Math 5110/6830
Instructor: Alla Borisyuk
Homework 10.1
Due: November 17
Considered the Fitzhugh-Nagumo modle that we discussed in class. Notice
that here x is representative of ”voltage”, but its values are reversed – the spike
will appear as x going down, not up as is conventional for voltage. Consequently,
the current z that needs to be injected to bring the cell to firing is negative in
this model.
x3
dx
= c[y + x −
+ z]
dt
3
x − a + by
dy
=−
dt
c
2
with 1 − 2b
3 < a < 1, 0 < b < 1, b < c .
a) Plot the nullclines with z = 0. (You can choose a = .9, b = 0.3, c = 0.8).
Add direction field arrows.
b) Find Jacobian at a general steady state (x∗ , y ∗ ) for general values of parameters and write the characteristic equation. What are the conditions on
Trace and Determinant of the matrix that ensure stability? Show that these
conditions mean that the steady state is stable unless it falls in the range
−γ < x∗ < γ
where γ = (1 − b/c2 )1/2 . Notice that the point can only be unstable when
|x∗ | < γ < 1, i.e. ”between the humps” of the cubic.
c). What happens to the nullclines and the fixed point when z is set to -0.2?
Draw in the phase plane for z = −0.2 the solution with the initial condition at
the fixed point for z = 0. For the same solution draw the time course of x
d) What happens to the nullclines and the fixed point when z is set to -3 instead?
Plot the solution in the phase plane and the time course of x with the same
initial condition as before.
e) (extra credit) Sketch bifurcation diagram with z as the parameter
f ) (extra credit, MATLAB) Solve these equations numerically in MATLAB
(convert equations to discrete as we discussed in class, or use ode45 built-in
solver) for z equal to -3. Compare time course of x to what you predicted in d).
What happens if you vary c keeping the condition b < c2 ?
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