nonlinearity f(u)=a*u*(u^3-u)/3. Set for AUTO calculations
using applied current i as parameter. |
| Two Pool Calcium Model |
TwoPool.ode |
|
2-pool model of
intracellular calcium.
In the model u=cytosolic calcium and v=calcium concentration
in an internal store with calcium inactivated calcium release (CICR).
The calcium in the IP3 sensitive store is assumed constant.
Initial conditions are set for an AUTO calculation. |
| Chay-Cook beta-cell Model |
Chay_Cook_88.ode |
|
a model of beta
cell
electrical activity. For different parameter values the model exhibits
a variety of qualitatively different bursting behaviors
(See Bull. Math. Biol. Vol 57, 00. 413-439, 1995 for a detailed
account). The fast subsystem for the model with s=sinf(v,c) and c
a bifurcation parameter can be downloaded by clicking
here. |
| Polynomial Model for Bursting |
Burst.ode |
|
For different parameter values
model
exhibits a variety of bursting patterns. Initial parameters are set for
bursting patterns analogous to those in pancreatic beta-cells.
(see: SIAM J. Appl. Math. Vol 54, pp. 814-832, 1994). The fast
subsystem
for the model can be obtained by clicking here. |
Old Computer Labs:
- Lab
1 Michaelis-Menten Model
- Lab
2 Chemical Oscillations, Hopf Bifurcations and an
"AUTO"
introduction
- Lab
3 Hodgkin-Huxley model of electrical activity in the
squid giant axon with an introduction to the concepts
of excitability and
refractoriness.
- Lab
4 Fast and slow dynamics in models
which exhibit bursting electrical activity
- Lab
5 Wave phenomena in the Fitz-Hugh Nagumo model
(on the finite interval) using different stimuli.