Good questions.

Generally what people mean by "the HH model" is one of two models proposed by Hodgkin and Huxley. One was a model of the "membrane action potential" by which they meant a spike that occurred in an isopotential patch of squid axon membrane; the other was for initiation and propagation of a spike in the squid giant axon, i.e. its chief difference from the membrane action potential model was addition of the cable equation. The HH models--or model, if you prefer to lump them together--occupy a special place in computational neuroscience, because they are the only empirically-based model that have achieved iconic status. That status can't be because they explain everything that is known about the mechanism of the squid axon spike, because they don't--for one thing, they doesn't account for the effect of hyperpolarization on the time course of potassium conductance activation (the Cole-Moore effect, see

@Article{Moore2015,

author = {John W. Moore},

title = {Enhancing the Hodgkin-Huxley equations: simulations based on the first publication in the Biophysical Journal},

journal = {Biophysical Journal},

year = {2015},

volume = {109},

number = {7},

pages = {1317 - 1320},

pmid = {26445431},

}

). Maybe it's because the HH models' authors received the Nobel prize so long ago that nobody has the energy to argue about them any more.

I am assuming that there presently exists a model for spontaneous oscillations in neurons (spontaneous in a manner similar to the clock-like activity of the hearts SA node albeit subthreshold) that is incorporated into the usual governing equations which already includes terms for the HH model, passive cell properties and cable properties.

Now we get to a zone of dissent. There are lots of mechanistic models of excitable cells, some of which include cable properties, but most neuroscientists would not say that any of them is an HH model unless it reuses the originally proposed HH model's equations (updated to the convention that the extracellular potential is 0). Many of these models use the same ODE formalism as the HH equations, or a mathematically-equivalent set of state transitions with voltage-dependent rate constants. But to neuroscientists they're not "HH models." And there is no general agreement that any of them is "the" model of any particular cell. Finally, many models have an underdamped response to subthreshold perturbations--the HH model included. Drive such a model with occasional miniature psps, (or channel noise) and you have a cell that appears to generate spontaneous subthreshold oscillations. There may be one or more published models that specifically address spontaneous subthreshold oscillations, but I don't recall seeing any.

if said model for spontaneous oscillations is dependent upon nonlinear terms (idon't know that it is or isn't) then a linearization might preclude spontaneous behavior.

In the real world there is no linear macroscopic oscillator that produces stable spontaneous oscillations. Linearization, however, is a useful tool for investigating the stability of a nonlinear system's equilibrium points.