Oscillators that sync and swarm

Collective decision making by embodied neural agents

Collective decision making using simple social interactions has been studied in many types of multiagent systems, including robot swarms and human social networks. However, existing multiagent studies have rarely modeled the neural dynamics that underlie sensorimotor coordination in embodied biological agents. In this study, we investigated collective decisions that resulted from sensorimotor coordination among agents with simple neural dynamics. We equipped our agents with a model of minimal neural dynamics based on the coordination dynamics framework, and embedded them in an environment with a stimulus gradient. In our single-agent setup, the decision between two stimulus sources depends solely on the coordination of the agent's neural dynamics with its environment. In our multiagent setup, that same decision also depends on the sensorimotor coordination between agents, via their simple social interactions. Our results show that the success of collective decisions depended on a balance of intra-agent, interagent, and agent-environment coupling, and we use these results to identify the influences of environmental factors on decision difficulty. More generally, our results illustrate how collective behaviors can be analyzed in terms of the neural dynamics of the participating agents. This can contribute to ongoing developments in neuro-AI and self-organized multiagent systems.

Bifurcations and collective states of Kuramoto oscillators with higher-order interactions and rotational symmetry breaking

We study a network of identical Kuramoto oscillators with higher-order interactions that also break the rotational symmetry of the system. To gain analytical insights into this model, we use the Watanabe-Strogatz Ansatz, which allows us to reduce the dimensionality of the original system of equations. The study of stability and bifurcations of the reduced system reveals a codimension two Bogdanov-Takens bifurcation and several other associated bifurcations. Such analysis is corroborated by numerical simulations of the associated Kuramoto system, which, in turn, unveils a variety of collective behaviors such as synchronized motion, oscillation death, chimeras, incoherent states, and traveling waves. Importantly, this system displays a case where alternating chimeras emerge in an indistinguishable single population of oscillators, which may offer insights into the unihemispheric slow-wave sleep phenomenon observed in mammals and birds.

Patterns of neuronal synchrony in higher-order networks

Synchrony in neuronal networks is crucial for cognitive functions, motor coordination, and various neurological disorders. While traditional research has focused on pairwise interactions between neurons, recent studies highlight the importance of higher-order interactions involving multiple neurons. Both types of interactions lead to complex synchronous spatiotemporal patterns, including the fascinating phenomenon of chimera states, where synchronized and desynchronized neuronal activity coexist. These patterns are thought to resemble pathological states such as schizophrenia and Parkinson's disease, and their emergence is influenced by neuronal dynamics as well as by synaptic connections and network structure. This review integrates the current understanding of how pairwise and higher-order interactions contribute to different synchrony patterns in neuronal networks, providing a comprehensive overview of their role in shaping network dynamics. We explore a broad range of connectivity mechanisms that drive diverse neuronal synchrony patterns, from pairwise long-range temporal interactions and time-delayed coupling to adaptive communication and higher-order, time-varying connections. We cover key neuronal models, including the Hindmarsh-Rose model, the stochastic Hodgkin-Huxley model, the Sherman model, and the photosensitive FitzHugh-Nagumo model. By investigating the emergence and stability of various synchronous states, this review highlights their significance in neurological systems and indicates directions for future research in this rapidly evolving field.

Strategy to control synchronized dynamics in swarmalator systems

Synchronization forms the basis of many coordination phenomena in natural systems, enabling them to function cohesively and support their fundamental operations. However, there are scenarios where synchronization disrupts a system's proper functioning, necessitating mechanisms to control or suppress it. While several methods exist for controlling synchronization in nonspatially embedded oscillators, to the best of our knowledge, no such strategies have been developed for swarmalators (oscillators that simultaneously move in space and synchronize in time). In this work, we address this gap by introducing a control strategy based on Hamiltonian control theory to suppress synchronization in a system of swarmalators confined to a one-dimensional space. The numerical investigations we performed demonstrate that the proposed control strategy effectively suppresses synchronized dynamics within the swarmalator population. We studied the impact of the number of controlled swarmalators and the strength of the control term in its original form and a simplified one.

Synchronization and self-assembly of free capillary spinners

Chiral active particles are able to draw energy from the environment to self-propel in the form of rotation. We describe an experimental arrangement wherein chiral objects, spinners, floating on the surface of a vibrated fluid rotate due to emitted capillary waves. We observe that pairs of spinners can assemble at quantized distances via the mutually generated wavefield, phase synchronize and, in some circumstances, globally rotate about a point midway between them. A mathematical model based on wave-mediated interactions captures the salient features of the assembly and synchronization while a qualitative argument is able to rationalize global rotations based on interference and radiation stress associated with the wavefield. Extensions to larger collections are demonstrated, highlighting the potential for this tabletop system to be used as an experimental system capable of synchronizing and swarming.

Modeling dynamics on the dance floor with directional swarmalators

Understanding collective behavior in both biological and social contexts, such as human interactions on dance floors, is a growing field of interest. Spatiotemporal dynamics of collective behavior have previously been modeled, for instance, with swarmalators, which are dynamical units that exhibit both swarming behavior and synchronization, combining spatial movement and entrainment. In our current study, we have expanded the swarmalator concept to encompass gaze direction as a representation of visual attention. We employ the newly developed directional swarmalator model for simulating the complex spatiotemporal dynamics observed on dance floors. Our model aims to reflect the complex dynamics of collective movement, as well as rhythmic synchronization and gaze alignment. It establishes a quantitative framework to dissect how individuals on dance floors self-organize and generate emergent patterns in response to both musical stimuli and visual perception of other dancers. The inclusion of gaze direction allows for the simulation of realistic scenarios on dance floors, mirroring the dynamic interplay of human movement in rhythm-driven environments. The model is initially tested against motion capture recordings of two groups dancing in a silent disco, however, it is theoretically adaptable to a variety of scenarios, including varying group sizes, adjustable degrees of auditory and visual coupling, as well as modifiable interaction ranges, making it a generic tool for exploring collective behavior in musical settings. The development of the directional swarmalator model contributes to understanding social dynamics in shared music and dance experiences.

