I am writing a book on the theory of action potentials. As I haven't published any new chapter for about 6 months, I will give a brief update on the progress of the book.
It turns out that writing a book is difficult (!). As I write the book, I get to question some aspects I take for granted and I realize they are actually not that simple. That I have learned a lot about biophysics. In turn, this tends to make the book more complicated, and so it requires some additional digestion work. I also realize that some important questions are just unresolved and require some research work. Finally, it is quite difficult to write a book while continuing the normal course of research. I started with a one hour per day habit, but this may not be optimal; I tend to spend the first half-hour trying to get back to the matter of the book. I am starting a new routine with two mornings twice a week. We will see how it goes!
These last months I have been working on the 4th chapter, on excitability of an isopotential membrane. This will contain in particular some of the material in Platkiewicz & Brette (2010) on the relation between biophysical parameters and threshold. I wanted to use the same theoretical approach to apply it to other aspects of the action potential (speed, peak etc), so that needed some more work. I realized that some approximations we had done could be enhanced, but the math is slightly more complicated. It is a challenge to keep this chapter simple. I also wanted to apply the theory to the original Hodgkin-Huxley model, but unfortunately it works moderately well. One reason is that the model was fitted to the full action potential and not to initiation (as in fact essentially all empirical neuron models). So in particular, H&H experiments show that the Na+ conductance depends exponentially on voltage at low voltage, but their model doesn't (or approximately does with a different factor). Another reason is the K+ channel has less delay than in the actual squid axon (which they acknowledge), so there is some interaction with the initial Na+ current. So I will go with a simpler approach, using a simplified excitability model. Neurons are not isopotential anyway.
I am also planning to reorganize and partly rewrite chapters 2 and 3. I find the current chapter 3 (action potential of an isopotential membrane) a bit too technical. I also want to change chapter 2 (membrane polarization) to talk about alternative theories of the membrane potential (Tasaki, Ling, etc). Then I want to insert a chapter on the ionic channel theory of action potentials, which explains the theory and discusses empirical evidence, before the chapter on the Hodgkin-Huxley model. Generally, I want to simplify the exposition. But given my rather slow progress on the book so far, I will probably postpone this and first write drafts of the following chapters.
Finally, I have worked a bit on energy consumption and pumps, and I found out that the current treatment in the literature is not entirely satisfactory (see for example my comments on a classical paper on the subject). It turns out that it is a pretty complicated problem (especially systems of pumps).
In brief, I am trying to finish a first version of the chapter on excitability of an isopotential membrane, hopefully within one month.