Membrane capacitance estimation

[dario.sannino97]

Hi everyone!

I have 8 months experience with NEURON and some question are starting to arise in my mind.
In particular, I am trying to calculate the membrane capacitance Cm of a multi-compartment conductance-based neuron model.
I calculate the membrane capacitance developing a voltage clamp protocol in the soma: from -70 mV to -80 mV. Then I calculate the charge Q integrating the area under the curve of the capacitive current. Once found the charge Q, I obtain Cm from the equation Cm*Vm= Q

Now, I am asking myself if it really the TOTAL membrane capacitance of the cell, or if it is something that should be called differently...
The value that I obtain as Cm from the protocol make me think that what I obtain is a membrane capacitance that is valid for the confined region (soma) that I am studying and its neighbouring regions.

Let's say that my specific membrane capacitance is 1 uF/cm^2 in all the sections of my model. If my neuron has great dimension (for instance a long axon), do you think that distal regions really impact on the experimental calculations (or the one that I've cited before with voltage clamp)?

These doubts arose because I believed that the total membrane capacitance could be described as:
Cm = cm*A(soma) + cm*A(dendrites) + cm*A(axon) + ...
where cm is the specific membrane capacitance of each section and A is the area of the section.

Any advice?

[ted]

Excellent questions. Good to see that you're thinking seriously about this. So have many others, for a long time. Would you be surprised to discover that the experimental literature contains more than a few relevant articles? What turns up when you do a Google or Duck Duck Go search for

experimental measurement membrane capacitance

?

After you have a chance to read a bit, let's continue the discussion.

[dario.sannino97]

Reading the paper at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2775376/, it seems that the voltage-clamp protocol underestimates the retrieved value of membrane capacitance Cm. This is valid experimentally, but I believe that the same could happen in silico.

Is there a function in NEURON that tells me the total area of my neuronal model. If I set the specific membrane capacitance constant I can calculate the total membrane capacitance as Cm = Atot * cm.

Otherwise, in the abovementioned paper, the authors state that a good method to calculate the membrane capacitance is to inject a DC current and describe the membrane potential change through a series of exponential terms. Honestly, I am still trying to understand why they propose this.

Do you have any advice?

[ted]

Is there a function in NEURON that tells me the total area of my neuronal model.

Read about area in NEURON's Python or hoc documentation at nrn.readthedocs.io. Usage in pseudocode:

Code: Select all

totalarea = 0
for each section in your model
  for each segment in this section
    totalarea += area of this segment

print totalarea

Python example that assumes you have a cell that has a section called soma:

Code: Select all

totalarea = 0
for sec in soma.wholetree():
  for seg in sec:
    totalarea += seg.area()

Be sure to read the documentation of Topology https://nrn.readthedocs.io/en/latest/py ... l#topology

If I set the specific membrane capacitance constant I can calculate the total membrane capacitance as Cm = Atot * cm.

You could, but you don't have to assume that cm is uniform over the entire cell. After all, cm is a range variable, which means that any segment foo has a value of cm that is called foo.cm. Time for another Python example:

Code: Select all

totalcm = 0
for sec in soma.wholetree():
  for seg in sec:
    totalcm += seg.area() * seg.cm

Could it be any easier?

Otherwise, in the abovementioned paper, the authors state that a good method to calculate the membrane capacitance is to inject a DC current and describe the membrane potential change through a series of exponential terms. Honestly, I am still trying to understand why they propose this.

Do you have any advice?

Yes, but first a comment about Golowasch et al. 2009. That paper is pretty good, even though this particular sentence is not quite correct:
"When a cell is nonisopotential, an equalizing process related to current flow between electrical compartments of the cell takes place"
A real cell doesn't have any electrical compartments. At the anatomical and temporal scales relevant to biological neurons, it is continuous in space and time. The "equalizing process" is a redistribution of charge throughout the cell that is driven by any spatial nonuniformity of membrane potential.

a series of exponential terms . . . I am still trying to understand why they propose this.

That's an important point. Time to read the references that they cite to support that statement: Rall 1977, Holmes et al. 1992, and Major 1993a. (Spoiler alert!) The bottom line is that, in the few situations where the cable equation for a finite structure can be solved analytically, that solution consists of the sum of an infinite series of exponential functions. This might remind you of the fact that the Fourier series representation of a finte duration waveform is a sum of an infinite series.

[dario.sannino97]

Thank you for the elucidation and for the hints Ted. I adjusted the code, so that the units are respected in my case. I obtain an area in um^2 and I have a specific membrane capacitance in uF/cm^2. Thus, I converted um^2 in cm^2 and I obtained a membrane capacitance in the units of uF, which finally can be converted in pF.

What I am understanding is that probably the voltage clamp protocol for calculating the membrane capacitance does not really gives you the total membrane capacitance, even if it is widely used in Neuroscience. What we obtain is an underestimation of the total membrane capacitance.

I think that probably it is difficult to say that a neuron is clamped at one voltage in all its surface in few ms. This might mean that you are not actually charging all your big (neural) capacitor. What do you think about this reasoning Ted?
Thank you!

[ted]

I adjusted the code, so that the units are respected in my case.

Here's a couple of simple rules. For a cell with surface area 100 um2,
total membrane current in nA is numerically equal to membrane current density in mA/cm2
and
total membrane capacitance in pF is numerically equal to cm in uF/cm2

So just divide surface area in um2 by 100 and multiply by cm in uF and you have total capacitance in pF.


WRT experimental estimates of total cell capacitance--trying to estimate total capacitance from voltage clamp measurements will work fine if you have perfect space clamp (so the clamp can charge all of the cell membrane by the same amount) AND the membrane has a long time constant (so when the clamp returns membrane potential to rest, you have plenty of time to pull that charge back out of the cell). But if space clamp is imperfect, then
1. if you displace v at the clamp's location by 1 mV from rest, the displacement at other locations will be < 1 mV, no matter how long the charging pulse lasts. That guarantees you'll get an underestimate of total membrane capacitance.
and
2. after the end of your charging pulse, if membrane time constant is fast, a lot of that charge won't be able to get back to the clamp--instead, it will leak out of the cell before it can get back to the clamp's electrode. And that means a further reduction in the experimental estimate of total membrane capacitance.

You can do the experiment yourself with a computational model of a passive cell.