Wilson-Cowan Neural Population Dynamics

τ_E dE/dt = -E + S(w_EE·E - w_IE·I + P_E), τ_I dI/dt = -I + S(w_EI·E - w_II·I + P_I). Excitatory (E) and inhibitory (I) populations with sigmoid firing rate S(x)=1/(1+e^{-x}).

w_EE 12

w_IE 4

w_EI 13

w_II 11

P_E (input) 0.5

P_I (input) 0.5

Left: E (purple) and I (green) activity over time. Right: phase portrait with nullclines (dE/dt=0 in purple, dI/dt=0 in green) and trajectory. Fixed points, limit cycles, and oscillations emerge from coupling.