Wilson-Cowan equations: how cortex switches between order and chaos

Added: 2026-05-27

[00:00:00]The cortex is full of two kinds of neurons, excitatory and inhibitory tangled together. The excitatory cells drive the inhibitory ones, which then push back. We want to capture this push and pull with just two numbers, the average activity of each population.

[00:00:15]Here are the Wilson-Cowan equations.

[00:00:17]Each population decays toward an input passed through a sigmoid saturation. The weights W set how strongly each population drives the other. Change those weights and the whole behavior changes.

[00:00:30]Three regimes, three phase portraits.

[00:00:33]With weak coupling, every trajectory spirals into a single resting state. The network settles down no matter where it starts. This is the quiet brain.

[00:00:43]Turn up the coupling and the fixed point loses stability. Trajectories settle onto a closed loop, a limit cycle.

[00:00:51]The populations now oscillate forever, taking turns rising and falling.

[00:01:44]Push the coupling harder and the orbit never quite repeats. The trajectory folds back on itself, never crossing, never closing.

[00:01:53]Same equations, just stronger weights.

[00:01:55]And now the dynamics are chaotic.

[00:02:00]Two equations, one knob. Slide the coupling up and the same cortical circuit moves from silence to rhythm to chaos.

[00:02:09]That is the power of Wilson-Cowan.