Phase Planes, Nullclines & Oscillations
With two variables we get a powerful new picture: the phase plane. Instead of plotting each rate against time, we plot inhibition against excitation — and the system traces a path you can actually see spiral or orbit.
Two special lines help read it — the nullclines. A nullcline is where one variable stops changing (its d/dt = 0). Setting each derivative to zero gives a line; where the two lines cross is the stable point — the calm center the system orbits around. (It's the two-variable version of the equilibrium point from the population equation.)
Left = rates over time (the oscillation). Right = the phase plane with the dashed nullclines and the orbit. Now slide w_EE (the excitation self-connection) and watch how one number reshapes everything:
Two populations — excitation and inhibition — chase each other, producing oscillations. The dashed nullclines (where each rate stops changing) cross at the stable point the trajectory orbits. Slide w_EE: one weight can make the orbit tighten, blow up, or settle — a small change changes everything.
Static view — paste your animation code and I'll make the phase-plane orbit animate.
Think it through (hints below):
• What happens to the oscillations as w_EE grows?
• Which is more intuitive — the time plot or the phase plane?
💡 Hints: As w_EE increases, the excitatory group feeds itself more, so the orbit spirals outward — oscillations grow and can become unstable (blow up). Turn it down and the spiral shrinks inward toward the stable point (oscillations die out). The time plot is intuitive for when things happen; the phase plane is better for seeing the shape and stability of the whole system at a glance.
🎯 The big message: in a system of equations, one parameter can flip the entire behavior — steady, oscillating, or exploding. This is why brains balance excitation and inhibition so carefully, and it's the heart of the 'Dynamical Systems' work later in NMA.