Saltatory Conduction: single AP or not?

atyy

The HH paper discussing AP propagation in an unmyelinated axon doesn't seem to be free online, unlike the others. I learnt 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 modelled 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.

DaleSpam

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/1...gdbfrom=pubmed


The 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

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

Granpa,
It is.

Attached Thumbnails

   

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. doesnt make much sense to me.

somasimple

Not at all.
Continuity 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

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

atyy

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).

In the data or in someone's model?

somasimple

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

I agree.
Edit:
http://www.pubmedcentral.nih.gov/art...?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?

Attached Thumbnails

   

somasimple

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

Why an inductance?

granpa

because inductance is the electrical equivalent of mass.