This 2D companion drops the anatomical sphere-cloud entirely and shows the same Kuramoto mean-field system through two genuinely 2D-native views: a phase circle, where every neuron is a point on the unit circle at its own phase θi (exactly the geometric object the order parameter is defined on), and a scrolling synchrogram — a raster of every sampled neuron's phase over time, the same kind of actogram-style plot real circadian labs use to read out SCN slice recordings.
dθ_i/dt = ω_i + K·r·sin(ψ − θ_i)
r·e^(iψ) = (1/N) Σ_j e^(iθ_j) (mean-field order parameter)
θi is each neuron's circadian phase, ωi = 2π / periodi its intrinsic angular frequency, drawn from a normal distribution of period width σ (real cell-to-cell variability of ~1–4 h). On the phase circle, full synchrony means every dot collapses onto a single point sweeping around the rim once per circadian cycle; the white vector from the center is literally the complex order parameter r·eiψ plotted as a vector, its length equal to r.
- Why modest K is enough: σ is a spread in period (hours), but the coupling term acts on the spread in angular frequency, ω = 2π/period. For a ~24 h period, a few hours of period spread is only a few hundredths of a radian/hour of frequency spread — so, unlike a naive "Kc ≈ 2σ" reading of σ in hours, the tissue's real critical coupling is far smaller and modest VIP/GABA signaling is genuinely enough to entrain it, matching how robustly real SCN slices synchronize.
- Light pulse — models retinohypothalamic input: nudges every neuron's phase toward CT 0 (subjective dawn), sharpening an already-forming cluster on the phase circle and pushing a bright band toward the bottom edge of the synchrogram.
- Scramble phases — models an SCN lesion or VIP-receptor knockout: phases are re-randomized, so the phase circle explodes back into a uniform ring and the synchrogram shows a burst of incoherent color before re-locking.
Color encodes each neuron's own phase: gold at subjective dawn (CT 0), deep violet at subjective midnight (CT 12) — identical convention to the 3D scene, just read off a circle and a timeline instead of an anatomical cloud.