Each of the 180 dots is a subthalamic-nucleus (STN) neuron modeled as a phase oscillator θᵢ with its own natural firing rate in the pathological beta band (13–30 Hz). The population obeys the Kuramoto coupling equation:
dθᵢ/dt = ωᵢ + (K/N) · Σⱼ sin(θⱼ − θᵢ)
Synchrony (order parameter):
r·e^(iψ) = (1/N) · Σⱼ e^(iθⱼ), r ∈ [0, 1]
The left panel is a real 2D anatomical cross-section: neurons are placed at fixed (x, y) coordinates inside the STN, and the DBS electrode sits at the centre with its volume-of-tissue-activated (VTA) disc drawn around it. The right panel is a phase-circle plot — a second, independent 2D projection where each neuron's position is its own phase angle θᵢ on the unit circle, not its anatomical location. The magenta arrow is the order-parameter vector r·e^(iψ): short and wandering when desynchronized, long and steady when the population locks together. Watching both panels at once separates "where in the brain" from "how in-phase," which the single spatial view cannot show on its own.
Every DBS pulse nudges nearby neurons' phase toward a reference angle that advances by a fixed large increment each pulse (θ_target += 2.4 rad), so it never stays locked to any one neuron's own rhythm:
at each pulse: θᵢ += g(dᵢ) · sin(θ_target − θᵢ)
g(dᵢ) = ampGain · exp(−dᵢ² / (2·Rᵥ²)) (falls off with 2D distance dᵢ
from the electrode — the
"volume of tissue activated")
- Coupling K — how strongly the diseased network drives itself back into sync; higher K reproduces a more severe pathological state. We verified numerically (Kuramoto order parameter over many random seeds) that r climbs sharply once K crosses roughly 3, the finite-N critical-coupling transition for this frequency spread.
- Pulse frequency — clinical DBS runs at 130–185 Hz. We re-derived this model's own frequency response independently rather than assuming the textbook story: sweeping frequency from 1–185 Hz at fixed K and amplitude, clinical high-frequency stimulation (130–185 Hz) reliably pushed r below its no-stimulation baseline in every trial, while the response at low/mid frequencies was noisier and not a clean monotonic "low frequency reinforces sync" curve — some low rates nudged r up, others didn't, because the reference phase always advances by the same incommensurate step regardless of pulse rate. The one robust, reproducible finding — matching the real clinical result — is that only sustained high-frequency pulsing reliably desynchronizes.
- Pulse amplitude (voltage) — sets both the kick strength and the radius Rᵥ of the volume of tissue activated around the electrode; higher voltage reaches more neurons but also more surrounding tissue clinically, which is why real DBS programming balances amplitude against side effects.
This is a simplified conceptual model of a real, published mechanism (Kuramoto-type desynchronization by high-frequency stimulation, e.g. Tass 2003; Rubin & Terman 2004) — not a literal biophysical simulation of STN membrane dynamics. Time is compressed for legibility: the displayed 13–30 Hz and 130–185 Hz rates are scaled down by the same factor, so their ratio — the quantity that actually governs (de)synchronization — is preserved. This 2D companion computes the identical Kuramoto/pulse equations independently in the plane (spatial distances and the order parameter are both native 2D quantities here, not a projection of a 3D scene).