Archive for the ‘neurology’ Category

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.

The FitzHugh-Nagumo model

March 30, 2009

As mentioned in a previous post the FitzHugh-Nagumo model is a simplified version of the Hodgkin-Huxley model describing the propagation of signals in nerve cells. In this post only the spatially homogeneous case is discussed. It is possible to consider a corresponding system including a diffusion term for one of the unknowns (x in the system below), a subject touched on in a previous post. A different ODE reduction can be obtained by looking at traveling wave solutions. The equations in the homogeneous case form a system of two ODE. It is supposed to capture the essential qualitative behaviour of the system of four ODE defining the Hodgkin-Huxley model while otherwise being as simple as possible. Here the notation of the original paper of FitzHugh (Biophys. J. 1, 445) will be used.The system is

\dot x=c(y+x-x^3/3+z)

\dot y=-(x-a+by)/c

with constant parameters a, b and c. The quantity z is in general a prescribed function of t. The unknown x corresponds to the voltage in the Hodgkin-Huxley system while y corresponds to the other variables. The parameter z plays the role of the external current in the Hodgkin-Huxley system. The relation to the van der Pol oscillator, as written in the previous post, is easily seen. The parameters a and b and the function z are new. If they are set to zero we get the van der Pol equation in slightly different variables. FitzHugh gives an intuitive description of the dynamics with phase plane pictures. Depending on the values of the parameters there may be a stable stationary solution which is a global attractor, or an unstable stationary solution together with a stable periodic solution. There is more discussion of the qualitative behaviour in Murray’s book ‘Mathematical Biology’.

Theorems about the van der Pol oscillator can be found in many textbooks. Corresponding material on the FitzHugh-Nagumo model seems to be much rarer. There is, nevertheless, a big literature out there. There is a thesis of Matthias Ringqvist online which collects a lot of interesting material on the subject and can serve as a point of entry to the literature for the uninitiated, like myself. He considers the properties of a dynamical system which contains the FitzHugh-Nagumo model as a special case. He motivates this generalization by noting that the more general system includes a number of different systems of interest in different problems in applied mathematics. He discusses the presence of periodic solutions in various parameter regimes and the bifurcations they are involved in. Hopf bifurcations play an important role. Another type of bifurcation which occurs in this context is the Bautin bifurcation. This differs from Hopf case in that one of the non-degeneracy conditions (the one which does not only depend on the linearization at the point of interest) fails. There can be coexistence of more than one periodic solution.

To what extent does the FitzHugh-Nagumo model capture the dynamics present in the Hodgkin-Huxley model? Ringqvist mentions numerical work of Guckenheimer and Oliva from 2002 which suggests that the HH model exhibits chaotic dynamics.The authors do not claim to have proved rigorously that chaos is present but this is a warning that it would be foolish to think that the dynamics of the HH system might be ‘essentially understood’. Chaos is observed in the forced van der Pol oscillator, i.e. the system obtained from the van der Pol oscillator by adding a prescribed function of time. It is more impressive for me to see chaos coming up in a system which is autonomous and which occurs naturally in an application. Of course the Lorenz system, one of the icons of chaos, satisfies the first condition and, at least in a weak sense, the second.

Migrating ion channels

November 11, 2008

It is common in people suffering from multiple sclerosis that the distance they can walk without a rest is a lot less than what a healthy person can do. While walking a certain kind of fatigue arises which is different from the ordinary tiredness of muscles. Continuing for long enough leads to a state where the muscles just do not seem to respond any more. If this state has been approached too closely it can take many hours for normality to be restored. What is the mechanism of this kind of fatigue? It does not seem to be mentioned in most texts on the subject.

Some years ago I read an account of a patient suffering from the neuromuscular disease myasthenia gravis. She could walk normally in the morning but was confined to a wheelchair in the afternoon. Superficially this sounds like an extreme version of the fatigue in MS mentioned above. In this case more is known about what is going on. Myasthenia gravis is an autoimmune disease where the target of the attack mounted by the immune system is the acetylcholine receptors of muscles. Normally acetylcholine released by nerve cells carries the signal to the muscle cells that they should contract. This is interfered with by antibodies against the receptors. The result of this is that when the nerves responsible for directing the action of muscles are stimulated too often there is not enough acetylcholine present. It then takes a long time before the system can recover. It is clear that this cannot be the mechanism working in MS but I wondered whether in that case neurotransmitters could be depleted in some other way. In fact it seems that this is not the right explanation. In the book ‘McAlpine’s Multiple Sclerosis’ I found another alternative for which there is some evidence. It has to do with ion channels.

