Draft of chapter 6, Spike initiation with an initial segment

I have just uploaded an incomplete draft of chapter 6, "Spike initiation with an initial segment". This chapter deals with how spikes are initiated in most vertebrate neurons (and also some invertebrate neurons), where there is a hotspot of excitability close to a large soma. This situation has a number of interesting implications which make spike initiation quite different from the situation investigated by Hodgkin and Huxley, that of stimulating the middle of an axon. Most of the chapter describes the theory that I have developed to analyze this situation, called "resistive coupling theory" because the axonal hotspot is resistively coupled to the soma.

The chapter is currently unfinished, because a few points require a little more research, which we have not finished. The presentation is also a bit more technical than I would like, so this is really a draft. I wanted nonetheless to release it now, as I have not uploaded a chapter for a while and it could be some time before the chapter is finished.

Technical draft for chapter 5, Propagation of action potentials

I have just uploaded a technical draft on chapter 5 of my book on action potentials: Propagation of action potentials. This draft introduces the cable equation, and how conduction velocity depends on axon diameter in unmyelinated and myelinated axons. There is also a short section on the extracellular potential. There are a few topics I want to add, including branching and determinants of conduction velocity (beyond diameter). There is also (almost) no figure at the moment. Finally, it is likely that the chapter is reorganized for clarity. I wanted to upload this chapter nonetheless so as to move on to the next chapter, on spike initiation with an initial segment.

New chapter : Excitability of an isopotential membrane

I have just uploaded a new chapter of my book on the theory of action potentials: Excitability of an isopotential membrane. In this chapter, I look mostly at the concept of spike threshold: the different ways to define it, its quantitative relation to different biophysical parameters (eg properties of sodium channels), and the conditions for its existence (eg a sufficient number of channels). This is closely related to my previous work on the threshold equation (Platkiewicz and Brette, 2010). It also contains some unpublished work (in particular updates of the threshold equation).

I am planning to extend this chapter with:

  • A few Brian notebooks.
  • A section on excitability types (Hodgkin classification).
  • Some experimental confirmations of the threshold equation that are under way (you will see in section 4.4.2 that current published experimental data do not allow precise testing of the theory).

I am now planning to work on the chapter on action potential propagation.

All comments are welcome.

Update on the book

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.

General bibliography on action potential theory

Two general introductory biology textbooks, covering the excitability of neurons and muscles are (Matthews, 2002) and (Keynes et al., 2011). The biophysics and modeling of neurons are covered in (Johnston and Wu, 1994) and (Sterratt et al., 2011). Both are quite accessible and include all essential material including compartmental modeling of dendrites.

There is a great book that covers many topics in cell biology from a physicist perspective, including excitability: Physical biology of the cell (Phillips et al., 2008).

There are two excellent reviews by Hodgkin that are particularly useful to understand the experimental basis of Hodgkin-Huxley theory, including myelinated axons: (Hodgkin, 1951, 1964).

The reference textbook for the biophysics ionic channels is (Hille, 2001). (Johnston and Wu, 1994) also includes some material about stochastic analysis of channels.

Linear cable theory is covered in great detail in (Tuckwell, 1988) and (Koch, 1999). An excellent review by one of the historical figures of cable theory is (Rall, 2011). (Jack et al., 1975) also covers cable theory including active axonal conduction, and it also includes muscle APs and their propagation and classic theory of excitability (threshold).

Theory of electro-osmosis (interaction between osmosis and electrical field) is treated in (Hoppensteadt and Peskin, 2004).


Hille B (2001) Ion Channels of Excitable Membranes. Sinauer Associates.

Hodgkin AL (1951) The Ionic Basis of Electrical Activity in Nerve and Muscle. Biological Reviews 26:339–409.

Hodgkin AL (1964) The conduction of the nervous impulse. C. C. Thomas.

Hoppensteadt FC, Peskin C (2004) Modeling and Simulation in Medicine and the Life Sciences, 2nd edition. New York: Springer.

Jack JB, Noble D, Tsien R (1975) Electric Current Flow in Excitable Cells. Oxford: OUP Australia and New Zealand.

Johnston D, Wu SM-S (1994) Foundations of Cellular Neurophysiology, 1 edition. Cambridge, Mass: A Bradford Book.

