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