Archive for April, 2009

Respecting the matter in general relativity

April 25, 2009

General relativity is a theory which describes space, time and the gravitational field in terms of a Lorentzian metric g_{\alpha\beta}. A complete understanding of the gravitational field requires an understanding of the matter sources which generate it. In the Einstein equations, G_{\alpha\beta}=8\pi T_{\alpha\beta}, the left hand side depends only on g_{\alpha\beta} and is a feature of the geometry alone. On the other hand the right hand side, the energy-momentum tensor, depends not only on the metric but also on some matter fields. The right hand side of Einstein’s equations seems to have suffered from bad press coverage from an early stage. Einstein himself is often quoted as having said that the left hand side of his equations is made of marble while the right hand side is made of wood. I do not have a source for this quote – if anyone reading this does I would be grateful to hear about it. In this post I want to suggest treating that humble right hand side with more respect. If I lived in a palace made of marble with beautiful wooden furniture then I might be more impressed by the marble than by the wood. I would nevertheless do my best to prevent little boys from carving their initials into the furniture with penknives or the cat (much as I love cats) from using it as an accessory for the care of its claws.

The left hand side of the Einstein equations is universal within general relativity – it is always the same, no matter which type of physical situation is to be described. On the other hand the nature of the matter fields depends very much on what physical situation is to be described and what aspects of it are to be included in the description. It is necessary to make a choice of matter model. What is remarkable is that there is a large variety of choices which, in conjunction with the Einstein equations, lead to a consistent closed system of equations which bears no traces of the fact that other physical effects have been omitted. In fact there are three related choices which have to be made to set up the mathematical model in any given case. The first is the matter fields themselves – what kind of geometrical objects are they? The second is the expression which defines the energy-momentum tensor in terms of the matter fields and the metric. The third is the system of equations of motion which describe the dynamics of the matter. Note that in general the energy-momentum tensor depends explicitly on the metric. It is not possible to define an energy-momentum tensor unless the spacetime geometry is given. The same is true in the case of the equations of motion of the matter. They also contain the metric explicitly. Without the metric even the nature of the matter fields themselves can become ambiguous. Which positions should we choose for the indices of a tensor occurring in the description of the matter fields? From a physical point of view it is clear why the metric is necessary in so many ways. The mathematical model must be given a physical interpretation which involves the consideration of measurements. In the absence of a given geometry there is no way to talk about measurements.

When a matter model has been chosen the basic equations which are to be solved are the Einstein-matter equations, i.e the Einstein equations coupled to the equations of motion for the chosen type of matter. The unknowns are the metric and the basic matter fields. For any reasonable choice of matter fields the energy-momentum tensor has zero divergence as a consequence of the equations of motion. However the equations of motion in general contain more information than the divergence-free property of the energy-momentum tensor. For more discussion of these things together with examples see Chapter 3 of my book. I emphasize that solving the equations describing the physical situation within the given model means solving both the Einstein equations and the equations of motion of the matter. This is too often neglected in the literature. A particular danger occurs when the solutions under consideration are of low regularity. If the Einstein equations do not make sense pointwise then it should be checked that they hold in the sense of distributions. For solutions which lack regularity on a hypersurface this is expressed in the junction conditions and it is common in the literature to check that they hold. The equations of motion should also be satisfied in the sense of distributions and this is often ignored. When I use the phrase ‘in the sense of distributions’ here this is just a shorthand since the equations are nonlinear. The correct statement is that it is necessary to think carefully about the sense in which the equations are satisfied.

An example may help to make the importance of the issue clear. At the GR12 conference in Boulder in 1989 there was a heated discussion of the question, whether colliding plane waves can give rise to spontaneous creation of matter. (I emphasize that this discussion was in a purely classical context. Quantum theory was not being taken into account.) This kind of creation of matter sounds ridiculous from a physical point of view. Nevertheless people exhibited ‘solutions’ which showed this type of effect. Their mistake was that they had only verified those things which I said above were usual in the literature. They had not considered whether the equations of motion of matter were satisfied. If the equations of motion are ignored, it is not surprising that arbitrary things can happen.


Models for saltatory nerve conduction

April 18, 2009

As mentioned in a previous post signals in the nerves of human beings, in contrast to those in the squid giant axon, propagate in a very non-uniform way, jumping from one node of Ranvier to the next. The propagation of a signal in the squid axon can be modelled as a travelling wave solution of the Hodgkin-Huxley system. To model the saltatory conduction in human nerves a different type of model is necessary. This kind of conduction is often studied experimentally in amphibians (specifically, frogs). A model of such phenomena was introduced by FitzHugh (Biophys J. 2, 11). The model couples copies of the HH model, one for each node, to a diffusion equation with damping describing the propagation of the voltage along the membrane in each internode (i.e. the section of the axon between two adjacent nodes).The coupling is achieved by imposing transition conditions at the nodes. At one of these, where the axon is stimulated electrically, an inhomogeneous Neumann boundary condition is assumed. At the others it is assumed that the potential is continuous and that its spatial derivative satisfies a transition condition. The central themes of the paper are a numerical solution of the equations and a comparison of the results with experimental data.

