Central theme: Synchronous oscillatory activity is a universal phenomenon that occurs in biological systems ranging from the level of intracellular dynamics to population dynamics across thousands of kilometers. The study of synchrony from a mathematical standpoint has had a very long history going back at least as far as Huygens in the 1600's. However, there are still many unanswered questions involving synchronization that are of central biological importance.
In order to discuss synchronization dynamics of coupled biological oscillators, we need to introduce a few basic concepts. The phase of a periodic oscillator can be defined as a function of time by mapping the state of the system at each point along its limit cycle onto a circle and thus getting a phase between 0 and 2π. Two oscillators are said to be phase locked if the phase difference between the two oscillators remains fixed in time. If the phase difference is zero, the oscillators are synchronized. The temporal dynamics of the phase difference will depend on both the intrinsic dynamics of each oscillator and also on the form of coupling between the two oscillators. There are obvious (and not so obvious) extensions of these concepts to large systems of coupled oscillators for which the dynamics of distributions of phase differences can be considered. The overall theme of the workshop focused on the dynamics of these phase differences.
The dynamics of synchrony has significant biological implications across a range of fields. Because synchrony is such a striking phenomenon, explanations of synchrony can be used to understand the forces controlling biological dynamics, as in studies of spatiotemporal dynamics of childhood diseases. In ecology, the absence of spatial synchrony is often thought to play an important role in persistence of species. In neural systems, synchrony plays a role in the coordination of locomotion and respiration, in the dynamics of diseases such as Parkinson’s and epilepsy, and in sensory processing and cognition. Synchronous oscillations in systems of coupled oscillators are also prevalent in the fields of circadian rhythms, intracellular dynamics, as well as various areas of physics and chemistry and engineering. The importance of synchrony in these wide range of fields has led to large bodies of literature on synchrony that have surprisingly little cross-referencing.
Much of the mathematical theory on synchronization is in special limiting cases. For example, one theoretical framework to study synchronization assumes that input due to coupling is pulsatile and the system quickly returns to its normal periodic cycle before subsequent input arrives. Another framework assumes that coupling is weak (i.e. intrinsic dynamics dominate the effects of weak coupling). While most theories can be extended to include heterogeneity and noise, they are assumed to be sufficiently weak. These assumptions are not appropriate for many biological systems.
The overall goal of the workshop was twofold:
The workshop focused on several interconnected issues.
Theoretical issues: What is the present state of theory in each represented field? How can existing theory be use to obtain a better understanding of how the biological properties of individual oscillators and the coupling dynamics interact to produce observed synchronization patterns? What are the limitations of existing theory and how can the theories be extended? For example, synchrony in biological systems needs to take into account the considerable heterogeneity and stochasticity that are present. Current mathematical theory is well developed for systems of oscillators that are weakly coupled or oscillators that are coupled in a regular fashion. However, real biological systems can have a large number of oscillating components, connected in irregular fashions, with coupling of various strengths. For example, the spatiotemporal dynamics of childhood diseases in England and Wales have been well studied and in this case important sources of heterogeneity include city size and complex movement patterns. Can we develop general theories that could apply to these kinds of heterogeneities in order to explain observed dynamical patterns?
Data sets and Measurements of Synchrony: What data sets addressing synchronization exist? How can these data sets be best analyzed? What are the fundamental limitations in data collection? In what ways could data collection be improved so that it would complement theoretical models and statistical analysis?
The statistical issues associated with spatiotemporal aspects of synchrony are daunting. Describing synchrony is relatively straightforward for pairs of oscillating systems. For larger numbers of oscillators, the Kuramoto index (or vector strength) provides a simple measure of synchrony, but does not take into account explicit spatial aspects of dynamics, nor is it ideal for anything but the simplest description. Can we identify existing statistical approaches or develop new statistical approaches that could build a more comprehensive description of dynamics in systems of coupled oscillators? Also, what are the best ways to link the spatiotemporal statistics of experimental data to dynamical models?
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