The chimera state in colloidal phase oscillators with hydrodynamic interaction

Abstract

The chimera state is the incongruous situation where coherent and incoherent populations coexist in sets of identical oscillators. Using driven non-linear oscillators interacting purely through hydrodynamic forces at low Reynolds number, previously studied as a simple model of motile cilia supporting waves, we find concurrent incoherent and synchronised subsets in small arrays. The chimeras seen in simulation display a "breathing" aspect, reminiscent of uniformly interacting phase oscillators. In contrast to other systems where chimera has been observed, this system has a well-defined interaction metric, and we know that the emergent dynamics inherit the symmetry of the underlying Oseen tensor eigenmodes. The chimera state can thus be connected to a superposition of eigenstates, whilst considering the mean interaction strength within and across subsystems allows us to make a connection to more generic (and abstract) chimeras in populations of Kuramoto phase oscillators. From this work, we expect the chimera state to emerge in experimental observations of oscillators coupled through hydrodynamic forces.

Synchronization of mammalian motile cilia in the brain with hydrodynamic forces

“…This colloidal particle model, which we have been studying in the last ten years (18), is based on the physical intuition that, in a coarse-grained fashion, the degrees of freedom of the complex cilium shapes and activity can be captured by a rower's driving potential. The main advantage of this approach is that it greatly simplifies the calculation of drag forces, both those acting on the individual object and the force induced by one object on another (44,45). The bead moves away from the trap vertex until it reaches the switch point A + xs, where the trap is reflected and the bead reverses direction.…”

“…These values were chosen to be similar to the ones that found experimentally tracking a single cilium (supplementary Figure S7). Simulations of chains were performed using a Brownian Dynamics code as in (44) varying the number of N rowers, keeping nearest neighbors separated on average by a distance d. We explored the set of N = [1, 2, 4, 6, 8, 10, 20, 30] and d = [0.4, 0.5, 0.6, 1.2] µm (see Figure 4c). The rowers are coupled through the hydrodynamic forces via a Blake tensor, with the oscillations occurring at a fixed distance z wall = 7 µm above the no-slip boundary (44).…”

“…Simulations of chains were performed using a Brownian Dynamics code as in (44) varying the number of N rowers, keeping nearest neighbors separated on average by a distance d. We explored the set of N = [1, 2, 4, 6, 8, 10, 20, 30] and d = [0.4, 0.5, 0.6, 1.2] µm (see Figure 4c). The rowers are coupled through the hydrodynamic forces via a Blake tensor, with the oscillations occurring at a fixed distance z wall = 7 µm above the no-slip boundary (44). For sufficiently small d, the chain of rowers synchronizes through hydrodynamic forces and beads oscillate at a common frequency f > f0, as seen in Figure 4c.…”

Interpreting the synchronisation of driven colloidal oscillators via the mean pair interaction

“…This converts the complex configuration and historydependent interactions of a rower pair synchronising to a simple one-dimensional nonlinear system. The fixed points of the system and their stability can be investigated from this perspective, where before it was interpreted in terms of eigenstates of the hydrodynamic tensor and their decay rate [33,35,36].…”

“…It is not feasible to extend the iterative map to systems of many rowers, but the eigenstate approach can be applied to larger collections of rowers. Through this lens the equilibrium behaviour of rowers in rings or chains was discussed in terms of eigenstate growth and decay between switches [33,35]. This culminated in predicting the steady state of planar arrays when varying their positions and alignment [36].…”