Math 8540
Topics in Mathematical Biology
Spring 2004, University of Minnesota

Mathematical Modeling of Neurons and Neural Networks

General Information

Lecture Time: MWF 12:20PM-1:10PM
Classroom: Vincent Hall 1

Instructor: Duane Nykamp
Office: 202 Vincent Hall
Phone: 625-0338
Office hours: MW 2:00PM-3:30PM, or by appointment

See the course description for an overview of topics covered in this course.

Grading will be based on homework assignments and a final project.  Sorry, there will be no tests or final exam.

Biophysical network

Network of complicated neurons

Simplified network

Network of simplified neurons


Homework will be assigned periodically throughout the course.  I encourage even auditors of the course to attempt the homework as it will help solidify the concepts discussed in lecture.

The homework will consist of small computing assignments.  It can be done in whatever software you choose.  Some of the assignments will be most easily done using phase plane software such as XPP.  XPP is especially nice for its incorporation of AUTO, a bifurcation analysis program.  However, XPP requires either Unix/Linux or an X-Windows client running on top of Microsoft windows. XPP is installed on the math Linux computers and can be run via the command
A tutorial for XPP can be found here.

Some assignments may be easier using general purpose mathematical software. I would recommend using Matlab, which is available for Macintosh, Windows and Unix/Linux.  You are also welcome to program everything yourself, if you are so inclined.

Here is an XPP file demonstrating two coupled integrate-and-fire neurons.


There will be no textbook for this class. Books and research papers relevant to this course will be announced in class and posted on this website.


  • Steven H. Strogatz. Nonlinear Dynamics and Chaos. Perseus Books, Reading, MA, 1984.

    This book is a good introduction to the mathematical tools we will use for much of this course.

  • Christof Koch and Idan Segev, editors. Methods in Neuronal Modeling: From Ions to Networks. MIT Press, Cambridge, MA, second edition, 1998.

    We'll be discussing selected topics from this book, especially material from chapters 7 and 13.

    Chapter 7 is available in interactive form on the web. If you are technically savy, you can set up XPP to open up the example files, or even ftp the source and run it locally.

  • Daniel Johnston and Sameul Wu. Foundations of Cellular Neurophysiology. MIT Press, Cambridge, MA, 1995.

    This book provides derivations for elements of the base model used at the beginning of the course.
Population network

Network of neuronal populations

Bonehead Blackbox

The Bonehead Black Box approach

Research Papers

  • B. Ermentrout. Reduction of conductance-based models with slow synapses to neural nets. Neural Comp., 6:679-695, 1994.

  • J. J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci.-Biol., 79:2554-2558, 1982.

  • J. J. Hopfield. Neurons with graded response have collective computational properties like those of 2-state neurons. Proc. Natl. Acad. Sci.-Biol., 81:3088-3092, 1984.

  • D. H. Hubel and T. N. Wiesel. Receptive fields and functional architecture of monkey striate cortex. Journal of Physiology (London), 195:215--245, 1968.

  • D. Kleinfeld, F. Raccuia-Behling, and H. Chiel. Circuits constructed from identified aplysia neurons exhibit multiple patterns of persistent activity. Biophys. J., 57:697-715, 1990.

  • D. Q. Nykamp. Reconstructing stimulus-driven neural networks from spike times. Preprint.

  • D. Q. Nykamp and D. Tranchina. A population density approach that facilitates large-scale modeling of neural networks: analysis and an application to orientation tuning. J. Comp. Neurosci., 8:19-50, 2000.

  • D. J. Pinto, J. C. Brumberg, D. J. Simons, and G. B. Ermentrout. A quantitative population model of whisker barrels: re-examining the Wilson-Cowan equations. J. Comp. Neurosci., 3:247-264, 1996.

  • J. Rinzel and P. Frankel. Activity patterns of a slow synapse network predicted by explicitly averaging spike dynamics. Neural Comp., 4:534-545, 1992.

  • M. Shelley and D. McLaughlin. Coarse-grained reduction and analysis of a network model of cortical response: I. drifing grating stimuli. J. Comp. Neurosci., 12:97-122, 2002.

  • H. R. Wilson and J. D. Cowan. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J., 12:1-24, 1972.