The FitzHugh-Nagumo model

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 ( 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

with constant parameters , and . The quantity is in general a prescribed function of . The unknown corresponds to the voltage in the Hodgkin-Huxley system while corresponds to the other variables. The parameter 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 and and the function 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.

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Shilnikov’s theorems on bifurcation from a homoclinic orbit

If is a stationary solution of a dynamical system a homoclinic orbit based at is the image of a solution which converges to both as and . Homoclinic orbits are typically unstable. In other words, if the given dynamical system is embedded into a family depending smoothly on a parameter then generically the qualitative behaviour of solutions for is different from that of the solutions of the original system. The homoclinic loop breaks. Consider first the case of a two-dimensional dynamical system and a hyperbolic stationary solution. Then it can be shown (Andronov-Leontovich theorem) that generically there exist periodic solutions near . Information can also be obtained about the stability of these periodic solutions. One of the genericity assumptions is that the sum of the two eigenvalues of the linearization is non-zero. In higher dimensions generalizations of these phenomena were studied in detail by Leonid Shilnikov in the 1960’s. Here I present some of the ideas involved. My main source for this is chapter 6 of the book ‘Elements of applied bifurcation theory’ by Y. Kuznetzov. Consider a homoclinic orbit based at a hyperbolic stationary solution where all eigenvalues of the linearization are real and of multiplicity one. Suppose that exactly one of them is positive. Label them in decreasing order , , . The eigenvalue is referred to as the leading eigenvalue. Let . It is called the saddle quantity. Under some genericity assumptions, which include it can be shown that a periodic orbit bifurcates from the homoclinic orbit. When the periodic orbit is stable while for positive it is of saddle type. When all eigenvalues are real higher dimensional systems may be reduced to the three-dimensional case by a kind of centre manifold construction.

Kuznetsov’s book discusses applications of the above general constructions to the example of travelling wave solutions of the FitzHugh-Nagumo system mentioned in a previous post. Different types of bifurcations can be used to classify different patterns of firing of neurons. This was already done by Hodgkin and is discussed in the book ‘Dynamical Systems in Neuroscience’ by E. Ishikevich, of which the first chapter is available on the Internet. The bifurcations which play a central role are different from those discussed above. One of them, called SNIC bifurcation by Ishikevich, arises from a homoclinic orbit based at a non-hyperbolic stationary solution and is discussed (under a different name) in Chapter 7 of Kuznetzov. There is a distinction between Class 1 and Class 2 excitability which reminded me of type I and type II critical collapse (see below) due to the similarity of the corresponding diagrams. I doubt if there is a deeper connection – the quantities plotted in the two cases are of quite a different nature.

The type of bifurcations studied by Shilnikov have been implicated in phenomena observed in a field of study in general relativity known as critical collapse. This concerns the gravitational collapse of matter. A typical scenario is as follows. A small amount of matter (‘case of small data’) disperses to infinity since its gravitational field is too weak to hold it together. A very large amount of matter collapses to a black hole since the effect of the gravitational field dominates. What happens for data of intermediate size? This question was investigated intensively using numerical methods by M. Choptuik in the case a self-gravitating scalar field. Since then many authors have investigated many related cases. The results are almost entirely numerical and it would be nice to have some theorems. Choptuik found that there is a transition between the small data and large data behaviour where there is a unique attractor solution giving rise to universal properties of the evolution. There is still no proof of the existence of ‘the Choptuik solution’ but there is strong numerical evidence pointing to an object of this kind. It is discretely self-similar (DSS). Equivalently, it is periodic when expressed in dimensionless variables. With other choices of matter model attractors are encountered which are continuously self-similar (CSS). In terms of dimensionless variables these become stationary solutions. These objects are called critical solutions. Both of these correspond to what is called type II critical collapse. The mass of the black hole formed approaches zero as the threshold between collapse and dispersion is approached. In type I collapse it tends to a non-zero value and the critical solution is stationary in the original variables. For more information see the review article of Gundlach and Martín-García.

There are examples where the existence of CSS critical solutions has been proved, although it has not been proved that they are attractors. In the case of DSS critical solutions, on the other hand, there is not a single existence theorem available. How might a theorem of this kind be proved? One possible idea would be to obtain a DSS solution from a CSS solution by bifurcation in a one-parameter family. The simplest scenario would be a Hopf bifurcation but no example of this kind has yet been found. A more complicated possibility is a bifurcation of the Shilnikov type described above. In work of Aichelburg, Bizon and Tabor (gr-qc/0512136) the authors gave numerical evidence for the occurrence of this type of bifurcation in a model of critical collapse where the matter model is a wave map with values in the three-sphere. Perhaps this could serve as a basis for an existence theorem. There are however major obstacles. It is, for instance, not clear how a rigorous formulation as a dynamical system should be set up. Moreover it is clear that whatever this dynamical system is it must be infinite-dimensional. In the most optimistic case some kind of centre manifold or Lyapunov-Schmidt reduction might be used to get back to a finite-dimensional (even low-dimensional) system. Another problem, and perhaps the most serious one, is that this is a non-local bifurcation which cannot be identified by local calculations at the stationary point.

