Math 428

49-41943, cowen@purdue.edu

Office Hours:

TuTh 2:30-3:30 or by appointment

Some slides from Pulfrich Effect talk

This course is intended for advanced undergraduate students or graduate students in the biological or mathematical sciences.

MATH 490N meets TuTh 12:30 - 1:20 and Tu 1:30 - 2:20 in REC 122.

BIOL 595N meets TuTh 12:30 - 1:20 and Th 1:30 - 2:20 in REC 122.

That is, the courses meet together two hours per week and separately one hour per week; this schedule will enable the course both to serve different audiences and to enable interaction between them.

The prerequisite for BIOL 595N is two semesters of Calculus, such as Math 223 and 224 or Math 161 and 162.

As would be expected, the BIOL 595N section will emphasize more of the biological issues involved and the MATH 490N section will emphasize more of the mathematical issues involved. For example, BIOL 595N will consider the interpretation of mathematical models and their relationship with the results of biological experiments and implications for future modeling or experimentation. On the other hand, MATH 490N will include more work on phase plane analysis and bifurcations and this work will be supported computationally by XPPAUT.

Acknowledgement: The development of this course was supported by the Purdue Mathematics Department, the Purdue Biology Department, and by an IGMS grant from the National Science Foundation, DMS-0308897.

For more information on the NSF's Interdisciplinary Grants in the Mathematical Sciences, check the IGMS webpage.

In addition, MATH 490N will use Non-linear Dynamics and Chaos, by S. H. Strogatz, Westview Press, 1994.

An Introduction to Dynamical Systems, Continuous and Discrete, by R. Clark Robinson, Prentice-Hall, 2004.

MATH 490N has been approved for graduation credit counting toward the Math major in the Core, CS, and Applied Math options and will count toward a Math minor. See your counselor for further clarification.

XPPAUT is a program developed by mathematical biologist Bard Ermentrout at the University of Pittsburgh that solves systems of ODE's, plots the phase diagrams, and (more unusual) plots the bifurcation diagrams. It is idiosyncratic, but useful! XPP Crib Sheet

Textbook's website has .ode files for many of the illustrations of the book.

NEURON is a simulation program developed at Yale University that can be used to model neurons and networks of neurons. It works from a biological description of the network and the differential equations are hidden from the user. It is also available at a second site at Duke University. Interestingly, while the software is the same at both sites, the reference materials available at the two sites are different.

SNNAP, the "Simulator for Neural Networks and Action Potentials," is a more modest program than NEURON developed at the University of Texas at Houston Medical College. SNNAP appears to be simpler to use, and apparently requires less programming, than NEURON.

We will discuss these programs in class.

This outline will be dynamic, updated as the topics and activities are
decided. This page will be kept as up to date as possible.

For each date, the first entry is the common class meeting. Entries beginning
with (Math) are meetings of Math 490N and entries beginning with (Biol)
are meetings of Biol 595N. You are welcome to come to class meetings
of both courses if you wish.

