Saltatory Conduction: single AP or not?

[atyy]

Why the capacity is omitted since it is 40 time greater than at node?

Capacity is not omitted - they are discussing resistor and capacitor in parallel as a model for the internode.

 

[Dale]

HH model for unmyelinated axon:
-Describes AP at a point and its propagation.
-Considers active and passive circuit components uniformly distributed along the axon.
-Is a wave equation with a well-defined propagation velocity which matches experiement.
-Reduces in a certain limit to the linear passive cable equation which does not have a well defined velocity.

Hi atyy, do you by any chance have a link for this? None of the variants of the HH models that I have seen have any spatial terms, but it has been years since I studied this stuff. I would be very interested to see a single model that includes the voltage-gated channels and spatial terms.

 

[atyy]

Hi atyy, do you by any chance have a link for this? None of the variants of the HH models that I have seen have any spatial terms, but it has been years since I studied this stuff. I would be very interested to see a single model that includes the voltage-gated channels and spatial terms.

The HH paper discussing AP propagation in an unmyelinated axon doesn't seem to be free online, unlike the others. I learned about this from somasimple, haven't read it, but looks sensible on a quick scan: http://butler.cc.tut.fi/~malmivuo/bem/bembook/ .

I'll summarise the argument presented by Koch (Biophysics of Computation, OUP 1999) [V_xx is second partial of V wrt x, I haven't bothered about correct signs]:

1. i_m~V_xx

2. i_m~V_t+F(V), where F(V) represents the HH model for the AP at a point, including terms that look like dp/dt~f(p)

3. So V_t~V_xx+F(V)

"no general analytical solution is known ... Hodgkin and Huxley only had access to a very primitive hand calculator ... Instead they considered a particular solution to these equations ... postulated the existence of a wave solution ... V_xx~V_tt ... [more steps until an ordinary DE is also obtained] ... Hodgkin and Huxley iteratively solved this equation until they found a value of u leading to a stable propagating wave solution. In a truly remarkable test of the power of their model, they estimated 18.8 m/s at (18.3^oC) ... a value within 10% of the experimental value of 21.2 m/s ..."

" ... more than 10 years later that Cooley, Dodge and Cohen solved the full partial differential equation numerically ..."

It boggles my mind they did that with a "primitive hand calculator"?!

 

[atyy]

I think I may finally understand somasimple's "discontinuity" objection - it makes sense to me if "discontinuous" means "non-analytic".

Linear passive cable equation: V_xx~V_t, which is a linear parabolic partial differential equation.

HH equation: V_xx~V_t+F(V,p,dp/dt), where p are the HH point conductance parameters. The considerations in its derivation are the same as in deriving the cable equation, but it is not parabolic. This is usually called the HH equation only if p is not a function of x, but I will refer to it as the HH equation even for p(x).

For an unmyelinated axon, some parameter like the density of sodium channels p_n is spatially constant.

For a myelinated axon, the spatial distribution of sodium channels can presumably be modeled by p_n(x), which if analytic will approach zero only asymptotically, and the equation will not be exactly parabolic for any axon segment, and we cannot do an exact separation into "active" and "passive" compartments (HS discuss this, but in different language, they say the internode may be active, but not active enough for current to lead voltage).

If p_n(x) is smooth but not analytic, then it can be exactly zero over some internode segment, and the equation will reduce exactly to the cable equation. In this case we can do an exact separation into "active" and "passive" compartments.

Presumably since the full analytical solution is not known, whether one chooses the parameter to be smooth and analytic, or smooth but not analytic, will be a matter of numerical convenience, since the difference will probably not be experimentally detectable.

 

[Dale]

3. So V_t~V_xx+F(V)

OK, that makes sense. The V_t~V_xx part is the cable equation and the F(V) part is what I knew as the HH model. I just hadn't seen them put together like that, but it is pretty obvious when someone else points it out for you

 

[granpa]

just thought this was interesting:

http://www.ncbi.nlm.nih.gov/pubmed/314337

Using a special albumin technique, nodes of Ranvier have been examined within frog skeletal muscle, sciatic nerve and rat and frog cerebrum. Initial segments have been examined in cerebrum of frog and rat. Mictotubules usually run longitudinally through these regions, but within the bare area of the intramuscular node of Ranvier, annular or helical bundles of microtubules run in a marginal band at right angles to the more centrally placed longitudinal microtubules. These nodal bare areas show a pronounced convexity and it is suggested that the annular microtubules serve to maintain this convexity during muscle contraction.

