Migrating ion channels, part 2

In the following I continue the discussion of fatigue in multiple sclerosis and ion channels as promised at the end of a previous post. I start with a few more details concerning the mechanism of ordinary nerve conduction.In the resting state there is a potential difference across the cell membrane.There are sodium-potassium pumps in the cell membrane which simultaneously move potassium to the inside of the cell and sodium to the outside. This process is charge neutral. The membrane also contains ‘leak channels’ which allow diffusion of potassium ions. This process continues until the concentration difference is balanced by the charge difference leading to a zero electrochemical potential. Then the resting membrane potential has been established. If this potential is reduced by a sufficiently large amount at some point on the membrane voltage-gated sodium channels open and sodium ions flow into the cell, depolarizing the membrane. In fact the sign of the potential difference is even reversed. After some time the sodium channels enter a temporary inactive state where they do not allow ions to pass and do not react to voltage changes. In the meantime voltage-gated potassium channels open, also as a result of the depolarization, and lead to the membrane potential rapidly returning to its equilibrium value. Finally the sodium channels return to a state where they are closed but active. This completes the cycle. An action potential in an axon without myelin, like that of the squid, is a travelling wave where this cycle happens successively at successive spatial points.

In the case of a myelinated axon the depolarization of one node of Ranvier propagates to the next node at a very high speed, essentially the speed of light in the given medium, and the greatest part of the total time of propagation of the nerve impulse is spent near the nodes. The mechanisms described above are then only relevant in the region close to a node. Under these circumstances it is not surprising that the ion channels involved are concentrated near the node. Less obvious is the fact that the sodium channels sit at the node while the potassium channels are concentrated a small distance away. An axon which is demyelinated often conducts signals more slowly than an intact one and sometimes suffers a complete block of conduction. It is these problems which presumably cause fatigue. (Here I am talking about fatigue in the context of physical activity, or possibly vision, and not the mental fatigue which also affects MS sufferers and which is harder to pin down precisely.) A demyelinated neuron is often not able to fire repeatedly at more than a certain frequency or even to fire a few times in a row. These things are usually studied experimentally in the context of peripheral nerves since the central nervous system is so difficult of access.

The key question is now which mechanisms lie behind the malfunctioning of the demyelinated axons. A better understanding of this question could have implications for therapeutic strategies. One hypothesis is that the problem is that the membrane potential is too large due to an excessive concentration of sodium ions outside the cell. This is proposed in a paper of Bostock and Grafe (J. Physiol. 365, 239). They describe how repeated firing can lead to hyperpolarization in normal axons. In more detail, an initial reduction of the intracellular sodium concentration stimulates the sodium-potassium pumps which finally bring the extracellular sodium concentration to a level which is too high. In the case of demyelinated axons these mechanisms might lead to harmful effects. There the depolarization of one node produced by a depolarization of the previous one is dangerously close to the threshold for activation. Thus a small change of the membrane potential can produce failure. In this paper some work on numerical computations of the properties of demyelinated nerve fibres by Koles and Rasminsky (J. Physiol. 227, 351) is quoted. The description of the model given in that paper is not very self-contained and so understanding it would require going to the previous literature. In a later paper of Vagg. et. al. on this subject (J. Physiol. 507, 919) the repeated firing of neurons producing the effect is due to a sustained voluntary contraction of a muscle. This underlines how this type of mechanism could explain difficulties in maintaining muscular contraction.

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In defence of cosmic censorship

Today Helmut Friedrich told me about an article by Pankaj Joshi in the February 2009 issue of Scientific American on the subject of cosmic censorship. I consider this article one-sided and consequently misleading and I find it sad that it appeared in such a prominent place. Here I present an alternative view.

General relativity predicts the occurrence of spacetime singularities where the gravitational force and/or matter density become unboundedly large, implying that known physical laws break down. If a singularity of this type can influence distant regions – this is referred to as a naked singularity – then the predictability, and hence the viability, of general relativity is undermined. Thus there is a strong motivation for ruling out this type of scenario. The cosmic censorship hypothesis of Roger Penrose suggests that the dynamics of the theory contrive to hide the effects of singularities from faraway observers – this is the cosmic censor. In fact there are two variants of this, called weak and strong cosmic censorship. In the first case the singularities are hidden because they are covered by the event horizon of a black hole while in the second they are made invisible to any observer outside the singularity.

