Introduction to Computational Neuroscience

Spring 2004

Professor Carl Cowen
Math 428
Office Hours:
TuTh 2:30-3:30 or by appointment

Approximate Syllabus



Some slides from Pulfrich Effect talk

Course Overview

Introduction to Computational Neuroscience is being offered Spring 2004 as both MATH 490N and BIOL 595N.
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 MATH 490N is Math 366, Differential Equations.
The prerequisite for BIOL 595N is two semesters of Calculus, such as Math 223 and 224 or Math 161 and 162.

General Information

Leaders in the National Institutes of Health and the National Science Foundation believe that computational and mathematical methods increasingly will provide the foundation for advances in the biological sciences. This course is intended to provide an introduction to mathematical modeling of the biological processes involved in neuroscience. The course will begin with a brief introduction to differential equations and the basic biology underlying the electrical processes in neurons. Classical systems of differential equations, such as those of Hodgkin-Huxley, FitzHugh-Nagumo, and Morris-Lecar, used to describe firing of action potentials in neurons and their propagation through networks will be developed and analyzed. These ideas and these models describe a diverse set of biological systems and organisms, from action potentials in the giant axon of the squid, to control of insulin production in pancreatic beta cells, to understanding the effect of dopamine in the thalamus of Parkinson's patients. The course will introduce ideas from dynamical systems to understand the behavior of these models, especially the ways in which the behavior changes as the inputs and biological parameters change. Since systems of differential equations of biological importance do not (usually) have closed form solutions, software packages Neuron and XPPAUT will be used to do modeling and computations with the resulting models. The course will emphasize setting up the models of neural systems and interpreting the computed solutions in the context in which the models arose and the dependence of the predicted behavior on the inputs. An important goal of the course is to help prepare students to work in an interdisciplinary environment that includes both biological and mathematical scientists.

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.


BIOL 595N and MATH 490N will both use Computational Cell Biology, by C. P. Fall, E. S. Marland, J. M. Wagner, and J. J. Tyson, editors, Springer, 2002.

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

Recommended References

An Introduction to the Mathematics of Biology, by E. K. Yeargers, R. W. Shonkwiler, and J. V. Herod, Birkhauser, 1996.

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


The course grade will be based on a midterm exam, a final exam, homework assignments that include computation using the packages Neuron and XPPAUT, and a group report on a published model (chosen by the group members) that was not covered in the lectures.


This course does NOT assume that students bring both mathematical and biological sophistication to the course, but it is assumed that students are at the Junior level or above in one of these areas. It is expected that students will gain an appreciation for the kinds of information that mathematical and computational approaches can add to understanding the functioning of a neural system, for example, to realize that some systems are inherently more sensitive to changes in the input parameters than others. It is hoped that students who have completed the course will be more willing and more able to incorporate mathematical or computational approaches into their own biological work or see ways in which their own mathematical work can be used in the biological sciences.

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.

Software for the Course

There are several pieces of software that we will use in the course. ITAP has made XPPAUT and NEURON available on both the PC's and the Mac's in the ITAP labs. SNNAP is a Java program so it can be downloaded as a "jar" file and accompanying documents into your home directory on ITAP machines and then run by Java on those machines. In addition, because all three programs are free, you can download them onto your own computers for the "price" of doing the installation. There are other programs that can solve systems of differential equations and do other kinds of related computations and graphics that are not free but are available on the ITAP machines. These programs include MATLAB, MAPLE, and MATHEMATICA.

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.

Approximate Course Outline

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

   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)

   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

   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

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