Global synchronization theorem for coupled swarmalators

The global stability of oscillator networks has attracted much recent attention. Ordinarily, the oscillators in such studies are motionless; their spatial degrees of freedom are either ignored (e.g., mean field models) or inactive (e.g., geometrically embedded networks like lattices). Yet many real-world oscillators are mobile, moving around in space as they synchronize in time. Here, we prove a global synchronization theorem for a simple model of such swarmalators where the units move on a 1D ring. This can be thought of as a generalization from oscillators connected on random networks to oscillators connected on temporal networks, where the edges are determined by the oscillators' movements.

Statistical description of mobile oscillators in embryonic pattern formation

Synchronization of mobile oscillators occurs in numerous contexts, including physical, chemical, biological, and engineered systems. In vertebrate embryonic development, a segmental body structure is generated by a population of mobile oscillators. Cells in this population produce autonomous gene expression rhythms and interact with their neighbors through local signaling. These cells form an extended tissue where frequency and cell mobility gradients coexist. Gene expression kinematic waves travel through this tissue and pattern the segment boundaries. It has been shown that oscillator mobility promotes global synchronization. However, in vertebrate segment formation, mobility may also introduce local fluctuations in kinematic waves and impair segment boundaries. Here, we derive a general framework for mobile oscillators that relates local mobility fluctuations to synchronization dynamics and pattern robustness. We formulate a statistical description of mobile phase oscillators in terms of probability density. We obtain and solve diffusion equations for the average phase and variance, revealing the relationship between local fluctuations and global synchronization in a homogeneous population of oscillators. Analysis of the probability density for large mobility identifies a mean-field onset, where locally coupled oscillators start behaving as if each oscillator was coupled with all the others. We extend the statistical description to inhomogeneous systems to address the gradients present in the vertebrate segmenting tissue. The theory relates pattern stability to mobility, coupling, and pattern wavelength. The general approach of the statistical description may be applied to mobile oscillators in other contexts, as well as to other patterning systems where mobility is present.

Effects of coupling range on the dynamics of swarmalators

We study a variant of the one-dimensional swarmalator model where the units' interactions have a controllable length scale or range. We tune the model from the long-range regime, which is well studied, into the short-range regime, which is insufficiently studied, and find diverse collective states: sync dots, where the swarmalators arrange themselves into k>1 delta points of perfect synchrony; q-waves, where the swarmalators form spatiotemporal waves with winding number q>1; and an active state where unsteady oscillations are found. We present the phase diagram and derive most of the threshold boundaries analytically. These states may be observable in real-world swarmalator systems with low-range coupling, such as biological microswimmers or active colloids.

Biased Ensembles of Pulsating Active Matter

We discover unexpected connections between packing configurations and rare fluctuations in dense systems of active particles subject to pulsation of size. Using large deviation theory, we examine biased ensembles which select atypical realizations of the dynamics exhibiting high synchronization in particle size. We show that the order emerging at high bias can manifest as distinct dynamical states featuring high to vanishing pulsation current. Remarkably, transitions between these states arise from changing the system geometry at fixed bias and constant density. We rationalize such transitions as arising from the change in packing configurations which, depending on box geometry, may induce highly ordered or geometrically frustrated structures. Furthermore, we reveal that a master curve in the unbiased dynamics, correlating polydispersity and current, helps predict the dynamical state emerging in the biased dynamics. Finally, we demonstrate that deformation waves can propagate under suitable geometries when biasing with local order.

Collective dynamics of pulse-coupled swarmalators

Collective motion in nature, from the murmurations of starlings to firefly flashes, showcases a remarkable level of self-organization. Swarmalator models are an emerging paradigm for describing collective behavior of oscillators which sync and swarm. Traditionally, most studies have focused on the self-organizing dynamics of swarmalators with continuous coupling. However, a key aspect of many biological systems, pulsed interactions, remains unexplored within the framework of swarmalators. In this paper, we analyze the collective behavior of pulse-coupled swarmalators with different characteristics of pulse and phase response curves. We report several collective states facilitated by the pulsatile interactions including bump state, active bump state, partial synchronized state, radial wave state, and death state. The stationary nature of spatial and phase dynamics characterizes the death state. Further, we provide an analytical estimate for the threshold coupling strength for the death state's existence by analyzing the minimal model's dynamics with two coupled swarmalators.

Swarm Autonomy: From Agent Functionalization to Machine Intelligence

Swarm behaviors are common in nature, where individual organisms collaborate via perception, communication, and adaptation. Emulating these dynamics, large groups of active agents can self-organize through localized interactions, giving rise to complex swarm behaviors, which exhibit potential for applications across various domains. This review presents a comprehensive summary and perspective of synthetic swarms, to bridge the gap between the microscale individual agents and potential applications of synthetic swarms. It is begun by examining active agents, the fundamental units of synthetic swarms, to understand the origins of their motility and functionality in the presence of external stimuli. Then inter-agent communications and agent-environment communications that contribute to the swarm generation are summarized. Furthermore, the swarm behaviors reported to date and the emergence of machine intelligence within these behaviors are reviewed. Eventually, the applications enabled by distinct synthetic swarms are summarized. By discussing the emergent machine intelligence in swarm behaviors, insights are offered into the design and deployment of autonomous synthetic swarms for real-world applications.