http://www.ncbi.nlm.nih.gov/pubmed/...nkpos=2&log$=relatedarticles&logdbfrom=pubmedThe relationship between the degree of nodal narrowing and the changes in the structure of the axonal cytoskeleton was studied in 53 fibres of mouse sciatic nerve. Nodal narrowing increased with increasing fibre calibre to reach about 20% of the internodal area in the thicker fibres. The narrowing corresponded quantitatively to a decreased number of nodal neurofilaments. Nodal microtubule numbers varied greatly, and a majority of fibres had considerably (approximately 55%) more microtubules in their nodal profile than in the internode

 

[somasimple]

Capacity is not omitted - they are discussing resistor and capacitor in parallel as a model for the internode.

http://butler.cc.tut.fi/~malmivuo/bem/bembook/21/21.htm
Sorry, but I do not see it.

Granpa,
It is.

 

[granpa]

you do realize that the attached image in your last post, the one from the website here:
http://butler.cc.tut.fi/~malmivuo/bem/bembook/21/21.htm

(in the case of dc, and neglecting the fh at the node, and using the water analogy for current) is just a description of a long empty and leaky pipe. you turn on the water and it takes a while before any comes out the other end.

it says the internode is just modeled as a resistor. the capacitors are for the nodes. doesn't make much sense to me.

 

[somasimple]

I think I may finally understand somasimple's "discontinuity" objection - it makes sense to me if "discontinuous" means "non-analytic".

Not at all.
http://www.sosmath.com/calculus/limcon/limcon05/limcon05.html" is a prerequisite for an electrical signal in a wire/cable.
There is discontinuities at internode/node junctions when the signal leaves the internode entering in the node and when it leaves the node entering to the next internode.

 

[somasimple]

it says the internode is just modeled as a resistor.

That is the problem I'm pointing out.
Normally the nodes are connected to external milieu.

 

[atyy]

http://butler.cc.tut.fi/~malmivuo/bem/bembook/21/21.htm
Sorry, but I do not see it.

The above doesn't even model most nodes as active. HS discuss resistance and capacitance of the internode, and it is very important for them to come to the conclusion that the internode is passive, or at least much less active than the nodes (p328 bottom paragraph through p329).

There is discontinuities at internode/node junctions when the signal leaves the internode entering in the node and when it leaves the node entering to the next internode.

In the data or in someone's model?

 

[somasimple]

The above doesn't even model most nodes as active. HS discuss resistance and capacitance of the internode, and it is very important for them to come to the conclusion that the internode is passive, or at least much less active than the nodes (p328 bottom paragraph through p329).

Adding a capacitor doesn't change the passivity but it is missing (I added the table 2)

In the data or in someone's model?

Both.
Edit: In the model a node is connected to 2 internodes and must be at the same potential.
In data: the end of an internode is not at the same potential than the beginning of the next internode.

 

[somasimple]

HS discuss resistance and capacitance of the internode, and it is very important for them to come to the conclusion that the internode is passive, or at least much less active than the nodes (p328 bottom paragraph through p329).

I agree.
Edit:
http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1473353
see figure 1 for a more appropriate electric model.

 

[granpa]

" The conduction velocity also is relatively insensitive to the internodal length"

i like that.

 

[somasimple]

Here is the problem:
And, active node or not, it does not change the passive internodes, does it?

 

[somasimple]

" The conduction velocity also is relatively insensitive to the internodal length"

i like that.

Me too. It is normal in a body that moves and thus stretches or shrinks nerves: The message must be delivered (safety factor) and insensitivity to internal motion.

 

[granpa]

if the impulse does indeed move at or just below the speed of sound in water or is even just limited by the speed of sound in water then that would mean that significant amounts of water are being moved. the mass of the water would add an inductance to the equivalent circuit. or so it seems to me.

 

[somasimple]

if the impulse does indeed move at or just below the speed of sound in water or is even just limited by the speed of sound in water then that would mean that significant amounts of water are being moved. the mass of the water would add an inductance to the equivalent circuit. or so it seems to me.

Why an inductance?

 

[granpa]

because inductance is the electrical equivalent of mass.

 

[granpa]

just think of a sound wave as passing through a series of masses connectedby springs. the mass effect becomes obvious.

 

[atyy]

Edit: In the model a node is connected to 2 internodes and must be at the same potential.
In data: the end of an internode is not at the same potential than the beginning of the next internode.

