Mathematics 562
Homework Assignment 4
Due Thursday Mar 17, 2016
Effects of different currents on electrical activities of excitable cells
Question 1:
The attached file “Butera NaP.ode” is the model by Butera er al on bursting in preBötzinger
neurons in which the extremely slowly inactivating persistent sodium current IN aP plays the
role of generating a rhythm that gives rise to the slow underlying cycles in the bursting
behaviour of the model.
(a) Integrate the system using different values el: -65, -60, -56, -52. Summarize the differences between the solutions.
(b) In order to understand the ability of IN aP to generate oscillations, change parameter
gna = 0 to eliminate the effect of the fast N a+ channel. Plot the bifurcation diagram
of the system against the parameter el on the interval [−65, −50]. Plot the period of
the oscillatory branch. Also, plot the shape of a typical time series of oscillations in
this case for a fixed value of parameter el within the oscillatory range.
(c) Generate a new ode file of the same model in which the quasi-steady state n = n∞ (V )
is used in K + current: IK = gK n4∞ (V )(V − EK ) while keeping gna = 0. Do phase
plane analysis of the model for different values of el. Also, calculate the bifurcation
diagram of this system against parameter el.
(d) Now, generate another ode file for the fast subsystem by adding h as a parameter and
eliminate the equation for h. This is because h varies so slowly as compared to the
fast variables, we can treat it as a constant when the fast dynamics is studied. Keep
gna = 28 and use el = −58. Use both XPP and numerical simulation to find the
behaviour of the system for h = 0 and h = 1. Calculate a bifurcation diagram of the
steady state solution starting from h = 0 using h as the parameter. Now, calculate
the periodic branch starting from h = 1. Superimpose the two diagrams and use the
result to explain how bursting occurs when h is allowed to very slowly.
Question 2:
The attached file “T-type.ode” is a model involving a T-type calcium current ICaT that,
depending on the resting membrane potential, trigger the phenomenon of hyperpolarizationinduced release. Set gna = gk = 0 so that the effects of the fast N a+ and K + currents are
eliminated. Only the leak and the T-type calcium currents are at work.
(a) First set el = −65 which results in Vrest ≈ −61 mV . Run “I” and “G” followed by “I”
and “L” several times to ensure that the steady state is reached. Now, set parameter
t1=2. Run “I” and “G”, this should cause a transient depolarization of the potential
V . Freeze the curve. Then, set parameter t1=-2. Run “I” and “G” again. This should
cause a transient hyperpolarization in V followed by a depolarization pulse upon the
release of the current injection. Plot a graph of the two responses.
(b) Now, change el = −80 , run “I” and “G” followed by “I” and “L” several times to
ensure that the steady state is reached. Now, set parameter t1=2. Run “I” and “G”,
this should cause a a depolarization pulse in V . Then, set parameter t1=-2. Run
“I” and “G” again. This should cause a transient hyperpolarization in V followed
by a depolarization pulse upon the release of the current injection. Therefore, both
depolarizing and hyperpolarizing injections can cause a pulse in this case. Plot a graph
of the two responses.
Question 3:
The attached file “A-current.ode” is a model involving a A-type potassium current IA .
(a) First set ga = 0. Run “I” and “G” followed by “I” and “L” several times to ensure
that the steady state is reached. Now, set parameter t1=20. Run “I” and “G”, this
should simulate the response to a current injection with an amplitude of 20 that is
switched on at t = 50. Freeze this curve. Now, change ga = 50. Repeat the previous
simulation. One should see the response of the system with an A-current to a current
injection with an amplitude of 20 that is switched on at t = 50. There should be a
delay in the occurrence of spikes in the presence of the A-current. Print a copy of the
graph.
(b) Now, recover the default parameter. Set ga = 40, use AUTO to generate a bifurcation
diagram against the parameter i0 in the interval [0, 20]. Generate a plot of both the
steady state and periodic branches.
(c) Now, set ga = 47.7, repeat (b) to generate a bifurcation diagram of both the steady
state and periodic branches. You might have to start the periodic branch using a
numerically calculated periodic solution.
(d) Compare the results in (b) and (c), and discuss the differences.
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Question 4:
The attached file “M-current.ode” is a model involving a M-type potassium current IM .
(a) First run “I” and “G” followed by “I” and “L” several times to ensure that the steady
state is reached. Now, set parameter I=6. Run “I” and “G”, this should generate a
train of spikes with frequency adaptation. Print a copy of the file. Can you explain
why the peaks appear to have different amplitude? This is an artifact. Why?
(b) Draw bifurcation diagrams for 3 different values of gm: 0, 1, and 2 against the parameter I on the interval [-6, 8]. For each case, it is better to start from I = 8 and use a
negative ds to calculate the diagram backwards. Restrict the range of V between -75
and -35.
(c) In two parameter space gkm vs I, draw the continuation of all SN and HB points in
this space. Use this diagram to explain the effects of M-current.
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