Action Potential: Passive Spread Current

[somasimple]

Hi All,

The Action Potential propagation involves a passive event called: Passive Spread or Electrotonic conduction.

Here is some references:
http://www.ncbi.nlm.nih.gov/books/bv.fcgi?rid=mcb.figgrp.6138
http://www.ncbi.nlm.nih.gov/books/bv.fcgi?rid=.0zyfzkapx787Lxyk2TNcPpbCOnVmwIAZMxK6R2
http://www.ncbi.nlm.nih.gov/books/bv.fcgi?rid=mcb.figgrp.6145
http://butler.cc.tut.fi/~malmivuo/bem/bembook/03/03.htm

It is defined a Constant Length that enables this passive event.
What are the normalized values of $$\lambda$$ for unmeylinated fibers that have 1 m/s and 20 m/s (such as the giant squid axon)?

 

[somasimple]

I brought these examples because they are the rare that have temporal values.
http://hawk.med.uottawa.ca/public/reprints/PropagationAP.pdf
http://www.ncbi.nlm.nih.gov/books/bv.fcgi?rid=mcb.figgrp.6145
This one says that the AP (of 2 ms duration) travels/runs at 1 mm/ms. So it must be a thin fiber since it runs at 1m/s.
The space constant of such a fiber is around 0.1/0.2 mm but the "length" of the AP is 2mm. Centered on the peak value of the AP, the space constant is too short to activate any next patch of membrane.
BTW, this example is perhaps not very good. The famous squid axon has a conduction velocity around 20/25 m/s. If we take an AP of the same duration, we get a length that is 40 mm. 4 cm. All experiments recorded a space constant around 5/6 mm => 0.5 cm. It is just 10 fold less the necessary value.

And it is a fact. The facts exclude a passive spread at such a distance. The positive loop involved in the HH model may be discussed. The cable proterties, too.

 

[somasimple]

A second disproof comes from myelinated fibers.
The propagation is an extension of the previous model. Unfortunately there is no room for any travel at the node of Ranvier. It is just a few µ large and can't stand a complete AP "length".
If the AP doesn't move in myelinated fibers, it is not the same phenomenon as in unmyelinated fibers.

 

[somasimple]

An AP (Action Potential) is traveling all along the axon for an unmyelinated fiber.
Its speed (conduction velocity) is variable and depends of the diameter of the fiber.
A Squid axon has a common speed around 20ms-1.
The duration of an AP is variable ans it is around 2/4 ms for the giant squid axon.
A phenomenon, which has a duration and travels at a known speed, occupies a length that is the result of duration*speed.
In our example we have: 40 mm (0.04 m).

If we record the AP at location Z, we get have at time t1, the value A, t2 => B and so on.
But at time t0, all values exist on the axon but located at x1, x2...
This means its exist sequences ABC, DEF or GHJ.
These sequences may be explained by the positive loop of the HH model. The HH models fits only the second one. And all exist at the same time.

The length of the phenomenon excludes also the local current process. the space constant is too short to enable any next patch.

 

[somasimple]

If we take, now, a common myelinated fiber (a cat one: values come from Bullock).
A fiber that has a 75ms-1 average speed. Its AP duration is measured: 1 ms.

The HH model expects the second AP (Node 2) is initiated when the AP, recorded at Node 1, reaches the A value.
This means a new AP is instantiated every 0.3 ms.

Since we know that an internode length is comprised between 1 and 2 mm, there is 500/1000 per meter.

In the worst case, we get:
1000*0.3E-3 => 0.3sm-1 => 3.3 ms-1
That is far from the expected 75ms-1

In the best case, we get:
500*0.3E-3 => 0.15sm-1 => 6.67 ms-1
That is far from the expected 75ms-1

 

[somasimple]

It is easy to find a solution that fits the measured values (facts).
If speed is 75ms-1 => AP travels at 75mm/ms.
If internode length is 1.5 mm, then a new AP appears at Node 51, just after 1ms.
Thus, it is mandatory that 50 AP exist at the same time.
Since the internode length is presumed constant then each AP is separated by 20 µs and the initiating value A must be much below the expected value.

