Oscillators that sync and swarm

Phase slips in coupled oscillator systems

Phase slips are a typical dynamical behavior in coupled oscillator systems: the route to phase synchrony is characterized by intervals of constant phase difference interrupted by abrupt changes in the phase difference. Qualitatively similar to stick-slip phenomena, analysis of phase slip has mainly relied on identifying remnants of saddle-nodes or "ghosts." We study sets of phase oscillators and by examining the dynamics in detail, offer a more precise, quantitative description of the phenomenon. Phase shifts and phase sticks, namely, the temporary locking of phases required for phase slips, occur at stationary points of phase velocities. In networks of coupled phase oscillators, we show that phase slips between pairs of individual oscillators do not occur simultaneously, in general. We consider additional systems that show phase synchrony: one where saddle-node ghosts are absent, one where the coupling is similarity dependent, and two cases of coupled chaotic oscillators.

Light-Fueled Synchronization of Two Coupled Liquid Crystal Elastomer Self-Oscillators

The synchronization and group behaviors of self-excited coupled oscillators are common in nature and deserve to be explored, for self-excited motions have the advantages of actively collecting energy from the environment, being autonomous, making equipment portable, and so on. Based on light-powered self-excited oscillators composed of liquid crystal elastomer (LCE) bars, the synchronization of two self-excited coupled oscillators is theoretically studied. Numerical calculations show that self-excited oscillations of the system have two synchronization modes, in-phase mode and anti-phase mode, which are mainly determined by their interaction. The time histories of various quantities are calculated to elucidate the mechanism of self-excited oscillation and synchronization. For strong interactions, the system always develops into in-phase synchronization mode, while for weak interaction, the system will evolve into anti-phase synchronization mode. Furthermore, the effects of initial conditions, contraction coefficient, light intensity, and damping coefficient on the two synchronization modes of the self-excited oscillation are investigated extensively. The initial condition generally does not affect the synchronization mode and its amplitude. The amplitude of self-oscillation always increases with increasing contraction coefficient, gravitational acceleration, and light intensity, while it decreases with the increasing damping coefficient. This work will deepen people's understanding of the synchronization behaviors of self-excited coupled oscillators, and the theoretical framework could be extended to scenarios involving large-scale synchronization of the systems with numerous interacting oscillators.

Synchronization and alignment of model oscillators based on Quincke rotation

Colloidal spheres in weakly conductive fluids roll back and forth across the surface of a plane electrode when subject to strong electric fields. The so-called Quincke oscillators provide a basis for active matter based on self-oscillating units that can move, align, and synchronize within dynamic particle assemblies. Here, we develop a dynamical model for oscillations of a spherical particle and investigate the coupled dynamics of two such oscillators in the plane normal to the field. Building on existing descriptions of Quincke rotation, the model describes the dynamics of the charge, dipole, and quadrupole moments due to charge accumulation at the particle-fluid interface and particle rotation in the external field. The dynamics of the charge moments are coupled by the addition of a conductivity gradient, which describes asymmetries in the rates of charging near the electrode. We study the behavior of this model as a function of the field strength and gradient magnitude to identify the conditions required for sustained oscillations. We investigate the dynamics of two neighboring oscillators coupled by far field electric and hydrodynamic interactions in an unbounded fluid. Particles prefer to align and synchronize their rotary oscillations along the line of centers. The numerical results are reproduced and explained by accurate low-order approximations of the system dynamics based on weakly coupled oscillator theory. The coarse-grained dynamics of the oscillator phase and angle can be used to investigate collective behaviors within ensembles of many self-oscillating colloids.

Directional switches in network-organized swarming systems with delay

Coordinated directional switches can emerge between members of moving biological groups. Previous studies have shown that the self-propelled particles model can well reproduce directional switching behaviors, but it neglects the impact of social interactions. Thus, we focus on the influence of social interactions on the ordered directional switching motion of swarming systems, in which homogeneous Erdös-Rényi networks, heterogeneous scale-free networks, networks with community structures, and real-world animal social networks have been considered. The theoretical estimation of mean switching time is obtained, and the results show that the interplay between social and delayed interactions plays an important role in regulating directional switching behavior. To be specific, for homogeneous Erdös-Rényi networks, the increase in mean degree may suppress the directional switching behaviors if the delay is sufficiently small. However, when the delay is large, the large mean degree may promote the directional switching behavior. For heterogeneous scale-free networks, the increase of degree heterogeneity can reduce the mean switching time if the delay is sufficiently small, while the increasing degree heterogeneity may suppress the ordered directional switches if the delay is large. For networks with community structures, higher communities can promote directional switches for small delays, while for large delays, it may inhibit directional switching behavior. For dolphin social networks, delay can promote the directional switching behavior. Our results bring to light the role of social and delayed interactions in the ordered directional switching motion.

Generalized frustration in the multidimensional Kuramoto model

The Kuramoto model describes how coupled oscillators synchronize their phases as the intensity of the coupling increases past a threshold. The model was recently extended by reinterpreting the oscillators as particles moving on the surface of unit spheres in a D-dimensional space. Each particle is then represented by a D-dimensional unit vector; for D=2 the particles move on the unit circle and the vectors can be described by a single phase, recovering the original Kuramoto model. This multidimensional description can be further extended by promoting the coupling constant between the particles to a matrix K that acts on the unit vectors. As the coupling matrix changes the direction of the vectors, it can be interpreted as a generalized frustration that tends to hinder synchronization. In a recent paper we studied in detail the role of the coupling matrix for D=2. Here we extend this analysis to arbitrary dimensions. We show that, for identical particles, when the natural frequencies are set to zero, the system converges either to a stationary synchronized state, given by one of the real eigenvectors of K, or to an effective two-dimensional rotation, defined by one of the complex eigenvectors of K. The stability of these states depends on the set eigenvalues and eigenvectors of the coupling matrix, which controls the asymptotic behavior of the system, and therefore, can be used to manipulate these states. When the natural frequencies are not zero, synchronization depends on whether D is even or odd. In even dimensions the transition to synchronization is continuous and rotating states are replaced by active states, where the module of the order parameter oscillates while it rotates. If D is odd the phase transition is discontinuous and active states can be suppressed for some distributions of natural frequencies.

