Saltatory Conduction: single AP or not?

[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)
$$c^2 \frac{\partial ^2f}{\partial x^2}=\frac{\partial ^2f}{\partial t^2}$$ (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.