Chimera states in a system of stationary and flying-through deterministic particles with an internal degree of freedom

We consider the effect of the emergence of chimera states in a system of coexisting stationary and flying-through in potential particles with an internal degree of freedom determined by the phase. All particles tend to an equilibrium state with a small number of potential wells, which leads to the emergence of a stationary chimera. An increase in the number of potential wells leads to the emergence of particles flying-through along the medium, the phases of which form a moving chimera. Further, these two structures coexist and interact with each other. In this case, an increase in the local synchronization degree of the chimera is observed in the areas of the synchronous cluster location.

Dynamics of swarmalators in the presence of a contrarian

Swarmalators are entities that combine the swarming behavior of particles with the oscillatory dynamics of coupled phase oscillators and represent a novel and rich area of study within the field of complex systems. Unlike traditional models that treat spatial movement and phase synchronization separately, swarmalators exhibit a unique coupling between their positions and internal phases, leading to emergent behaviors that include clustering, pattern formation, and the coexistence of synchronized and desynchronized states, etc. This paper presents a comprehensive analysis of a two-dimensional swarmalator model in the presence of a predatorlike agent that we call a contrarian. The positions and the phases of the swarmalators are influenced by the contrarian and we observe the emergence of intriguing collective states. We find that swarmalator phases are synchronized even with negative coupling strength when their interaction with the contrarian is comparatively strong. Through a combination of analytical methods and simulations, we demonstrate how varying these parameters can lead to transitions between different collective states.

Metastable Oscillatory Modes as a Signature of Entropy Management in the Brain

Entropy management, central to the Free Energy Principle, requires a process that temporarily shifts brain activity toward states of lower or higher entropy. Metastable synchronization is a process by which a system achieves entropy fluctuations by intermittently transitioning between states of collective order and disorder. Previous work has shown that collective oscillations, similar to those recorded from the brain, emerge spontaneously from weakly stable synchronization in critically coupled oscillator systems. However, direct evidence linking the formation of collective oscillations to entropy fluctuations is lacking. In this short communication, we demonstrate how the emergence of Metastable Oscillatory Modes (MOMs) is directly associated with a temporary reduction in entropy in the ongoing dynamics. We apply Shannon entropy to the distribution of eigenvalues of phase covariance over sliding time windows, capturing the temporal evolution of entropy at the level of the entire dynamical system. By demonstrating how the formation of MOMs impacts a system's entropy levels, we bridge theoretical works on the physics of coupled oscillators with the FEP framework, supporting the hypothesis that brain rhythms recorded experimentally are a signature of entropy management.

High-Mode Coupling Yields Multicoherent-Phase Phenomena in Nonlocally Coupled Oscillators

Capturing the intricate dynamics of partially coherent patterns in coupled oscillator systems is vibrant and one of the crucial areas of nonlinear sciences. Considering higher-order Fourier modes in the coupling, we discover a novel type of clustered coherent state in phase models, where inside the coherent region oscillators are further split into q dynamically equivalent subgroups with a 2π/q phase difference between two neighboring subgroups, forming a multicoherent-phase (MUP) chimera state. Both a self-consistency analysis and the Ott-Antonsen dimension reduction techniques are used to theoretically derive these solutions, whose stability are further demonstrated by spectral analysis. The universality of MUP effects is demonstrated by generalized twisted states and higher-order spatial swarm chimera states of swarmalators which are beyond pure phase models since oscillators can move in space.

Spatial Transcriptomics Reveals the Temporal Architecture of the Seminiferous Epithelial Cycle and Precise Sertoli-Germ Synchronization

Spermatogenesis is characterized by the seminiferous epithelial cycle, a periodic pattern of germ cell differentiation with a wave-like progression along the length of seminiferous tubules. While key signaling and metabolic components of the cycle are known, the transcriptional changes across the cycle and the correlations between germ cell and somatic lineages remain undefined. Here, we use spatial transcriptomics via RNA SeqFISH+ to profile 2,638 genes in 216,090 cells in mouse testis and identify a periodic transcriptional pattern across tubules that precisely recapitulates the seminiferous epithelial cycle, enabling us to map cells to specific timepoints along the developmental cycle. Analyzing gene expression in somatic cells reveals that Sertoli cells exhibit a cyclic transcriptional profile closely synchronized with germ cell development while other somatic cells do not demonstrate such synchronization. Remarkably, in mouse testis with drug-induced ablation of germ cells, Sertoli cells independently maintain their cyclic transcriptional dynamics. By analyzing expression data, we identify an innate retinoic acid cycle, a network of transcription factors with cyclic activation, and signaling from germ cells that could interact with this network. Together, this work leverages spatial geometries for mapping the temporal dynamics and reveals a regulatory mechanism in spermatogenesis where Sertoli cells oscillate and coordinate with the cyclical progression of germ cell development.

Collective dynamics of swarmalators driven by a mobile pacemaker

Swarmalators, namely, oscillators with intrinsic frequencies that are able to self-propel to move in space, may undergo collective spatial swarming and meanwhile phase synchronous dynamics. In this paper, a swarmalator model driven by an external mobile pacemaker is proposed to explore the swarming dynamics in the presence of the competition between the external organization of the moving pacemaker and the intrinsic self-organization among oscillators. It is unveiled that the swarmalator system may exhibit a wealth of novel spatiotemporal patterns including the spindle state, the ripple state, and the trapping state. Transitions among these patterns and the mechanisms are studied with the help of different order parameters. The phase diagrams present systematic scenarios of various possible collective swarming dynamics and the transitions among them. The present study indicates that one may manipulate the formation and switching of the organized collective states by adjusting the external driving force, which is expected to shed light on applications of swarming performance control in natural and artificial groups of active agents.

