Math 4600, Homework 5
1. One of the very influential neuronal models is the highly idealized
Fitzhugh-Nagumo system:
dv
= − v(v − a)(v − 1) − w
dt
dw
= (v − γw).
dt
In this model v can be thought of the scaled membrane potential of the cell with rest at v = 0 and the
threshold at v = a. Alternatively, v can be thought of as the fraction of open (active) sodium channels
(recall that activation of sodium channels is the main player in the generation of action potentials). The
variable w represents the slow blocking of sodium channels. Notice that w is positively affected by v (if
sodium activation or the voltage is high, then the blocking mechanisms turns on); w is negatively affected
by itself, i.e. if the blocking mechanism is on, it is trying to turn itself off; and small is representative
of the fact that w changes more slowly than v.
For the rest of this problem set = .1, γ = 2.5, and a = 0.3.
a. Find steady states and determine their type analytically
b. Draw nullclines, equilibria and direction arrows in the phase plane
c. Sketch solution in the phase plane with v(0) = 0.1, w(0) = 0. Plot time course of the same solution.
Describe the behavior of solution in words. This is what happens when the initial voltage perturbation
is small (below the threshold).
d. Repeat part c) with v(0) = 0.4. This is what happens when the initial voltage perturbation is above
threshold, and this solution is representative of the spike (action potential) in Fitzhugh -Nagumo
model.
2. In the Beta Bursting Model, we saw that the equilibrium E3 underwent a Hopf bifurcation. The goal
of this problem is for you to study the normal form (the simplest representative example) of a Hopf
bifurcation. The normal form (with µ as the parameter) is:
x˙1 = − x2 + x1 (µ − x21 − x22 )
x˙2 = x1 + x2 (µ − x21 − x22 )
a. Find the equilibrium of this system, and calculate the Jacobian matrix for this equilibrium. Use the
eigenvalues of the Jacobian to draw the local dynamics of the system (in the x1 − x2 plane) when
µ < 0 and µ > 0
b. Convert the system to one involving polar coordinates (use x1 = rcos(θ), x2 = rsin(θ)). Get equations
for ṙ and θ̇ in terms of r and θ only. A limit cycle now corresponds to ṙ = 0, meaning that the radius
of oscillation is not changing. Find what values of µ allow a limit cycle to exist. Is this limit cycle
stable (will solutions head towards it, or away from it)? How big is the limit cycle when it exists?
c. Use your answer from part b to sketch the bifurcation diagram.
3. (computing)
a. Plot nullclines of the two-dimensional V − n system for c0 = 0.2. In the same plot, graph several
solutions, with different initial conditions (using provided file) to explore the stability of the steady
state(s). What do you find?
b. Repeat part a) for c0 = 0.5, 1.2
4. (computing) Using the R file for the full model, test the effect of changing the glucose level. The effect
of glucose in this original model (proposed by Atwater, 1980) is to increase the parameter kc , the removal
of calcium from the system.
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a. Simulate the full bursting model with kc = (.001, .04) and describe three qualitatively different behaviors in this range; note the values of kc for which they occur. Describe the biology of how this
theoretical cell is responding to glucose levels, using the fact that insulin is released during the active
phase of bursting.
b. Look at the solution of the full bursting model. What does c do during the silent and active phase?
These dynamics were important in refuting the claim that Calcium acted as the variable responsible
for the slow oscillation (we’ll talk about this next week).
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