## La vie oscillatoire

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.