Probabilistic modeling reveals coordinated social interaction states and their multisensory bases

Social behavior across animal species ranges from simple pairwise interactions to thousands of individuals coordinating goal-directed movements. Regardless of the scale, these interactions are governed by the interplay between multimodal sensory information and the internal state of each animal. Here, we investigate how animals use multiple sensory modalities to guide social behavior in the highly social zebrafish (Danio rerio) and uncover the complex features of pairwise interactions early in development. To identify distinct behaviors and understand how they vary over time, we developed a new hidden Markov model with constrained linear-model emissions to automatically classify states of coordinated interaction, using the movements of one animal to predict those of another. We discovered that social behaviors alternate between two interaction states within a single experimental session, distinguished by unique movements and timescales. Long-range interactions, akin to shoaling, rely on vision, while mechanosensation underlies rapid synchronized movements and parallel swimming, precursors of schooling. Altogether, we observe spontaneous interactions in pairs of fish, develop novel hidden Markov modeling to reveal two fundamental interaction modes, and identify the sensory systems involved in each. Our modeling approach to pairwise social interactions has broad applicability to a wide variety of naturalistic behaviors and species and solves the challenge of detecting transient couplings between quasi-periodic time series.

Probabilistic modeling reveals coordinated social interaction states and their multisensory bases

Social behavior across animal species ranges from simple pairwise interactions to thousands of individuals coordinating goal-directed movements. Regardless of the scale, these interactions are governed by the interplay between multimodal sensory information and the internal state of each animal. Here, we investigate how animals use multiple sensory modalities to guide social behavior in the highly social zebrafish (Danio rerio) and uncover the complex features of pairwise interactions early in development. To identify distinct behaviors and understand how they vary over time, we developed a new hidden Markov model with constrained linear-model emissions to automatically classify states of coordinated interaction, using the movements of one animal to predict those of another. We discovered that social behaviors alternate between two interaction states within a single experimental session, distinguished by unique movements and timescales. Long-range interactions, akin to shoaling, rely on vision, while mechanosensation underlies rapid synchronized movements and parallel swimming, precursors of schooling. Altogether, we observe spontaneous interactions in pairs of fish, develop novel hidden Markov modeling to reveal two fundamental interaction modes, and identify the sensory systems involved in each. Our modeling approach to pairwise social interactions has broad applicability to a wide variety of naturalistic behaviors and species and solves the challenge of detecting transient couplings between quasi-periodic time series.

Self-organization of anti-aligning active particles: Waving pattern formation and chaos

Recently, it has been shown that purely anti-aligning interaction between active particles may induce a finite wavelength instability. The formed patterns display intricate spatiotemporal dynamics, suggesting the presence of chaos. Here, we propose a quasi-one-dimensional simplification of the particle interaction model. This simplified model allows us to deduce amplitude equations that describe the collective motion of the active entities. We show that these equations exhibit chaotic orbits. Furthermore, via direct numerical simulations of the particle's system, we discuss the pertinence of these amplitude equations approach for describing the particle's self-coordinated motions.

Revealing the mechanism and function underlying pairwise temporal coupling in collective motion

Coordinated motion in animal groups has predominantly been studied with a focus on spatial interactions, such as how individuals position and orient themselves relative to one another. Temporal aspects have, by contrast, received much less attention. Here, by studying pairwise interactions in juvenile zebrafish (Danio rerio)-including using immersive volumetric virtual reality (VR) with which we can directly test models of social interactions in situ-we reveal that there exists a rhythmic out-of-phase (i.e., an alternating) temporal coordination dynamic. We find that reciprocal (bi-directional) feedback is both necessary and sufficient to explain this emergent coupling. Beyond a mechanistic understanding, we find, both from VR experiments and analysis of freely swimming pairs, that temporal coordination considerably improves spatial responsiveness, such as to changes in the direction of motion of a partner. Our findings highlight the synergistic role of spatial and temporal coupling in facilitating effective communication between individuals on the move.

Amplitude responses of swarmalators

Swarmalators are entities that swarm through space and sync in time and are potentially considered to replicate the complex dynamics of many real-world systems. So far, the internal dynamics of swarmalators have been taken as a phase oscillator inspired by the Kuramoto model. Here we examine the internal dynamics utilizing an amplitude oscillator capable of exhibiting periodic and chaotic behaviors. To incorporate the dual interplay between spatial and internal dynamics, we propose a general model that keeps the properties of swarmalators intact. This adaptation calls for a detailed study, which we present in this paper. We establish our study with the Rössler oscillator by taking parameters from both chaotic and periodic regions. While the periodic oscillator mimics most of the patterns in the previous phase oscillator model, the chaotic oscillator brings some fascinating states.

Exotic swarming dynamics of high-dimensional swarmalators

Swarmalators are oscillators that can swarm as well as sync via a dynamic balance between their spatial proximity and phase similarity. Swarmalator models employed so far in the literature comprise only one-dimensional phase variables to represent the intrinsic dynamics of the natural collectives. Nevertheless, the latter can indeed be represented more realistically by high-dimensional phase variables. For instance, the alignment of velocity vectors in a school of fish or a flock of birds can be more realistically set up in three-dimensional space, while the alignment of opinion formation in population dynamics could be multidimensional, in general. We present a generalized D-dimensional swarmalator model, which more accurately captures self-organizing behaviors of a plethora of real-world collectives by self-adaptation of high-dimensional spatial and phase variables. For a more sensible visualization and interpretation of the results, we restrict our simulations to three-dimensional spatial and phase variables. Our model provides a framework for modeling complicated processes such as flocking, schooling of fish, cell sorting during embryonic development, residential segregation, and opinion dynamics in social groups. We demonstrate its versatility by capturing the maneuvers of a school of fish, qualitatively and quantitatively, by a suitable extension of the original model to incorporate appropriate features besides a gallery of its intrinsic self-organizations for various interactions. We expect the proposed high-dimensional swarmalator model to be potentially useful in describing swarming systems and programmable and reconfigurable collectives in a wide range of disciplines, including the physics of active matter, developmental biology, sociology, and engineering.