Optimal synchronization in pulse-coupled oscillator networks using reinforcement learning

Spontaneous synchronization is ubiquitous in natural and man-made systems. It underlies emergent behaviors such as neuronal response modulation and is fundamental to the coordination of robot swarms and autonomous vehicle fleets. Due to its simplicity and physical interpretability, pulse-coupled oscillators has emerged as one of the standard models for synchronization. However, existing analytical results for this model assume ideal conditions, including homogeneous oscillator frequencies and negligible coupling delays, as well as strict requirements on the initial phase distribution and the network topology. Using reinforcement learning, we obtain an optimal pulse-interaction mechanism (encoded in phase response function) that optimizes the probability of synchronization even in the presence of nonideal conditions. For small oscillator heterogeneities and propagation delays, we propose a heuristic formula for highly effective phase response functions that can be applied to general networks and unrestricted initial phase distributions. This allows us to bypass the need to relearn the phase response function for every new network.

Solvable Dynamics of Coupled High-Dimensional Generalized Limit-Cycle Oscillators

We introduce a new model consisting of globally coupled high-dimensional generalized limit-cycle oscillators, which explicitly incorporates the role of amplitude dynamics of individual units in the collective dynamics. In the limit of weak coupling, our model reduces to the D-dimensional Kuramoto phase model, akin to a similar classic construction of the well-known Kuramoto phase model from weakly coupled two-dimensional limit-cycle oscillators. For the practically important case of D=3, the incoherence of the model is rigorously proved to be stable for negative coupling (K<0) but unstable for positive coupling (K>0); the locked states are shown to exist if K>0; in particular, the onset of amplitude death is theoretically predicted. For D≥2, the discrete and continuous spectra for both locked states and amplitude death are governed by two general formulas. Our proposed D-dimensional model is physically more reasonable, because it is no longer constrained by fixed amplitude dynamics, which puts the recent studies of the D-dimensional Kuramoto phase model on a stronger footing by providing a more general framework for D-dimensional limit-cycle oscillators.

Co-evolution of synchronization and cooperation with multi-agent Q-learning

Cooperation is a widespread phenomenon in human society and plays a significant role in achieving synchronization of various systems. However, there has been limited progress in studying the co-evolution of synchronization and cooperation. In this manuscript, we investigate how reinforcement learning affects the evolution of synchronization and cooperation. Namely, the payoff of an agent depends not only on the cooperation dynamic but also on the synchronization dynamic. Agents have the option to either cooperate or defect. While cooperation promotes synchronization among agents, defection does not. We report that the dynamic feature, which indicates the action switching frequency of the agent during interactions, promotes synchronization. We also find that cooperation and synchronization are mutually reinforcing. Furthermore, we thoroughly analyze the potential reasons for synchronization promotion due to the dynamic feature from both macro- and microperspectives. Additionally, we conduct experiments to illustrate the differences in the synchronization-promoting effects of cooperation and dynamic features.

Diverse behaviors in non-uniform chiral and non-chiral swarmalators

We study the emergent behaviors of a population of swarming coupled oscillators, dubbed swarmalators. Previous work considered the simplest, idealized case: identical swarmalators with global coupling. Here we expand this work by adding more realistic features: local coupling, non-identical natural frequencies, and chirality. This more realistic model generates a variety of new behaviors including lattices of vortices, beating clusters, and interacting phase waves. Similar behaviors are found across natural and artificial micro-scale collective systems, including social slime mold, spermatozoa vortex arrays, and Quincke rollers. Our results indicate a wide range of future use cases, both to aid characterization and understanding of natural swarms, and to design complex interactions in collective systems from soft and active matter to micro-robotics.

Pinning in a system of swarmalators

We study a population of swarmalators (swarming/mobile oscillators) which run on a ring and are subject to random pinning. The pinning represents the tendency of particles to stick to defects in the underlying medium which competes with the tendency to sync and swarm. The result is rich collective behavior. A highlight is low dimensional chaos which in systems of ordinary, Kuramoto-type oscillators is uncommon. Some of the states (the phase wave and split phase wave) resemble those seen in systems of Janus matchsticks or Japanese tree frogs. The others (such as the sync and unsteady states) may be observable in systems of vinegar eels, electrorotated Quincke rollers, or other swarmalators moving in disordered environments.

multiSyncPy: A Python package for assessing multivariate coordination dynamics

In order to support the burgeoning field of research into intra- and interpersonal synchrony, we present an open-source software package: multiSyncPy. Multivariate synchrony goes beyond the bivariate case and can be useful for quantifying how groups, teams, and families coordinate their behaviors, or estimating the degree to which multiple modalities from an individual become synchronized. Our package includes state-of-the-art multivariate methods including symbolic entropy, multidimensional recurrence quantification analysis, coherence (with an additional sum-normalized modification), the cluster-phase 'Rho' metric, and a statistical test based on the Kuramoto order parameter. We also include functions for two surrogation techniques to compare the observed coordination dynamics with chance levels and a windowing function to examine time-varying coordination for most of the measures. Taken together, our collation and presentation of these methods make the study of interpersonal synchronization and coordination dynamics applicable to larger, more complex and often more ecologically valid study designs. In this work, we summarize the relevant theoretical background and present illustrative practical examples, lessons learned, as well as guidance for the usage of our package - using synthetic as well as empirical data. Furthermore, we provide a discussion of our work and software and outline interesting further directions and perspectives. multiSyncPy is freely available under the LGPL license at: https://github.com/cslab-hub/multiSyncPy , and also available at the Python package index.

Decision Making in Networks: A Model of Awareness Raising

This work investigates how interpersonal interactions among individuals could affect the dynamics of awareness raising. Even though previous studies on mathematical models of awareness in the decision making context demonstrate how the level of awareness results from self-observation impinged by optimal decision selections and external uncertainties, an explicit accounting of interaction among individuals is missing. Here we introduce for the first time a theoretical mathematical framework to evaluate the effect on individual awareness exerted by the interaction with neighbor agents. This task is performed by embedding the single agent model into a graph and allowing different agents to interact by means of suitable coupling functions. The presence of the network allows, from a global point of view, the emergence of diffusion mechanisms for which the population tends to reach homogeneous attractors, and, among them, the one with the highest level of awareness. The structural and behavioral patterns, such as the initial levels of awareness and the relative importance the individual assigns to their own state with respect to others’, may drive real actors to stress effective actions increasing individual and global awareness.

