Math
591-592: Mathematical Biology I-II
(Fall
2012/Spring 2013)
| Textbook |
There is
no required text.
Material will be drawn from several sources but a few
decent texts are:
Mathematical Biology, J. D.
Murray
Mathematical Physiology, J.
Keener
Computational Cell Biology,
Fall, Marland,
Wagner, Tyson
Biophysics of Computation,
Koch
Mathematical Models in
Biology,
Edelstein-Keshet
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| Instructor |
Mark
Pernarowski |
| Grading |
Your grade will
be based on homework assignments and a term project presentation.
HW = % of sum of raw scores of all HW
assignments
TP = % assigned for term
project
Then the course grade is P=0.75*HW+0.25*TP and the letter grade from:
| A |
A- |
B+ |
B |
B- |
C+ |
C |
C- |
D |
| 90-100 |
87-89 |
84-86 |
80-83 |
77-79 |
74-76 |
70-73 |
67-69 |
60-66 |
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Your term project will be an oral presentation (20-25min) on a topic in
Mathematical biology. Preferably the topic should be new (with ten
years) and should include some discussion of the mathematics used to
infer something about the system. I will say more later on this.
Many of the HW problems will be purely mathematical/computational. Thus
the emphasis will be on the mathematical techniques used to analyze the
models. |
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| Homework |
Homework 1
: 1D-population models, 1rst order
PDE |
Due: Sept 14 |
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Homework 2
: Linear systems and Predator prey |
Due: Oct 10 |
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Homework 3:
Competition, Dulac Criterion, Poincare-Bendixson |
Due:
Nov 7 |
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Homework 4:
Mutualism, Michaelis Menten, Glycolytic Oscillator |
Due:
Dec 5 |
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Homework 5:
Electrodiffusion, Channel dynamics, Cell electrical models |
Due:
Feb 1 |
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| Notes |
Single Species
Continuous
Population Models
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Single Species
Age Structured Models |
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Bacteria/Chemostat
model - an introduction to planar dynamics |
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Dynamics
101 - Equlibria stability, flow, planar systems |
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Predator
Prey systems |
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Competition
models |
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Dynamics
101 - Continuation, Periodic orbits, omega-limit sets
(Hopf
Pts , Dynamics
Notes) |
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SIR,
SIS, SIRS epidemic models -
introduction |
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Reaction
Kinetics - Law of Mass Action |
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Reaction
Kinetics -
Buffering/Michaelis-Menten |
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Reaction
Kinetics - Inhibition, Cooperativity, Passive Transport
(SERCA-pumps) |
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Reaction
Kinetics - Brusselator
Glucose-Insulin
Oscillations (Brusselator with
diffusion: 1 2
3) |
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Transmembrane
(spatial)
transport models - Passive and Facilitated Diffusion |
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Transmembrane
Ionic - Nernst
potential, Nernst and GHK currents, PHP-equations |
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Ionic
Channel Models |
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Hodgkin
Huxley model: Blockers, Voltage Clamps, Whole neuron model |
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Ionic
Single Cell
Electrical Activity Models:
| Paper |
Paper
Cell Type(s) |
Description |
Figures |
Media |
| Morris-Lecar
81 |
Barnacle
Muscle Fiber |
Very
early low dimensional
model.
First to model muscle. Often used
as a "generic" model. |
1 2 |
*.ode |
| Rasmusson_90 |
Bullfrog
Heart SA-node
pacemaker neurons |
Controls
electrical waves of
heart muscle activity |
1 2 3 4 5 6 |
movie |
| DeSchutter_94 |
Cerebellar
Purkinje Neurons |
Cerebellum
is small part of brain
that controls motor function, fear,
pleasure and some cognitive functions.
Is layered like the cortex. |
1 2 3 4 |
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| Chay_96 |
Pancreatic
beta-cell |
The
cells that make insulin (hormone)
hence an "endocrine" cell
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1 2 3 4 5
6 7 |
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| LeBeau_97 |
Corticotrophs |
Pituitary
gland secretes many
hormones.
(Anterior) Corticotrophs cells secretes the Adrenocorticotropic Hormone
(ACTH) which
promotes corticosteroid production to cope
with stress due to trauma: regulates
carbohydrate, fat and protein metabolism.
|
1 2 3 4 5 |
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| Pospischil_08 |
Cortical
and Thalamic Neurons |
Cortex
and thalamic neurons involved in higher brain functions. Basically
controls
everything. |
1 2 3 4 5
6
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FitzHugh
Naumo Model of Excitability |
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Bistability
and Fast-Slow subsytem approximations
(Review Paper
- Izhikevich) |
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Linear Cable Equations - Passive
dendrites |
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