Bifurcation Structure and Cross Nuclei Universality Govern Frequency-Selective Deep Brain Stimulation
High-frequency deep brain stimulation (DBS, >90 Hz) reliably suppresses Parkinsonian motor symptoms, whereas sub-therapeutic frequencies (<60 Hz) worsen them, yet the circuit mechanism underlying this frequency selectivity remains unresolved. We develop an analytically tractable excitatory-inhibitory continuous attractor neural network with threshold-linear transfer functions. The model predicts a boundary equilibrium bifurcation (BEB) at a critical DBS frequency fth that simultaneously accounts for two observations that prior models have treated separately: abrupt excitatory suppression and a concurrent linear rise in GABAergic output with stimulation rate. A separate, orthogonal bifurcation condition -- the Hopf boundary [Formula]-- governs endogenous beta oscillations independently of stimulation frequency; DBS suppresses these oscillations by destroying the active fixed point via the BEB, not by crossing the Hopf boundary. The predicted signature -- an oscillation frequency that remains at f0 up to the suppression threshold, then drops discontinuously -- is testable with existing intraoperative protocols. The sub-critical spectral sharpness index C= 2f0/{Delta}fFWHM diverges at the Hopf boundary and provides a parameter-free biomarker of pathological synchrony computable from any local field potential recording. Fitting the model to intraoperative VIM single-unit recordings across nine stimulation frequencies yields physiologically plausible parameters consistent with the observed transient and steady-state dynamics. Frequency-response data from four DBS nuclei (STN, SNr, VIM, RT) each conform to a rectified quadratic derived from the mean-field equations; nucleus pairs sharing circuit role collapse onto common master curves (R2 [≥] 0.988 per nucleus, collapsed R2 [≥] 0.988) after a two-parameter rescaling, with no further free parameters. SignificanceHigh-frequency deep brain stimulation (DBS) relieves Parkinsons disease motor symptoms, while sub-therapeutic frequencies worsen them; the circuit origin of this [~]90 Hz threshold has lacked explanation. We show that one bifurcation -- a boundary equilibrium bifurcation (BEB) at a critical frequency fth -- jointly explains the abrupt loss of excitatory firing and the concurrent rise in GABAergic output. The Hopf boundary governing endogenous beta oscillations is independent of stimulation frequency, so DBS suppresses beta by destroying the active state through the BEB, not by shifting the Hopf boundary: disease progression and therapy act on separate axes. Frequency-response curves from four DBS nuclei collapse onto two universality-class master curves after a two-parameter rescaling, yielding cross-nucleus predictions without further fitting.