Lecture 9 Notes, Thursday 22 September 2005

  • Fall et al. Chap. 2, pp. 27-43
  • Review Strogatz, Chap. 3.1, pp. 45-50
  • Homework due: Thur 29 Sept 2005

    • Today we will back up and review the properties of the saddle-node bifurcation in 1D, in order to then analyze the behavior of our simple cell model bistable.ode. You should be very comfortable with the saddle-node in its normal form (x′ = r - x2), and when the equations contains additional terms (e.g., x′ = x2 - 2x - μ) that modify the vector field. We will study the phase portraits (phase lines) for the system for critical values of the parameter μ and develop a bifurcation diagram. We will also look in detail at the qualities of the vector field, in particular examining the size of the change vectors near fixed points, as compared to far away from fixed points. After the saddle-node bifurcation there are no fixed points, but the influence of the saddle node is still present, in the form of a "ghost" or "bottleneck" in the dynamics, which influences the magnitude of the change vectors.

    We will then go back to the bistable cell model and study the saddle-node bifurcations in that context, where Iapp is the bifurcation parameter. We will introduce the concept of hysteresis, where the dynamics depend on history, or previous states of the system. This is especially important for models with bistability.

    We will use several versions of the bistable cell model: bistable.ode, bistable2.ode, bistable3.ode, bistable3PP.ode, bistable3_IV.ode.

    Here you can download some of the figures we discussed in class: simulations from 3 initial conditions, two pulse protocol, ramp stimuli, bistable responses to ramp stimuli, bifurcation diagram, phase plane, the IV curve.

    Homework (Due Thur 29 Sept 2005) answer key for Homework #5

    (1) In class we discussed the bifurcation diagram for the model bistable.ode. Your task here is to use XPPAUT to analyze this bifurcation structure and produce the bifurcation diagram. Once you have the printout of the bifurcation diagram you are to label the saddle-node bifurcations as well as stable and unstable steady-state branches of the bifurcation structure.

  • Open the file bistable.ode in XPP
  • Do Initialconds|(G)o. At the end of this run, do Initialconds|(L)ast, which allows the solution to continue where it left off, and thus V approaches very close to it's steady state V=-70 mV after two full runs.
  • Open the AUTO continuation package using commands File|Auto.
  • Use Axes|hI-lo, make sure you have the following settings: Y-axis:V, Main Parm:iapp, Xmin=-12, Ymin=-300, Xmax=5, Ymax=100. For these settings, Xmin and Xmax tell AUTO to vary the bifurcation parameter iapp from -12 to +5. Ymin and Ymax tell AUTO to show V* (fixed point solutions) from -300 mV to +100 mV on the y-axis.

    • First we will track the stable steady-state branch as we decrease iapp.

  • Go to Numerics and change to these settings: Ds:-0.02, Par Min:-12, Par Max:5. Other settings should be at default values: Ntst:15, Nmax:200, NPr:50, Dsmin:0.001, Ncol:4, EPSL:0.0001, Dsmax:0.5, Norm Min:0, Norm Max:1000, EPSU:0001, EPSS:0.0001. The important thing here is that Ds tells AUTO to incrementally decrease the value of iapp.
  • Do Run|steady state. The output should be line moving down and to the left. There should be 5 distinct points.
  • Do Grab, and tab 4 times to reach the endpoint (#5) and then press enter to "grab" it, which enters it as the initial conditions for the next phase of continuation. Note: point #5 should be at iapp=-4.873 and V=-167.5.
  • Do Run again, which should track the branch of stable steady states to point #9 which lies at iapp = -9.749 at V=-265.
  • Do Grab again, and then tab to endpoint #1, which lies at iapp=0 at V=-69.95.

    • Now we will track the rest of the bifurcation diagram.

  • Go to Numerics, change to Ds:0.02. Click OK.
  • Do Run again. The curve should be S-shaped and pass through the saddle-node bifurcation, and then track the middle branch of unstable fixed points (note the line is a thinner one here because the fixed points are unstable, whereas stable fixed points are drawn in thick bold lines). Using Grab with tabbing, you should also see the second turning point (i.e., 2nd saddle-node bifurcation) at point #14 with iapp=-7.955 and V=15.29.
  • Use Grab to tab to the endpoint (#15) at iapp=-7.129 and V=25.75, press enter to "grab" it for use in further continuation.
  • Do Run again, which should now continue to track the upper branch of stable steady states, and should run off the edge of the diagram in the upper right near iapp=5 and V=100.
    • Use File|Postscript to save a copy of your bifurcation diagram. As stated at the top of this question --> (a) print out your bifurcation diagram, (b) label the saddle-node points, and finally (c) label the branches for stable and unstable steady states.
    Lecture 9 Top - Cellular Biophysics and Modeling - Del Negro Homepage - Applied Science - The College of William and Mary