 

[Dale]

Hi somasimple,

If you have a patch of membrane that is in the "resting state" what event is required to initiate a rising potential cascade at that patch?

Can you use the cable conduction theory rigorously to demonstrate that the required event will not occur at a resting patch next to a rising patch? (I am willing to accept numerical solutions to the partial differential equation)

 

[somasimple]

Can you use the cable conduction theory rigorously to demonstrate that the required event will not occur at a resting patch next to a rising patch? (I am willing to accept numerical solutions to the partial differential equation)

I can't because my little computation is based upon registered facts. These facts are only facts and I can't contest them.
In my first example: If a AP "takes" 40 mm, what is a good length for a membrane patch? 1.3 cm? We must admit that a good patch size is shorter because a patch of 1.3 exhibits too many values.
Since a complete AP exists and is 40 mm long, how do you solve the sequences ABC, then CDE then DEF, then FGH and finally GHJ with a single positive loop?

I'll bring another example for the myelinated fiber based on the http://www.ncbi.nlm.nih.gov/pubmed/...el.Pubmed_DefaultReportPanel.Pubmed_RVDocSum"r.

 

[Cincinnatus]

Somasimple, you CAN perform the calculation that DaleSpam suggested. If you did, then you would see that cable theory predicts that action potentials do propagate down an axon. Of course, you wouldn't accept this anyway since we all know that you don't accept cable theory.

For anyone following along with these silly discussions, the best reference that I know of on cable theory in neuroscience is Cristof Koch's book: Biophysics of Computation.

 

[somasimple]

The velocity of conduction in the isolated part of the fibre is the reciprocal of the mean slope of either of the two lowest graphs in Fig. 7. It is found to be
23.2 m./sec. The detailed analysis described later was carried out on records obtained from this fibre and from three others which gave velocities of 22.2, 24.3 and 23.1 m./sec. These values are normal for frog fibres of 12-15 ,u. diameter, at temperatures of 18-20° C.

We will round the 23.2 to 23 ms-1.
From figure 6, the AP duration is around 0.5 ms.
The apparent length/duration of the AP is thus 11.5 mm.
And because the AP began at time 0, at time +1ms, an AP must begin at a distance of 23 mm.

If the internode is 2 mm => 11 nodes
1.5 => 15
1 mm => 23

Then divide an AP that is 0.5 ms by our 11 equidistant intervals.
Make a graph and solve the differential equation by readind the values...

 

[somasimple]

Cincinnatus
Read the pages of Prof Bezanilla =>
http://nerve.bsd.uchicago.edu/med98d.htm#Propagation
And explain how a rigorous electric current is able to flow in a wire or a resistance with opposite direction?
http://www.somasimple.com/images/cells_axon/propagation800.gif

 

[somasimple]

Somasimple, you CAN perform the calculation that DaleSpam suggested.

I did with this software =>
http://www.spectrum-soft.com/index.shtm

 

[somasimple]

If you have a patch of membrane that is in the "resting state" what event is required to initiate a rising potential cascade at that patch?

I made a mistake because I didn't take account of the first spike where there is a latency. You're right. This delay enables a propagation with higher values. BTW, it doesn't change anything about space constant.

 

[somasimple]

I did with this software =>
http://www.spectrum-soft.com/index.shtm

The cable model is a simple cascade of low pass filters.
It is normal that each cell fills itself with some virtual delay (at rising phase).
But if the current is shut the cells are all emptied at the same time.
It means that the duration of the initial pulse is reduced with each added cell.
This does not reflect the real situation since AP duration remains unchanged.
And, and there is no propagated delay!

 

[Dale]

I can't because my little computation is based upon registered facts. These facts are only facts and I can't contest them.

I agree that experimental facts can't be contested, but when you are trying to disprove a theory you must also demonstrate that the theory fails to predict a given experimental fact.

