Wilson-Cowan Neural Oscillator
Excitatory (E) and Inhibitory (I) coupled neural populations
Phase portrait E-I with nullclines
Time series E(t) and I(t)
Wilson-Cowan (1972): τ_E dE/dt = −E + S(w_EE·E − w_EI·I + P), τ_I dI/dt = −I + S(w_IE·E − w_II·I + Q).
S(x) = 1/(1+e^(−x)) sigmoid. Nullclines shown in red (dE/dt=0) and blue (dI/dt=0). Their intersection = fixed points.
Oscillation (limit cycle) emerges via Hopf bifurcation when the fixed point loses stability.