Order, chaos, and dimensionality transition in a system of swarmalators

Similarly to sperm, where individuals self-organize in space while also striving for coherence in their tail swinging, several natural and engineered systems exhibit the emergence of swarming and synchronization. The arising and interplay of these phenomena have been captured by collectives of hypothetical particles named swarmalators, each possessing a position and a phase whose dynamics are affected reciprocally and also by the space-phase states of their neighbors. In this work, we introduce a solvable model of swarmalators able to move in two-dimensional spaces. We show that several static and active collective states can emerge and derive necessary conditions for each to show up as the model parameters are varied. These conditions elucidate, in some cases, the displaying of multistability among states. Notably, in the active regime, the system exhibits hyperchaos, maintaining spatial correlation under certain conditions and breaking it under others on what we interpret as a dimensionality transition.

Influence of anomalous agents on the dynamics of an active system

Swarming behavior in systems of self-propelled particles, whether biological or artificial, has received increased attention in recent years. Here, we show that even a small number of particles with anomalous behavior can change dramatically collective dynamics of the swarming system and can impose unusual behavior and transitions between dynamic states. Our results pave the way to practical approaches and concepts of multiagent dynamics in groups of flocking animals: birds, insects, and fish, i.e., active and living soft matter.

A Cellular Potts Model of the interplay of synchronization and aggregation

We investigate the behavior of systems of cells with intracellular molecular oscillators ("clocks") where cell-cell adhesion is mediated by differences in clock phase between neighbors. This is motivated by phenomena in developmental biology and in aggregative multicellularity of unicellular organisms. In such systems, aggregation co-occurs with clock synchronization. To account for the effects of spatially extended cells, we use the Cellular Potts Model (CPM), a lattice agent-based model. We find four distinct possible phases: global synchronization, local synchronization, incoherence, and anti-synchronization (checkerboard patterns). We characterize these phases via order parameters. In the case of global synchrony, the speed of synchronization depends on the adhesive effects of the clocks. Synchronization happens fastest when cells in opposite phases adhere the strongest ("opposites attract"). When cells of the same clock phase adhere the strongest ("like attracts like"), synchronization is slower. Surprisingly, the slowest synchronization happens in the diffusive mixing case, where cell-cell adhesion is independent of clock phase. We briefly discuss potential applications of the model, such as pattern formation in the auditory sensory epithelium.

Collective chemomechanical oscillations in active hydrogels

We report on the collective response of an assembly of chemomechanical Belousov-Zhabotinsky (BZ) hydrogel beads. We first demonstrate that a single isolated spherical BZ hydrogel bead with a radius below a critical value does not oscillate, whereas an assembly of the same BZ hydrogel beads presents chemical oscillation. A BZ chemical model with an additional flux of chemicals out of the BZ hydrogel captures the experimentally observed transition from oxidized nonoscillating to oscillating BZ hydrogels and shows this transition is due to a flux of inhibitors out of the BZ hydrogel. The model also captures the role of neighboring BZ hydrogel beads in decreasing the critical size for an assembly of BZ hydrogel beads to oscillate. We finally leverage the quorum sensing behavior of the collective to trigger their chemomechanical oscillation and discuss how this collective effect can be used to enhance the oscillatory strain of these active BZ hydrogels. These findings could help guide the eventual fabrication of a swarm of autonomous, communicating, and motile hydrogels.

Synchronous behaviors of three coupled liquid crystal elastomer-based spring oscillators under linear temperature fields

Self-oscillating coupled systems possess the ability to actively absorb external environmental energy to sustain their motion. This quality endows them with autonomy and sustainability, making them have application value in the fields of synchronization and clustering, thereby furthering research and exploration in these domains. Building upon the foundation of thermal responsive liquid crystal elastomer-based (LCE-based) spring oscillators, a synchronous system comprising three LCE-based spring oscillators interconnected by springs is established. In this paper, the synchronization phenomenon is described, and the self-oscillation mechanism is revealed. The results indicate that by varying system parameters and initial conditions, three synchronization patterns emerge, namely, full synchronous mode, partial synchronous mode, and asynchronous mode. For strongly interacting systems, full synchronous mode always prevails, while for weak interactions, the adjustment of initial velocities in magnitude and direction yields the three synchronization patterns. Additionally, this study explores the impact of several system parameters, including LCE elasticity coefficient and spring elasticity coefficient, on the amplitude, frequency, and synchronous mode of the system. The findings in this paper can enhance our understanding of the synchronization behavior of multiple mutually coupled LCE-based spring oscillators, with promising applications in energy harvesting, soft robotics, signal monitoring, and various other fields.

Motility-induced phase separation and frustration in active matter swarmalators

We introduce a two dimensional system of active matter swarmalators composed of elastically interacting run-and-tumble active disks with an internal parameter ϕ_{i}. The disks experience an additional attractive or repulsive force with neighboring disks depending upon their relative difference in ϕ_{i}, making them similar to swarmalators used in robotic systems. In the absence of the internal parameter, the system forms a motility-induced phase separated (MIPS) state, but when the swarmalator interactions are present, a wide variety of other active phases appear depending upon whether the interaction is attractive or repulsive and whether the particles act to synchronize or ant-synchronize their internal parameter values. These phases include a gas-free gel regime, arrested clusters, a labyrinthine state, a regular MIPS state, a frustrated MIPS state for attractive antisynchronization, and a superlattice MIPS state for attractive synchronization.

