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 f_th 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 J_0* -- 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 f_0 up to the suppression threshold, then drops discontinuously -- is testable with existing intraoperative protocols. The sub-critical spectral sharpness index C = 2f_0/Delta f_FWHM 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 (R^2 >= 0.988 per nucleus, collapsed R^2 >= 0.988) after a two-parameter rescaling, with no further free parameters.