Diameter of a Squid Giant Axon Model

[mi2n]

Hello everyone, when you create a new section in NEURON, the default length of the section will be 100 um, the diameter will be 500 um, and the temperature will be 6.3 degrees Celsius. I guess this is in accordance with the properties of squid giant axon. But then why do the dimensions in the "Squid axon (Hodgkin, Huxley 1952) (NEURON)": https://modeldb.science/5426?tab=2&file=hh/mosinit.hoc have diam = 10 um and L = 10 / PI ?

I see the exact dimensions in HH patch model in neurondemo. But I don't quite understand the significance of these dimensions, which seem so different from the actual scenario. Forgive me if I am asking something very trivial.

[ted]

That's not a trivial question at all, even though the answer may seem obvious in retrospect. The "HH patch model" is a model of a patch of squid axon membrane. The L and diam values make the patch have an area of 100 um2. Why 100 um2? Because for this area of membrane, if some ion X has a current density of j mA/cm2, the total current of that ion is j nA (and that's because 100 um2 is 1e-6 * 1 cm2). Likewise, given a patch of membrane with area 100 um2, if specific membrane capacitance is j uf/cm2, total capacitance of that patch is j pf. And if specific membrane conductance for some ion X is j S/cm2, the patch's total conductance for ion X is j uS.

[mi2n]

Thanks, this makes sense now.

I have one more question regarding squid axon behavior (not sure if I should post it as a separate post in the forum). When I use the default HH axon (L=100, diam = 500, celsius = 6.3) and stimulate it with a step current (IClamp, amp = 1, delay = 1000, dur = 300), I get an oscillatory voltage response similar to an underdamped LC oscillator. As the temperature increases the damping also increases (because Rm increased?) and, finally at around 35°C, the oscillations disappear. How does an inductor component appear in the cell membrane? I thought it was strictly an RC circuit. Also, how should one estimate the time constant of a cell membrane when it is behaving like an underdamped LC oscillator?

[ted]

Great questions. They should be part of the qualifying exams for anyone studying membrane biophysics or the dynamics of biological neurons. How many would pass such an exam? That's a different question.

Phenomenological inductance in excitable membranes is a rather old observation. You'll want to read this paper by K.S. Cole, which is freely available
Cole KS. RECTIFICATION AND INDUCTANCE IN THE SQUID GIANT AXON. J Gen Physiol. 1941 Sep 20;25(1):29-51. doi: 10.1085/jgp.25.1.29. PMID: 19873257; PMCID: PMC2142026.
https://rupress.org/jgp/article-pdf/25/ ... 238/29.pdf

As for the origin of this phenomenological inductance, here is another paper that is also freely available
Mauro A, Conti F, Dodge F, Schor R. Subthreshold behavior and phenomenological impedance of the squid giant axon. J Gen Physiol. 1970 Apr;55(4):497-523. doi: 10.1085/jgp.55.4.497. PMID: 5435782; PMCID: PMC2203007.
https://www.ncbi.nlm.nih.gov/pmc/articl ... df/497.pdf

Welcome to the domain of nonlinear dynamical analysis. If you have the time, interest, and opportunity, you might find a formal course in this topic to be useful. One of my favorite applied math texts is
Andronov A.A., Vitt A.A., and Khaikin S.E.. Theory of Oscillators. 2nd edition.
which was originally published in russian in 1959, and subsequently became available in english translation from various sources. If you want, you can download a pdf of the 1959 russian edition free from
http://booksshare.net/books/physics/and ... iy1959.pdf

how should one estimate the time constant of a cell membrane when it is behaving like an underdamped LC oscillator?

That's a real problem, isn't it. If it is very underdamped, time constant isn't even particularly useful--instead it is better to think in terms of resonant frequency and Q or damping ratio. An experimentalist would probably suggest eliminating the phenomenological inductance by using intracellular tetraethylammonium (TEA) to block potassium channels. There's a long and interesting literature on experimental estimation of membrane biophysical properties.

[mi2n]

Woww! This is going into deep waters. But yeah, I got the idea of why the oscillations happen. Thank you once again for your quick and informative reply as always.

So, the perceived inductance is not real, the oscillation is caused by the involvement of active channels, huh! I tried the experimentalist's TEA approach in NEURON (axon.gkbar_hh = 0.0), but some overshoot remained. It vanished when I blocked the sodium channel also (axon.gnabar_hh = 0.0).

PS: Thanks for the references. I hope I will be able to give them the time they deserve.

[ted]

Here's a strategy for empirical estimation of the "phenomenological membrane time constant" of an underdamped neuron. It is an experimentalist's equivalent of "linearizing" a set of equations in the vicinity of an operating point. The idea is to analyze the cell's response to small perturbations that keep the cell within that vicinity.

1. With the cell at rest, inject a brief current pulse and see what happens to membrane potential (this is a rough estimate of what is called the "impulse response"). "Brief" means much shorter than the fastest component of the phenomena that you are studying. In this case, you want to estimate the membrane time constant, which will be on the order of several ms, and you need to separate that from the damped oscillation of membrane potential, which has a period that is also on the order of several ms. A 0.1 ms current pulse would probably be sufficiently short; its amplitude should be large enough to produce a response that is about 1 mV peak to peak. Do this once with a depolarizing current, and once with a hyperpolarizing current. Ideally the detailed time courses of these two responses should be identical but opposite in sign; check this by adding them together and seeing if the result is a flat line.

Be sure to wait at least 10 ms before applying the current pulse because the HH model does not actually rest at -65 mV, and there is a small but noticeable spontaneous fluctuation of membrane potential during the first few ms of a simulation. The response should be recorded for 10 or more ms because the membrane time constant and the period of the damped oscillation are on the the order of 10s of ms.

2. Assume that the cell's response can be regarded as the superposition of two components that start at the beginning of the current pulse: a damped sinusoid of the form
Asin*sin(w*T)*exp(T/tausin)
where w = 2*PI*f,
and a decaying exponential of the form
Aexp*e((T)/tauexp)
where T = 0 at the start of the current pulse, and Asin, w, tausin, Aexp, and tauexp are to be determined from the recording. Use a standard function fitting algorithm to estimate their values.