BMA 771 Biomathematics I

Tue/Thur 1.30-2.45

    Class-room: 2225 SAS Hall

    Instructor: Mette S Olufsen
    Office: SAS 3216
    Office Hours: By appointment via email.
    Phone Number: 515-2678
    Email address: msolufse@ncsu.edu

Course Information

This course provides an introduction to modeling and analysis of dynamic biological systems. We focus on analysis of models that employ ordinary differential equations (ODEs). The aim is to develop an appreciation for the formulation, analysis, interpretation and criticism of dynamic mathematical models in biology.

The course will develop the theory of ordinary differential equations, worrying less about the more esoteric mathematical aspects of the theory than would be the case in an applied math course on ODEs, and instead focusing on biological motivation and interpretation.

By the end of the course you should be able to construct, analyze, interpret and criticize ordinary differential equation models of biological systems. You should be able to use appropriate analytic, geometric and numerical techniques to study a particular model. You should be able to use computer packages (such as Maple or Matlab) to facilitate goal. You should also have an appreciation of the broad range of behaviors that these models can exhibit and how model behavior can change (for instance as a parameter is varied).

Although this is the first course in a two semester sequence, BMA 771 can be taken as a stand-alone course. BMA 772 (Biomathematics II) goes on to look at stochastic models (ones that incorporate randomness). Beyond this, BMA 773 also looks at stochastic models and BMA 774 considers partial differential equation models (ones where there is more than one independent variable, such as space and time).

Text: Nonlinear Dynamics and Chaos by Steven Strogatz.  Focus will cover the first 8 chapters, this book focuses mainly on the analysis of models, rather than their formulation and criticism. Therefore, our emphasis will be slightly different in places. Supplemental material will be given in the form of  handouts or reserve reading. Computer tools will be used, e.g. Matlab, Maple, Mathematica, and XXPAUT.


F
ormat: During the course a number of smaller and larger homework projects will be assigned. Students should be prepared to discuss results of homework problems on the board or using electronic media in class. While homework problems can be worked out in a group, everybody should be able to discuss the solution to a given homework. In addition to homework problems, one in-class (mid-term) and one take-home (final) test will be given. Results of the take-home test should be typed up and handed in electronically.

Evaluation criteria:  The course will be graded based on the work performed in homework (50% of the final grade) and in the one-two tests (midterm 20% and final 30%).

Prerequisites: This course is suitable for grad students and undergrads, depending on your math background. You are expected to have knowledge of calculus corresponding to two semesters  courses (e.g. MA 141/241 or MA 131/231). In addition you should have basic knowledge of arithmetic involving complex numbers, and some knowledge of linear algebra would be useful (we will need to work with eigenvalues and eigenvectors of matrices), but we shall review the material we need.  Note: A course in differential equations is not required; BMA 771 is an introductory differential equations course. If you are at all worried about having the necessary mathematical background (particularly with regard to linear algebra), please come and see me. It will be possible to fill in some gaps during the semester, and I will do what I can to help.


Homework:
I encourage you to work with others on the homework, but remember - if you work with someone else, put all names on the paper for a common grade. If computers are used for solving aspects of an homework problem (e.g., Maple, Matlab, XPP, or Mathematica) you must indicate so on the homework. Nicely commented computer code should be attached as part of the homework. Points will be deducted for homework handed in late - graded homework can be reworked for some extra credit. Some homework problems will be discussed in class, and I will call on you to go through your solution on the board. For homework, anyone in the group can be called to discuss the solution of any problem.


NOTES

    1.    Syllabus
    2.    Course policies
    3.    Modeling
    4.    Introduction to eigenvalues and eigenvectors
    5.    Notes: Introduction to eigenvalues and eigenvectors
    6.    Phase portraits pplane: http://math.rice.edu/~dfield/dfpp.html
    7.    Numerical solutions of ODEs: Excel example, Matlab ex 1, Matlab ex 2
    8.    Runge Kutta methods
    9.    Bifurcation plots in matlab: Ex dx = r - x^2, Ex dx = r - x^2, Ex dx = r -x - exp(-x)
  10.    Maple code for sketching solutions and implicit 3d plots (spruce budworm)
  11.    Matlab for spruce budworm
  12.    Syncrhonous fireflies Smokey Mountains, video 1, video 2, video 3 
  13.    Notes from Edelstein-Keshet 2D system
  14.    Notes from Edelstein-Keshet Chemostat model
  15.    Chemostat code
  16.    Van der Pol oscillator code
  17.    Test limit cycle code
  18.    Population models
  19.    Oscillating population models

  20.    Neural modeling
  21.    FitzHugh Nagumo Lecture
  22.    FtizHugh Nagumo Matlab
  23.    Hopf bifurcation
  24.    Neural network

