Hodgkin-Huxley models and Markov equivalents

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!

  1. There’s a recent set of papers updating the original papers for modern conventions.
  2. I’ve sketched how (in the case of ‘powered’ gates) different equivalent Markov models can be written down for the same Hodgkin-Huxley model.
  3. A widely followed but rarely-expressed convention for the Markov diagrams.

Updated Papers

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:

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.

HH_and_Markov_equivalence

Figure 1: Two equivalent Markov Model structures for a Hodgkin-Huxley squared gate. Firstly two Hodgkin-Huxley gates multiplied together can be represented as a square Markov diagram, with the second gating process acting identically regardless of whether the first gating process was ‘closed’ or ‘open’. The rest of the diagram shows a simplification that can be made due to symmetry if these gating processes are identical (i.e. a squared gate like m^2) to reduce the Markov diagram to a linear chain which is one state smaller with related rates and states.

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.

Shows the direction of increasing rates with voltage in Markov diagrams

Figure 2: a convention to lay out these Markov diagrams such that rates which increase with increasing voltage go from left to right and bottom to top. This is useful as one can tell at a glance “if voltage is high, the states towards the top right will be more occupied” and “if voltage is low the states towards the bottom left will be more occupied”.

So if you have a choice* try to present the diagrams such that increasing voltage pushes you right and up!

*sometimes, for reasons of clarity, it is nice to present bits of the diagram as mirror images, in which case I’ll let you off.
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