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Let'a study a famous neural network model, the Wilson-Cowan
equations. Here is the ODE file:
# wilson cowan
u'=-u+s(a*u-b*v+c)
v'=-v+s(d*u-e*v+f)
par c=0,f=-3,a=12,b=12,d=12,e=0
s(u)=1/(1+exp(-u))
init u=.182,v=.307
done

It is a two-dimensional system. Run XPP and get yourself on a solid
fixed point by tapping `i g` and then `i l` a few times. Now
run AUTO. The parameter we want to vary is `c`. Set up the Axes so
that the Y range is 0-1.1 and the X-range 0-10. In the Numerics menu
set Par Max to 10 and Dsmax to 0.2.
Run from a steady state. Find the Hopf point and
then find the periodic orbit. Explain the diagram. How is the periodi
orbit born? Is it super- or sub-critical? How does it die? What
happens for very large inputs, *c*? Is this system bistable?

*G. Bard Ermentrout*

*1999-09-15*