The mechanism of propagation of electrical signals in nerve cells was discovered in the early 1950’s by Alan Hodgkin and Andrew Huxley. It got them a Nobel prize in 1963. The basis of the phenomenon are flows of potassium and sodium ions across the cell membrane. In the resting state of the axon there is an electrical potential across the cell membrane which results from the difference in concentration of potassium ions on both sides. When a nerve signal (action potential) passes the permeability of the membrane to sodium and potassium ions changes in response to the changes in potential. This is a non-trivial dynamical process. A natural mathematical model would be a system of reaction-diffusion equations which admits travelling wave solutions. The study of this may be simplified by going to the “space-clamped” case. This comes down to studying solutions which only depend on time. It gives rise to a system of nonlinear ordinary differential equations. A possibility of studying this situation experimentally was found in the giant axon of the squid Loligo. This was used by Hodgkin and Huxley to get information about the coefficients of the ODE system. Once they had that information they had to solve the equations numerically in order to compare theory with experiment. In those days this numerical work had to be done by hand although a couple of years later they were able to apply some of the first ever computers, then being developed in Cambridge. In the Nobel lecture of Huxley we find a vivid description of doing this calculation. He says: “This was a laborious business … a propagated action potential took a matter of weeks. But it was often quite exciting. … an important lesson I learned from these manual computations was the complete inadequacy of one’s intuition in trying to deal with a system of this degree of complexity.” The Hodgkin-Huxley model is one of the most notable successes of mathematical biology. It is an example of the concept of an excitable system mentioned in a previous post. Now we know that the changes of permeability of the membrane are due to ion channels which regulate the movement of ions through the cell membrane. Ion channels are made by molecules embedded in the membrane which undergo conformational changes as a result of various stimuli. In the case of nerve conduction the stimuli are electrical. These changes are not deterministic. What changes with the applied potential is the probability of a channel being open, closed or inactivated.

The propagation speed of nerve impulses increases with the diameter of the axon and the unusually large diameter of the axon in the squid is its method of achieving fast signalling. Vertebrates like ourselves have found another method, which is to insulate the axon using myelin. It is the myelin which is damaged in MS. The rate of travel of nerve signals along the axon is not uniform. There are small regions along the axon, the nodes of Ranvier, where myelin is absent. The nerve impulse jumps from one node to the next. Associated to this is the fact that under normal circumstances most of the sodium and potassium channels are concentrated near the nodes of Ranvier. They are kept there by the oligodendrocytes which are also the cells which produce myelin. In fact myelin consists of layers of cell membrane of the oligodendrocytes.

In MS myelin is removed from the axons and nerve conduction no longer works properly, if it works at all in a given axon. There may be remyelination but this generally only produces a thin layer of myelin whose insulating properties are limited. Now it seems, and here I come back to what I read in ‘McAlpine’s Multiple Sclerosis’, that the nerve cells have developed another strategy to restore conduction to some extent. This is that ion channels migrate along the axon from the nodes of Ranvier. The general mechanism of conduction is then a different one from that of the fully myelinated axon. One disadvantage is that the nerve conduction is not so fast. The other is that ions may accumulate on one side of the cell membrane and the resting potential cannot be reestablished in an efficient way. Each firing of the nerve cell only results a small change in the concentration of ions after it has passed. If, however, these small changes are not being corrected for regularly they can add up to a serious change. This is a possible explanation for the fatigue. I would not claim that this is definitive explanation. On p. 628ff of ‘McAlpine’s Multiple Sclerosis’ a zoo of different ion channels is dicussed. From the short section on fatigue on p. 643 it seems clear that there are a lot of theories around. I would like to penetrate into the matter further.

