Models of the Neuron (580.439/639)

Fall 2013

Course overview

This course discusses single neuron modeling, including molecular models of channels and channel gating, Hodgkin-Huxley style models of membrane currents, non-linear dynamics as a way of understanding membrane excitability, neural integration through cable theory, and network computation. The goal of the course is to understand how neurons work as biological computing elements and also to give students experience with modeling techniques as applied to complex biological systems.

The course meets Mondays, Tuesdays, Wednesdays, and Fridays from 9:00-10:00 a.m. on the Homewood campus. The MWF classes are lectures and will be held in Hodson 311. The Tuesday classes are recitations held in Hodson 301, where homework will be discussed and help with questions provided; Tuesday recitation is required for undergraduates and optional for graduate students. The course is taught by Eric Young, 505 Traylor at the Medical School, telephone 410-955-3164 (; the T.A. is Kristin Hageman, telephone (609) 651-7936 ( TA office hours will be on Wednesdays from 10AM-12PM in Clark 110 (if no one shows up by 10:30AM office hours will be cancelled for the day unless a prior appointment has been made). The prerequisites are mathematics through linear algebra and differential equations and an introduction to neuroscience (e.g. 580.422, 080.205, or 080.304); introductory signal and system theory (e.g. 580.222 or 520.213-214) is helpful.

There is no required text, although Biophysics of Computation by C. Koch is an excellent book that covers most of the material in the course. Foundations of Cellular Neurophysiology by D. Johnston and S. Wu covers some of the material in a more elementary fashion and P. Dayan and L.F. Abbott Theoretical Neuroscience provides a more modern view of some topics. Several chapters from Methods in Neuronal Modeling (2nd ed.) edited by C. Koch and I. Segev will be used. An excellent and comprehensive book on membrane physiology is Ionic Channels of Excitable Membranes, (3rd ed.) by B. Hille. This book covers ion channels in more depth than those above; it is required reading for anyone seriously interested in this subject. Additional references include the following: G.M. Shepherd, The Synaptic Organization of the Brain (4th ed.) is a good introduction to neural systems for persons with no previous experience. S.H. Strogatz Nonlinear Dynamics and Chaos cover aspects of nonlinear dynamics and network theory that will be discussed in the course; some material from H.R. Wilson, Spikes Decisions and Actions will also be used. The book Dynamical Systems in Neuroscience by E.M. Izhikevich provides a useful view of 2nd-order nonlinear systems of the type used in neuroscience. A good overview of network theory is J. Hertz, A Krogh, and R.G. Palmer, Introduction to the Theory of Neural Computation. Finally, J.J.B. Jack, D. Noble, and R.W. Tsien, Electric Current Flow in Excitable Cells contains detailed discussions of older work, especially useful for cable theory. All of these are on reserve in Eisenhower library.

Weekly homework assignments will be given. Solutions should be handed in during class on the due date and will be graded. Two computer modeling projects will be assigned during the term. The grade for graduate students will be based on the midterm (20%), final (30%), the modeling projects (40%), and the homework (10%). Undergraduate's grades will be based on the midterm (30%), first modeling project (30%), final (30%), and homework (10%). Students are encouraged to discuss homework problems with colleagues, but the final product that is handed in should be the student's own work. Modeling projects must be done individually. A conscientious homework record will contribute to raising marginal grades.

Course schedule Updated August 19, 2013

Lectures are MWF 9-10 in Hodson 311. Parentheses indicates no class meeting on that day. The problem session meets T 9-10 in Hodson 301.


Sep. 4, 6 - Introduction; review of neurophysiology and thermodynamics.

9, 11, 13 - Equilibria, electrodiffusion I-V relationships; cellular steady state. Biological membranes, ion transporters, channels.

16, 18, 20 - Kcsa and similar channels. Barrier models of channel permeation. Kcsa permeation model.

23, 25, 27 - Transporter models, voltage clamp analysis, gating; Hodgkin-Huxley and similar models.

Sep. 30, Oct. 2, 4 - Phase-plane analysis of nonlinear systems; model reduction, linearization; classification of behavior near equilibrium points

7, 9, 11 - Simulation methods; limit cycles; bifurcations.

(14), 15, 16, 18 - Bursting; role of calcium; varieties of channels; (Class on the 14th is moved to the 15th for fall break). Bursting neurons, corticothalamic neurons; regulation of ion channel density.


21, 23, 25 - Synaptic transmission and neuromodulation. Dendritic trees, distribution of inputs on dendrites. Cable equation for dendritic trees.

28, 30, Nov. 1 - Solutions to the cable equation; finite cylinders.


Nov. 4, 6, 8 - Transfer functions in dendritic trees; Equivalent cylinder.

11, 13, 15 - Real dendritic trees, synaptic coupling to the soma, arrangement of synapses, subunits.

18, 20, 22 - Spines and calcium. Plasticity; neural integration.

25, (27, 29) - feedforward neural networks. (No class Nov. 27, 29 for Thanksgiving vacation)

Dec.2, 4, 6 - Network learning rules; Hopfield network; Liapunov functions and the Cohen-Grossberg theorem.

FINAL EXAM DEC 14, 2013-9:00 A.M.-Noon in Hodson 311



Homework assignments will be given weekly, and are generally due on Fridays by the end of class. They can be submitted in class or dropped off to the TA (notify TA if doing so). Homework will not be accepted after solutions are posted. The links below will return pdf files of the homework sets and solutions. The pdf files can be viewed with the free Adobe Acrobat Reader. Solution sets will be available for downloading after the due-date of the homework.

Homework 1

Homework 1 Solutions

Homework 2

Homework 2 Solutions

Homework 3

Homework 3 Solutions

Homework 4

Homework 4 Solutions

Homework 5

Homework 5 Solutions

Homework 6

Homework 6 Solutions

Homework 7

Homework 7 Solutions

Homework 8

Homework 8 Solutions

Homework 9

Homework 9 Solutions



Course notes

  1. Review of circuit theory and differential equations
  2. Thermodynamics and Transport
  3. Channels, Selectivity, and Permeation
  4. Nonlinear Dynamics and Hodgkin Huxley Equations
  5. Cable Theory
  6. Papers on Dendritic Processing

Class lectures

Other relevant notes

Modeling projects

Two computer modeling projects will be assigned. All work on the modeling projects must be done individually.


Project #1 : Due October 30, 2013 by 5:00 P.M.

Project 1 Assignment

Project 1 Code Download


Project #2 : Due December 20, 2013 by 5:00 P.M.

Project 2 Assignment


Previous exams

Copies of previous midterms and finals are posted below, along with solutions.

Midterm exam, 2013

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Final exam, 2013

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