Noise-induced collective actuation in active solids

Collective actuation describes the spontaneous synchronized oscillations taking place in active solids when the elasto-active feedback, which generically couples the reorientation of the active forces and the elastic stress, is large enough. In the absence of noise, collective actuation takes the form of a strong condensation of the dynamics on a specific pair of modes and their generalized harmonics. Here we report experiments conducted with centimetric active elastic structures, where collective oscillation takes place along the single lowest energy mode of the system, gapped from the other modes because of the system's geometry. Combining the numerical and theoretical analysis of an agent-based model, we demonstrate that this form of collective actuation is noise-induced. The effect of the noise is first analyzed in a single-particle toy model that reveals the interplay between the noise and the specific structure of the phase space. We then show that in the continuous limit, any finite amount of noise turns this new form of transition to collective actuation into a bona fide supercritical Hopf bifurcation.

Influence of human motion patterns on epidemic spreading dynamics

Extensive real-data indicate that human motion exhibits novel patterns and has a significant impact on the epidemic spreading process. The research on the influence of human motion patterns on epidemic spreading dynamics still lacks a systematic study in network science. Based on an agent-based model, this paper simulates the spread of the disease in the gathered population by combining the susceptible-infected-susceptible epidemic process with human motion patterns, described by moving speed and gathering preference. Our simulation results show that the emergence of a hysteresis loop is observed in the system when the moving speed is slow, particularly when humans prefer to gather; that is, the epidemic prevalence of the systems depends on the fraction of initial seeds. Regardless of the gathering preference, the hysteresis loop disappears when the population moves fast. In addition, our study demonstrates that there is an optimal moving speed for the gathered population, at which the epidemic prevalence reaches its maximum value.

Directional synchrony among self-propelled particles under spatial influence

Synchronization is one of the emerging collective phenomena in interacting particle systems. Its ubiquitous presence in nature, science, and technology has fascinated the scientific community over the decades. Moreover, a great deal of research has been, and is still being, devoted to understand various physical aspects of the subject. In particular, the study of interacting active particles has led to exotic phase transitions in such systems which have opened up a new research front-line. Motivated by this line of work, in this paper, we study the directional synchrony among self-propelled particles. These particles move inside a bounded region, and crucially their directions are also coupled with spatial degrees of freedom. We assume that the directional coupling between two particles is influenced by the relative spatial distance which changes over time. Furthermore, the nature of the influence is considered to be both short and long-ranged. We explore the phase transition scenario in both the cases and propose an approximation technique which enables us to analytically find the critical transition point. The results are further supported with numerical simulations. Our results have potential importance in the study of active systems like bird flocks, fish schools, and swarming robots where spatial influence plays a pertinent role.

Order parameter dynamics in complex systems: From models to data

Collective ordering behaviors are typical macroscopic manifestations embedded in complex systems and can be ubiquitously observed across various physical backgrounds. Elements in complex systems may self-organize via mutual or external couplings to achieve diverse spatiotemporal coordinations. The order parameter, as a powerful quantity in describing the transition to collective states, may emerge spontaneously from large numbers of degrees of freedom through competitions. In this minireview, we extensively discussed the collective dynamics of complex systems from the viewpoint of order-parameter dynamics. A synergetic theory is adopted as the foundation of order-parameter dynamics, and it focuses on the self-organization and collective behaviors of complex systems. At the onset of macroscopic transitions, slow modes are distinguished from fast modes and act as order parameters, whose evolution can be established in terms of the slaving principle. We explore order-parameter dynamics in both model-based and data-based scenarios. For situations where microscopic dynamics modeling is available, as prototype examples, synchronization of coupled phase oscillators, chimera states, and neuron network dynamics are analytically studied, and the order-parameter dynamics is constructed in terms of reduction procedures such as the Ott-Antonsen ansatz, the Lorentz ansatz, and so on. For complicated systems highly challenging to be well modeled, we proposed the eigen-microstate approach (EMP) to reconstruct the macroscopic order-parameter dynamics, where the spatiotemporal evolution brought by big data can be well decomposed into eigenmodes, and the macroscopic collective behavior can be traced by Bose-Einstein condensation-like transitions and the emergence of dominant eigenmodes. The EMP is successfully applied to some typical examples, such as phase transitions in the Ising model, climate dynamics in earth systems, fluctuation patterns in stock markets, and collective motion in living systems.

Swarmalators with delayed interactions

We investigate the effects of delayed interactions in a population of "swarmalators," generalizations of phase oscillators that both synchronize in time and swarm through space. We discover two steady collective states: a state in which swarmalators are essentially motionless in a disk arranged in a pseudocrystalline order, and a boiling state in which the swarmalators again form a disk, but now the swarmalators near the boundary perform boiling-like convective motions. These states are reminiscent of the beating clusters seen in photoactivated colloids and the living crystals of starfish embryos.

Attractive and repulsive interactions in the one-dimensional swarmalator model

We study a population of swarmalators, mobile variants of phase oscillators, which run on a ring and have both attractive and repulsive interactions. This one-dimensional (1D) swarmalator model produces several of collective states: the standard sync and async states as well as a splaylike "polarized" state and several unsteady states such as active bands or swirling. The model's simplicity allows us to describe some of the states analytically. The model can be considered as a toy model for real-world swarmalators such as vinegar eels and sperm which swarm in quasi-1D geometries.

Large Coherent States Formed from Disordered k-Regular Random Graphs

The present work is motivated by the need for robust, large-scale coherent states that can play possible roles as quantum resources. A challenge is that large, complex systems tend to be fragile. However, emergent phenomena in classical systems tend to become more robust with scale. Do these classical systems inspire ways to think about robust quantum networks? This question is studied by characterizing the complex quantum states produced by mapping interactions between a set of qubits from structure in graphs. We focus on maps based on k-regular random graphs where many edges were randomly deleted. We ask how many edge deletions can be tolerated. Surprisingly, it was found that the emergent coherent state characteristic of these graphs was robust to a substantial number of edge deletions. The analysis considers the possible role of the expander property of k-regular random graphs.

