Wilson-Cowan Neural Field Theory

Excitatory-Inhibitory Dynamics · Traveling Waves · Turing Patterns

Controls

Wilson-Cowan Parameters

τ_E ∂E/∂t = −E + S(w_EE*E − w_EI*I + I_ext + D·∇²E)
τ_I ∂I/∂t = −I + S(w_IE*E − w_II*I + D·∇²I)
2.0
1.5
2.5
0.5
5
10
0.5
0.5

Phase Plane

Physics

Wilson-Cowan (1972): canonical E-I neural mass model
S(x) = 1/(1+exp(−βx)) — sigmoidal firing rate
Turing instability: diffusion + slow inhibitor
→ spatial patterns (neural activity bumps)
Traveling waves: propagation at v ≈ √(D/τ)
Gamma oscillations ≈ 40Hz: E→I→E loop
WC model predicts EEG rhythms α,β,γ,δ