How is membrane capacitance related to the increased speed of saltatory conduction?

Question

Here is the original question which inspired my question. As explained by the answers there, the reason saltatory conduction in myelinated neurons is faster than non-myelinated conduction is because the capacitance of the membrane is lowered by reducing the number of channels (Channel density) or equivalently, increasing the spacing between channels. I also did a preliminary study of membrane electrodynamic modelling here and here.

From what I gathered from the linked question, the decreased capacitance overcompensates the effect of absence of channels reinforcing the sodium current, and on the whole, increases the speed of conduction by allowing the depolarizing potential to travel to the adjacent node faster than in case of an unmyelinated fibre. In view of this, I have a question:-

Why does lower capacitance increase "the effectiveness of nearby nodes" or allow the depolarizing voltage to "travel not by ion diffusion, but as an electric field"? I am comfortable with capacitors and related physics, but why would lower capacitance allow propagation of the changing voltage as an electric field, is still unclear to me? (The links I have placed also help quantify the problem mathematically. This allows us to say that $\lambda$, or the rate of spatial decay of potential, increases, decreasing the effective length of the neuron. Why should its decrease help faster conduction?)

And another question is, how would reduced density of sodium voltage gated channels cause a decreased capacitance?

Answer 1

Very nice question! I'll go through your three questions sequentially.

Q1: Why does lower capacitance increase "the effectiveness of nearby nodes" or allow the depolarizing voltage to "travel not by ion diffusion, but as an electric field"?

A: Capacitance basically results in sequestering of charge of opposite polarities along the cell membrane, which basically results in a neutralization of charge differences. The effect of this is explained on the website of Amrita and I quote:

[...] a higher capacitance results in a lower potential difference. In a cellular sense, increased capacitance requires a greater ion concentration difference across the membrane.

What myelin does is insulating the neuron, thereby decreasing its capacitance. You could say that by increasing the thickness of the membrane, the negative insides of the cell do not attract positive charge outside the cell. It's a bit simplified, yet it effectively describes what myelin does (see cambridge web page on capacitance).

So in a myelinated axon, when Na⁺ enters the cell in a node of Ranvier, the positive charge entering the cell is not counterbalanced by outside negative charge and hence, the charge is not or at least less counterbalanced. This allows for the depolarizing potential to be transmitted by electric charge. If there were no myelin, the depolarizing potential would fade out pretty quickly along the axon by neutralizing charges outside the cell. Without myelin the opening of more sodium channels in the direct vicinity are necessary not only to transmit the action potential across the axon, but it is also necessary to amplify the signal to prevent it from dying out. In a myelinated axon, hence, the depolarizing potential reaches much farther and in answer to your next question:

Q2:This allows us to say that λ, or the rate of spatial decay of potential, increases, decreasing the effective length of the neuron. Why should its decrease help faster conduction?)

One could say that myelination effectively decreases the length of the axon (the lamba parameter) as the depolaring potential reaches further along the axon.

The fact that the depolarizing potential reaches further means that voltage-operated sodium channels can be activated at larger distances from a certain depolaring potential. Hence, adjacent nodes of Ranvier can be spaced by as much as 1.5 mm. Due to the fact that the next node is activated by passive spread of an electric field, which is pretty much instant, it skips the intervening distance with about light speed, greatly enhancing conduction velocity.

Q3: And another question is, how would reduced density of sodium voltage gated channels cause a decreased capacitance?

Basically there are no ion channels below a myelin sheath as they are totally useless there. It is the myelin that decreases capacitance and there happen to be no channels underneath.

Answer 2

The Hodgkin-Huxley model:

$$I=C_m\frac{dV}{dt} + g_k(V_m - V_k) + g_{Na}(V_m - V_{Na}) + g_l(V_m- V_l)$$

Where $C_m$ is membrane capacitance per unit area and $g_i$ are membrane conductances.

Reducing the number of channels does not affect capacitance; it basically reduces membrane conductance.

Myelination causes reduction of number of channels (concentrating them only at the nodes of Ranivier) and also increases the effective membrane thickness.

Capacitance decreases as an inverse function of "inter-plate distance" (of parallel plates) which is the membrane thickness. This reduces the capacitative current.It also prevents the build up charge and thereby allowing it to propagate forward (Longitudinal current). IMO, the effect of myelination of capacitance would be much less than that of its effect on conductance.