Here, you have compared the length constant with the AP length and determined that the length constant is shorter. So what? As far as I know the cable theory does not predict that the length constant will be the same size as the AP length. Your cited observation therefore does not appear to be in contradiction of the theory.

By the way, the required event in the question I posed above is "a depolarization greater than the threshold". So, in order to demonstrate that the cable model fails to predict AP propagation, you need to rigorously use the full cable theory to demonstrate that the next patch never gets a supra-threshold depolarization.

 

[somasimple]

Here, you have compared the length constant with the AP length and determined that the length constant is shorter. So what? As far as I know the cable theory does not predict that the length constant will be the same size as the AP length. Your cited observation therefore does not appear to be in contradiction of the theory.

http://hawk.med.uottawa.ca/public/reprints/PropagationAP.pdf

Any time an active response (last set of notes) is
produced, in one place, a passive electrotonic response spreads quickly
from that area. If the electrotonic response causes sufficient depolarization around
voltage-gated channels, they will open and produce an active response. Since this
opening of the gated channel involves an allosteric change and it also takes time.
But notice that the result is that the action potential now generated in a
place adjacent to where it was previously. IT IS PROPAGATING OR
"MOVING"

So what?
A passive spread that is under theses conditions is unable to promote a propagation.
The given graphs come from the cable theory. The electric schema fails to produce any delay.
So what?

 

[Dale]

A passive spread that is under theses conditions is unable to promote a propagation.

That is precisely what you have not yet demonstrated rigorously using the cable theory.

 

[atyy]

BTW, this example is perhaps not very good. The famous squid axon has a conduction velocity around 20/25 m/s. If we take an AP of the same duration, we get a length that is 40 mm. 4 cm. All experiments recorded a space constant around 5/6 mm => 0.5 cm. It is just 10 fold less the necessary value.

Maybe the space constant measurement is wrong?

Since we know that an internode length is comprised between 1 and 2 mm

And anyway, the internode distance is less than the space constant, so it's not obvious there's a problem.

The point of the node of Ranvier is that it increases the "space constant" of the axon, because it is an "active" process.

All that is needed is that the AP can travel passively from one node to the next, and then it is regenerated "actively" at each node. The problem with passive travel is not so much the speed, but rather that the amplitude of the action potential dies down during passive travel.

 

[atyy]

If we record the AP at location Z, we get have at time t1, the value A, t2 => B and so on.
But at time t0, all values exist on the axon but located at x1, x2...
This means its exist sequences ABC, DEF or GHJ.
These sequences may be explained by the positive loop of the HH model. The HH models fits only the second one. And all exist at the same time.

A single action potential can travel down the axon. It does not exist at multiple points on the axon at the same time. The "active" part of the HH model describes the generation of an action potential at a single point. The "passive" part describes how the action potential travels down the axon. The main problem of passive travel is that the action potential dies down (after one space constant, the action potential will be very small). So nodes of Ranvier have to be spaced much shorter than the space constant in order to regenerate the amplitude of the action potential.

 

[atyy]

It is easy to find a solution that fits the measured values (facts).
If speed is 75ms-1 => AP travels at 75mm/ms.
If internode length is 1.5 mm, then a new AP appears at Node 51, just after 1ms.
Thus, it is mandatory that 50 AP exist at the same time.
Since the internode length is presumed constant then each AP is separated by 20 µs and the initiating value A must be much below the expected value.

OK, I see your point. But now it seems that the AP is traveling too fast, not too slow?

So first it seemed the AP was too slow. Now its too fast. So maybe that indicates there's no problem - it seems that some adjustment of parameters would produce a value between the slowest and fastest estimates, and our calculation is just too simple?

 

[atyy]

For anyone following along with these silly discussions, the best reference that I know of on cable theory in neuroscience is Cristof Koch's book: Biophysics of Computation.

And, and there is no propagated delay!