The basic equations of general relativity are the Einstein equations which describe the gravitational field coupled with matter equations which describe the sources of the gravitational field. There are mathematical proofs of the existence of singularities in general relativity under rather general physically reasonable conditions. These stem from the singularity theorems of Penrose and Hawking which appeared starting in the mid 1960’s. It is not a priori clear how to define a singularity in this context but a mathematically clear definition has emerged which is useful and widely accepted. It is formulated in terms of geodesic incompleteness. The mathematics entering the proofs of these theorems is differential geometry and ordinary differential equations. The input is less than the full Einstein-matter equations with a definite choice of matter model – rather only certain inequalities on the energy-momentum tensor, the right hand side of the Einstein equations, are required. These inequalities are the so-called energy conditions.

In contrast there is no proof or disproof of cosmic censorship and even finding a precise formulation is difficult. There a couple of concrete obstacles involved:

1. There are very special classes of solutions known, for instance some with high symmetry, which do contain naked singularities. To avoid these a genericity assumption is required.

2. There are certain matter models with pathological properties which can lead to singularities even in the absence of gravity. The most notorious of these is dust, a fluid without pressure. A formulation of cosmic censorship which has a chance of being true must include some restriction on the matter model.

I like to hope that cosmic censorship is true. I strongly believe that it is impossible to prove it using differential geometry and ordinary differential equations alone. A proof will require taking the Einstein-matter equations seriously as a system of partial differential equations. It so happens that proving results about solutions of partial differential equations is much harder than proving analogous results about ordinary differential equations. Thus it is not surprising that progress in proving cosmic censorship has been limited, even taking the optimistic view that it is true. Another difficulty is that relevant parts of PDE theory simply do not belong to the usual mathematical repertoire of people working in general relativity.

Now I come back to the article of Joshi. The author appears rather fond of naked singularities and has mentioned a lot of work tending to support their existence. My aim here is not to give a detailed criticism of the work he cites. Instead I will describe important work providing support for cosmic censorship which is completely ignored in the article. As already mentioned, there is no proof of cosmic censorship. The character of the results I will mention here is that they concern model problems (with some assumptions on symmetry and the choice of matter model) where the mechanisms which are invoked to make cosmic censorship work can be seen in action. The first mechanism is genericity. I give two examples. The first is the spherically symmetric scalar field, as studied by Christodoulou (Ann. Math. 149, 183 (1999)). He showed that in this model naked singularities do occur but that they are eliminated by a genericity assumption. The second is the case of vacuum Gowdy spacetimes where a conceptually similar result was obtained by Ringström (to appear in Ann. Math., see http://www.math.kth.se/~hansr). The second mechanism is the influence of the choice of matter model. Gerhard Rein, Jack Schaeffer and I showed that a large class of naked singularities (the non-central ones) in solutions of the Einstein-dust equations can be removed by replacing dust by collisionless matter. There are many other papers tending to support cosmic censorship. Here I have just mentioned some which illustrate important aspects well. For a valuable account of the development of ideas about cosmic censorship I recommend the Prologue to the recent book of Christodoulou “The formation of black holes in general relativity”. A preprint version is available as gr-qc/0805.3880.

The jury is still out on cosmic censorship. I think that the appropriate expert witnesses who might be called to open the way to a just verdict are the specialists in partial differential equations.

A text of Grothendieck

Starting in the autumn of 1994 I spent a year at the Institut des Hautes Etudes Scientifiques (IHES) in Bures sur Yvette near Paris, which for me at that time was an institution which seemed almost mythical. While there I discovered a bound volume of more than 1000 typewritten pages in the library. I do not know why I opened it – perhaps in those days I was tempted to open any thick book in a good library. In any case, once I had opened it I was soon captured by the text. I did not read it all, but I read most of it. The text was ‘Récoltes et Semailles’ by Alexandre Grothendieck. (This is not exactly the text occurring in the title of this post.) In English the title is ‘Reaping and Sowing’. From time to time in what follows I give English versions of text where the original is in French. These are my spontaneous translations, and I have tried to convey the essential meaning as I understand it rather than to be absolutely literal.

