As mentioned in a previous post signals in the nerves of human beings, in contrast to those in the squid giant axon, propagate in a very non-uniform way, jumping from one node of Ranvier to the next. The propagation of a signal in the squid axon can be modelled as a travelling wave solution of the Hodgkin-Huxley system. To model the saltatory conduction in human nerves a different type of model is necessary. This kind of conduction is often studied experimentally in amphibians (specifically, frogs). A model of such phenomena was introduced by FitzHugh (Biophys J. 2, 11). The model couples copies of the HH model, one for each node, to a diffusion equation with damping describing the propagation of the voltage along the membrane in each internode (i.e. the section of the axon between two adjacent nodes).The coupling is achieved by imposing transition conditions at the nodes. At one of these, where the axon is stimulated electrically, an inhomogeneous Neumann boundary condition is assumed. At the others it is assumed that the potential is continuous and that its spatial derivative satisfies a transition condition. The central themes of the paper are a numerical solution of the equations and a comparison of the results with experimental data.
In fact the conduction mechanisms at the node of Ranvier seem to somewhat different from those in the squid axon. This was studied in detail in the frog Xenopus by Frankenhaeuser and Huxley. They introduced a modified version of the HH model in order to incorporate these differences (J. Physiol. 171, 302). In the HH model the variables other than the voltage are three gating variables which can be interpreted with hindsight as corresponding to the processes by which voltage-gated sodium channels open and enter the inactive state and the process by which voltage-gated potassium channels open. In the Frankenhaeuser-Huxley model a fourth gating variable is added. I do not know if hindsight has provided it with a clear interpretation. It is referred to in the original paper as a ‘non-specific current’. Goldman and Albus (Biophys. J. 8, 596) presented an improved version of Fitzhugh’s model for saltatory conduction where the HH model was replaced by the Frankenhaeuser-Huxley model. Apart from numerical calculations they did a dimensional analysis, leading to a derivation of the relation between conduction speed and the diameter of the nerve fibre.
Koles and Rasminsky (J. Physiol. 227, 351) investigated the effect of (partial) demyelination on nerve conduction in the framework of a model of the Goldman-Albus type. In particular, they studied numerically the effects of temperature and sodium concentration on the effectiveness of nerve conduction and the occurrence of a conduction block, obtaining results broadly compatible with experimental data. They also compared results of demyelination close to the nodes or further away from them.
In a paper of Bostock and Grafe (J. Physiol. 365, 239) it was argued that the poor functioning of demyelinated nerves after repeated stimulation is due to hyperpolarization of the cell membrane and that this results from the activity of the sodium-potassium pump. This hyperpolarization occurs in normal nerve fibres but has no major effects on the function of the nerve. In general it brings the size of the depolarization required to fire a node closer to that actually produced by the previous node. The problem in the demyelinated case is that these two quanitities can be close to each other to start with so that a small change due to hyperpolarization can cause firing to fail. Note that this type of effect cannot be captured by the mathematical models discussed above since as they stand they do not include the sodium-potassium pump.
Now I have completed the homework I left myself at the end of the post ‘Migrating ion channels‘ and I have come back to my starting point having gained some height in terms of my level of understanding.