Spontaneous vortex formation by microswimmers with retarded attractions

Collective states of inanimate particles self-assemble through physical interactions and thermal motion. Despite some phenomenological resemblance, including signatures of criticality, the autonomous dynamics that binds motile agents into flocks, herds, or swarms allows for much richer behavior. Low-dimensional models have hinted at the crucial role played in this respect by perceived information, decision-making, and feedback, implying that the corresponding interactions are inevitably retarded. Here we present experiments on spherical Brownian microswimmers with delayed self-propulsion toward a spatially fixed target. We observe a spontaneous symmetry breaking to a transiently chiral dynamical state and concomitant critical behavior that do not rely on many-particle cooperativity. By comparison with the stochastic delay differential equation of motion of a single swimmer, we pinpoint the delay-induced effective synchronization of the swimmers with their own past as the key mechanism. Increasing numbers of swimmers self-organize into layers with pro- and retrograde orbital motion, synchronized and stabilized by steric, phoretic, and hydrodynamic interactions. Our results demonstrate how even most simple retarded interactions can foster emergent complex adaptive behavior in small active-particle ensembles.

Fluid circulation driven by collectively organized metachronal waves in swimming T. aceti nematodes

Recent experiments have shown that the nematode T. aceti can assemble into collectively undulating groups at the edge of fluid drops. This coordinated state consists of metachronal waves and drives fluid circulation inside the drop. We find that the circulation velocity is about 2 mm/s and nearly half the speed of the metachronal wave. We develop a quasi-two-dimensional hydrodynamics model using the Stokes flow approximation. The periodic motion of the nematodes constitute our moving boundary condition that drives the flow. Our model suggests that large-amplitude excursions of the nematode tails produce the fluid circulation. We discuss the constraints on containers that would enhance fluid motion, which could be used in the future design of on demand flow generating systems.

Sync and Swarm: Solvable Model of Nonidentical Swarmalators

We study a model of nonidentical swarmalators, generalizations of phase oscillators that both sync in time and swarm in space. The model produces four collective states: asynchrony, sync clusters, vortexlike phase waves, and a mixed state. These states occur in many real-world swarmalator systems such as biological microswimmers, chemical nanomotors, and groups of drones. A generalized Ott-Antonsen ansatz provides the first analytic description of these states and conditions for their existence. We show how this approach may be used in studies of active matter and related disciplines.

Effect of mobility on collective phase dynamics of nonlocally coupled oscillators with a phase lag

Nonlocally coupled oscillators with a phase lag self-organize into various patterns, such as global synchronization, the twisted state, and the chimera state. In this paper, we consider nonlocally coupled oscillators that move on a ring by randomly exchanging their positions with the neighbors and investigate the combined effects of phase lag and mobility on the collective phase dynamics. Spanning the whole range of phase lag and mobility, we show that mobility promotes synchronization for an attractive coupling, whereas it destroys coherence for a repulsive coupling. The transition behaviors are discussed in terms of the timescales of synchronization and diffusion of the oscillators. We also find a novel spatiotemporal pattern at the border between coherent and incoherent states.

Transition from inhomogeneous limit cycles to oscillation death in nonlinear oscillators with similarity-dependent coupling

We consider a system of coupled nonlinear oscillators in which the interaction is modulated by a measure of the similarity between the oscillators. Such a coupling is common in treating spatially mobile dynamical systems where the interaction is distance dependent or in resonance-enhanced interactions, for instance. For a system of Stuart-Landau oscillators coupled in this manner, we observe a novel route to oscillation death via a Hopf bifurcation. The individual oscillators are confined to inhomogeneous limit cycles initially and are damped to different fixed points after the bifurcation. Analytical and numerical results are presented for this case, while numerical results are presented for coupled Rössler and Sprott oscillators.

Matrix coupling and generalized frustration in Kuramoto oscillators

The Kuramoto model describes the synchronization of coupled oscillators that have different natural frequencies. Among the many generalizations of the original model, Kuramoto and Sakaguchi (KS) proposed a frustrated version that resulted in dynamic behavior of the order parameter, even when the average natural frequency of the oscillators is zero. Here, we consider a generalization of the frustrated KS model that exhibits new transitions to synchronization. The model is identical in form to the original Kuramoto model but written in terms of unit vectors and with the coupling constant replaced by a coupling matrix. The matrix breaks the rotational symmetry and forces the order parameter to point in the direction of the eigenvector with the highest eigenvalue, when the eigenvalues are real. For complex eigenvalues, the module of order parameter oscillates while it rotates around the unit circle, creating active states. We derive the complete phase diagram for the Lorentzian distribution of frequencies using the Ott-Antonsen ansatz. We also show that changing the average value of the natural frequencies leads to further phase transitions where the module of the order parameter goes from oscillatory to static.

Bayesian optimization of distributed neurodynamical controller models for spatial navigation

Dynamical systems models for controlling multi-agent swarms have demonstrated advances toward resilient, decentralized navigation algorithms. We previously introduced the NeuroSwarms controller, in which agent-based interactions were modeled by analogy to neuronal network interactions, including attractor dynamics and phase synchrony, that have been theorized to operate within hippocampal place-cell circuits in navigating rodents. This complexity precludes linear analyses of stability, controllability, and performance typically used to study conventional swarm models. Further, tuning dynamical controllers by manual or grid-based search is often inadequate due to the complexity of objectives, dimensionality of model parameters, and computational costs of simulation-based sampling. Here, we present a framework for tuning dynamical controller models of autonomous multi-agent systems with Bayesian optimization. Our approach utilizes a task-dependent objective function to train Gaussian process surrogate models to achieve adaptive and efficient exploration of a dynamical controller model's parameter space. We demonstrate this approach by studying an objective function selecting for NeuroSwarms behaviors that cooperatively localize and capture spatially distributed rewards under time pressure. We generalized task performance across environments by combining scores for simulations in multiple mazes with distinct geometries. To validate search performance, we compared high-dimensional clustering for high- vs. low-likelihood parameter points by visualizing sample trajectories in 2-dimensional embeddings. Our findings show that adaptive, sample-efficient evaluation of the self-organizing behavioral capacities of complex systems, including dynamical swarm controllers, can accelerate the translation of neuroscientific theory to applied domains.

