Synchronization and Aggregation

Dr. Tilmann Glimm
Professor of Mathematics
Western Washington University

The phenomenon of spontaneous synchronization in networks of interacting oscillators (‘clocks’) has fascinated mathematical modelers since the 1980s.
Examples include flashing of fireflies, synchronous firing of neurons, or rhythmic applause in concert audiences. Typically, these networks are assumed to be static. The interplay of motion and synchronization is much less well studied. Motivated by examples from developmental biology and from the behavior of organisms on the threshold to multicellularity, we present and investigate models of moving cells with intracellular clocks in this talk. Cells undergo random motion and adhere to each other. Crucially, the adhesion strength between neighbors depends on their clock phases. Their oscillators are linked via Kuramoto-type local interactions. The corresponding PDE model is a nonlocal advection-diffusion equation. We classify the possible emerging patterns depending on the model parameters.Combining these results with numerical simulations, we determine a range of possible far-from equilibrium patterns when baseline adhesion strength is zero: These include aggregation into separate synchronized clusters, global synchronization without aggregation, and unexpectedly a “phase wave” pattern characterized by spatial gradients of clock phases. Discrete Cellular-Automata type models confirms and illustrate these results.