Hodgkin & Huxley did some incredible work in the 1930s-50s on ion currents flowing through biological membranes. Despite not knowing what an ion channel was, they managed to work out an incredibly accurate predictive mathematical model for the currents that flow through them, and solved the differential equations numerically on a hand calculator. These models are still fundamental to a lot of electrophysiology work – we are still publishing Hodgkin-Huxley style models of particular currents (we just tried to parameterise them better)!
So there are a few points to raise about good old Hodgkin-Huxley models in this blog post!
- There’s a recent set of papers updating the original papers for modern conventions.
- I’ve sketched how (in the case of ‘powered’ gates) different equivalent Markov models can be written down for the same Hodgkin-Huxley model.
- A widely followed but rarely-expressed convention for the Markov diagrams.
Recently my colleague Angus M Brown at the University of Nottingham published some updates to the landmark (and Nobel-prize winning) series of papers published in the Journal of Physiology in the 1940s-1950s by Hodgkin & Huxley:
We now have updated classic #electrophysiology papers by Hodgkin, Huxley and Katz, so that the topics are much more understandable for students. Check out all the translations now: https://t.co/VLlNfKclRH pic.twitter.com/qvjyZwzisL
— Journal of Physiology (@JPhysiol) July 15, 2019
I think this is great – there have been changes in conventions since the original papers were published (in particular the sign of Voltage/membrane potential, which is also now relative to earth rather than relative to resting potential). The changes are discussed in the editorial accompanying the papers.
So there is an updated set of equations for the squid axon action potential model in the ‘translated’ 1952 paper. I’d strongly recommend pointing students and colleagues towards this translation instead of the original paper, as I think it resolves a lot of confusions that can arise in trying to do all these convention changes (which you might not even be aware of!) in your head.
Equivalent Markov Models for Hodgkin-Huxley Structures
Quite a few people have asked me how the equivalence between Hodgkin-Huxley gating variables and Markov models works. So I thought I’d sketch it out – my effort is in Figure 1.
The same procedure holds for higher powers, so that an m^3 Markov model has three closed states and rates of 3α, 2α, α on the top and β, 2β, 3β on the bottom.
Now you haven’t really gained anything by re-casting like this (as it adds an equation in the case I’ve showed above). But if you come to modify the model so that something (like a drug) is interacting with just one of the states and breaking all the independence and symmetry in a Hodgkin-Huxley model, then being able to work out the Markov Model is necessary to simulate what happens then.
A convention for Markov diagrams of voltage-gated ion channels
It took me quite a while working in the field of cardiac electrophysiology to realise that an implicit (and, as such, not always followed!) convention in these diagrams is to have voltage underlying the axes, as I’ve sketched in Fig 2.
So if you have a choice* try to present the diagrams such that increasing voltage pushes you right and up!