Programmable, Spatiotemporal Control of Colloidal Motion Waves via Structured Light

Traveling waves in a reaction-diffusion system are essential for long-range communication in living organisms and inspire biomimetic materials of similar capabilities. One recent example is the traveling motion waves among photochemically oscillating, silver (Ag)-containing colloids. Being able to manipulate these colloidal waves holds the key for potential applications. Here, we have discovered that these motion waves can be confined by light patterns and that the chemical clocks of silver particles are moved forward by reducing local light intensity. Using these discoveries as design principles, we have applied structured light technology for the precise and programmable control of colloidal motion waves, including their origins, propagation directions, paths, shapes, annihilation, frequency, and speeds. We have also used the controlled propagation of colloidal waves to guide chemical messages along a predefined path to activate a population of micromotors located far from the signal. Our demonstrated capabilities in manipulating colloidal waves in space and time offer physical insights on their operation and expand their usefulness in the fundamental study of reaction-diffusion processes. Moreover, our findings inspire biomimetic strategies for the directional transport of mass, energy, and information at micro- or even nanoscales.

Synchronization of a Passive Oscillator and a Liquid Crystal Elastomer Self-Oscillator Powered by Steady Illumination

Self-oscillators have the advantages of actively harvesting energy from external steady environment, autonomy, and portability, and can be adopted as an engine to drive additional working equipment. The synchronous behavior of self-oscillators and passive oscillators may have an important impact on their functions. In this paper, we construct a self-oscillating system composed of a passive oscillator and an active liquid crystal elastomer self-oscillator powered by steady illumination, and theoretically investigate the synchronization of two coupled oscillators. There exist three synchronous regimes of the two coupled oscillators: static, in-phase, and anti-phase. The mechanisms of self-oscillations in in-phase and anti-phase synchronous regimes are elucidated in detail by calculating several key physical parameters. In addition, the effects of spring constant, initial velocity, contraction coefficient, light intensity, and damping coefficient on the self-oscillations of two coupled oscillators are further investigated, and the critical conditions for triggering self-oscillations are obtained. Numerical calculations show that the synchronous regime of self-oscillations is mainly determined by the spring constant, and the amplitudes of self-oscillations of two oscillators increase with increasing contraction coefficient, light intensity, and spring constant, while decrease with increasing damping coefficient. This study deepens the understanding of synchronization between coupled oscillators and may provide new design ideas for energy harvesters, soft robotics, signal detection, active motors, and self-sustained machinery.

From the origin of life to pandemics: emergent phenomena in complex systems

When a large number of similar entities interact among each other and with their environment at a low scale, unexpected outcomes at higher spatio-temporal scales might spontaneously arise. This non-trivial phenomenon, known as emergence, characterizes a broad range of distinct complex systems-from physical to biological and social-and is often related to collective behaviour. It is ubiquitous, from non-living entities such as oscillators that under specific conditions synchronize, to living ones, such as birds flocking or fish schooling. Despite the ample phenomenological evidence of the existence of systems' emergent properties, central theoretical questions to the study of emergence remain unanswered, such as the lack of a widely accepted, rigorous definition of the phenomenon or the identification of the essential physical conditions that favour emergence. We offer here a general overview of the phenomenon of emergence and sketch current and future challenges on the topic. Our short review also serves as an introduction to the theme issue Emergent phenomena in complex physical and socio-technical systems: from cells to societies, where we provide a synthesis of the contents tackled in the issue and outline how they relate to these challenges, spanning from current advances in our understanding on the origin of life to the large-scale propagation of infectious diseases. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Basin sizes depend on stable eigenvalues in the Kuramoto model

We show that for the Kuramoto model (with identical phase oscillators equally coupled), its global statistics and size of the basins of attraction can be estimated through the eigenvalues of all stable (frequency) synchronized states. This result is somehow unexpected since, by doing that, one could just use a local analysis to obtain the global dynamic properties. But recent works based on Koopman and Perron-Frobenius operators demonstrate that the global features of a nonlinear dynamical system, with some specific conditions, are somehow encoded in the local eigenvalues of its equilibrium states. Recognized numerical simulations in the literature reinforce our analytical results.

Suppression of discontinuous phase transitions by particle diffusion

We investigate the phase transitions of the q-state Brownian Potts model in two dimensions (2D) comprising Potts spins that diffuse like Brownian particles and interact ferromagnetically with other spins within a fixed distance. With extensive Monte Carlo simulations we find a continuous phase transition from a paramagnetic to a ferromagnetic phase even for q>4. This is in sharp contrast to the existence of a discontinuous phase transition in the equilibrium q-state Potts model in 2D with q>4. We present detailed numerical evidence for a continuous phase transition and argue that diffusion generated dynamical positional disorder suppresses phase coexistence leading to a continuous transition.

Microrobot collectives with reconfigurable morphologies, behaviors, and functions

Mobile microrobots, which can navigate, sense, and interact with their environment, could potentially revolutionize biomedicine and environmental remediation. Many self-organizing microrobotic collectives have been developed to overcome inherent limits in actuation, sensing, and manipulation of individual microrobots; however, reconfigurable collectives with robust transitions between behaviors are rare. Such systems that perform multiple functions are advantageous to operate in complex environments. Here, we present a versatile microrobotic collective system capable of on-demand reconfiguration to adapt to and utilize their environments to perform various functions at the air-water interface. Our system exhibits diverse modes ranging from isotropic to anisotrpic behaviors and transitions between a globally driven and a novel self-propelling behavior. We show the transition between different modes in experiments and simulations, and demonstrate various functions, using the reconfigurability of our system to navigate, explore, and interact with the environment. Such versatile microrobot collectives with globally driven and self-propelled behaviors have great potential in future medical and environmental applications.

