A review of swarmalators and their potential in bio-inspired computing
AA review of swarmalators and their potential in bio-inspired computing
Kevin O’Keeffe and Christian Bettstetter Senseable City Lab, Massachusetts Institute of Technology, Cambridge, MA 02139 Institute of Networked and Embedded Systems, University of Klagenfurt, 9020, Austria
From fireflies to heart cells, many systems in Nature show the remarkable ability to spontaneouslyfall into synchrony. By imitating Nature’s success at self-synchronizing, scientists have designed cost-effective methods to achieve synchrony in the lab, with applications ranging from wireless sensornetworks to radio transmission. A similar story has occurred in the study of swarms, where in-spiration from the behavior flocks of birds and schools of fish has led to ’low-footprint’ algorithmsfor multi-robot systems. Here, we continue this ’bio-inspired’ tradition, by speculating on the tech-nological benefit of fusing swarming with synchronization. The subject of recent theoretical work,minimal models of so-called ’swarmalator’ systems exhibit rich spatiotemporal patterns, hinting atutility in ’bottom-up’ robotic swarms. We review the theoretical work on swarmalators, identifypossible realizations in Nature, and discuss their potential applications in technology.
I. INTRODUCTION
In 1967 Winfree proposed a model for coupled oscilla-tors that spontaneously synchronize [1]. Beyond a criticalcoupling strength, the oscillators overcome the disorder-ing effect of the dissimilarities in their natural frequenciesand spontaneously lock their cycles. Kuramoto later sim-plified Winfree’s model and solved it exactly [2]. Sincethen, the study of sync has matured into a vibrant field[3–5]. On the theoretical side, theorists have twisted Ku-ramoto’s model in various ways resulting in rich phenom-ena like glassy behavior [6–9] and chimeras states [10–12].On the applied side, coupled oscillators have found usein neurobiology [13–16], cardiac dynamics [17–19], andthe bunching of school buses [20]. An interesting appli-cation of sync is the design of ‘bio-inspired’ algorithms.Here, by aping the simplicity of coupled oscillator models,researchers have devised resource-efficient algorithms toprocure synchrony in the lab, useful for networked com-puting and robotics [21–24].A story with many parallels to the sync story hasevolved in the study of swarms. In 1995 Vicsek proposeda simple model of swarming agents [25], which – like thesync transition of coupled oscillators – showed a transi-tion from disorder to order: beyond a critical couplingstrength, the agents switched from a gas-like, incoher-ent state to one in which the agents moved as a coher-ent flock. The novelty of the out-of-equilibrium natureof the flocking transition – the system is out of equilib-rium since agents constantly consume energy to propelthemselves – piqued the minds of physicists and othertheorists, which in turn helped give rise to the field ofactive matter [26–28], a field perhaps more vibrant thanthe field of synchronization. In a final mirroring of thesync story, the bio-inspired community has also mimickedthe minimalism of Vicsek’s and other models of swarm-ing to design novel algorithms for optimization [29–34]and robotic swarms [33, 35–38].These stories demonstrate swarming and synchroniza-tion are intimately related. In a sense, the two effects are‘spatiotemporal opposites’: in synchronization the units self-organize in time but not in space; in swarming theunits self-organize in space but not in time. Given thisconceptual twinship, it is natural to wonder about thepossibility of units which can self-organize in both spaceand time – that is, to wonder how swarming and syn-chronization might interact. And more pertinent to thiswork, to wonder how a mix of swarming and synchro-nization could be useful in technology.Several researchers have started to address these ques-tions by analyzing systems that both sync and swarm.Von Brecht and Unimsky have generalized swarming par-ticles by endowing them with an internal polarizationvector [39], equivalent to an oscillator’s phase. Othershave attacked the problem from the other direction, byconsidering synchronizing oscillators able to move aroundin space [40–43]. In these