Lecture 17 Notes, Thursday 27 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 03 Nov 2005

    • Today we continue our discussion of the Morris-Lecar model and the emergence of limit cycle oscillations at a Hopf bifurcation. We also discuss the criteria for loss of stability at the equilibrium point, which occurs when ∂/∂V[Iion(V,w)] + Φ/&Tau(V) = 0. We examine the effects of changing Φ on the point where the fixed point destabilizes and on the qualitative shape of the oscillations in the phase plane.

    Homework (Due Thur 03 Nov 2005) answer key for Homework #8

    (1) If we express the Morris-Lecar equations as V′=f(V,w) and w′=g(V,w), then the Jacobian matrix of partial derivatives contains these terms: ∂f/∂V, ∂f/∂w, ∂g/∂V and ∂g/∂w. Use the Quotient rule to determine the partial derivatives: ∂g/∂V, and ∂g/∂w. Recall that the Jacobian is meant to be evaluated at the critical point where (hint) w=w∞.(V)

    (2) Prof. Ermentrout's has posted useful exercises on his website IV. Two dimensional systems, which uses the Morris-Lecar model. Do this tutorial and its associated exercises up to, but not including, Homework 2.3 (i.e., stop after producing the frequency vs. Iapp diagram). I have tried to encapsulate the exercises in the following bulleted list:

  • The first "excitability exercise," which asks you to produce a plot of v vs t for three different initial conditions, such as v(0) = -60, -20, -10 mV.
  • Under More plotting tricks you should create a graph showing Ica and Ik for a subthreshold and a superthreshold response, e.g., using v(0) = -20 and -10 mV.
  • "Homework 2.1" asks you to find the most positive value of Ik and the most negative value of Ica. Write your response giving proper units.
  • Under Periodics and phase-plane analysis you should produce a phase-plane graph showing at least two trajectories, super- and sub-threshold, such as v(0) = -20 and -10 mV.
  • Verify the stability of the fixed point, ensure that you find a stable point with two negative complex eigenvalues.
  • Try using the Mouse to specify initial conditions.
  • Show a plot of ten runs using the Range command, where v(0) ranges from -20 to -10 mV.
  • change Iapp to 120 ľA/cm2. Now plot v vs t using v(0) = -60, -20, -10 mV, to verify that the stable resting state no longer exists, but instead we now have a globally atttracting limit cycle.
  • Try out the kinescope feature as described. Print out a 3D picture of the trajectory in V, w, t space.
  • "Homework 2.2" asks you the nature of the fixed point when I=80 and 120 ľA/cm2. Using the Sing pts feature of XPP, find the stability and types of eigenvalues for these points and write your answer.
  • Under Hopf bifurcation theorem -- AUTO III you will go through the steps to produce the bifurcation diagram we studied in class today, i.e., the emergence of stable oscillations (limit cycles) via a subcritical Hopf bifurcation. There will be three graphs for you to hand-in for this last, and most important, part of HW#8 (Due on 11/3/05). First, follow the instructions and produce the main bifurcation diagram showing v on the y-axis and Iapp on the x-axis. Second, follow the instructions and use File Import to take unstable (UPO) and stable (SPO) periodic orbits from AUTO and import them into the phase plane in the main XPP window. Make a graph showing the UPO and the SPO with the nullclines also shown. Third, Use the Axes fRequency feature of AUTO to produce the diagram showing that stable oscillations emerge via the Hopf bifurcation with finite, non-zero frequency, and also terminate at high Iapp also with finite, non-zero frequency.

    • A complete homework assignment will constitute seven well-labeled graphs (meaning axes are labeled correctly with proper units and multiple traces are distinguished with labels). You should also hand in at least one page of explanations for determining the partial derivatives in (1) and "Homework 2.1" and "Homework 2.2" as listed on the site.

    The goal of this assignment is to get you to become a master with XPP and to become very familiar with the Morris-Lecar model by reproducing many of the figures appearing in the chapter for yourself.

    Lecture 17 Top - Cellular Biophysics and Modeling - Del Negro Homepage - Applied Science - The College of William and Mary