Date Activity 1/13 Course Organization, Introduction, Mathematical Models (slides) 1/13 (Math) Cells, Neurons, the Nervous System (slides) 1/15 Differential Equations 1/15 (Biol) Math Refresher, Some Simple Differential Equations Homework 1, Due Tuesday, 1/27 1/20 More Differential Equations, Phase Portraits 1/20 (Math) Linear Systems and Linearization of Non-linear Systems Strogatz, Chapter 2 1/22 Bifurcations, Bifurcation Diagrams 1/22 (Biol) Graphing for Differential Equations Strogatz, Chapter 3 1/27 Intro to Computational Modeling in Biology Voltage Gated Ion Channels 1/27 (Math) Description of Activation and Inactivation of Channels Fall, et al., Chapters 1, 2 1/29 Voltage Gated Ion Channels, Morris-Lecar Models 1/29 (Biol) Morris-Lecar on the Barnacle Giant Muscle (slides) Fall, et al., Chapter 2 Homework 2, Due Tuesday, 2/10 2/3 More on the Morris-Lecar Model for the Barnacle Giant Muscle 2/3 (Math) Using Fast and Slow Equations 2/5 Synapses and Ion Channels (Professor Christie Sahley) 2/5 (Biol) Synapses and Ion Channels (Professor Christie Sahley) 2/10 Pictures from XPP-Auto 2/10 (Math) More on Using XPP 2/12 Synapses and Ion Channels (Professor Christie Sahley) 2/12 (Biol) Synapses and Ion Channels (Professor Christie Sahley) 2/17 Calcium control in Cell and ER (Text, Section 5.1) 2/17 (Math) Phase Planes and more on using XPP Homework 3 Due Tuesday, 2/24: From Strogatz Use XPP to get bifurcation diagram for problems 3.1.2, 3.2.1, 3.2.2, 3.4.1, 3.4.7 2/19 Calcium control in Cell and ER, II (slides) 2/19 (Biol) Papers on oscillations in the bullfrog ganglion 2/20 Special! Brian A. Wandell, Stanford University, "Computational Neuroimaging: Cortical Color Responses in Human and Macaque" 3:30p Stewart, Room 202 2/24 Calcium control in Cell and ER, III BFSG closed system ode file 2/24 (Math) Some Differential Equations Theory 2/26 Calcium control in Cell and ER, IV BFSG open system ode file 2/26 (Biol) Biological interpretions from the XPP data 3/2 Calcium control in Cell and ER, V BFSG open system, reduced, ode file Calcium in the Pituitary Gonadotrophs, I (slides) (Math) More on Differential Equations Theory a sample problem 3/4 Calcium in the Pituitary Gonadotrophs, II Pituitary Gonadotroph, closed system, ode file (Biol) Calcium oscillations and exocytosis in the Pituitary Gonadotrophs (slides) 3/4 Special! David Crews, "Evolution of Neuroendocrine Mechanisms Controlling Male-and Female-Typical Sexual Behaviors" 3:30p Stewart, Room 214D 3/9 Review (Math) More on Differential Equations Theory Solution of System Example 3/11 Midterm Test 12:30 - 1:20 Topic List 3/11 Extra time to complete Midterm Test 1:20 - 2:20 3/12 Special! Alan Slater, University of Exeter, "Vision in the Young Infant: From Sensation to Perception to Representation" 3:30p Stewart, Room 202 3/16 SPRING BREAK! No Classes 3/18 SPRING BREAK! No Classes 3/22 Registrar: Last day to drop a course, 5:00pm. 3/23 Discussion of Midterm Test (Math) Existence and uniqueness, stability (from Robinson) 3/25 Report topics due for approval 3/30 No Class 4/1 Gap junction connected Morris-Lecar Neurons A (ode file), B odefile. 4/6 Synaptic, mutually excitatory Morris-Lecar Neurons excitatory (ode file), inhibitory (ode file). (Math) Relating stability of non-linear system to linearization 4/8 No Class 4/13 Discussion on BOLD, Cortical Activity in Movie Watching SEE Reading Assignment (Math) no meeting 4/15 Integrate-and-Fire Models; the Central Pattern Generator for swimming in Tritonia diomedea (slides) (Biol) Getting the parameters to support computational models 4/20 Group Presentations, Group 5, then Group 2, (Math) Periodic orbits (Robinson, Chapter 6) 4/22 Group Presentations, Group 1, then Group 4, (Biol) Parkinson's disease and modeling of activity in the globus pallidus and subthalamic nucleus (slides) Terman files 4/27 Group Presentations, Group 3 Course evaluation (Math) More on periodic orbits (Robinson, Chapter 6) Poincare Bendixson example ode file 4/29 Review Written Reports Due Group 1: Paths to Diabetes (Refs omitted) Group 2: Dopaminergic Modulation of Na+ Currents Group 3: Leech Heart CPG Group 4: Retinal Ganglion Group 5: Binocular Rivalry (Biol) No official class meeting Optional presentation on Pulfrich effect and axonal transport 5/7 Final Exam: Friday, May 7, 1:00p - 3:00p, REC 122 Topic List

Last Update: May 2, 2004