Phase Transitions in Large Oscillator Lattices

This thesis deals with large networks of limit cycle oscillators. A limit cycle oscillator is a dynamical system that has a periodic attractor in phase space, and is defined in continuous time. To each such oscillator one can associate a natural frequency. Virtually all biological systems that show periodic oscillations can be seen as limit cycle oscillators. The same is true for oscillating mechanical or electrical systems that are driven and damped. We study lattices of limit cycle oscillators in the thermodynamic limit where the number of oscillators goes to infinity. The natural frequencies are assigned randomly in the lattice from a given density function. If the width s of this density function is small enough, and the coupling strength g between the oscillators is large enough, it may happen that a non-zero portion r of the oscillators attain the same frequency. A phase transition towards temporal order takes place when a system parameter (e.g. s or g) is changed so that r becomes non-zero. Such a phase transition can be seen, for example, in an applauding theatre audience. Suddenly everyone may find themselves clapping in unison. Another example is the onset of an epileptic fit. Then an abnormally large portion of the brain cells synchronise their electrical activity.

We study one-dimensional oscillator chains for two different types of oscillator models. One type applies generally in the limits of small s and small g. The other type applies for many kinds of oscillators that interact with short pulses. In both cases we prove analytically that there is a critical coupling strength g (at a given s), at which the system switches from no frequency order (r = 0) to perfect order (r = 1). We also study two-dimensional oscillator lattices numerically. The oscillators interact with short pulses. We find that there is one phase transition to partial frequency order (0 < r < 1) as g increases, and a second transition to perfect frequency order (r = 1). Between these phase transitions the system seems critical, with spatial self-similarity and infinite transient time.

A more applied part of the study deals with the sinus node. The sinus node is the natural pacemaker of the heart. It consists of millions of cells, each of which is a limit cycle oscillator that fires electrical signals with its own natural frequency. All these cells attain the same working frequency in the healthy heart, and thus stimulate the cardiac muscle to contract regularly. By means of simulations we investigate which cardiac arrhythmias may arise from a backward phase transition at which this perfect frequency order is lost. We find that most cardiac rhythm disorders associated with a malfunctioning sinus node can be produced by this condition. We also put forward the hypothesis that some features of the sinus node have evolved to protect it from going through such a backward phase transition. We support this hypothesis by means of simulation and argumentation.