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

Abstract

To explore the generic features of synchronisation facilitated by hydrodynamic interactions dominated by viscous forces, we investigate small systems of rowers; rowers are a highly simplified model for motile cilia, with each cilium approximated by a rigid sphere driven by a geometrically updated force. We introduce a new framework to analyse rowers, in which we average the pair-wise interaction of rowers by converting to the natural phase. In doing so the function describing the interaction becomes continuous, dramatically simplifying the system and allowing standard dynamical system techniques to be applied. Through inspection of phase portraits, we capture the broad features of rowers driven by power law forces and quantify their coupling strength. This approach is not limited to power law potentials, and can be applied to any monotonically increasing function. When implemented in systems with more rowers, the phase portraits show more diverse behaviour. By exploring the phase space, small systems of rowers can be designed to demonstrate specific features.

Entrainment of mammalian motile cilia in the brain with hydrodynamic forces

“…Simulations. The simulation scheme builds on previous work (36,49,50) but implements external flow on an array of active phase-free oscillators. Rowers are driven by repulsive harmonic traps, with the potential U(xr) = kx 2 r .…”

“…This type of model 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 (49,50). The sphere moves away from the trap vertex until it reaches the switch point A + xs , where the trap is reflected and the sphere reverses direction.…”

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.…”

Changes in geometrical aspects of a simple model of cilia synchronization control the dynamical state, a possible mechanism for switching of swimming gaits in microswimmers

“…The alternative minimal models are ‘rotors’, which we investigated recently in [ 43 ]; both rowers and rotors produce general synchronisation features in viscously coupled systems. The two classes of models can in fact both be mapped onto the same coarse-grained description of pairwise phase interaction [ 44 ], so both can be tuned to exhibit the same collective properties. Previously, rower models have been used to investigate generic features of oscillator pairs or small systems [ 42 , 44 – 48 ].…”

“…The two classes of models can in fact both be mapped onto the same coarse-grained description of pairwise phase interaction [ 44 ], so both can be tuned to exhibit the same collective properties. Previously, rower models have been used to investigate generic features of oscillator pairs or small systems [ 42 , 44 – 48 ]. Questions concerning alignment, metachronal waves, and the effect of heterogeneity in spatial layout have all been studied in larger system using the model [ 45 , 49 – 51 ].…”