Lucid dreams

A lucid dream is one in which the dreamer is conscious that he or she is dreaming. The name arose by translation from the French ‘rêve lucide’, a concept which was introduced by Léon d’Hervey de Saint-Denys. When I was a student I was very interested in dreams and I came across the idea of lucid dreams in a book by Celia Green. I would not recommend that book since it contains too much parapsychology for my taste. In 1992 I spent a term as assistant professor at Syracuse University and I enjoyed exploring the university library. There I found a scientific book on lucid dreaming by Stephen LaBerge. There it is described how this phenomenon has been studied under controlled conditions. A technique available in that context is monitoring the movement of the eyes. A correlation was discovered between the actual movements of the eyes of a dreaming person and the visual impressions they see. After a subject had reported dreaming about watching a game of ping-pong it was noticed that during this time the dreamer’s eyes had shown regular side to side motions. This opened up a possibility of the dreamer to signal to the outside world. It could be agreed that under certain circumstances (for instance when a lucid period began) they would give a signal by consciously moving their eyes in a certain regular way. This could then be recognized on the trace of the machine recording the eye movements. At that time I spent a lot of time concentrating on my dream life and managed to have a number of lucid dreams. Now I have had none for a long time, which is probably due to lack of attention.

During my stay in Bures sur Yvette in 1994-95 I was a frequent visitor in the library of the Centre Pompidou. I was happy to find a copy of the book ‘Les rêves et les moyens de les diriger’ (dreams and ways to direct them) by d’Hervey de Saint-Denys. I knew his name from my previous reading on dreams. d’Hervey de Saint-Denys studied his own dreams intensively starting at a young age. In 1855 the ‘Académie des Sciences morales et politiques’ in Paris had a prize competition on the subject of sleep and dreams. He did not take part but his dissatisfaction with the essays submitted led him to write a book criticizing them and explaining his own ideas. Later the book was reprinted and I was able to buy a copy. It is available from www.oniros.fr. d’Hervey de Saint-Denys was a well-known sinologist in his day and I find it remarkable how few traces he left. (I now learned from Wikipedia that his translations of Chinese poems were the source of the texts of the songs in ‘Das Lied von der Erde’ by Mahler. I also discovered that he is mentioned in Proust’s ‘Sodome et Gomorrhe’, in connection with China rather than dreams.) It is no doubt typical that it is very difficult to get information about any individual from a hundred or more years ago apart from a rather small number of exceptions. With the computer age it will presumably happen that, concerning individuals, the time we can see into the past will increase steadily.

The van der Pol oscillator

The van der Pol oscillator is a model for an electrical circuit which is a simple ODE with a stable periodic solution. The equation is where is a parameter. Only non-negative values of will be considered here. The sign of can be changed by replacing by . For the system reduces to the harmonic oscillator. It is common to reduce the equation to a first order system by defining . The resulting system is , . It is an example of the Liénard system which in general reads . We just need to choose and . In the general case defining a new variable where gives the first order system , . There is a good general theory of the existence of unique stable periodic solutions for Liénard systems. This shows that for any positive value of the van der Pol oscillator has a unique stable limit cycle. The origin is an unstable stationary point. It is of interest to consider the limits and . In the first limit the periodic solution converges to the circle of radius two. In the other limit it converges to what is called a relaxation oscillation. This is a closed curve which consists of two pieces of the nullcline of the original equation and two straight lines of constant joining them. To get an intuitive explanation of this take the quotient of the two equations to get . The only way to make this consistent with tending to infinity is to either let the expression in the brackets tend to zero or to let tend to zero. For large the two straight line pieces are traversed much faster than the other two. Thus the solution spends most of its time on the curved pieces and the period is determined in leading order by the time spent on these pieces. On the curved pieces it is possible (at least formally) to derive a simplified equation by substituting the relation into the evolution equation for . The result is . This equation, determines the limiting behaviour of the period. The limiting value can be determined explicitly. Detailed information can also be obtained about the asymptotic behaviour in the limit . This can be done by the method of averaging which actually originated in the work of van der Pol which was developed further by Krylov and Bogoliubov.

How is the existence and uniqueness of the limit cycle for the van der Pol system proved? It is relatively elementary to prove that solutions repeatedly cross one of the axes at arbitrarily late times. Consider the mapping from one crossing point to the next. A periodic solution corresponds to a fixed point of this mapping. It is shown that the difference in the value is positive for certain initial data and negative for others. It follows by continuity that it is zero for some choice of data and this gives the existence of a periodic solution. Its uniqueness is shown by using a discrete symmetry of the equation.

The van der Pol equation appears to occupy a central position in the field of nonlinear oscillations. For many years now the book ‘Mathematical Biology’ of Murray has been on my bookshelf. I have only read small parts of it in any detail and now I realized that a variety of ideas related to those which are important for the van der Pol equation are explained in a very helpful way there. The van der Pol equation was a basis for the FitzHugh-Nagumo equation, a system of two ODE which is a kind of caricature of the Hodgkin-Huxley model of nerve conduction. Another system which shows relaxation oscillations is the Field-Noyes model. It is a three-dimensional system and so analytically harder to handle than FitzHugh-Nagumo. It is a model for the Belousov-Zhabotinski model of oscillatory chemical reactions. I intend to return to the FitzHugh-Nagumo model in a future post.