Cell Responsiveness to Physical Energies: Paving the Way to Decipher a Morphogenetic Code

We discuss emerging views on the complexity of signals controlling the onset of biological shapes and functions, from the nanoarchitectonics arising from supramolecular interactions, to the cellular/multicellular tissue level, and up to the unfolding of complex anatomy. We highlight the fundamental role of physical forces in cellular decisions, stressing the intriguing similarities in early morphogenesis, tissue regeneration, and oncogenic drift. Compelling evidence is presented, showing that biological patterns are strongly embedded in the vibrational nature of the physical energies that permeate the entire universe. We describe biological dynamics as informational processes at which physics and chemistry converge, with nanomechanical motions, and electromagnetic waves, including light, forming an ensemble of vibrations, acting as a sort of control software for molecular patterning. Biomolecular recognition is approached within the establishment of coherent synchronizations among signaling players, whose physical nature can be equated to oscillators tending to the coherent synchronization of their vibrational modes. Cytoskeletal elements are now emerging as senders and receivers of physical signals, "shaping" biological identity from the cellular to the tissue/organ levels. We finally discuss the perspective of exploiting the diffusive features of physical energies to afford in situ stem/somatic cell reprogramming, and tissue regeneration, without stem cell transplantation.

Thermally Driven Self-Rotation of a Hollow Torus Motor

Self-oscillating systems based on thermally responsive polymer materials can realize heat-mechanical transduction in a steady ambient temperature field and have huge application potential in the field of micro-active machines, micro-robotics and energy harvesters. Recently, experiments have found that a torus on a hot surface can rotate autonomously and continuously, and its rotating velocity is determined by the competition between the thermally induced driving moment and the sliding friction moment. In this article, we theoretically study the self-sustained rotation of a hollow torus on a hot surface and explore the effect of the radius ratio on its rotational angular velocity and energy efficiency. By establishing a theoretical model of heat-driven self-sustained rotation, its analytical driving moment is derived, and the equilibrium equation for its steady rotation is obtained. Numerical calculation shows that with the increase in the radius ratio, the angular velocity of its rotation monotonously increases, while the energy efficiency of the self-rotating hollow torus motor first increases and then decreases. In addition, the effects of several system parameters on the angular velocity of it are also extensively investigated. The results in this paper have a guiding role in the application of hollow torus motor in the fields of micro-active machines, thermally driven motors and waste heat harvesters.

Noise resistant synchronization and collective rhythm switching in a model of animal group locomotion

Biology is suffused with rhythmic behaviour, and interacting biological oscillators often synchronize their rhythms with one another. Colonies of some ant species are able to synchronize their activity to fall into coherent bursts, but models of this phenomenon have neglected the potential effects of intrinsic noise and interspecific differences in individual-level behaviour. We investigated the individual and collective activity patterns of two Leptothorax ant species. We show that in one species (Leptothorax sp. W), ants converge onto rhythmic cycles of synchronized collective activity with a period of about 20 min. A second species (Leptothorax crassipilis) exhibits more complex collective dynamics, where dominant collective cycle periods range from 16 min to 2.8 h. Recordings that last 35 h reveal that, in both species, the same colony can exhibit multiple oscillation frequencies. We observe that workers of both species can be stimulated by nest-mates to become active after a refractory resting period, but the durations of refractory periods differ between the species and can be highly variable. We model the emergence of synchronized rhythms using an agent-based model informed by our empirical data. This simple model successfully generates synchronized group oscillations despite the addition of noise to ants' refractory periods. We also find that adding noise reduces the likelihood that the model will spontaneously switch between distinct collective cycle frequencies.

Synchronized oscillations in swarms of nematode Turbatrix aceti

There is a recent surge of interest in the behavior of active particles that can at the same time align their direction of movement and synchronize their oscillations, known as swarmalators. While theoretical and numerical models of such systems are now abundant, no real-life examples have been shown to date. We present an experimental investigation of the collective motion of the nematode Turbatrix aceti that self-propel by body undulation. We discover that these nematodes can synchronize their body oscillations, forming striking traveling metachronal waves, which produces strong fluid flows. We uncover that the location and strength of this collective state can be controlled through the shape of the confining structure; in our case the contact angle of a droplet. This opens a way for producing controlled work such as on-demand flows or displacement of objects. We illustrate this by showing that the force generated by this state is sufficient to change the physics of evaporation of fluid droplets, by counteracting the surface-tension force, which allow us to estimate its strength. The relatively large size and ease of culture make Turbatrix aceti a promising model organism for experimental investigation of swarming and oscillating active matter capable of producing controllable work.

Self-Sustained Collective Motion of Two Joint Liquid Crystal Elastomer Spring Oscillator Powered by Steady Illumination

For complex micro-active machines or micro-robotics, it is crucial to clarify the coupling and collective motion of their multiple self-oscillators. In this article, we construct two joint liquid crystal elastomer (LCE) spring oscillators connected by a spring and theoretically investigate their collective motion based on a well-established dynamic LCE model. The numerical calculations show that the coupled system has three steady synchronization modes: in-phase mode, anti-phase mode, and non-phase-locked mode, and the in-phase mode is more easily achieved than the anti-phase mode and the non-phase-locked mode. Meanwhile, the self-excited oscillation mechanism is elucidated by the competition between network that is achieved by the driving force and the damping dissipation. Furthermore, the phase diagram of three steady synchronization modes under different coupling stiffness and different initial states is given. The effects of several key physical quantities on the amplitude and frequency of the three synchronization modes are studied in detail, and the equivalent systems of in-phase mode and anti-phase mode are proposed. The study of the coupled LCE spring oscillators will deepen people’s understanding of collective motion and has potential applications in the fields of micro-active machines and micro-robots with multiple coupled self-oscillators.

Mechanical Torque Promotes Bipolarity of the Mitotic Spindle Through Multi-centrosomal Clustering

Intracellular forces shape cellular organization and function. One example is the mitotic spindle, a cellular machine consisting of multiple chromosomes and centrosomes which interact via dynamic microtubule filaments and motor proteins, resulting in complicated spatially dependent forces. For a cell to divide properly, it is important for the spindle to be bipolar, with chromosomes at the center and multiple centrosomes clustered into two 'poles' at opposite sides of the chromosomes. Experimental observations show that in unhealthy cells, the spindle can take on a variety of patterns. What forces drive each of these patterns? It is known that attraction between centrosomes is key to bipolarity, but what prevents the centrosomes from collapsing into a monopolar configuration? Here, we explore the hypothesis that torque rotating chromosome arms into orientations perpendicular to the centrosome-centromere vector promotes spindle bipolarity. To test this hypothesis, we construct a pairwise-interaction model of the spindle. On a continuum version of the model, an integro-PDE system, we perform linear stability analysis and construct numerical solutions which display a variety of spatial patterns. We also simulate a discrete particle model resulting in a phase diagram that confirms that the spindle bipolarity emerges most robustly with torque. Altogether, our results suggest that rotational forces may play an important role in dictating spindle patterning.

