## measured vs specific Rm

Dax42
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Joined: Mon Feb 06, 2006 12:18 pm

### measured vs specific Rm

Hi all,

I would like to see how conduction velocity changes in response to changes in membrane resistance. Not just theoretically though, but using experimentally determined values. From whole-cell voltage clamp experiments I have estimates of the conduction velocity, membrane resistance and membrane capacitance, and thus tau.

Here's how I got so far:
If conduction velocity theta = 2*lambda/tau
and lambda = sqroot(Rm/Ri)
then I could determine the only missing value, the internal resistance Ri by
Ri = Rm/(theta * tau / 2)^2

My thinking was that I could then use the same equation to see what happens to the conduction velocity with different values of Rm.
However, the membrane resistance that I measured is for the whole cell, not the specific resistance per unit area. Is there some way to get around that problem? Do I really need to make some assumptions about the surface area of the cell or use one of the models from your ModelDB for what seems to be a relatively simple problem?

Any ideas or pointers in the right direction would be most appreciated.
Thanks!
ted
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### Re: measured vs specific Rm

Dax42 wrote:I would like to see how conduction velocity changes in response to changes in membrane resistance.
In myelinated or nonmyelinated axon? Prediction for nonmyelinated axon: Rm has no effect as long as membrane isn't grossly leaky.
If conduction velocity theta = 2*lambda/tau
Interesting. Don't recall seeing that before; where's that from?
and lambda = sqroot(Rm/Ri)
the membrane resistance that I measured is for the whole cell, not the specific resistance per unit area.
You mean what you measured is the input resistance of the whole cell. Since "membrane resistance of the whole cell" has no generally accepted meaning, what did you mean by this phrase?
Is there some way to get around that problem?
You mean the problem that "input resistance of a cell" is not the same as specific membrane resistance (a term that has a well-defined meaning)? You could start with an experimentally determined estimate of membrane time constant, assume some value for specific membrane capacitance, and then calculate specific membrane resistance from that. Would be useful to review the relevant literature on estimates of specific membrane capacitance.

Of course this assumes that membrane time constant is the same in the axon as in the rest of the cell. There is experimental evidence that membrane resistance is NOT uniform over the entire cell, so the only way tau_m can be uniform is if specific membrane capacitance varies in exactly the opposite direction, by exactly the right amount, from the spatial variation of membrane resistance--but what's the chance of that? And if tau_m is not uniform over the entire cell, what exactly does an experimental estimate of tau_m mean?
Dax42
Posts: 5
Joined: Mon Feb 06, 2006 12:18 pm

### Re: measured vs specific Rm

Hi Ted,

Thanks very much for getting back to me and sorry for not clarifying enough.
In myelinated or nonmyelinated axon? Prediction for nonmyelinated axon: Rm has no effect as long as membrane isn't grossly leaky.
In nonmyelinated axons. Why do you say that it won't have any effect? Surely tau is going to change with a change in Rm and that should influence conduction velocity…? I see a decrease in R_input of around a third (my experimental measure of "membrane resistance", sorry for not being clear enough on this earlier).
If conduction velocity theta = 2*lambda/tau
Interesting. Don't recall seeing that before; where's that from?
It's from the Foundations of Cellular Neurophysiology book (Johnston and Wu), for passive membranes. lambda = sqroot(Rm/Ri) with Rm the specific membrane resistance and Ri the internal resistance. Thanks for the suggestion of assuming something for the specific membrane capacitance. In fact it seems it is often just assumed to be 1uf/cm^2, but I will try to find a value for the type of cell I am studying. Out of interest, if you don't recall seeing the above definition of theta before, how would you define it?
And if tau_m is not uniform over the entire cell, what exactly does an experimental estimate of tau_m mean?
Indeed. But I've got to start somewhere! Also, the conduction velocity I am interested in is only over a relatively short stretch of a process, so I'm assuming a more or less uniform cable. Which I then want to compare to the same stretch of cable, just with a lower Rm. Doesn't sound like it should be too difficult somehow!

As before, any help would be most appreciated.
Thanks!
ted
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### Re: measured vs specific Rm

Dax42 wrote:
Prediction for nonmyelinated axon: Rm has no effect as long as membrane isn't grossly leaky.
In nonmyelinated axons. Why do you say that it won't have any effect?
Because the rate of spike propagation depends on the rate at which the part of the axon that is depolarized above rest can drive depolarization of "downstream" membrane. The half width of a spike is only about 1 ms, much shorter than membrane time constant, so almost all of the charge delivered from the depolarized part of the axon to downstream membrane is deposited on downstream membrane capacitance, and almost none is lost through downstream membrane's resistance. This means that propagation velocity will be independent of Rm, but will depend on specific membrane capacitance, cytoplasmic resistivity, and fiber diameter, in particular reflecting the facts that the rate of depolarization of downstream membrane will be:
(1) reduced by increasing specific membrane capacitance or cytoplasmic resistivity
(2) increased by increasing fiber diameter (which reduces axial resistance faster than it increases net downstream capacitance)
I see a decrease in R_input of around a third
(Under what condition?) That's substantial, but (1) how much of a decrease of axonal membrane resistance does it imply (could be more, or less), and (2) is it enough to have an experimentally measurable effect on conduction velocity?
If conduction velocity theta = 2*lambda/tau
Interesting. Don't recall seeing that before; where's that from?
It's from the Foundations of Cellular Neurophysiology book (Johnston and Wu)
Good book. In section 6.4 on page 156 I see theta = sqrt(K*a/(2*Ri*Cm)), where K is ~ 10.47/ms based on experimental observations (presumably one of the papers cited in the back of J&W). Notice that Rm doesn't appear. Where exactly in J&W is the formula theta = 2*lambda/tau?
if you don't recall seeing the above definition of theta before, how would you define it?
Ordinarily it wouldn't occur to me to define it. I'll accept the partially empirical formula on p. 156 of J&W. How did you arrive at theta = 2*lambda/tau, and how can predictions from it be reconciled with the very different predictions that come from the formula in J&W?
Effect of Cm: inversely proportional to Cm vs. inversely proportional to the square root of Cm
Effect of Rm: inversely proportional to the square root of Rm vs. no effect
These are big differences, and it should be relatively easy to empirically determine which is closer to what actually happens.

