Homework 4.2
1. (dV 4.5.5) Consider the following model for signal tranduction in the
axon:
ut = uxx + u(1 − u)(u − 1/2),
where u represents the membrane potential. We study this model on 0 ≤ x ≤ 1
with homogeneous Neumann boundary conditions:
ux (0, t) = 0,
ux (1, t) = 0.
a) Determine the system of two ODEs which describe the steady-state solution
of the PDE
b) Find equilibrim points of the system of (a) and study their stability
c) Show that
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1
1
1
H(u, ux ) = (ux )2 − u4 + u3 − u2
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4
2
4
is a Hamiltonian function for the system you found in (a)
d) Sketch a phase portrait in the (u, ux ) plane
e) Find the steady state solution that satisfies the Neumann boundary conditions
and sketch them as a function of x
f) Give a biological interpretation of these steady-state solutions
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