Effects of topological characteristics on rhythmic states of the D-dimensional Kuramoto model in complex networks

Synchronization is a ubiquitous phenomenon in engineering and natural ecosystems. While the dynamics of synchronization modeled by the Kuramoto model are commonly studied in two dimensions and the state of dynamic units is characterized by a scalar angle variable, we studied the Kuramoto model generalized to D dimensions in the framework of a complex network and utilized the local synchronous order parameter between the agent and its neighbors as the controllable variable to adjust the coupling strength. Here, we reported that average connectivity of networks affects the time-dependent, rhythmic, cyclic state. Importantly, we found that the level of heterogeneity of networks governs the rhythmic state in the transition process. The analytical treatment for observed scenarios in a D-dimensional Kuramoto model at D=3 was provided. These results offered a platform for a better understanding of time-dependent swarming and flocking dynamics in nature.

The Myosin Myocardial Mesh Interpreted as a Biological Analogous of Nematic Chiral Liquid Crystals

There are still grey areas in the understanding of the myoarchitecture of the ventricular mass. This is despite the progress of investigation methods since the beginning of the 21st century (diffusion tensor magnetic resonance imaging, microcomputed tomography, and polarised light imaging). The objective of this article is to highlight the specificities and the limitations of polarised light imaging (PLI) of the unstained myocardium embedded in methyl methacrylate (MMA). Thus, to better differentiate our method from other PLI modes, we will refer to it by the acronym PLI-MMA. PLI-MMA shows that the myosin mesh of the compact left ventricular wall behaves like a biological analogous of a nematic chiral liquid crystal. Results obtained by PLI-MMA are: the main direction of the myosin molecules contained in an imaged voxel, the crystal liquid director n, and a regional isotropy index RI that is an orientation tensor, the equivalent of the crystal liquid order parameter. The vector n is collinear with the first eigenvector of diffusion tensor imaging (DTI-MRI). The RI has not been confounded with the diffusion tensor of DTI that gives information about the three eigenvectors of the ellipsoid of diffusion. PLI-MMA gives no information about the collagen network. The physics of soft matter has allowed the revisiting of Streeter’s conjecture on the myoarchitecture of the compact left ventricular wall: “geodesics on a nested set of toroidal surfaces”. Once the torus topology is understood, this characterisation of the myoarchitecture is more accurate and parsimonious than former descriptions. Finally, this article aims to be an enthusiastic invitation to a transdisciplinary approach between physicists of liquid crystals, anatomists, and specialists of imaging.

Coupled liquid crystalline oscillators in Huygens' synchrony

In the flourishing field of soft robotics, strategies to embody communication and collective motion are scarce. Here we report the synchronized oscillations of thin plastic actuators by an approach reminiscent of the synchronized motion of pendula and metronomes. Two liquid crystalline network oscillators fuelled by light influence the movement of one another and display synchronized oscillations in-phase and anti-phase in a steady state. By observing entrainment between the asymmetric oscillators we demonstrate the existence of coupling between the two actuators. We qualitatively explain the origin of the synchronized motion using a theoretical model and numerical simulations, which suggest that the motion can be tuned by the mechanical properties of the coupling joint. We thus anticipate that the complex synchronization phenomena usually observed in rigid systems can also exist in soft polymeric materials. This enables the use of new stimuli, featuring an example of collective motion by photo-actuation.

Coupling disorder in a population of swarmalators

We consider a population of two-dimensional oscillators with random couplings and explore the collective states. The coupling strength between oscillators is randomly quenched with two values, one of which is positive while the other is negative, and the oscillators can spatially move depending on the state variables for phase and position. We find that the system shows the phase transition from the incoherent state to the fully synchronized one at a proper ratio of the number of positive couplings to the total. The threshold is numerically measured and analytically predicted by the linear stability analysis of the fully synchronized state. It is found that the random couplings induce the long-term state patterns appearing for constant strength. The oscillators move to the places where the randomly quenched couplings work as if annealed. We further observe that the system with mixed randomnesses for quenched couplings shows the combination of the deformed patterns understandable with each annealed average.

Unveiling the morphogenetic code: A new path at the intersection of physical energies and chemical signaling

In this editorial, we discuss the remarkable role of physical energies in the control of cell signaling networks and in the specification of the architectural plan of both somatic and stem cells. In particular, we focus on the biological relevance of bioelectricity in the pattern control that orchestrates both developmental and regenerative pathways. To this end, the narrative starts from the dawn of the first studies on animal electricity, reconsidering the pioneer work of Harold Saxton Burr in the light of the current achievements. We finally discuss the most recent evidence showing that bioelectric signaling is an essential component of the informational processes that control pattern specification during embryogenesis, regeneration, or even malignant transformation. We conclude that there is now mounting evidence for the existence of a Morphogenetic Code, and that deciphering this code may lead to unprecedented opportunities for the development of novel paradigms of cure in regenerative and precision medicine.

Why animals swirl and how they group

We report a possible solution for the long-standing problem of the biological function of swirling motion, when a group of animals orbits a common center of the group. We exploit the hypothesis that learning processes in the nervous system of animals may be modelled by reinforcement learning (RL) and apply it to explain the phenomenon. In contrast to hardly justified models of physical interactions between animals, we propose a small set of rules to be learned by the agents, which results in swirling. The rules are extremely simple and thus applicable to animals with very limited level of information processing. We demonstrate that swirling may be understood in terms of the escort behavior, when an individual animal tries to reside within a certain distance from the swarm center. Moreover, we reveal the biological function of swirling motion: a trained for swirling swarm is by orders of magnitude more resistant to external perturbations, than an untrained one. Using our approach we analyze another class of a coordinated motion of animals-a group locomotion in viscous fluid. On a model example we demonstrate that RL provides an optimal disposition of coherently moving animals with a minimal dissipation of energy.

