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General bibliography on action potential theory

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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.

 

On the misuses of statistics in biology, medicine, neuroscience and psychology

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These days, it seems that the scientific community is concerned about the lack of reproducibility of experimental results in biology and related fields (e.g. medicine, neuroscience, psychology), in relation with statistics. Say, some study claims that there is a significant correlation between X (eating potatoes) and Y (developing lung cancer), with p<0.05. The next study tries to replicate it and finds no significant correlation. This happens all the time. Every week we read a media report about a peer-review study finding a significant correlation between something we eat and some disease. But in general those reports are not taken too seriously, at least by scientists and doctors. But why is that? Isn't “statistically significant” precisely supposed to mean that the observed outcome was not a matter of chance and that we should trust it? If yes, then why shouldn't we take those reports seriously? And if not, then why do we keep on backing up scientific claims with statistics?

I have to say I am a bit surprised that the biology fields generally don't seem to fully draw the implications of this rather obvious observation. Paper after paper, X and Y are “significantly different” or “not significantly different”, and some scientific conclusions seem to be drawn from this statistical statement.

Let us briefly recall what “significantly different with p<0.05” means. You have two sets of quantitative observations, obtained in two different conditions. You calculate the difference of their means, and you want to know whether the result is just due to irrelevant variability or whether it really reflects a difference due to the different conditions. To say that the difference is “significant with p<0.05” essentially means that if the two sets of observations were drawn from the same distribution (with the same mean), then there would be a 5% chance that you would observe that difference. For example, if you do the same experiments 20 times, then you should expect to find a “statistically significant” difference, even if the experimental condition has actually no effect at all on what you are measuring. For the biologists: every 20 experiments, you will find something statistically significant even when there is actually nothing interesting. I bet only that experiment will be published. And that is if only one particular condition is monitored. For the epidemiologists: if you monitor 20 correlations, chances are that one of them will be statistically significant by pure chance.

It is ironic that this replicability crisis comes at the same time as the hype around “big data”. Michael Jordan, one of the leading experts in machine learning, pointed out very clearly that we don't have the statistical tools to deal with big data. Looking for statistically significant correlations among piles of data is nonsense. Of course will find them, tons of them. And how will you know which ones are meaningful?

As I pointed out above, in biology there is a huge selection bias, ie you publish only statistically significant results. Some say that we should also publish negative results, and not just positive results. But the problem is that “not statistically significant” is not a negative result. It's no result at all. It says that we haven't seen anything. It doesn't mean there is nothing. Maybe we would see something with more observations. We just can't know. I often read in biology papers that “X and Y are not significantly correlated” as if it was a result, i.e. X and Y have nothing to do with each other. But that is not true at all! It's a lack of result, and neither a positive nor a negative result.

So some have argued for increasing the number of observations (ie cells/animals/subjects), so that we can say something like: X and Y are significantly different with p<0.005. I think this misses the point and reveals a deeper problem, which is of epistemological nature, not just statistical. Here is a quote I like by Ernest Rutherford, who was a Nobelized physicist and chemist: "If your experiment needs statistics, you ought to have done a better experiment". That's a bit exaggerated, probably, but quite true in my opinion. If you need statistics, it's because the result is not obvious, in other words the difference between X and Y might be “statistically significant” but it's tiny. Statistical significance is not at all about significance in the usual sense of the word. A difference of 0.1% can be statistically significant, if you have enough observations. But think about it: in a complex system, like a living organism for example, would you expect that one part of the system is absolutely uncorrelated with another part of the system? That would basically mean that the two parts are unconnected and therefore not part of the same system. For example, I'm pretty sure that eating potatoes is positively or negatively correlated with developing lung cancers. There might be a correlation between potatoe eating and revenue, and clearly a correlation between revenue and any kind of disease. I'm not even mentioning the effect of potatoes on metabolism, which certainly has some slight correlation with cancer development or the immune system. However tiny these correlations might be, you will still find a significant correlation between potatoe eating and lung cancers, if you look at enough cases.

So in reality, in a complex system, one should expect that every single pair of variables are correlated, and any condition that affects the system should affect all its components in various, possible tiny, amounts. Therefore “significantly different” and “significantly correlated” are really not very useful statistical concepts for biology. The first thing to do should be to start reporting how big those differences or correlations actually are, and not just if they exist. A useful statistical concept, for example, is effect size: the difference between the means of the two groups of observations divided by their standard deviation. So for example, effect size of 1 tells you that the mean difference you observe is of the same order as the intrinsic variability in each group, so that would be considered a quite strong effect. Effect size is much closer to what we would naturally mean by “significant” than statistical significance. If we reported effect sizes for all the observed cognitive differences between men and women that have been publicized in media and books, we would find in most cases that they are statistically significant but their effect size is very small. As Dindia put it concisely, “Men are from North Dakota, women are from South Dakota”.

