Math 5110/6830
Homework 10
1. In the network model of developing spinal cord assume find steady states
for n equal 0.3, 0.5 and 0.9 in the following 2 ways to see that they both give
you the same results:
a) plot the right hand side for a equation and see where it crosses 0;
b) as we did in class, find where 2 components of the right hand side intersect.
Use
a∞ (I) = 1/(1 + e−(I−θ)/Ka )
θ = 0.18, ka = 0.05.
c) Make enough plots of phase line to convince yourself (and us) that there are
2 saddle-node bifurcation happening
2. Now allow n to vary as a dynamic variable.
a) Discretize the (a, n) system by an Euler method, as we discussed a few
lectures ago, and solve the discrete system by iteration. Use ∆t=0.1, τa = 1,
n∞ (a) = 1/(1 + e(a−0.5)/kn ), τn = 40, kn = 0.05.
b) Sketch the bifurcation diagram as I did in class (you don’t have to have
the exact values, just get the shape rigt) and explain your numerical results
from part a) by walking through the dynamics of the system on the bifurcation
diagram
c) (Extra credit) What happens to the behavior of the system if kn = .5.
Why?
1