This is the 2D companion to the 3D tACS ring visualizer, computed independently rather than flattened from a camera view. It runs the exact same population of coupled cortical oscillators, dθᵢ/dt = ωᵢ + Kc·R·sin(ψ−θᵢ) + Ks·sin(θs−θᵢ), but renders it two ways that only make sense in 2D:
R e^(iψ) = (1/N) Σᵢ e^(iθᵢ) (order parameter, plotted directly on the unit circle)
Kc_crit = 2 / (π g(0)) = σ · 2√(2π)/π (Kuramoto sync threshold, Gaussian ωᵢ)
- Top: phase circle — every oscillator plotted at (cos θᵢ, sin θᵢ) on the unit circle in the complex plane, the population's actual state space. The cyan arrow is the order-parameter vector R·e^(iψ) itself (length = R), not a proxy for it — the same quantity the 3D scene has to fake with an ArrowHelper and a bar height. The gold hand rotates at the stimulation frequency.
- Bottom: Arnold-tongue map — an independently-run mini-ensemble is integrated to steady state at every (detuning Δf, stimulation amplitude Ks) grid point, for the coupling/dispersion you currently have set, and coloured by its settled R. The crosshair marks your current sliders live inside that landscape, so you can see the entrainment boundary widen and narrow as you move Kc and σ — a genuinely 2D, parameter-space view with no 3D analogue.
- Cortical coupling Kc — above the labelled critical value the population self-synchronizes with no stimulation at all (the classic Kuramoto transition for a Gaussian-dispersed population).
- Frequency dispersion σ — wider tissue variability raises that critical coupling and narrows the Arnold tongue, both shown live in the readouts and the map.
Real-world relevance: this is the accepted mechanistic model for non-invasive tACS neuromodulation — the map view is exactly the kind of stimulation-parameter chart used when choosing a frequency and current amplitude likely to entrain a target cortical rhythm.