Stability of Kuramoto networks subject to large and small fluctuations from heterogeneous and spatially correlated noise

Oscillatory networks subjected to noise are broadly used to model physical and technological systems. Due to their nonlinear coupling, such networks typically have multiple stable and unstable states that a network might visit due to noise. In this article, we focus on the assessment of fluctuations resulting from heterogeneous and spatially correlated noise inputs on Kuramoto model networks. We evaluate the typical, small fluctuations near synchronized states and connect the network variance to the overlap between stable modes of synchronization and the input noise covariance. Going beyond small to large fluctuations, we introduce the indicator mode approximation that projects the dynamics onto a single amplitude dimension. Such an approximation allows for estimating rates of fluctuations to saddle instabilities, resulting in phase slips between connected oscillators. Statistics for both regimes are quantified in terms of effective noise amplitudes that are compared and contrasted for several noise models. Bridging the gap between small and large fluctuations, we show that a larger network variance does not necessarily lead to higher rates of large fluctuations.

Swarmalators on a ring with uncorrelated pinning

We present a case study of swarmalators (mobile oscillators) that move on a 1D ring and are subject to pinning. Previous work considered the special case where the pinning in space and the pinning in the phase dimension were correlated. Here, we study the general case where the space and phase pinning are uncorrelated, both being chosen uniformly at random. This induces several new effects, such as pinned async, mixed states, and a first-order phase transition. These phenomena may be found in real world swarmalators, such as systems of vinegar eels, Janus matchsticks, electrorotated Quincke rollers, or Japanese tree frogs.

Synchronization in collectively moving inanimate and living active matter

Whether one considers swarming insects, flocking birds, or bacterial colonies, collective motion arises from the coordination of individuals and entails the adjustment of their respective velocities. In particular, in close confinements, such as those encountered by dense cell populations during development or regeneration, collective migration can only arise coordinately. Yet, how individuals unify their velocities is often not understood. Focusing on a finite number of cells in circular confinements, we identify waves of polymerizing actin that function as a pacemaker governing the speed of individual cells. We show that the onset of collective motion coincides with the synchronization of the wave nucleation frequencies across the population. Employing a simpler and more readily accessible mechanical model system of active spheres, we identify the synchronization of the individuals' internal oscillators as one of the essential requirements to reach the corresponding collective state. The mechanical 'toy' experiment illustrates that the global synchronous state is achieved by nearest neighbor coupling. We suggest by analogy that local coupling and the synchronization of actin waves are essential for the emergent, self-organized motion of cell collectives.

The evolution of social timing

Sociality and timing are tightly interrelated in human interaction as seen in turn-taking or synchronised dance movements. Sociality and timing also show in communicative acts of other species that might be pleasurable, but also necessary for survival. Sociality and timing often co-occur, but their shared phylogenetic trajectory is unknown: How, when, and why did they become so tightly linked? Answering these questions is complicated by several constraints; these include the use of divergent operational definitions across fields and species, the focus on diverse mechanistic explanations (e.g., physiological, neural, or cognitive), and the frequent adoption of anthropocentric theories and methodologies in comparative research. These limitations hinder the development of an integrative framework on the evolutionary trajectory of social timing and make comparative studies not as fruitful as they could be. Here, we outline a theoretical and empirical framework to test contrasting hypotheses on the evolution of social timing with species-appropriate paradigms and consistent definitions. To facilitate future research, we introduce an initial set of representative species and empirical hypotheses. The proposed framework aims at building and contrasting evolutionary trees of social timing toward and beyond the crucial branch represented by our own lineage. Given the integration of cross-species and quantitative approaches, this research line might lead to an integrated empirical-theoretical paradigm and, as a long-term goal, explain why humans are such socially coordinated animals.

Antiphase synchronization in a population of swarmalators

Swarmalators are oscillatory systems endowed with a spatial component, whose spatial and phase dynamics affect each other. Such systems can demonstrate fascinating collective dynamics resembling many real-world processes. Through this work, we study a population of swarmalators where they are divided into different communities. The strengths of spatial attraction, repulsion, as well as phase interaction differ from one group to another. Also, they vary from intercommunity to intracommunity. We encounter, as a result of variation in the phase coupling strength, different routes to achieve the static synchronization state by choosing several parameter combinations. We observe that when the intercommunity phase coupling strength is sufficiently large, swarmalators settle in the static synchronization state. However, with a significant small phase coupling strength the state of antiphase synchronization as well as chimeralike coexistence of sync and async are realized. Apart from rigorous numerical results, we have been successful to provide semianalytical treatment for the existence and stability of global static sync and the antiphase sync states.

Phase transitions on a multiplex of swarmalators

Dynamics of bidirectionally coupled swarmalators subject to attractive and repulsive couplings is analyzed. The probability of two elements in different layers being connected strongly depends on a defined vision range r_{c} which appears to lead both layers in different patterns while varying its values. Particularly, the interlayer static sync π has been found and its stability is proven. First-order transitions are observed when the repulsive coupling strength σ_{r} is very small for a fixed r_{c} and, moreover, in the absence of the repulsive coupling, they also appear for sufficiently large values of r_{c}. For σ_{r}=0 and for sufficiently small values of r_{c}, both layers achieve a second-order transition in a surprising two steps that are characterized by the drop of the energy of the internal phases while increasing the value of the interlayer attractive coupling σ_{a} and later a smooth jump, up to high energy value where synchronization is achieved. During these transitions, the internal phases present rotating waves with counterclockwise and later clockwise directions until synchronization, as σ_{a} increases. These results are supported by simulations and animations added as supplemental materials.

