4 Ways a Neuron Starts to Fire — the Four Bifurcations, Animated in 2D and 3D

A neuron with two variables — membrane voltage V and a potassium gate n — has exactly four ways to go from resting to firing as you turn up the injected current. This animation shows each one in the phase plane at the current you're at (nullclines, equilibria, the threshold, the limit cycles, a cloud of trajectories, and one gold cell whose voltage runs top-right), starting at low current and ramping through the bifurcation, while the whole stack of phase planes along the current axis sits bottom-right as a 3D manifold with a plane sliding up through it. 1 · Saddle-node on an invariant circle (SNIC) — the resting node and the saddle annihilate on the spike loop itself; firing starts at arbitrarily low frequency (class 1, an integrator). 2 · Saddle-node off the circle — a limit cycle already exists while the cell rests (bistability); when rest disappears the state drops onto it at a finite frequency, and keeps firing when the current comes back down (hysteresis). 3 · Subcritical Andronov–Hopf — a stable focus inside an unstable cycle (the threshold ring) inside a large stable cycle; the ring shrinks onto the focus and firing starts at full amplitude, with hysteresis (a resonator). 4 · Supercritical Andronov–Hopf — the focus loses stability and a tiny cycle is born at the equilibrium and grows like the square root of the distance past the bifurcation (a resonator; small oscillations that swell into spikes). Model: Izhikevich's persistent-sodium plus potassium (I_Na,p + I_K) neuron — the same two equations with four parameter sets (Dynamical Systems in Neuroscience, chapters 4–6). Bifurcation currents in this model: 4.6 (SNIC and SN), 48.9 (subcritical Hopf), 14.6 (supercritical Hopf). CHAPTERS 0:00 The phase plane, and the current as a knob 0:39 1 · Saddle-node on an invariant circle (class 1) 1:52 2 · Saddle-node off the circle (bistable) 2:49 3 · Subcritical Hopf (resonator, hysteresis) 3:44 4 · Supercritical Hopf (oscillations that grow) 4:20 Four doorways from rest to rhythm CODE The MATLAB script that computes the 3D manifolds and the Python port that renders this animation: https://github.com/caldwbr/neuralDyna... Brad Caldwell · theoriesofconsciousness.com · bradcaldwell.substack.com #neuroscience #dynamicalsystems #bifurcation #Hopf #phaseplane #Izhikevich #computationalneuroscience #limitcycle