Wilson–Cowan E–I Dynamics

E–I Population Dynamics: the Wilson–Cowan Model

One excitatory population and one inhibitory population, each with a saturating input–output curve. Neither generates a rhythm alone. Coupled, with inhibition acting more slowly than excitation, they do.

τ_E dE/dt = −E + S(w_EEE − w_IEI + I_ext)
τ_I dI/dt = −I + S(w_EIE)
S(u) = 1 / (1 + e^{−10(u−0.5)})

w_EE recurrent excitation: 1.50

w_EI E drives I: 1.50

w_IE I inhibits E: 1.50

I_ext external drive: 0.30

τ_I inhibitory time constant: 2.00

Regime

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Fixed point

–

(E*, I*)

Period

–

time units per cycle

E amplitude

–

peak − trough

Time series (normalised firing rate)

The window rescales to show the approach to the attractor plus several cycles.

Excitatory (E)

Inhibitory (I)

Phase plane

Click anywhere in the plot to start the system from a different state.

Attractor

Transient

E-nullcline

I-nullcline

● stable ◯ unstable × saddle