Edit:
http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1473353
see figure 1 for a more appropriate electric model.

The data doesn't show a discontinuity, just a quick change in voltage over distance (HS Fig. 11). But I agree that if the node and internode are each modeled as a single compartment, it looks like there will be some discontinuity. I suppose more compartments can be added for both the node and internode, or it could also be taken care of by a partial differential equation in which parameters vary continuously over space.

Moore 1978 does look more appropriate. Some papers that cite their work are:

Hartline DK, Colman DR. Rapid conduction and the evolution of giant axons and myelinated fibers. Curr Biol. 2007 Jan 9;17(1):R29-35.
http://www.pbrc.hawaii.edu/~danh/PDFs/Hartline&Colman_2007.pdf

Richardson AG, McIntyre CC, Grill WM.
Modelling the effects of electric fields on nerve fibres: influence of the myelin sheath. Med Biol Eng Comput. 2000 Jul;38(4):438-46.

McIntyre CC, Richardson AG, Grill WM.
Modeling the excitability of mammalian nerve fibers: influence of afterpotentials on the recovery cycle. J Neurophysiol. 2002 Feb;87(2):995-1006.
http://jn.physiology.org/cgi/content/full/87/2/995

Hartline's site: http://www.pbrc.hawaii.edu/~danh/
Grill's site: http://fds.duke.edu/db/pratt/BME/faculty/warren.grill/publications

 

[somasimple]

because inductance is the electrical equivalent of mass.

http://en.wikipedia.org/wiki/Inductance
A neutral thing (water) seems unable to create electric field or voltage by itself.

 

[granpa]

this all reminds me so much of the equivalent circuits of microscopic straight wires in megahertz microprocessor design. I've been trying to find a diagram but I don't even know what to google.

and if its being driven close to its limit (the speed of sound in water) then that is also similar to microprocessor wires being driven close to the speed of light.

both are semi-dc.

 

[granpa]

http://en.wikipedia.org/wiki/Inductance
A neutral thing (water) seems unable to create electric field or voltage by itself.

it has mass and inductance is the electrical equivalent of mass. it is an 'equivalent circuit'.

 

[somasimple]

I suppose more compartments can be added for both the node and internode, or it could also be taken care of by a partial differential equation in which parameters vary continuously over space.

No, because adding a compartment does not change anything: Discontinuity will be... propagated,
And No because a model may be tortured until it fits your though but it is better when it sticks facts.
ps: I'll take a closer look to papers.

Granpa: Water may be a perfect silent actor.
Edit: I received "Biophysics of computation" By C Koch (it will help.)

 

[granpa]

http://www.edn.com/article/CA56702.html

http://www.ece.uci.edu/docs/hspice/hspice_2001_2-2874.jpg

http://www.ece.uci.edu/docs/hspice/hspice_2001_2-269.html

It is only during the initial surge of the voltage that a transmission line behaves as a constant impedance, with a value equal to its characteristic impedance. For this reason the characteristic impedance of a line is also called the surge impedance. The surge time during which the impedance is constant is the round trip time of flight, or twice the time delay. Reflections from the far end complicate the electrical behavior of the line after the surge time.
The instantaneous impedance measured at the front end of a transmission line is a complicated function of time. It depends on the nature of the terminations at the far end. When the line is shunted to ground with a resistor of value equal to the characteristic impedance of the line, there is no reflection back, and the front end of the line behaves as a resistive load. When the termination at the far end is open, the impedance at the front end starts out at the characteristic impedance and eventually, after multiple reflections, approaches an infinite impedance. During some periods the instantaneous impedance may be zero.

http://en.wikipedia.org/wiki/Impedance_mismatch

Impedance matching is the electronics design practice of setting the output impedance (ZS) of a signal source equal to the input impedance (ZL) of the load to which it is ultimately connected, usually in order to maximize the power transfer and minimize reflections from the load. This only applies when both are linear devices.
The concept of impedance matching was originally developed for electrical power, but can be applied to any other field where a form of energy (not just electrical) is transferred between a source and a load.

To prevent all reflections of the signal back into the source, the load (which must be totally resistive) must be matched exactly to the source impedance (which again must be totally resistive)

https://www.physicsforums.com/showpost.php?p=1873931&postcount=2

 

[somasimple]

Granpa,
I have already stated that the cable model has no inductance so it makes problem for an eventual delay.
BTW, having a delay is not a proof of an impedance existence.