Recently, during a conversation with David Klawonn, we got talking about Grothendieck and memories of my time at IHES naturally came back to me. David asked me if I knew why Grothendieck changed subject from functional analysis to algebraic geometry. I did not know and I was surprised that I had never asked myself this before. I was led to do a little research in the Internet and found some things contributing to an answer although I still feel that there is a lot I do not know. While I was doing this I came across another text with the title ‘Promenade à travers une oeuvre ou L’enfant et la Mère’ (Stroll through a work or The child and the Mother). I started reading and was rapidly absorbed by the prose, just as I had been by ‘Récoltes et Semailles’ so many years before. I am not sure of the relation
between the two texts – the one I am talking about here may even be a part of (a version of) the other. A large selection of writings by Grothendieck can be found at http://www.grothendieckcircle.org/. In this text Grothendieck describes his career as a mathematician, but with a wealth of digressions. Among these I found many fascinating, both due to the content and due to the wonderful use of language. I will only discuss a couple of them here.

At one point Grothendieck describes his passage from analysis to geometry in 1955. The reader should be warned that Grothendieck’s definitions of ‘analysis’ and ‘geometry’ may not be those commonly used. Among other things he says that it was as if he was leaving arid and harsh steppes to suddenly find himself in a kind of promised land. He wants to describe the experience by the German word ‘überwältigend’ and has difficulty finding a satisfactory French translation. I personally cannot imagine experiencing a transition in that direction in this way. Nevertheless at this point, and in many other cases, I found it difficult to avoid being carried along by Grothendieck’s language even when not naturally inclined to share his point of view on the subject in question. And in a way I wanted to be carried along. Grothendieck’s judgement of himself and his own work is a curious mixture of modesty and pride. On the one hand he emphasizes his lack of talent while at the same time suggesting that he finds his own work tremendously important. His idea of his own greatest strength seems to be that he has looked carefully at things which others regarded as too unimportant to merit their attention.

I am not qualified to give an assessment of the importance of Grothendieck’s work – my own interests in mathematics are too far away from his. I do, however, have the impression that he overrates the importance of his own discoveries for mathematics as a whole. At least I can appreciate his idea of ‘ordinary’ cohomology as a derived functor. This is a case where simplicity is impressive. Grothendieck’s ideas on the central importance of his work, in particular for geometry and topology, are now vulnerable to comparisons with the work of Hamilton and Perelman leading to the proof of the Poincaré conjecture, which are of a very different kind. In this context I could not help thinking of certain parallels between Grothendieck and Perelman. I now list some. Both of them
1. are exceptionally gifted mathematicians.
2. have had difficult relations with the mathematics community and have questioned the morality of that community.
3. have refused prestigious prizes (Grothendieck the Crafoord prize, Perelman the Fields medal).
4. isolated themselves from the mathematical community at some point.
5. are uncompromising about standing up for what they believe in.
6. have a considerable charisma.
In Grothendieck’s case I could only experience the charisma through his writings; in Perelman’s case I was fortunate enough to hear the seminar in Berlin where he first talked about ideas related to his now famous work on the Poincaré conjecture. Of course the points on the above list are not all independent. Perelman has not been isolated so long and we may yet be fortunate enough to see him present us with a new breakthrough.

It is important to mention that in his writings Grothendieck sometimes seems to leave the region which most of us would call sanity. In saying this I am thinking less of the text I am discussing here than of ‘Récoltes et Semailles’ and others. When reading ‘Récoltes et Semailles’ I could not help thinking of Rousseau’s ‘Confessions’. It has a similar attractiveness for me in its earlier parts and seems to go off the rails (as seen from a frame of reference of ordinary everyday life) towards the end.

The fact that I am writing about this subject here is certainly not just due to the mathematical content of the text in question. It affected me on an emotional level and shook the limits within which my thinking usually takes place. This is not an accident. Grothendieck writes: If in ‘Récoltes et Semailles’ I am addressing someone other than myself it is not a ‘public’. I am addressing you who are reading me as a person and as a person who is alone.