But even before doing a "real" or "computational" experiment, it's easy to do this Gedankenexperiment:
Suppose Rm is indeed an important determinant of conduction velocity.
What should happen to conduction velocity if Rm is increased?
Less of the charge that spreads from the depolarized zone to the downstream membrane will leak out through membrane resistance.
What happens to this charge that stays in the cell?
It must accumulate on local membrane capacitance.
So downstream membrane must charge faster, not slower.
This should accelerate spike propagation, not slow it down.
But theta = 2*lambda/tau predicts that a 10 fold increase of Rm should slow conduction velocity by a factor of about 3, which doesn't seem reasonable.
Dax42
Posts: 5
Joined: Mon Feb 06, 2006 12:18 pm

### Re: measured vs specific Rm

Hi Ted,

Thanks again for taking time to respond to my questions.
The half width of a spike is only about 1 ms, much shorter than membrane time constant
Actually, the time constants that I am seeing are on the same order as my AP half widths, around 1.5 ms (measured at the soma, granted).
Where exactly in J&W is the formula theta = 2*lambda/tau?
Page 73
What should happen to conduction velocity if Rm is increased?
Less of the charge that spreads from the depolarized zone to the downstream membrane will leak out through membrane resistance.
What happens to this charge that stays in the cell?
It must accumulate on local membrane capacitance.
So downstream membrane must charge faster, not slower.
That's interesting. My thinking was that if Rm increases, tau also increases, hence the charging of the downstream membrane will be slower! I feel like I must have misunderstood something completely here. I can't really explain the two different definitions of conduction velocity given by J&W. Somehow the 2*lambda/tau formula made intuitive sense to me though, as the membrane must charge before the depolarisation can spread further and how far away it starts to charge new bits of membrane will depend on the length constant.

It would be interesting to know what the constant K in J&W page 156 entails. It seems that (from the line before) I_L, the leak current, should come into play somewhere. They probably assume this to be constant, which is not the case in my cells (hence the decrease in R_input)...
ted
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### Re: measured vs specific Rm

Dax42 wrote:
The half width of a spike is only about 1 ms, much shorter than membrane time constant
Actually, the time constants that I am seeing are on the same order as my AP half widths, around 1.5 ms (measured at the soma, granted).
Very strange. What kind of cell? Patch clamp or sharp electrode?
Where exactly in J&W is the formula theta = 2*lambda/tau?
Page 73
That explains all. Decremental conduction. Quite different from conduction velocity of a regenerated waveform. For decremental conduction, the very definition of "conduction velocity" is somewhat fishy, because the waveform changes as the signal spreads. One convention is to base latency measurements on the time at which the local waveform reaches half of the local max V (otherwise how can one compare two very different waveforms?), and given that convention, one can indeed define a "theoretical" conduction velocity.* Similar analysis has been applied to estimating diffusion constant by observing the local time course of concentration at various distances from a source, and the result is rather useful in that context. If, however, one bases the latency measurements on the time at which the local max V occurs, the result is that conduction velocity decreases with distance.

Everything I wrote in previous posts pertained to regenerative propagation of a spike along an excitable axon. Everything you wrote pertained to decremental conduction. Apples vs. oranges. Big shaggy dog story. A good example of how the slovenliness of our language makes it easier for us to have foolish thoughts (thanks, G.O.).

Which makes me wonder exactly what definition of "time constant" you were using (see above). Might it have been what is sometimes called "equalization time constant," not membrane time constant? Just how do you measure it?
It would be interesting to know what the constant K in J&W page 156 entails. It seems that (from the line before) I_L, the leak current, should come into play somewhere.
Not to spike propagation in healthy axons.
Dax42
Posts: 5
Joined: Mon Feb 06, 2006 12:18 pm

### Re: measured vs specific Rm

Hi Ted,

I am performing patch-clamp experiments and the estimate of tau comes from a depolarisation step. This is a standard protocol called seal test or sometimes membrane test.

Thanks for raising a number of interesting points. I am trying to look into action potential backpropagation, hence I wasn't assuming a regenerative conduction. However even dendritic spikes are frequently reported, so I am wondering whether there is passive conductance anywhere in the neuron. I can see that APs backpropagating along the axon will still be regenerative (at least if antidromic, i.e. evoked distally).

To be honest, if I can assume that R_m won't impact conduction velocity, that certainly makes my life easier. I won't have to try and calculate whether the changes in R_input could cause a change in conduction velocity!
ted