In fact the conduction mechanisms at the node of Ranvier seem to somewhat different from those in the squid axon. This was studied in detail in the frog Xenopus by Frankenhaeuser and Huxley. They introduced a modified version of the HH model in order to incorporate these differences (J. Physiol. 171, 302). In the HH model the variables other than the voltage are three gating variables which can be interpreted with hindsight as corresponding to the processes by which voltage-gated sodium channels open and enter the inactive state and the process by which voltage-gated potassium channels open. In the Frankenhaeuser-Huxley model a fourth gating variable is added. I do not know if hindsight has provided it with a clear interpretation. It is referred to in the original paper as a ‘non-specific current’. Goldman and Albus (Biophys. J. 8, 596) presented an improved version of Fitzhugh’s model for saltatory conduction where the HH model was replaced by the Frankenhaeuser-Huxley model. Apart from numerical calculations they did a dimensional analysis, leading to a derivation of the relation between conduction speed and the diameter of the nerve fibre.

Koles and Rasminsky (J. Physiol. 227, 351) investigated the effect of (partial) demyelination on nerve conduction in the framework of a model of the Goldman-Albus type. In particular, they studied numerically the effects of temperature and sodium concentration on the effectiveness of nerve conduction and the occurrence of a conduction block, obtaining results broadly compatible with experimental data. They also compared results of demyelination close to the nodes or further away from them.

In a paper of Bostock and Grafe (J. Physiol. 365, 239) it was argued that the poor functioning of demyelinated nerves after repeated stimulation is due to hyperpolarization of the cell membrane and that this results from the activity of the sodium-potassium pump. This hyperpolarization occurs in normal nerve fibres but has no major effects on the function of the nerve. In general it brings the size of the depolarization required to fire a node closer to that actually produced by the previous node. The problem in the demyelinated case is that these two quanitities can be close to each other to start with so that a small change due to hyperpolarization can cause firing to fail. Note that this type of effect cannot be captured by the mathematical models discussed above since as they stand they do not include the sodium-potassium pump.

Now I have completed the homework I left myself at the end of the post ‘Migrating ion channels‘ and I have come back to my starting point having gained some height in terms of my level of understanding.

When is a dynamical system empty of information?

April 16, 2009

This post is concerned with dynamical systems in the sense of systems of ordinary differential equations. It is well-known that as soon as the dimension of the system is at least three general dynamical systems can include very complicated behaviour (chaos, strange attractors etc.) which defy detailed understanding in the absence of extra restrictions. In many applications of dynamical systems to the real world (for instance to biology) we encounter the following situation. A dynamical system is postulated which involves a lot of unknowns (in particular more than three) and depends on many parameters. There is only limited information about the parameters which means that in reality we are confronted with a large class of systems. Moreover there is only a limited amount of information about which initial data lead to solutions relevant to the application of interest. This means that in effect a large class of solutions is involved when an attempt is made to study the problem. The door is wide open to the daunting complexity of general solutions of general systems. The standard reaction is to make some choice of parameters and initial data and solve for the evolution on the computer. Since only a finite number of cases can be computed there is no guarantee that what comes out is typical of what happens in general or relevant for the application.

It might be hoped that general properties of dynamical systems which arise in particular wide classes of applications might reduce some of these difficulties. An example of a class of this type is defined by the systems which describe the competition of species in ecology. In 1976 Stephen Smale (J. Math. Biol. 3, 5) showed in a three-page paper that the restriction to this type of system does not, on its own, lead to any improvement whatsoever. In more detail what he did was the following. He considered a system of ODE of the form \dot x_i=x_if_i(x) where the f_i are smooth and \frac{\partial f_i}{\partial x_j}<0 for all i,j. The inequality expresses the competitive property while the factor x_i on the right hand side ensures that if the population of a species is initially zero it stays zero. The region where all x_i are non-negative, which is the biologically interesting region, is invariant. Finally he assumes that the functions f_i are negative outside a large ball so that solutions cannot escape to infinity. Let \Delta_1 be the set where all x_i are non-negative and \sum_i x_i=1. Smale shows that there is a dynamical system in the class under discussion here where \Delta_1 is an invariant subset, all solutions converge to \Delta_1 as t\to\infty and the dynamical system on \Delta_1 is arbitrary. Thus any complication of a general dynamical system of dimension n can be built into a model of the dynamics of n+1 competing species. There is no fundamental simplification if n is at least three. Soon afterwards an analogous result was proved for a class of systems in chemical physics describing continuous flow stirred tank reactors by Perelson and Wallwork (J. Chem. Phys. 66, 4390). A paper of Kaplan and Yorke (American Naturalist 111, 1030) mentions the relevance of Smale’s result to competitive exclusion, a topic mensioned in a previous post.

It is not my intention here to spread pessimism. I hope and believe that dynamical systems can be of immense value in many scientific applications and in biology and medicine in particular. I just want to emphasize that the act of writing down a dynamical system satisfying a few general assumptions is no guarantee that some interesting information about a scientific problem has been implemented. I would add that being able to calculate a few solutions on the computer is not a sufficient reason to be confident that something has been achieved. It should be a matter of priority to develop ways of ensuring that systems proposed do contain information.