Lecture 15 Notes, Thursday 20 October 2005

  • Strogatz. Chap. 6, pp. 145-180
  • Handout: Rinzel, J. and Ermentrout, B. Analysis of neural excitability and oscillations, pp. 251-291. From Methods in neuronal modeling, 2nd edition. Edited by Koch, C. and Segev, I. Cambridge: MIT Press, 1998.
  • Homework due: Thur 27 Oct 2005

    • In today's lecture we discuss the Morris-Lecar model of membrane excitation. For those who wish to simply copy the M-L equations to make an *.ODE file, you can use this text (below), but make sure you understand what the terms and the functions represent:

    # Morris-Lecar reduced model 
dv/dt=(i+gl*(vl-v)+gk*w*(vk-v)+gca*minf(v)*(vca-v))/c
dw/dt=lamw(v)*(winf(v)-w)
# where
minf(v)=.5*(1+tanh((v-v1)/v2))
winf(v)=.5*(1+tanh((v-v3)/v4))
lamw(v)=phi*cosh((v-v3)/(2*v4))
#
param vk=-84,vl=-60,vca=120
param i=0,gk=8,gl=2, gca=4, c=20
param v1=-1.2,v2=18,v3=2,v4=30,phi=.04
# for type II dynamics, use v3=2,v4=30,phi=.04
# for type I dynamics, use v3=12,v4=17,phi=.06666667
v(0)=-60.899
w(0)=0.014873
# track some currents
aux Ica=gca*minf(V)*(V-Vca)
aux Ik=gk*w*(V-Vk)
done

    Or you can use the link above to get the text. Keep in mind, the M-L model was widely used in a non-dimensional form in past years, which is not as easy to interpret and the equations are not compatible with the version we are now studying, which is dimensionalized with typical units for electrophysiology. You may have noticed that files named lecar.ode and ml1.ode were downloaded with XPPAUT, but both of these versions are non-dimensionalized, thus not useful.

    It is critical that you read the chapter in Strogatz, because here you will get a much more in-depth treatment of nonlinear two-dimensional ODE systems. Strogatz lays out many important principles, which we will apply to our analyses of the M-L model, including:

  • Existence and uniqueness theorem
  • Poincare-Bendixson theorem
  • Linearization of nonlinear equations to analyze the stability and characteristics of fixed points
  • Hyperbolic fixed points: nodes, spirals, saddles, but not centers or stars.
  • Hartman-Grobman theorem: local phase portrait near a hyperbolic fixed point is topologically equivalent to the fixed point of the linearized system, and captures the stability classification of the fixed point.
  • Example from 6.4 (Rabbits-Sheep): understand connections between fixed points as parts of the phase portrait, basins of attraction, basin boundaries, and separatrices.
  • Example 6.5.2: you should understand homoclinic orbits. Strogatz says they are rare in non-conservative systems, but actually they are fairly common in cellular neurophysiology - as we will see in the M-L model.
  • Example 6.6.1: you should also understand heteroclinic orbits, which also occur in neurophysiology and the M-L model in particular.
  • The damping of the pendulum example breaks the homoclinic orbits, compare Fig. 6.7.3 to 6.7.7. We will see a very similar dynamical mechanism in the M-L in upcoming classes.

    • Here is a plot of the voltage-dependent and kinetic properties of the M-L model currents

    Homework (Due Thur 27 Oct 2005) answer key for Homework #7

    Do the following problems in Strogatz Chap. 5

  • 5.1.3
  • 5.1.4
  • 5.1.5
  • 5.1.6
  • 5.1.7 (Use XPP to sketch the vector field )
  • 5.1.8 (Use XPP to sketch the vector field )
  • 5.1.9 part a (use XPP to sketch the vector field) and part c
  • 5.1.10 a - f
  • 5.2.1 a - d
  • 5.2.2 a and b
  • 5.2.3
  • 5.2.5
  • 5.2.7
  • 5.2.9

    • Many of these problems are very easy and quick to solve. For those problems that seem more challenging, don't try to simply solve them in your head or using pencil and paper. Instead, write simple *.ODE files and plot nullclines, vector fields, flow diagrams, and individual trajectories using XPP - that's why we have good software - to help us solve problems in dynamics and to make great looking plots and diagrams.

    Lecture 15 Top - Cellular Biophysics and Modeling - Del Negro Homepage - Applied Science - The College of William and Mary