Communication delays are common in many complex systems. It has been shown that these delays cannot be neglected when they are long enough compared to other timescales in the system. In systems of coupled phase oscillators discrete delays in the coupling give rise to effects such as multistability of steady states. However, variability in the communication times inherent to many processes suggests that the description with discrete delays maybe insufficient to capture all effects of delays. An interesting example of the effects of communication delays is found during embryonic development of vertebrates. A clock based on biochemical reactions inside cells provides the periodicity for the successive and robust formation of somites, the embryonic precursors of vertebrae, ribs and some skeletal muscle. Experiments show that these cellular clocks communicate in order to synchronize their behavior. However, in cellular systems, fluctuations and stochastic processes introduce a variability in the communication times. Here we account for such variability by considering the effects of distributed delays. Our approach takes into account entire intervals of past states, and weights them according to a delay distribution. We find that the stability of the fully synchronized steady state with zero phase lag does not depend on the shape of the delay distribution, but the dynamics when responding to small perturbations about this steady state do. Depending on the mean of the delay distribution, a change in its shape can enhance or reduce the ability of these systems to respond to small perturbations about the phase-locked steady state, as compared to a discrete delay with a value equal to this mean. For synchronized steady states with non-zero phase lag we find that the stability of the steady state can be altered by changing the shape of the delay distribution. We conclude that the response to a perturbation in systems of phase oscillators coupled with discrete delays has a sharper functional dependence on the mean delay than in systems with distributed delays in the coupling. The strong dependence of the coupling on the mean delay time is partially averaged out by distributed delays that take into account intervals of the past.

Communication delays are common in many complex systems. It has been shown that these delays cannot be neglected when they are long enough compared to other timescales in the system. In systems of coupled phase oscillators discrete delays in the coupling give rise to effects such as multistability of steady states. However, variability in the communication times inherent to many processes suggests that the description with discrete delays maybe insufficient to capture all effects of delays. An interesting example of the effects of communication delays is found during embryonic development of vertebrates. A clock based on biochemical reactions inside cells provides the periodicity for the successive and robust formation of somites, the embryonic precursors of vertebrae, ribs and some skeletal muscle. Experiments show that these cellular clocks communicate in order to synchronize their behavior. However, in cellular systems, fluctuations and stochastic processes introduce a variability in the communication times. Here we account for such variability by considering the effects of distributed delays. Our approach takes into account entire intervals of past states, and weights them according to a delay distribution. We find that the stability of the fully synchronized steady state with zero phase lag does not depend on the shape of the delay distribution, but the dynamics when responding to small perturbations about this steady state do. Depending on the mean of the delay distribution, a change in its shape can enhance or reduce the ability of these systems to respond to small perturbations about the phase-locked steady state, as compared to a discrete delay with a value equal to this mean. For synchronized steady states with non-zero phase lag we find that the stability of the steady state can be altered by changing the shape of the delay distribution. We conclude that the response to a perturbation in systems of phase oscillators coupled with discrete delays has a sharper functional dependence on the mean delay than in systems with distributed delays in the coupling. The strong dependence of the coupling on the mean delay time is partially averaged out by distributed delays that take into account intervals of the past.:Abstract i Acknowledgement iii I. INTRODUCTION 1. Coupled Phase Oscillators Enter the Stage 5 1.1. Adjusting rhythms – synchronization . . . . . . . . . . . . . . . . . . 5 1.2. Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3. Reducing variables – phase models . . . . . . . . . . . . . . . . . . . . 9 1.4. The Kuramoto order parameter . . . . . . . . . . . . . . . . . . . . . . 10 1.5. Who talks to whom – coupling topologies . . . . . . . . . . . . . . . . 12 2. Coupled Phase Oscillators with Delay in the Coupling 15 2.1. Communication needs time – coupling delays . . . . . . . . . . . . . . 15 2.1.1. Discrete delays consider one past time . . . . . . . . . . . . . . 16 2.1.2. Distributed delays consider multiple past times . . . . . . . . 17 2.2. Coupled phase oscillators with discrete delay . . . . . . . . . . . . . . 18 2.2.1. Phase locked steady states with no phase lags . . . . . . . . . 18 2.2.2. m-twist solutions: phase-locked steady states with phase lags 21 3. The Vertebrate Segmentation Clock – What Provides the Rhythm? 25 3.1. The clock and wavefront mechanism . . . . . . . . . . . . . . . . . . . 26 3.2. Cyclic gene expression on the cellular and the tissue level . . . . . . 27 3.3. Coupling by Delta-Notch signalling . . . . . . . . . . . . . . . . . . . . 29 3.4. The Delayed Coupling Theory . . . . . . . . . . . . . . . . . . . . . . . 30 3.5. Discrete delay is an approximation – is it sufficient? . . . . . . . . . 32 4. Outline of the Thesis 33 II. DISTRIBUTED DELAYS 5. Setting the Stage for Distributed Delays 37 5.1. Model equations with distributed delays . . . . . . . . . . . . . . . . . 37 5.2. How we include distributed delays . . . . . . . . . . . . . . . . . . . . 38 5.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6. The Phase-Locked Steady State Solution 41 6.1. Global frequency of phase-locked steady states . . . . . . . . . . . . . 41 6.2. Linear stability of the steady state . . . . . . . . . . . . . . . . . . . . 42 6.3. Linear dynamics of the perturbation – the characteristic equation . 43 6.4. Summary and application to the Delayed Coupling Theory . . . . . . 50 7. Dynamics Close to the Phase-Locked Steady State 53 7.1. The response to small perturbations . . . . . . . . . . . . . . . . . . . 53 7.2. Relation between order parameter and perturbation modes . . . . . 54 7.3. Perturbation dynamics in mean-field coupled systems . . . . . . . . 56 7.4. Nearest neighbour coupling with periodic boundary conditions . . . 62 7.4.1. How variance and skewness influence synchrony dynamics . 73 7.4.2. The dependence of synchrony dynamics on the number of oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.5. Synchrony dynamics in systems with arbitrary coupling topologies . 88 7.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8. The m-twist Steady State Solution on a Ring 95 8.1. Global frequency of m-twist steady states . . . . . . . . . . . . . . . . 95 8.2. Linear stability of m-twist steady states . . . . . . . . . . . . . . . . . 97 8.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9. Dynamics Approaching the m-twist Steady States 105 9.1. Relation between order parameter and perturbation modes . . . . . 105 9.2. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.Conclusions and Outlook 111 vi III. APPENDICES A. 119 A.1. Distribution composed of two adjacent boxcar functions . . . . . . . 119 A.2. The gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 124 A.3. Distribution composed of two Dirac delta peaks . . . . . . . . . . . . 125 A.4. Gerschgorin’s circle theorem . . . . . . . . . . . . . . . . . . . . . . . . 127 A.5. The Lambert W function . . . . . . . . . . . . . . . . . . . . . . . . . . 127 A.6. Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 B. Simulation methods 129