 

[Dale]

Not at all.
http://www.sosmath.com/calculus/limcon/limcon05/limcon05.html" is a prerequisite for an electrical signal in a wire/cable.
There is discontinuities at internode/node junctions when the signal leaves the internode entering in the node and when it leaves the node entering to the next internode.

This is wrong, there is no continuity requirement.

Consider the wave equation in 1 spatial dimension (e.g. an electrical signal in a wire)
c^2 \frac{\partial ^2f}{\partial x^2}=\frac{\partial<br /> ^2f}{\partial t^2} (1)

This has the solution
f = H(x-ct) (2)
where H is the Heaviside unit step function

Equation (2) is discontinuous in both time and space and it remains discontinuous even in the limit as c goes to infinity. And similarly discontinuous solutions exist for the wave equation in 3 spatial dimensions.

The above is not even including discontinuities in the medium which can lead to solutions with even more complicated discontinuities. There is simply nothing about Maxwell's equations or circuit theory that requires continuity.

 

[somasimple]

https://www.physicsforums.com/showpost.php?p=786028&postcount=7
A soliton is unfortunately... continuous since it is a traveling wave.
Edit;
http://en.wikipedia.org/wiki/Soliton
I'm wrong for sure...

 

[granpa]

http://www.ece.uci.edu/docs/hspice/hspice_2001_2-2874.jpg

the interesting thing aboutthis model is that if yoi suddenly apply a dc voltage (turn on the water) then only the leading edge is affected. behind the leading edge the pipe is already full of water so the capacitors are irrelevant and the current isn't changing so the inductance (mass of the water. an inductor would be modeled as a constriction in the pipe) is irrelevant. only at the leading edge do these have any effect. the speed of the leading edge is v=1/√(L*C)

and as long as the inductance of one portion matches the inductance of the next then there is no reflection. it all becomes quite simple to visualize.

it also works if the pipe is emptying.

 

[Dale]

https://www.physicsforums.com/showpost.php?p=786028&postcount=7
A soliton is unfortunately... continuous since it is a traveling wave.
Edit;
http://en.wikipedia.org/wiki/Soliton
I'm wrong for sure...

Check the definition of continuous that you cited earlier. There are no continuity constraints on solitons. H(x-ct) is a soliton (traveling wave) which is discontinuous in both x and t.

 

[granpa]

what is the maximum frequency at which the long myelinated axons of the spinal cord can transmit ap's?

 

[somasimple]

http://www.ece.uci.edu/docs/hspice/hspice_2001_2-2874.jpg

the interesting thing aboutthis model is that if yoi suddenly apply a dc voltage (turn on the water) then only the leading edge is affected. behind the leading edge the pipe is already full of water so the capacitors are irrelevant and the current isn't changing so the inductance (mass of the water. an inductor would be modeled as a constriction in the pipe) is irrelevant. only at the leading edge do these have any effect. the speed of the leading edge is v=1/√(L*C)

That is wrong. The charge of a capacitor isn't linear and its impedance changes from 0 to infinite => currunt is changing.

DaleSpam,
Are you masochist?
There is a quite soliton solution for unmyelinated axons and the function has a derivative at any portion => Continuity.
The case is totally different for myelinated axons => A discontinuity exists in regard of x.

How do you infer on the t variable? Are you able to rewind time or stop it...?
That is the fate of a temporal function: Time that inexorably flows without...discontinuity.

 

[somasimple]

what is the maximum frequency at which the long myelinated axons of the spinal cord can transmit ap's?

In mammals the CV speed is 120~150 ms-1 but the firing rate is often < 200 HZ

 

[granpa]

so more myelin or larger axon=less capacitance=greater voltage difference in signal=less delay at the node. (which kinda somehow makes sense)

the myelin doest have much effect on the speed of the wave across a single internode.

 

[granpa]

That is wrong. The charge of a capacitor isn't linear and its impedance changes from 0 to infinite => currunt is changing.
.

not sure what you are saying. but the capacitors behind the leading edge are already full and the voltage isn't changing (its only changing at the leading edge) so the capacitors have no further effect and can be ignored.

 

[somasimple]

so more myelin or larger axon=less capacitance=greater voltage difference in signal=less delay at the node. (which kinda somehow makes sense)

the myelin doest have much effect on the speed of the wave across a single internode.

We are disagreeing on this. Computations have shown the contrary (in regard to length).