Intercellular communication and the organization of simple multicellular animals

Animal cells are amazing examples of decentralized systems: By interchanging information about their position and internal state, cells coordinate their behavior and organize themselves in time and space. Examples of this behavior are the development of an embryo or of an organoid. In this work we have asked which are the "rules of intercellular relationship" that allow the organization of an abstract cell collective into structures similar to simple metazoans, without being specific about the (molecular, cellular or physical) nature of the processes involved. To do so, we have used a computational modeling approach following a modified version of the "Swarmalator" concept introduced by O'Keeffe, Hong and Strogatz (2017): a collection of interacting particles ("swarmalators"), each of which defined by a position in space and an internal state (a phase). The key feature is that swarmalators are coupled, so that their position and internal state are both affected by the position and state of all other swarmalators. This model can be easily analogized to biological systems, with "cells" being the swarmalators, and their phase the cell's internal state or "cell type". With this model we explore the conditions (represented by the coupling parameters) that would allow the organization of a multicellular "bioswarmer" and its dynamics along a sort of life cycle. Originally developed in 2D, we implement the model in 3D as well. We describe how changing the strength of intercellular communication can alter the structure and differentiation state of the bioswarmer, how internal polarization can arise and trigger collective directed migration, or how partly erasing the cellular memory of cell state is critical to allow bioswarmers to transit through different states. In addition, we show that the size of a multicellular ensemble might control the differentiation of its constituent cells without changing its rules of relationship.

Metachronal waves in concentrations of swimming Turbatrix aceti nematodes and an oscillator chain model for their coordinated motions

At high concentration, free swimming nematodes known as vinegar eels (Turbatrix aceti), collectively exhibit metachronal waves near a boundary. We find that the frequency of the collective traveling wave is lower than that of the freely swimming organisms. We explore models based on a chain of oscillators with nearest-neighbor interactions that inhibit oscillator phase velocity. The phase of each oscillator represents the phase of the motion of the eel's head back and forth about its mean position. A strongly interacting directed chain model mimicking steric repulsion between organisms robustly gives traveling wave states and can approximately match the observed wavelength and oscillation frequency of the observed traveling wave. We predict body shapes assuming that waves propagate down the eel body at a constant speed. The phase oscillator model that impedes eel head overlaps also reduces close interactions throughout the eel bodies.

Binary mixtures of locally coupled mobile oscillators

Synchronized behavior in a system of coupled dynamic objects is a fascinating example of an emerged cooperative phenomena which has been observed in systems as diverse as a group of insects, neural networks, or networks of computers. In many instances, however, the synchronization is undesired because it may lead to system malfunctioning, as in the case of Alzheimer's and Parkinson's diseases, for example. Recent studies of static networks of oscillators have shown that the presence of a small fraction of so-called contrarian oscillators can suppress the undesired network synchronization. On the other hand, it is also known that the mobility of the oscillators can significantly impact their synchronization dynamics. Here, we combine these two ideas-the oscillator mobility and the presence of heterogeneous interactions-and study numerically binary mixtures of phase oscillators performing two-dimensional random walks. Within the framework of a generalized Kuramoto model, we introduce two phase-coupling schemes. The first one is invariant when the types of any two oscillators are swapped, while the second model is not. We demonstrate that the symmetric model does not allow for a complete suppression of the synchronized state. However, it provides means for a robust control of the synchronization timescale by varying the overall number density and the composition of the mixture and the strength of the off-diagonal Kuramoto coupling constant. Instead, the asymmetric model predicts that the coherent state can be eliminated within a subpopulation of normal oscillators and evoked within a subpopulation of the contrarians.

Cost-effective circadian mechanism: rhythmic degradation of circadian proteins spontaneously emerges without rhythmic post-translational regulation

Circadian protein oscillations are maintained by the lifelong repetition of protein production and degradation in daily balance. It comes at the cost of ever-replayed, futile protein synthesis each day. This biosynthetic cost with a given oscillatory protein profile is relievable by a rhythmic, not constant, degradation rate that selectively peaks at the right time of day but remains low elsewhere, saving much of the gross protein loss and of the replenishing protein synthesis. Here, our mathematical modeling reveals that the rhythmic degradation rate of proteins with circadian production spontaneously emerges under steady and limited activity of proteolytic mediators and does not necessarily require rhythmic post-translational regulation of previous focus. Additional (yet steady) post-translational modifications in a proteolytic pathway can further facilitate the degradation's rhythmicity in favor of the biosynthetic cost saving. Our work is supported by animal and plant circadian data, offering a generic mechanism for potentially widespread, time-dependent protein turnover.

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Rhythmic degradation of circadian proteins lowers the cost of protein synthesis

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This rhythmic degradation emerges without rhythmic post-translational regulation

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Extra, yet steady post-translational modifications enhance degradation rhythmicity

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This mechanism hints at how organisms afford the price of daily biological rhythms

Designing temporal networks that synchronize under resource constraints

Being fundamentally a non-equilibrium process, synchronization comes with unavoidable energy costs and has to be maintained under the constraint of limited resources. Such resource constraints are often reflected as a finite coupling budget available in a network to facilitate interaction and communication. Here, we show that introducing temporal variation in the network structure can lead to efficient synchronization even when stable synchrony is impossible in any static network under the given budget, thereby demonstrating a fundamental advantage of temporal networks. The temporal networks generated by our open-loop design are versatile in the sense of promoting synchronization for systems with vastly different dynamics, including periodic and chaotic dynamics in both discrete-time and continuous-time models. Furthermore, we link the dynamic stabilization effect of the changing topology to the curvature of the master stability function, which provides analytical insights into synchronization on temporal networks in general. In particular, our results shed light on the effect of network switching rate and explain why certain temporal networks synchronize only for intermediate switching rate.

The ability of complex networks to synchronize themselves is limited by available coupling resources. Zhang and Strogatz show that allowing temporal variation in the network structure can lead to synchronization even when stable synchrony is impossible in any static network under the fixed budget.

