Phase plane representations of the Wilson-Cowan model during equilibrium, oscillatory, and limit-cycle activities. These plots correspond to the parameters in Figures 4, 10, and 11 of the original 1972 paper [29]. Note that in the equilibrium case there are two stable fixed points: one in a “down” and one in an “up” state. Also note that in these examples, oscillations are obtained by flattening the null-cline (magenta) and making the null-cline (orange) steeper. Sample trajectories are depicted in blue. (a) Three fixed points, two of them are stable and one (in between) is unstable. The eigenvalues of these points from left to right are as follows: −0.59, −1.13; 0.73, −1.6; −1.43, −2.88. (b) Two fixed points, one is clearly associated with a damped oscillation. The eigenvalues from left to right are as follows: −0.34, −0.98; . (c) The limit cycle with a fixed point inside. The eigenvalues of the fixed point are .