Christof Koch: The linear cable equation does not admit any wave solution ... the voltage will, in principle, respond infinitely fast to a change in current input an arbitrary distance away ... in general one cannot define a velocity ...Yet all is not lost ... Agmon-Snir and Segev (1993) ...

OK, I'm trying to understand what he says about Agmon-Snir's and Segev's work ...

BTW, the reason for being careful about using values from different experiments to calculate precisely, especially in this field, comes from Wilfred Rall who was one of the first people to study the application of electrical theory to neuronal experiments : From hindsight, we now know that those early transients were also made faster by membrane shunting at the electrode penetration site ... I did not know the magnitude of the leak until much later. (Dendrites, ed Stuart et al, OUP 2008)

Basically, the electrical properties all depend on an intact neuron. But we have to poke a hole in the neuron to make a measurement ... how do we know the size of the hole we poked?

 

[Dale]

it seems that some adjustment of parameters would produce a value between the slowest and fastest estimates, and our calculation is just too simple?

IMO, that is the real problem with HH and the cable theory model. There are just too many parameters, so it is too easy to get agreement with experiment by judicious "tweaking" of parameters.

If somasimple really wants to do some good theoretical work in this field he could develop a physically-motivated model that is as accurate as HH and cable theory and uses fewer parameters. He is really barking up the wrong tree trying to say that it doesn't agree with experiment.

 

[somasimple]

He is really barking up the wrong tree trying to say that it doesn't agree with experiment.

I do not. =>

I agree that experimental facts can't be contested, but when you are trying to disprove a theory you must also demonstrate that the theory fails to predict a given experimental fact.

I do not agree with theory but accept facts.

A single action potential can travel down the axon. It does not exist at multiple points on the axon at the same time.

That's true for an unmeylinated axon.
It becomes false for a myelinated one because it is traveling too. You mix nodes that must conduct slowly with fast patcthes that enhance the speed. Since this is a yet a traveling wave you can't have such a situation. The only way to preserve observed facts is to consider that there is multiple APs.
Prediction: the number of APs is proportional to speed.

 

[atyy]

That's true for an unmeylinated axon. It becomes false for a myelina

Yes, I saw your calculation and it's reasonable. I do suspect that the action potential is not at exactly the same point in its time course at different spatial points of the axon if you look on time scales much smaller than 1 ms. But I've never done this calculation or simulation carefully myself. Have you got a reference?

 

[somasimple]

Have you got a reference?

There are some that may be exploited in this direction.

 

[atyy]

The only way to preserve observed facts is to consider that there is multiple APs.
Prediction: the number of APs is proportional to speed.

One thing to take into account when making the prediction of the number of "events" (suitably defined) along a length of axon at a particular time is the refractory period. After a node has fired, it is unable to fire again within 1-2 ms.

 

[somasimple]

One thing to take into account when making the prediction of the number of "events" (suitably defined) along a length of axon at a particular time is the refractory period. After a node has fired, it is unable to fire again within 1-2 ms.

You're right. The refractory period limits the rate of possible events (the interval they are separated) but it doesn't change the prediction. It just tells us that "AP trains" can't collide.

 

[atyy]

You're right. The refractory period limits the rate of possible events (the interval they are separated) but it doesn't change the prediction. It just tells us that "AP trains" can't collide.

Well, let's say at t=0, we initiate 1 AP at the soma. And we take your estimate that at t=1 ms it activates nodes #1 to #51. Then nodes #1 through #51 become inactivated for 1-2 ms, and the APs within that segment have no boosting for that period of time. So although there was an AP at node #1 at t=1 ms, node #51 doesn't fire in response to that because by the time node #51 is ready to fire again, the effect of AP at node #1 at t=1 ms would have died down too much to cause node #51 to fire again. So in effect, node #51 only fires in response to the AP initiated at the soma at t=0.

I wouldn't be too surprised if 1 AP at the soma caused 2-3 APs to arrive at the axon terminal, but I would be surprised if 1 AP at the soma caused 50 APs to arrive at the axon terminal.