Keynes RD, Aidley DJ, Huang CL-H (2011) Nerve and Muscle, 4 edition. Cambridge ; New York: Cambridge University Press.

Koch C (1999) Biophysics of computation: Information processing in single neurons. Oxford University Press, USA.

Matthews GG (2002) Cellular Physiology of Nerve and Muscle, 4 edition. Osney Mead, Oxford ; Malden, MA: Wiley-Blackwell.

Phillips R, Kondev J, Theriot J (2008) Physical Biology of the Cell, 1 edition. New York: Garland Science.

Rall W (2011) Core Conductor Theory and Cable Properties of Neurons. In: Comprehensive Physiology. John Wiley & Sons, Inc.

Sterratt D, Graham B, Gillies DA, Willshaw D (2011) Principles of Computational Modelling in Neuroscience, 1 edition. Cambridge; New York: Cambridge University Press.

Tuckwell H (1988) Introduction to theoretical neurobiology, vol 1: linear cable theory and dendritic structure. Cambridge: Cambridge University Press.


A book on the theory of action potentials

Latest news

I am writing a book on the theory of action potentials. I will post chapters on this website as I write them. Note however that this is essentially preparatory work for the book, and I am probably going to rewrite and reorganize it quite extensively. So do not expect a very well organized and didactical text at this stage; only the essential content should remain. I would be happy to read your comments, in particular if you find errors or omissions: please let me know and I will reward you with your name in the acknowledgments!

The plan is to start with standard biophysics of excitability, and then I will expose more advanced topics, such as: how spikes initiate in real life (as opposed to when you stimulate an axon), how excitability changes on different time scales, and how a cell learns to spike. The book adopts a systemic viewpoint; that is, the goal is to understand how the coordination of channels creates and maintains functional action potentials. I would also like to give an epistemological flavor to it that I find is missing in most textbooks: what is a model, how is it built and tested, what is its empirical value, etc.

Why this book and who is it for? With this book, I am hoping to bring theoreticians to the field of neural excitability, and to give them the necessary material that is currently scattered over many references. Currently, the field is largely dominated by experimental biologists. Yet, as I will try to convey, this is a field where one can ask many key neuroscientific questions in a context where the link between structure and function is much less speculative than in neural network research, including questions of learning and adaptation, and where one can actually develop quantitative, testable theories. As a bonus, I would also be happy if I could manage to convey some elements of theory to biologists.

Prerequisites. In principle you do not need to know much about biology to read this book, as I will try to introduce the necessary information. I am expecting some mathematical skills, mostly calculus and basics of differential equations, but nothing very advanced. Regarding physics, electrophysiology is obviously a lot about electricity. In the current version, I am assuming the reader has some basic knowledge of electricity (what current and charges are, Ohm's law). But I am planning to add a primer on electricity.

Each chapter is accompanied by a set of examples using the Brian 2.0 simulator, in the form of Jupyter notebooks.

I am also compiling a general bibliography on action potential theory (books and reviews only).

Here is a tentative outline (available chapters are in bold):

  1. Action potentials. An overview of action potentials, their scientific history and their function. Brian notebooks for chapter 1Last update: 9.6.2016
  2. The membrane potential. The biophysical basis of membrane polarization. Brian notebooks for chapter 2Last update: 9.6.2016
  3. Action potential of an isopotential membrane. Basic biophysics of the squid giant axon and Paramecium, and the Hodgkin-Huxley model. Brian notebooks for chapter 3Last update: 19.7.2016
  4. Excitability of an isopotential membrane. Theoretical analysis of excitability in isopotential models [Some content to be added on excitability types]. Last update: 14.3.2017.
  5. Propagation of action potentials (technical draft). The cable equation; active propagation in unmyelinated and myelinated axons. Last update: 13.4.2017.
  6. Spike initiation in the axonal initial segment (incomplete draft). Excitation through the soma and AIS, as opposed to excitation of the middle of a squid axon. Last update: 5.4.2018.
  7. Dynamics of excitability. How excitability (spike threshold) changes at different time scales (adaptation and plasticity).
  8. Energy consumption.
  9. Learning to spike. How a cell builds and maintains a functional spiking system.