5.8.1 Give
an equation like Equation 5.8.2 describing a system with two
thresholds. The system has a stable equilibrium at a resting potential
at 0, but will be pushed to a higher positive equilibrium if the
potential is raised above a particular positive threshold, and to a
negative equilibrium if the potential is dropped below a particular
negative threshold. Draw a phase-line diagram and check that your
equations match it.
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5.8.2 Draw phase-line diagrams for Equation 5.8.2 in the following cases.
a. a = 0.01. How might this neuron malfunction?
b. a = 0.5.
c. a = 0.99. Why might this neuron work poorly? ...
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5.8.3 Use the method of separation of variables to solve ... (Equation 5.8.3) assuming that v is constant. Show how the dynamics are slowed when ε is small, but that ε does not affect the equilibrium.
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5.8.4 Assuming that w is constant (perhaps the potassium channels are jammed in a particular state) and a = 0.3, figure out the dynamics of ... (Equation 5.8.4) thought of as a one-dimensional differential equation. Try it with w = 0.01, w = 0.05, and w = 0.1.
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5.8.5 Draw direction arrows on the nullclines for the Fitzhugh- Nagumo equations.
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5.8.6 Sketch
the phase-plane trajectory and solutions of the Fitzhugh-Nagumo
equations when the initial stimulus is less than the threshold.
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5.8.7 Sketch
the phase-plane diagram and solution for the Fitzhugh-Nagumo equations
for the following values of ε. (Think of changing ε as changing the direction arrows: the arrows are nearly horizontal when ε is small because w, the vertical variable, changes slowly.)
a. ε very small.
b. ε rather large.
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5.8.8 With
positive applied current, there could be three intersections of the
nullclines (as in Figure 5.8.8). Draw a phase-plane diagram illustrating
this scenario, and take a guess at the dynamics. Try to make sense of
the results biologically.
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5.8.9 There could also be a single intersection with positive applied current, but on the rightmost decreasing part of the v-nullcline.
Draw such a phase plane. Assuming that this equilibrium is stable,
sketch the dynamics. Why might this cell also be thought of as excitable
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5.8.10 What
happens to the phase plane and the cell if the applied current is
negative? Can the cell lose its ability to respond if the applied
current is negative and large?
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5.8.11 Suppose that a higher applied current ... in
Equation 5.8.6 produces faster bursting. How might the body use this
ability to translate signal strength into response speed?
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5.8.12 Use a computer to study a cell that is forced by an external current ... that oscillates. Try different periods and amplitudes of the oscillation. What happens? Do you see any strange behaviors?
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5.8.13
The behaviors observed in the previous problem can occur when an
object that naturally oscillates at one frequency is forced at a
different frequency. For example, a spring following the equation
... Study this equation with the following values of T , trying a range of values of A from 0.1 to 10.0.
a. T =2π.
b. T = ...
c. T =4π.
d. T = 3.0.
e. T = 4.0.
f. T = 3.14.
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