The blood-brain barrier

August 27, 2008

Paul Ehrlich can be considered the founder of immunology. He discovered much about the variety of different blood cells using staining techniques. In 1885 he observed that a dye injected into the bloodstream stained all body tissues with the notable exception of the central nervous system (CNS). This was later interpreted as an indication that there is a kind of partition, the blood-brain barrier (BBB), which prevents the dye from entering the brain. This was later confirmed by showing that a dye injected into the cerebrospinal fluid stains only the tissues of the CNS and nothing else.

The central nervous system consists of the brain, the spinal cord and certain particular nerves such as the optic nerve. All the other nerves in the body constitute the peripheral nervous system. What is this BBB which encloses the CNS? The walls of normal capillaries, the smallest blood vessels, consist of endothelial cells. These walls allow many kinds of chemical substances and cells to move rather freely between the bloodstream and the tissues. The smallest blood vessels adjacent to the CNS are different. There the endothelial cells are stuck together in a much stronger way through so-called tight junctions. The walls of these vessels let a much more restricted variety of cells and substances pass. This is the basis of the BBB but in fact it is a much more complicated structure. For instance, it is supported on the side facing the CNS by processes of astrocytes called ‘feet’. An idea of the complexity of the matter can be got from a review article of Hawkins and Davis (Pharmacol. Rev. 57, 173). The BBB acts to prevent pathogens such as bacteria entering the brain. It also keeps many elements of the immune system out. Breakdown of the normal function of the BBB seems to be a key element in multiple sclerosis. It results in immune cells getting access to the CNS and causing damage there. Some drugs use to treat MS have the property of sealing the BBB (see the previous post) Breakdown of the BBB has also been suggested to play a role, whether as cause or effect of the main events, in other illnesses such as Alzheimer’s disease and stroke. The BBB also has an important role to play in the treatment of diseases of the brain such as tumours. In some cases doctors would like to administer certain drugs to tissues within the brain and this is hindered by the BBB. Thus there is interest in finding controlled ways of increasing its permeability.

Balo’s concentric sclerosis

June 6, 2008

A mathematical model for this rare disease was presented in a paper by Khonsari and Calvez in the journal PLoS ONE, abbreviated as [KC] from now on) and this is my primary source of information for this post. In addition I have used the standard work ‘McAlpine’s Multiple Sclerosis‘ as a reference. The disease was first described by Balo in 1927 and he himself already mentioned a possible connection to Liesegang rings. It is relatively common in the Philippines compared to other parts of the world. The model proposed in [KC] is a system of PDE similar to those used to describe chemotaxis, e.g. the Keller-Segel model. An important difference from reaction diffusion equations is the presence of cross diffusion, i.e. the flux of one diffusive species depends on the gradient of another. The system of [KC] also contains an ODE describing the density of oligodendocytes (the cells responsible for production and maintenance of myelin) which is analogous to the equation describing the formation of a precipitate in certain models of Liesegang rings. Computer simulations using this model show the production of rings for certain values of the parameters involved and not for others. The authors of [KC] compare their model to one based on another mechanism (which has discontinuous coefficients) and argue that their model performs better.

In considering the relation of Balo’s disease to MS an interesting related question is whether MS is one disease or many. A classification into four types has been suggested with different pathological mechanisms. Balo’s disease most closely resembles Type III. If this distinction could be pushed further it might lead to a global increase in understanding. Perhaps different forms require different therapies. Characteristic features of Type III include destruction of oligodendrocytes and signs that the oxygen supply in the area where damage is taking place is not sufficient. Two different general pictures of the formation of the rings of demyelinated tissue have been proposed in the literature. In one of these the demyelination only takes place in certain rings while in the other there is demyelination everywhere followed by remyelination in certainrings. Remyelination, a process by which the myelin sheaths of nerve cells are repaired is known to play an important role in MS. The quality of the new coatings is generally poorer than that of the original ones. Understanding the dynamics of the interplay of demyelination and remyelination is a crucial issue in understanding the progression of MS.

Another disease related to MS (or a special form of it, the question arises here too, it resembles Type II) is neuromyelitis optica (NMO), also known as Devic’s disease. This disease is automimmune in nature and a specific target of autoimmune attacks has been identified, the protein aquaporin 4. This knowledge is useful in distinguishing NMO from typical MS by determining the concentration of antibodies against aquaporin 4 in the blood. For more details see this article in PloS Medicine.


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