Here is another aspect of discontinuity:

 

[granpa]

your messages have a tendency to be something more than cryptic. its not clear to me what you mean by 'in regard to length'. if your just saying that myelin does speed ip internode travel then ok. it may speed it up some but still the internode speed is so great that it hardly matters. it seems to me that it is the delay at the node that pretty much determines the net speed of the ap over many nodes.

http://www.pubmedcentral.nih.gov/pagerender.fcgi?artid=1392492&pageindex=8

what do you mean x1 and x2 are undefined? that article you originally linked to actually measured internode values. that's what started this conversation.

 

[somasimple]

And here is a graph that show the number of active nodes during a single propagated AP.
It becomes obvious that the energetic cost is dependent of speed and spike duration.
It is another proof of discontinuity.

 

[somasimple]

your messages have a tendency to be something more than cryptic. its not clear to me what you mean by 'in regard to length'.

It was described earlier:
https://www.physicsforums.com/showthread.php?t=258168

what do you mean x1 and x2 are undefined? that article you originally linked to actually measured internode values. that's what started this conversation.

Onto the right of the picture:
x1 and x2 are taken on the apparent AP given by the apparent conduction velocity.
Thus you see an apparent AP at nodes (or wherever you want if you respect a constant interval) that is the result of synchrony of multiple firing nodes.
This apparent AP looks like an overlapping of time axis.

 

[granpa]

those are actual measurements of an actual ap by electrodes placed 3 to an internode along a single axon. one by the node. another in the middle of the internode. the third by the node at the opposite end. this is repeated for several consecutive nodes.

they even go against what was expected. look at the backward propagating signal in curve c

notice the phrase 'from the same records as fig 6'

 

[somasimple]

granpa,
we don't care at all.
If an AP has a duration and a speed then it has a "length".
This last value determines the # of active nodes and their values (curve fitting).
you can't mix nodes with internodes if you respect a constant interval = internode lenght.

 

[somasimple]

they even go against what was expected. look at the backward propagating signal in curve C

This is not really a backward propagation since the peak stays at its place but a shortening of the falling phase.

 

[granpa]

This is not really a backward propagation since the peak stays at its place but a shortening of the falling phase.

absolutely. exactly right. its not the peak that's moving backward. its the falling phase. a nort of anti-ap that is moving backward (from the next node at coincidentally exactly the same time that that node fires its ap forward) returning the axon to its resting state.

 

[somasimple]

absolutely. exactly right. its not the peak that's moving backward. its the falling phase. a nort of anti-ap that is moving backward (from the next node at coincidentally exactly the same time that that node fires its ap forward) returning the axon to its resting state.

Hmm, it is now, impossible to find a passive and electrical solution...

 

[granpa]

Hmm, it is now, impossible to find a passive and electrical solution...

not if the node from which from which the anti-ap propagates becomes (just before it fires) completely impassable to all ap's and anti-ap's. then its very simple

 

[atyy]

so more myelin or larger axon=less capacitance=greater voltage difference in signal=less delay at the node. (which kinda somehow makes sense)

The part I don't understand about this is membrane capacitance seems to always appear with membrane resistance in the equations: RmCm.

Myelin decreases capacitance, but increases resistance, so it does nothing to RmCm.

It would make a lot more sense to me if the standard explanation emphasized that myelin increases Rm to increase the length constant, in which case decreased Cm is needed to keep RmCm the same.

 

[granpa]

I don't know anything about the standard explanation. and very little about time and length constants. (just a little about rlc circuits)

all i know is that myelin (and increased axon diameter) increases net speed of conduction which is apparently (http://www.pubmedcentral.nih.gov/pagerender.fcgi?artid=1392492&pageindex=8) mainly controlled by the delay at the node. I'm assuming passive conduction through the internode and virtually no leakage (cells can't be that leaky. they wouldn't survive). the only reason for the myelin then would be to increase capacitance and that means that fewer ions moving across the membrane produces a larger voltage difference.

 

[somasimple]

Hmm, it is now, impossible to find a passive and electrical solution...

Two currents are added in this configuration and it can't give such a solution, Granpa.
I insist.
see:
www.somasimple.com/flash_anims/node_01.swf

Myelin decreases capacitance, but increases resistance, so it does nothing to RmCm.

A Nothing that increases speed is THE explanation.

 

[granpa]

Two currents are added in this configuration and it can't give such a solution, Granpa.
I insist.

I'm afraid I have no idea what you mean by that.