Archive for February, 2012

Entrainment by oscillations

February 25, 2012

In the book of Goldbeter which I have mentioned in several recent posts a concept which occurs repeatedly is that of entrainment. While looking for some more information about this topic I found a paper of Russo, di Bernardo and Sontag (PloS Computational Biology 4, e1000739) which gives an insightful treatment of the subject. The basic idea is to consider two systems which are coupled in some way and to consider the influence of oscillations in one system on the behaviour of the other. It is easy to see how this might be translated into a problem expressed in terms of dynamical systems. A classical example related to this is contained in a story about Christiaan Huygens who was, among other things, the inventor of the pendulum clock in the mid 17th century. Apparently he did not construct clocks himself but had them made by others according to his plans. The well-known story is that he noticed that when two pendulum clocks were placed next to each other the phase of their oscillations became synchronized with, say, one always at the leftmost point of its swing when the other was at the rightmost. Another example is that of the circadian rhythm. There is a 24 hour rhythm in our body and it is interesting to know whether it comes from an intrinsic oscillator or not. Experiments with subjects isolated from the usual rhythm of day and night show that there is an intrinsic oscillator but that its period is closer to 25 hours. Under normal circumstances its period is brought to 24 hours due to the cycle of day and night by entrainment.

The particular mathematical set-up considered in the paper of Russo et. al. is the following. Consider an autonomous dynamical system containing some parameters. Now replace one or more of those parameters by functions of time with period $T$. If solutions of the original system have a suitable tendency to converge to a stationary solution for a given choice of the parameters then solutions of the resulting non-autonomous system converge to periodic solutions of period $T$. In the papers there are nice plots of numerical simulations which give a striking picture of this behaviour. The central result of the paper is a theorem which guarantees this type of behaviour under certain hypotheses. As pointed out in the paper verifying these hypotheses has some similarity to finding a Lyapunov function for an autonomous system. The positive side is that if it can be done it is possible to get strong conclusions. The negative side is that verifying the hypotheses is generally a matter of trial and error. There is no algorithm available for doing that.

The criterion is dependent on the choice of a matrix norm. This is used to define a quantity called the matrix measure $\mu(A)$ of a matrix $A$. The criterion is that the Jacobian of the function defining the dynamical system should have a matrix measure which is bounded above by a negative constant. In that case the system is said to be infinitesimally contracting. The matrix measure is defined by a limiting procedure, $\mu(A)=\lim_{h\to 0}\frac{1}{h}(\|I+hA\|-1)$, but for particular choices of the matrix norm it is possible to calculate in a purely algebraic way. I have no intuitive feeling for what this definition means.

La vie oscillatoire

February 16, 2012

I have continued reading the book ‘La Vie Oscillatoire’ and I have learned many interesting things. Some of them were things I was aware of on some level already but they have now become clearer. Others were quite new to me. Some of them have to do with biology, some with mathematics. Chapter 5 is concerned with the secretion of hormones. I had the naive view that the effect of a hormone was due to its overall level. In reality frequency encoding is important for many hormone signals. For instance the triggering of ovulation is dependent on having the right kind of oscillatory signals and there is a therapy for infertility based on delivering a hormone in an appropriate oscillatory way. The menstrual cycle makes it natural to think about oscillations in that context but the oscillations just mentioned are on timescales of one hour rather than one month. Similar control mechanisms seem to apply to many other hormones. In the book biological systems are often compared across very different animals or other living organisms. One interesting conclusion is that the fact that the human menstrual cyle takes about one month is an accident. This deals a blow to more or less mystical ideas relating this cycle with the moon. A recurring theme in the book is the way in which frequency modulated signals are decoded in biological systems. Here I feel that I am at the edge of a part of the theory of dynamical systems which I should learn a lot more about.

The theme of Chapter 6 is rhythms in the brain. Elsewhere in this blog I have written about the Hodgkin-Huxley model several times. This can be used to describe the propagation of an action potential along an axon. However this is not its only application. It can be also be used to describe the oscillatory behaviour of individual neurons. The basic phenomena involved are the flow of sodium and potassium ions across the cell membrane. Calcium ions also play a role in some cases. A mathematical phenomenon which comes up in this discussion is that of bursting oscillations. Since I was not previously familiar with that I now read up on it a bit. My main source was the book ‘Mathematical Physiology’ by Keener and Sneyd. A variable in a dynamical system is said to display bursting oscillations when it has the following type of behaviour. There is a period where it changes very little followed by a period where it oscillates with high frequency and fairly constant amplitude. Then it returns to the quiescent phase it started in. It goes through this cycle repeatedly. The minimum ingredients required for a dynamical system to show this type of behaviour are dimension at least three and equations with different timescales. One example occurs in a model for the production of insulin by the $\beta$-cells of the pancreas. In this case there are one slow and two fast variables. Heuristically the slow variable is first thought of as a parameter. As it is varied the dynamics of the fast system changes. For certain parameter values the fast system has a stable steady state. Starting from this point the slow variable changes in such a way that this steady vanishes in a fold bifurcation. The solution then moves quickly to being close to a limit cycle of the fast system and stays there for a while. The slow variable then changes in such a way that the limit cycle is destroyed in a homoclinic bifurcation. By that time the stable steady state has reappeared and the solution can jump back to it. Burst oscillations are classified into three types and the scenario just sketched is Type I. Coming back to the brain (or at least to neurons) a popular experimental system is the sea slug Aplysia. An isolated neuron of this organism can exhibit bursting oscillations. In this case (Type II) there are two slow variables. The oscillatory phase starts and ends with a homoclinic bifurcation. In Type III a period of oscillations is bounded by two subcritical Hopf bifurcations.

Another class of oscillations arises from the collective behaviour of a small number (up to about thirty) neurons. This has been studied for instance in the context of the motion of the snail Lymnaea. In the human brain there is a great variety of oscillations, producing characteristic traces in the EEG. They include a number of type of waves named after letters of the Greek alphabet, starting with $\alpha$, some of which have entered popular culture. I would like to learn more about them sometime, but the day for that has not yet arrived.