Each of the 180 spheres 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]
When the recurrent coupling K is strong — as in the beta-hypersynchronized basal ganglia–thalamocortical loop seen in Parkinson's disease — the neurons lock into a single rotating cluster and r climbs toward 1. This is the model's stand-in for the excessive beta-band synchrony that correlates clinically with bradykinesia and rigidity.
Deep brain stimulation delivers a periodic pulse train from the electrode at the center. Every 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 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.
- Pulse frequency — clinical DBS runs at 130–185 Hz. At these rates the reference phase sweeps through many, effectively incommensurate, angles within one neuron's own cycle, so successive pulses pull different neurons toward different targets and the population decorrelates (r falls). Low-frequency pulses (a few Hz, comparable to the neurons' own rate) instead entrain the population to the stimulus and can reinforce synchrony — matching the clinical finding that only high-frequency DBS reliably suppresses pathological beta activity.
- 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.