How the speed of the potassium current sets the onset of firing: certified codimension-two bifurcations in the Morris–Lecar neuron

The kinetics of the potassium current, which in conductance-based models is scaled by a temperature-like rate factor $\phi$, can change how a neuron starts to fire without moving its resting states. We study this effect in the Morris–Lecar model in the plane of applied current $I$ and rate factor $\phi$, for the two standard parameter sets. Because $\phi$ enters only the trace of the Jacobian, the whole Hopf locus is an explicit curve $\phi=h(V)$, $I=J(V)$ parametrised by the resting voltage, and every question about it reduces to one-dimensional problems that we settle with interval arithmetic. In the class II set we prove that Hopf bifurcations exist exactly for $\phi<\phi^*$, $\phi^*\in[0.3869298972,\,0.3869298973]$; that there are exactly two generalised Hopf (Bautin) points, at $\phi\in[0.3063450317,\,0.3063450318]$ and $\phi\in[0.2538563149,\,0.2538563150]$; and that the onset of firing switches from hard (subcritical) to soft (supercritical) at the first of them. In the class I set we prove two nondegenerate Bogdanov–Takens points, at $\phi\in[0.01181036804,\,0.01181036805]$ on the firing threshold and at $\phi\in[0.39096195618,\,0.39096195619]$, and one Bautin point. Simulations locate the folds of limit cycles and the two saddle-node-loop points that bound the class I window, $0.0162\lesssim\phi\lesssim0.1969$. Speeding up the potassium current therefore turns a bistable integrator-like onset into a gradual one, and class I excitability exists only in a finite window of potassium speeds. Contents of this record: the preprint (PDF, template of Peer Community In / Peer Community Journal) and a ZIP archive with the interval-arithmetic code, the certificates in JSON format, the simulation code, the figures, the scripts that reproduce every number and figure, and the LaTeX source of the article.