In this context, I don't find the call for increasing the number of observations in biology papers (ie of animals) very ethical. Reporting effect sizes would be the minimum that everyone should do in biology papers, and if it's tiny then why bother increasing the number of observations to show that indeed it is tiny, yes, but statistically significant?

This obviously doesn't solve all the issues. I'll mention just one more. Very often, groups are compared that differ not by one condition, but by many. This is typical in epidemiology for example. You are looking for the effect of obesity and heart disease, say, and you find a strong correlation. But you want to make sure that it isn't just due to the fact that obese people tend to do less exercise, for example. It's crucial because then diet would probably not be efficient. The standard way to deal with this issue is to do multilinear regression or analysis of variance, that is, to fit a statistical model that includes all the variables that you think might be important. These are almost always linear models, ie, you assume that the observation you are interested in scales linearly with every variable. Then you will read in the paper that the authors have taken into account the other possible factors and so they can be sure that those are not involved.

I find this sort of statement hilarious. Who would think that a living organism is a linear system? A linear system takes no decision. In a linear system, there is no state that is qualitatively different from any other state. There is no life and death. There is no cancer. If you inject a dose of anesthetics into a human and monitor say heart beat, I bet that you won't find a linear relation between dose and heart beat.

I am not saying that multilinear regressions and similar tools are completely useless (although possibly close to it in many cases); after all linear approximations often work in some limited ranges (but that would need to be demonstrated specifically!). I simply mean that you can't trust their results. They should be taken as suggestive at best. If you look at the epidemiology literature, or simply reviews and reports from WHO, you'll find that there are not so many cases where experts hold strong convinctions about causal relations between diet or habits and diseases. When they do, it is never only based on statistical correlations. It is a combination of epidemiological studies, which are difficult to interpret because of the issues I have mentioned, and also intervention studies (ie an experiment with a control group) and detailed biological knowledge about the physiopathology of the disease, ie biological experiments. Such is the case of the relation between smoking and lung cancer, for example. Hence the relevance of Rutherford's quote: if you need statistics, then you ought to have done a better experiment.

I would like to conclude that the problem with the use of statistics in biology and related field is not in my opinion due to a lack of mathematical (specifically, statistical) education. Rather, it is due to a lack of epistemological education or reflection, which is a much broader problem, not specific to biologists at all. The question is not to know about all statistical tests and how to use them, the question is to understand the epistemological value of results, whether statistical or not, ie: what does it tell me exactly about the system I am interested in, and what exact question, beyond the suggestive words (“significant”), does it provide an answer to?

This week's paper selection (6-13 Apr 2016)

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This week's paper selection (2-31 March 2016)

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After a few weeks away from the lab, here is a selection of papers:

This week's paper selection (24 Feb - 2 Mar 2016)

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This week I got interested in how a cell adapts too changes in size (for example, the membrane area of a neuron grows by several orders of magnitude during development).

 

A few ideas to improve the grant system

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In France, the success rate for national grants has gone below 10% (with the ANR). Not surprisingly since the budget has dropped in recent years. Unfortunately, there is not much else that is available for labs (apart from European grants, which have similar success rate). The situation is also quite difficult in the American's NIH (although not as bad). With success rates so low, many scientists complain about the time wasted on writing grants that eventually seem to be more or less arbitrarily selected. This is not just an expression of the frustration of losers: once again a study comes out showing that peer review of grants is no better than chance, when applied to the top grants. The public should be aware of this: writing a grant application takes a significant amount of time (not mentioning reviewers time), and 90% of this tax-payed timed is completely wasted, in a process that is not better than lottery. Does this really make sense?

How have we arrived to such an absurd situation? There are different factors. The first obvious error I see is to fund scientific projects almost entirely based on a competitive grant system. Only the best science should be funded, we hear. Makes sense. Really? Given that both the basic infrastructure (administration, space) and the salary of permanent scientists are paid for anyway, what this advertising slogan means concretely is that only a small proportion of paid scientists will be given the means to work normally. How does that make sense? So the first obvious thing to do is to ensure that there is a baseline that allows labs to work normally (eg pay for conferences, publications, basic equipment). Why would you permanently hire a scientist and then not give her the means to work? Absurd, but that's what the CNRS currently does (main French research organization). INSERM (medical research organization) gives a start-up package to junior scientists; it covers only two years though, but is certainly an improvement.

