How much Kv1 current makes a vestibular ganglion neuron fire transiently? A certified bifurcation map of a conductance-based model

Vestibular ganglion neurons fall into two groups: transient neurons fire one or two action potentials at the onset of a current step and then stay silent, while sustained neurons fire throughout the step. Experiments and simulations attribute the difference mainly to the density of the low-voltage-activated, Kv1-type potassium conductance $g_{KL}$, but how much of it is needed, and why, has not been established rigorously. We answer these questions for the conductance-based model of Hight and Kalluri (2016), a seven-dimensional system, by combining an exact reduction with computer-assisted proofs. Because every gate depends on voltage only, all eigenvalues that can destabilise an equilibrium are zeros of $C\lambda+Y(\lambda)$, where the membrane admittance $Y$ is affine in each maximal conductance; fold and Hopf loci therefore become explicit curves parametrised by the equilibrium voltage, and interval arithmetic turns statements about them into theorems. We prove that the transient neuron ($g_{KL}=1.1$, $g_{Na}=13\ \mathrm{mS\,cm^{-2}}$) has, for every applied current, a single equilibrium in the voltage window $[-100,-10]$ mV and that this equilibrium is always stable: its excitability is of class 3. The sustained neuron ($g_{KL}=0$) instead has two folds, at 2.557 pA and 0.883 pA, and a Hopf point at 9.680 pA. At the nominal sodium conductance every Hopf point disappears once $g_{KL}$ exceeds a value in $[0.1277,\,0.1280]\ \mathrm{mS\,cm^{-2}}$, about 12% of the transient value, and every fold once $g_{KL}$ exceeds a value in $[0.04041,\,0.04050]\ \mathrm{mS\,cm^{-2}}$. We certify this boundary on a grid of sodium conductances from 8 to 35 $\mathrm{mS\,cm^{-2}}$, prove that a region of the $(g_{KL},g_{Na})$ plane containing the transient neuron is entirely of class 3, and show that the transient neuron acquires a Hopf point only when $g_{Na}$ reaches a value in $[26.402,\,26.670]\ \mathrm{mS\,cm^{-2}}$. Simulations of current steps on the same plane agree with the certified map and show where subcritical oscillations extend sustained firing beyond it. This record contains the preprint (PDF) and an archive with the Python code (interval-arithmetic certification, simulations, figures), the certificates and numerical data, and the LaTeX sources.