Self-Vibration of a Liquid Crystal Elastomer Fiber-Cantilever System under Steady Illumination

A new type of self-oscillating system has been developed with the potential to expand its applications in fields such as biomedical engineering, advanced robotics, rescue operations, and military industries. This system is capable of sustaining its own motion by absorbing energy from the stable external environment without the need for an additional controller. The existing self-sustained oscillatory systems are relatively complex in structure and difficult to fabricate and control, thus limited in their implementation in practical and complex scenarios. In this paper, we creatively propose a novel light-powered liquid crystal elastomer (LCE) fiber-cantilever system that can perform self-sustained oscillation under steady illumination. Considering the well-established LCE dynamic model, beam theory, and deflection formula, the control equations for the self-oscillating system are derived to theoretically study the dynamics of self-vibration. The LCE fiber-cantilever system under steady illumination is found to exhibit two motion regimes, namely, the static and self-vibration regimes. The positive work done by the tension of the light-powered LCE fiber provides some compensation against the structural resistance from cantilever and the air damping. In addition, the influences of system parameters on self-vibration amplitude and frequency are also studied. The newly constructed light-powered LCE fiber-cantilever system in this paper has a simple structure, easy assembly/disassembly, easy preparation, and strong expandability as a one-dimensional fiber-based system. It is expected to meet the application requirements of practical complex scenarios and has important application value in fields such as autonomous robots, energy harvesters, autonomous separators, sensors, mechanical logic devices, and biomimetic design.

Heat-Driven Synchronization in Coupled Liquid Crystal Elastomer Spring Self-Oscillators

Self-oscillating coupled machines are capable of absorbing energy from the external environment to maintain their own motion and have the advantages of autonomy and portability, which also contribute to the exploration of the field of synchronization and clustering. Based on a thermally responsive liquid crystal elastomer (LCE) spring self-oscillator in a linear temperature field, this paper constructs a coupling and synchronization model of two self-oscillators connected by springs. Based on the existing dynamic LCE model, this paper theoretically reveals the self-oscillation mechanism and synchronization mechanism of two self-oscillators. The results show that adjusting the initial conditions and system parameters causes the coupled system to exhibit two synchronization modes: in-phase mode and anti-phase mode. The work conducted by the driving force compensates for the damping dissipation of the system, thus maintaining self-oscillation. The phase diagrams of different system parameters are drawn to illuminate the self-oscillation and synchronization mechanism. For weak interaction, changing the initial conditions may obtain the modes of in-phase and anti-phase. Under conditions of strong interactions, the system consistently exhibits an in-phase mode. Furthermore, an investigation is conducted on the influence of system parameters, such as the LCE elastic coefficient and spring elastic coefficient, on the amplitudes and frequencies of the two synchronization modes. This study aims to enhance the understanding of self-oscillator synchronization and its potential applications in areas such as energy harvesting, power generation, detection, soft robotics, medical devices and micro/nanodevices.

Synchronization of Sakaguchi swarmalators

Swarmalators are phase oscillators that cluster in space, like fireflies flashing in a swarm to attract mates. Interactions between particles, which tend to synchronize their phases and align their motion, decrease with the distance and phase difference between them, coupling the spatial and phase dynamics. In this work, we explore the effects of inducing phase frustration on a system of swarmalators that move on a one-dimensional ring. Our model is inspired by the well-known Kuramoto-Sakaguchi equations. We find, numerically and analytically, the ordered and disordered states that emerge in the system. The active states, not present in the model without frustration, resemble states found previously in numerical studies for the two-dimensional swarmalators system. One of these states, in particular, shows similarities to turbulence generated in a flattened media. We show that all ordered states can be generated for any values of the coupling constants by tuning the phase frustration parameters only. Moreover, many of these combinations display multistability.

Nanorobot-Mediated Synchronized Neuron Activation

Interactions between active materials lead to collective behavior and even intelligence beyond the capability of individuals. Such behaviors are prevalent in nature and can be observed in animal colonies, providing these species with diverse capacities for communication and cooperation. In artificial systems, however, collective intelligence systems interacting with biological entities remains unexplored. Herein, we describe black (B)-TiO2@N/Au nanorobots interacting through photocatalytic pure water splitting-induced electrophoresis that exhibit periodic swarming oscillations under programmed near-infrared light. The periodic chemical-electric field generated by the oscillating B-TiO2@N/Au nanorobot swarm leads to local neuron activation in vitro. The field oscillations and neurotransmission from synchronized neurons further trigger the resonance oscillation of neuron populations without synaptic contact (about 2 mm spacing), in different ways from normal neuron oscillation requiring direct contact. We envision that the oscillating nanorobot swarm platforms will shed light on contactless communication of neurons and offer tools to explore interactions between neurons.

A physics-based model of swarming jellyfish

We propose a model for the structure formation of jellyfish swimming based on active Brownian particles. We address the phenomena of counter-current swimming, avoidance of turbulent flow regions and foraging. We motivate corresponding mechanisms from observations of jellyfish swarming reported in the literature and incorporate them into the generic modelling framework. The model characteristics is tested in three paradigmatic flow environments.

Using neuronal models to capture burst-and-glide motion and leadership in fish

While mathematical models, in particular self-propelled particle models, capture many properties of large fish schools, they do not always capture the interactions of smaller shoals. Nor do these models tend to account for the use of intermittent locomotion, often referred to as burst-and-glide, by many species. In this paper, we propose a model of social burst-and-glide motion by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion. We first show that our model can capture the motion of a single fish swimming down a channel. Extending to a two-fish model, where visual stimulus of a neighbour affects the internal burst or glide state of the fish, we observe a rich set of dynamics found in many species. These include: leader-follower behaviour; periodic changes in leadership; apparently random (i.e. chaotic) leadership change; and tit-for-tat turn taking. Moreover, unlike previous studies where a randomness is required for leadership switching to occur, we show that this can instead be the result of deterministic interactions. We give several empirically testable predictions for how bursting fish interact and discuss our results in light of recently established correlations between fish locomotion and brain activity.