  X.    Selkov glycolysis paper
  X.    Glycolysis: http://www.emc.maricopa.edu/faculty/farabee/biobk/biobookglyc.html#Glycolysis
  X.    JSIM: http://www.physiome.org/jsim/models/webmodel/NSR/Selkov/
  X.    Glycolysis process
  X.    Bifurcations
  X.    Matlab supercritical Hopf bifurcation

TEST 

Test 1 (Due Thur 10/17)

PROJECT

Project presentations 12/3 and 12/5
Project report due 12/12 before 5pm

HOMEWORK

Homework 1 (Due Thur 8/29/2013)
1:  Identify 4 biological models and classify them according to definitions given in the pdf discussing models.
2:  Give 2 examples of mathematical models that have made an impact in biological sciences.
3:  Discuss early contributions to mathematical biology involving dynamics.
4:  For the population problem with juveniles and adults, calculate both eigenvalues and eigenvectors. Give the complete
      solution for the problem and compare results for the complete solution with those obtained by "time-stepping" in excel.
      Also graph the solution, the age distribution (the ratio of adults to juveniles), and the change of population in successive
      time periods (i.e. x(t+1)/x(t) and y(t+1)/y(t)). See specific questions in the pdf file (Note 4).
      
Homework 2 (Due Thur 9/5/2013)
1:  Solve problems 2.1.1, 2.1.2, 2.1.3, 2.1.5 from Strogatz.
2:  Solve problems 2.2.3 and 2.2.4 from Strogatz.
3:  Solve problems 2.3.1, 2.3.3, and 2.3.4 from Strogatz. Also solve this equation numerically using Eulers method, the 2-step Runge Kutta
     method, and Matlab's build in ODE solver ode45. Calculate the error between analytical and numerical solutions. What do you observe?
4:  Solve problem 2.4.8. Solve this equation using Eulers method, the 2-step Runge Kutta method, and using Matlab's build in ODE solver ode45.
     Calculate the error between analytical and numerical solutions. What do you observe?

Set parameter values and initial values such that the model equations represent real biological populations. Include a paragraph (with a literature reference)
describing the populations chosen.

Homework 3 (Due Thur 9/19/2013)
1:    Solve problems 3.1.1, 3.1.4, 3.2.2, 3.2.5, 3.4.2, 3.4.3, 3.5.7 from Strogatz.
2:    Solve one of problems (3.7.3 + 3.7.4) or 3.7.5 or 3.7.6.
       Use a computational tool to sketch the fix-points [plot (x vs dx)], directional fields, and solve equations numerically.
3:    Solve problem 4.5.1 from Strogatz.

Homework 4 (Due Tue 10/8/2013)
1:    In the notes for the Chemostat model solve problems 6-10.
2:    Solve the Chemostat model numerically find appropriate parameter values and use code given above. Discuss your choice for parameter values.

Homework 5 (Due Tue 11/5/2013)
1:    In class we discussed impact of changing the parameter alpha, investigate dynamics further by exploring changes in the other parameter values (epsilon and gamma).
2:    In class we had dv/dt = v ( v- a ) ( 1 - v ) - w, now assume that the neuron is stimulated by I(t)
         A) assume I(t) take the form of a step function I = 0 for t < t0; I = I0 for t > t0.
         B) assume I(t) take the form of a pulse I = I0, for t0 <= t <= t1; and I = 0 otherwise
         C) assume I(t) is constant I(t) = I0
        Explore dynamics of the system for each of these simulations.
3:    For the Fitz-Hugh Nagumo model, show how parameter values of a affect stability of the limit cycles, use the Hopf Bifurcation Theorem and the stability criterion (from notes).

Homework 6 (Due Tue 11/20/2013)
This homework is associated with section  8.4 in Strogatz
1:    For examples 1 (page 261), 2 (page 262), and 3 (page 263)
            A: Rewrite equation in cartesian coordinates
            B: Use pplane 8 to plot phase portraits for mu < muc, mu = muc, and mu > muc, note mu is always positive
            C: Plot solutions as functions of time (in cartesian coordinates) [ plot x and y, what is more interesting to study? ]
       For example 3 try to recreate figures shown in 8.4.3 note your figures will have more lines on them, which is fine, the book have only displayed important lines.
2:    Solve problems 8.4.1 to 8.4.4

Homework 7 (Due Tue 12/12/2013) - INDIVIDUAL
1:    Strogatz problem 8.1.6
2:    Strogatz problem8.2.8
3:    Strogatz problem 8.6.2 questions a, c, d


NOTE: IF YOU DO NOT HAVE MATLAB ON YOUR PERSONAL COMPUTERS, THE COMPUTERS IN SAS HALL LABS ALL HAVE MATLAB!


GRADES

    Homework 50%
    Test 20% (in-class)
    Project 30% (Presentation (15%) and report (15%))