It is only quite recently that there is a competitive grant system in France. Many people didn't (and still don't) like it, partly for the obvious reasons I just discussed. Personally, I don't think it's a bad thing in principle. It has one value, which is to shift the power from lab directors to individual scientists, who can be more independent (that is, if they get a grant). The problem is not so much that it exists, but that it entirely replaces any other means of funding.

But a competitive grant system doesn't necessarily need to be based on projects. Great if you have a particular project that requires specific expensive equipment, for example. Or a project that requires simultaneous funding of several partners. But in most cases I know, scientists use the grant system to fund the daily work of their lab, and pitch it into a “project”, sometimes assembling different arbitrary “partners” just because the system requires it. Everyone knows that the best way to get a grant is when you already have the results but have not published them yet (so-called “preliminary results” that grant reviewers like so much). Then you get the grant to fund the “preliminary results” of the next one. It is not always like that; sometimes you really do have a good project idea – although in most cases you will use the budget parly for things that are not in the grant anyway, since it's research, not an engineering project. This happens because the funding system is built this way. Scientists have no choice but be cynical, which also results in a huge waste of resources. For programmatic funding (ie funding political priorities, say research on energy), a project-based grant system might make sense. But for basic research, all we ask the funding system is to fund individual scientists or labs because it is believed that they will do good science. For experienced scientists, the best way to predict whether they will do good science is probably to just look at what they did before. Anyway, that's what grant reviewers partly do when they judge CVs and “preliminary results”, with the added bias of story-telling skills that the project system emphasizes. So, there should be a competitive system that allocates grants based on previous work. This system already exists: it is used for hiring and promotion; it also exists at the level of teams and labs, except it has little impact on budgets. Let us just expand it and combine it with some long-term budget, and a lot of resources will be saved.

One possible objection is that it is difficult to estimate budgets if there is no specific project. True, but in reality this is exactly the same problem for project-based grants. Reviewers, who are often international scientists, cannot and do not judge budgets. In practice, the institution generally sets orders of magnitude for budgets, and applicants just ask for the maximum budget (there are of course exceptions).

A second objection is that it doesn't work for junior scientists, since they don't have much to show in their CV. I disagree, because they are hired through a highly competitive process, which is not much based on projects but on their achievements. So by design, the junior scientist did go through a highly selective process and should be granted budget. This is just called a start-up package and should be generalized. There is a valid concern, however, which is to compare scientists that are at very different stages in their career. But this doesn't sound like an impossible difficulty; one could use for example a similar system as the ERC (starting/consolidator/advanced).

All this being said, I believe there is still a need for project-based grants, but with a smaller perimeter. Projects make sense when they require setting up a collaboration, or buying some specific equipment, or within programmatic constraints. Not so much in general to fund a single team or individual. I would like to make a final suggestion. If the project-based system still has a very low success rate, so low that selection is random, then there is a simple way to increase it and waste less ressource. One suggestion made in the abovementioned paper is to use a lottery. Indeed if the process is no better than chance, we might as well dispense ourselves of the review process and save some time for reviewers. This is an improvement, but it still wastes the time of applicants; I don't think it would be received very well. There is another way that also saves the time of applicants: let them apply only every other year (say, if you are born on an even year then you can apply only on even years). The number of funded grants would stay the same of course, if the budget is the same, but the success rate would automatically increase. Maybe not by a factor of two, but probably not far from it. Note that this is not the same as the individual strategy to decide to apply only every two years; when selection is random, it is in your best interest to apply every year, because each application is independent. There is not even any reason to modify your application between successive years. Whereas if success rate is forcefully increased by blocking part of the applications, the same individual success rate is obtained by applicants with half as many applications. In addition, it makes selection less random. This system is clearly better than lottery since you know in advance that you will be blocked and don't have to write a grant. You can also prepare better for the next round.

In summary, I have made four complementary propositions:

  1. Ensure a baseline budget for each individual scientist or team, which allows them to work normally (conferences, publication costs, basic equipment).
  2. Have a competitive grant system based on past rather than hypothetical future.
  3. Have a smaller competitive project-based grant system.
  4. Increase success rates of competitive systems (possibly both) above chance level by letting scientists apply only every two years.

None of these propositions depends on total budget – which is clearly too low in France, but that's an obvious point.

This week's paper selection (17-24 Feb 2016)

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A book on the theory of action potentials

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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 1. Last update: 9.6.2016
  2. The membrane potential. The biophysical basis of membrane polarization. Brian notebooks for chapter 2. Last 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 3. Last 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.