Introduction to Mathematical Neurophysiology

A series of five lectures for the
Fields Institute Summer School
Introduction to Mathematical Medicine
held at the University of Waterloo, July 21-26, 2003

FRONTMATTER
Title page, list of lectures, key references

PART I
Introduction: some terminology, some history, and the biophysics of the cell membrane

PART II
Modelling electrical activity: the Hodgkin and Huxley formalism

PART III
The geometry of excitability and oscillatory behaviour: phase plane analysis and toy models

PART IV
Intercellular communication: the construction of networks of electrically excitable cells

PART V
Introduction to bursting oscillations

Related XPPAUT Codes

The following files contain code for reproducing a number of the figures shown in the lectures with the software XPPAUT.  Knowledge of XPPAUT is assumed, and instructions below to reproduce the figures are minimal.  XPPAUT can be downloaded for free from the XPPAUT website.  Tutorial information can be found there as well.

• Hodgkin-Huxley equations, as presented on pages 33-34 of part II.
• Current of 2.4 applied between t=20 and 30 (exercise: determine the units of the values quoted).
• Integrate-Go to create a figure similar to the one shown on page 37 of Part II.
fhn.ode
• FitzHugh-Nagumo equations, as presented on page 10 of Part III.
• The main XPP window is set up to display the (v,w) phase plane.  Use Nullcline-New to plot the nullclines.  Change the value of Iapp, and see how the nullclines move.
• To make bifurcation diagram, with Iapp as the bifurcation parameter: 1) Use the default value for Iapp (-0.1), and Integrate-Go, followed with by a few Integrate-Last to get to the steady state.  2) Run-SteadyState in AUTO.  3) Continue on the first Hopf bifurcation.
hh-vnew-bif.ode
• Hodgin-Huxley equations again.  File set up to make a bifurcation diagram for the full system, using Iapp as the bifurcation parameter.
• To make the bifurcation diagram: 1) Integrate-Go, followed by a few Integrate-Last to get to the steady state.  2) Run SteadyState in AUTO.  3) Continue on the second Hopf bifurcation.
ml-parsetA.ode
• For reproducing figure A on page 19 of Part III (figure 7.2A on page 261 of the Methods in Neuronal Modeling book).
• To obtain the bifurcation diagram: 1) Integrate-Go followed by a few Integrate-Last to get to the steady state.  2) Run-SteadyState in AUTO.  3) Continue on the first Hopf bifurcation.
ml-parsetB.ode
• For reproducing figure A on page 22 of Part III (figure 7.6A on page 268 of the Methods in Neuronal Modeling book).
• Instructions as above.
NOTE the source for the following ml***.ode files: Electronic support pages for the Computational Cell Biology book.
• For reproducing the figure on page 9 of Part IV (figure 6.2 on page 143 of the Computational Cell Biology book).
• Integrate-Go reproduces the figure.  The white trace is for V1, the red one for V2.
ml143b.ode
• For reproducing the figure on page 11 of Part IV (figure 6.3 on page 143 of the Computational Cell Biology book).
• Integrate-Go reproduces the figure.  The white trace is for V1, the red one for V2.
ml152a.ode, ml152c.ode
• For reproducing figures A and C, respectively, on page 29 of Part IV (figures 6.6A and C on page 152 of the Computational Cell Biology book).
• Integrate-Go reproduces the figure.  The white trace is for V1, the red one for V2.
ml156.ode
• For reproducing the figure on page 37 of Part IV (figure 6.9 on page 156 of the Computational Cell Biology book).
• Integrate-Go will display V (white trace), Isyn1 (red trace), and Isyn2 (orange trace) in the main XPP window.
• Parameters are set for figure A.  Change TDEL from -40 to -10 to obtain figure B, and change it to 0 to obtain figure C.
ml158a.ode, ml158b.ode
• For reproducing figures A1-A3 and figures B1-B3 on pages 40 and 43, respectively, of Part IV (figures 6.10 A1-A3 and B1-B3 on page 158 of the Computational Cell Biology book).
• Integrate-Go will display the trace for the membrane potential of the appropriate cell (for figures A1-A3, this is cell V in network 2, which is equivalent to cell V1 in network 3; for figures B1-B3, this is cell V2 in network 3).
• Parameters are set for figures A1 and B1.  Change TDEL from -40 to -10 and 20 to obtain the remaining figures.