Discordant synchronization patterns on directed networks of identical phase oscillators with attractive and repulsive couplings

We study the collective dynamics of identical phase oscillators on globally coupled networks whose interactions are asymmetric and mediated by positive and negative couplings. We split the set of oscillators into two interconnected subpopulations. In this setup, oscillators belonging to the same group interact via symmetric couplings while the interaction between subpopulations occurs in an asymmetric fashion. By employing the dimensional reduction scheme of the Ott-Antonsen (OA) theory, we verify the existence of traveling wave and π-states, in addition to the classical fully synchronized and incoherent states. Bistability between all collective states is reported. Analytical results are generally in excellent agreement with simulations; for some parameters and initial conditions, however, we numerically detect chimera-like states which are not captured by the OA theory.

Non-reciprocal phase transitions

Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter^1-6, non-equilibrium systems^7-9, networks of neurons^10,11, social groups with conformist and contrarian members¹², directional interface growth phenomena^13-15 and metamaterials^16-20. Although wave propagation in non-reciprocal media has recently been closely studied^1,16-20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points²¹. We describe the emergence of these phases using insights from bifurcation theory^22,23 and non-Hermitian quantum mechanics^24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.

Chemical pumps and flexible sheets spontaneously form self-regulating oscillators in solution

The synchronization of self-oscillating systems is vital to various biological functions, from the coordinated contraction of heart muscle to the self-organization of slime molds. Through modeling, we design bioinspired materials systems that spontaneously form shape-changing self-oscillators, which communicate to synchronize both their temporal and spatial behavior. Here, catalytic reactions at the bottom of a fluid-filled chamber and on mobile, flexible sheets generate the energy to "pump" the surrounding fluid, which also transports the immersed sheets. The sheets exert a force on the fluid that modifies the flow, which in turn affects the shape and movement of the flexible sheets. This feedback enables a single coated (active) and even an uncoated (passive) sheet to undergo self-oscillation, displaying different oscillatory modes with increases in the catalytic reaction rate. Two sheets (active or passive) introduce excluded volume, steric interactions. This distinctive combination of the hydrodynamic, fluid-structure, and steric interactions causes the sheets to form coupled oscillators, whose motion is synchronized in time and space. We develop a heuristic model that rationalizes this behavior. These coupled self-oscillators exhibit rich and tunable phase dynamics, which depends on the sheets' initial placement, coverage by catalyst and relative size. Moreover, through variations in the reactant concentration, the system can switch between the different oscillatory modes. This breadth of dynamic behavior expands the functionality of the coupled oscillators, enabling soft robots to display a variety of self-sustained, self-regulating moves.

Demixing of two species via reciprocally concentration-dependent diffusivity

We propose a model for demixing of two species by assuming a density-dependent effective diffusion coefficient of the particles. Both sorts of microswimmers diffuse as active overdamped Brownian particles with a noise intensity that is determined by the surrounding density of the respective other species within a sensing radius r_{s}. A higher concentration of the first (second) sort will enlarge the diffusion and, in consequence, the intensity of the noise experienced by the second (first) sort. Numerical and analytical investigations of steady states of the macroscopic equations prove the demixing of particles due to this reciprocally concentration-dependent diffusivity. An ambiguity of the numerical integration scheme for the purely local model (r_{s}→0) is resolved by considering nonvanishing sensing radii in a nonlocal model with r_{s}>0.

Collective steady-state patterns of swarmalators with finite-cutoff interaction distance

We study the steady-state patterns of population of the coupled oscillators that sync and swarm, where the interaction distances among the oscillators have a finite-cutoff in the interaction distance. We examine how the static patterns known in the infinite-cutoff are reproduced or deformed and explore a new static pattern that does not appear until a finite-cutoff is considered. All steady-state patterns of the infinite-cutoff, static sync, static async, and static phase wave are repeated in space for proper finite-cutoff ranges. Their deformation in shape and density takes place for the other finite-cutoff ranges. Bar-like phase wave states are observed, which has not been the case for the infinite-cutoff. All the patterns are investigated via numerical and theoretical analyses.

Solitons in complex systems of chiral fields with Kuramoto interactions

We construct a complex system of N chiral fields, each regarded as a node or a constituent of a complex field-theoretic system, which interact by means of chirally invariant potentials across a network of connections. In the classical case, these interactions are identical or similar to Kuramoto interactions, leading to synchronization phenomena for the well-known Kuramoto model and its many extensions and generalizations to higher dimensions. We consider chiral systems of arbitrary size N, where each constituent carries a conserved charge of topological origin, which evolve according to a coupled system of second-order, Lorentz invariant, nonlinear partial differential equations. Stable soliton configurations occur as a consequence of the nonlinear network interactions, not necessarily from self-interactions of the fundamental fields. In 1+1 dimensions, these chirally invariant models allow for multi-soliton configurations that for N=2 are determined by the sine-Gordon equation and for N=3 reduce in special cases to the double sine-Gordon equation, which has exact double-kink static solutions consisting of solitons positioned at arbitrary locations. Planar and three-dimensional networked skyrmions appear in higher dimensions. Such configurations can be viewed for general N as bound states of the constituent fields, which exist together with the usual fundamental excitations. Whereas Kuramoto interactions in first-order systems lead to emergent classical phenomena such as synchronization, these same interactions in complex systems of chiral fields result in a rich variety of multi-soliton bound states.

Discontinuous Transitions and Rhythmic States in the D-Dimensional Kuramoto Model Induced by a Positive Feedback with the Global Order Parameter

From fireflies to cardiac cells, synchronization governs important aspects of nature, and the Kuramoto model is the staple for research in this area. We show that generalizing the model to oscillators of dimensions higher than 2 and introducing a positive feedback mechanism between the coupling and the global order parameter leads to a rich and novel scenario: the synchronization transition is explosive at all even dimensions, whilst it is mediated by a time-dependent, rhythmic, state at all odd dimensions. Such a latter circumstance, in particular, differs from all other time-dependent states observed so far in the model. We provide the analytic description of this novel state, which is fully corroborated by numerical calculations. Our results can, therefore, help untangle secrets of observed time-dependent swarming and flocking dynamics that unfold in three dimensions, and where this novel state could thus provide a fresh perspective for as yet not understood formations.