Edit: Actually, that seems to be in accord with colliding AP trains annihilating each other. The AP at node #51 at t=1 ms should also propagate backwards, and collide with the AP from node #49 at t=1 ms, and they will annihilate each other.

 

[somasimple]

Then nodes #1 through #51 become inactivated for 1-2 ms, and the APs within that segment have no boosting for that period of time.

Each AP is delayed thus the first becomes inactivated when the 50# is just activated. This delay permits to the refractory period to be delayed as well.

 

[atyy]

Each AP is delayed thus the first becomes inactivated when the 50# is just activated. This delay permits refractory period to be delayed as well.

I didn't understand the figure - could you give more detail?

Your prediction of approx 50 "events" along a length of axon seems reasonable, but I want to know whether you would also predict 50 APs arriving at the axon terminal - that wouldn't be reasonable, at least not if we only initiated 1 spike at the soma.

 

[somasimple]

but I want to know whether you would also predict 50 APs arriving at the axon terminal

Fact: Only one arrives. Nature plays with us!
Prediction:When an AP is initiated at a next node then the previous one must be lost. There is a simple and anatomical explanation.
I'm was actually working on the drawing...

 

[somasimple]

I'll bring another schema that proves the existence of multiple APs at the same time.

 

[atyy]

Fact: Only one arrives. Nature plays with us!
Prediction:When an AP is initiated at a next node then the previous one must be lost. There is a simple and anatomical explanation.
I'm was actually working on the drawing...

Good, that's fine. I think the APs are lost mainly through the same thing that destroys colliding APs.

There are backpropagating APs, but from the soma/axon (I'm not sure which) into the dendrites, and these are hypothesized to cause the synapse to strengthen, and thus to underlie memory formation.

I'll bring another schema that proves the existence of multiple APs at the same time.

I think this is reasonable from your rough calculation - I only wonder whether the number is 1 or 10 or 50 or 100 - so I'm not debating this point.

Edit: The reason why I don't know whether the multiple APs at one time are 1 or 100 is because first of all you have used a rough calculation, and one would need to do a fuller calculation on the HH model of AP propagation to know what that predicts. Actually I'd be surprised if that wasn't known. Secondly, your calculation involved parameters from many different sorts of neurons, which is completely reasonable for a rough calculation. However, different neurons have different sets of ion channels, and the HH equations cannot be applied to all neurons - they must be modified depending on which part of the brain you are in. For example, in the cortex, some neurons seem to only fire one spike in response to a stimulus, but other neurons fire with many spikes in response to the same stimulus. So I think we are at the limits of what a rough calculation can do here.

Edit: Experimental evidence may already exist about the number of "events" (what you're calling APs, but I'm avoiding that term for technical reasons) along an axon at one point in time. Koch mentions Waxman et al, "The Axon: Structure, Function and Pathophysiology", OUP, 1995.

 

[somasimple]

I think the APs are lost mainly through the same thing that destroys colliding APs.

It's simpler: the delay is constant => The previous AP lasts before the one that exists at next node.

I think this is reasonable from your rough calculation - I only wonder whether the number is 1 or 10 or 50 or 100 - so I'm not debating this point.

Its depends of AP duration and speed.

The model is universal but derivations are possible. As you said, channels are different and functions, too...

Experimental evidence may already exist about the number of "events"

Yes, plenty! Just look closer at this one (fig 7)
http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pubmed&pubmedid=16991863

 

[atyy]

Yes, plenty! Just look closer at this one (fig 7)
http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pubmed&pubmedid=16991863

The bottom panel shows the location of one AP along the axon at different times.

The top panel seems to be amplitudes of one AP along the axon at different times.

What we need is a plot of voltage or current along the axon at one time (not different times).

Edit: Perhaps Fig 6. In the 1 ms window, there are APs (each with a slightly different phase) at different positions along the axon. This remains true even for a 0.5 ms window. So that's in accord with the time resolution you used in your calculation - nice job! It also shows that although at fixed time there are multiple APs along the axon, as time is varied, only one AP propagates.