Abstract
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 nonzero 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 nonzero. 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 onedimensional 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 twodimensional 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 selfsimilarity 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.
We study onedimensional 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 twodimensional 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 selfsimilarity 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.
Original language  English 

Qualification  Doctor 
Awarding Institution 

Supervisors/Advisors 

Award date  2003 Nov 21 
Publisher  
ISBN (Print)  9162858904 
Publication status  Published  2003 
Bibliographical note
Defence detailsDate: 20031121
Time: 13:30
Place: Lecture Hall B, Dept. of Physics, Sölvegatan 14, Lund Institute of Technology, Lund, Sweden
External reviewer(s)
Name: Mosekilde, Erik
Title: Prof
Affiliation: The Technical University of Denmark, Lyngby

Article: Östborn P.Phase transition to frequency entrainmentin a long chain of pulsecoupled oscillators.Phys. Rev. E 66, 016105, 2002
Article: Östborn P., Åberg S., Ohlén G.Phase transitions towards frequency entrainmentin large oscillator lattices.Phys. Rev. E 68, 015104(R), 2003
Article: Östborn P.Frequency entrainment in long oscillator chainswith random natural frequencies in the weakcoupling limit.(Submitted to Phys. Rev. E)
Article: Östborn P., Wohlfart B., Ohlén G.Arrhythmia as a result of poor intercellularcoupling in the sinus node: a simulation study.J. Theor. Biol. 211(3), pp 201217, 2001
Article: Östborn P., Ohlén G., Wohlfart B.Simulated sinoatrial exit blocks explained bycircle map analysis.J. Theor. Biol. 211(3), pp 219227, 2001
Article: Östborn P.Functional role of the connective tissue and thegap junction distribution in the sinus node.(submitted to J. Theor. Biol.)
The information about affiliations in this record was updated in December 2015.
The record was previously connected to the following departments: Mathematical Physics (Faculty of Technology) (011040002), Classical archaeology and ancient history (015004001)
Subject classification (UKÄ)
 Physical Sciences
Free keywords
 Matematisk och allmän teoretisk fysik
 thermodynamics
 statistical physics
 phase transition
 limit cycle oscillator
 frequency entrainment
 synchronization
 sinus node
 classical mechanics
 Mathematical and general theoretical physics
 arrhythmia
 gravitation
 quantum mechanics
 relativity
 klassisk mekanik
 kvantmekanik
 relativitet
 statistisk fysik
 termodynamik
 Fysicumarkivet A:2003:Östborn