studies, however, the oscil-lators’ movements affect their phase dynamics, but notthe other way around; thus, the interaction betweenswarming and synchronization is only one-way. Two-way interaction between swarming and synchronizationhas also been considered: The pioneering work is theIwasa-Tanaka model of chemotactic oscillators [44, 45],i.e., oscillators which interact through a background dif-fusing chemical. More recent works were carried out byStarnini et al [46], Belovs et al [47], and O’Keeffe et al[48] who proposed ‘bottom-up’ toy models without refer-ence to a background medium. O’Keeffe et al called theelements of their systems ‘swarmalators’, to capture theirtwin identities as swarming oscillators, and to distinguishthem from the mobile oscillators mentioned above forwhich the coupling between swarming and synchroniza-tion is unidirectional.This paper reviews research on swarmalators andother systems that mix swarming and synchronization.Our main motivation is to identify if the interplay of syncand swarming can be useful for bio-inspired computingand related engineering domains. We outline the theoret-ical work on swarmalators, experimental realizations, andfinally conjecture on their technological utility. a r X i v : . [ n li n . AO ] M a r FIG. 1:
Swarmalator states . Scatter plots in the ( x, y ) plane, where the swarmalators are colored according totheir phase. (a) Static sync for (
J, K ) = (0 . , J, K ) = (0 . , − J, K ) = (1 , J, K ) = (1 , − . J, K ) = (1 , − . II. MODELS COMBINING SWARMING ANDSYNCHRONIZATION
To combine synchronization with swarming, we firstdefine what we mean by swarming. While to our knowl-edge there is no universally agreed on definition, swarm-ing systems typically have at least one of two key features:( i ) aggregation, arising from a balance between the units’mutual attraction and repulsion and ( ii ) alignment, re-ferring to the units’ tendency to align their orientation inspace and move in a flock. Synchronization can thus becombined with either aggregation alone, alignment alone,or with both aggregation and alignment. In what follows,we present classes of model based on these three ways tocombine sync and swarming. - - - - - K - J Static asyncStatic asyncActivephase wave
Splinteredphase wave
Staticsync
Staticphase wave
Staticsync
FIG. 2:
Phase diagram . Locations of states of themodel defined by equations (3) and (4) in the (
J, K )plane. The straight line separating the static async andactive phase wave states is a semi-analyticapproximation given by J ≈ . K – see Eq. (18) in [48].Black and red dots show simulation data. The reddashed line simply connects the red dots and wasincluded to make the boundary visually clearer. Note,this figure has been reproduced from [48]. A. Aggregation and Synchronization
The paradigmatic model of biological aggregation hasthe form˙ x i = 1 N N (cid:88) j (cid:54) = i I att ( x j − x i ) − I rep ( x j − x i ) , (1)where x i ∈ R d (usually with d ≤
3) is the i -th particle’sposition, I att captures the attraction between particles,and I rep captures the repulsion between them. The com-petition between I att and I rep gives rise to congregationsof particles with sharp boundaries, in accordance withmany biological systems (see [49, 50] for a review).The paradigmatic model in synchronization is the Ku-ramoto model [51]:˙ θ i = ω i + KN N (cid:88) j (cid:54) = i sin( θ j − θ i ) . (2)Here θ i ∈ S and ω i are the phase and natural frequencyof the i -th oscillator. The sine term captures the os-cillators’ coupling, where K >
K < K c , a fraction of oscillators overcome the dis-ordering effects imposed by their distributed natural fre-quencies ω i and spontaneously synchronize.To study the co-action of synchronization and aggre-gation, a natural strategy would be to stitch the aggre-gation model (1) and the Kuramoto model (2) together.This was the approach taken by O’Keeffe et al [48] whoproposed the following swarmalator model:˙ x i = 1 N N (cid:88) j (cid:54) = i (cid:34) x j − x i | x j − x i | (cid:16) J cos( θ j − θ i ) (cid:17) − x j − x i | x j − x i | (cid:35) (3)˙ θ i = ω i + KN N (cid:88) j (cid:54) = i sin( θ j − θ i ) | x j − x i | . (4)Equation (3) models phase-dependent aggregation andEquation (4) models position-dependent synchroniza-tion. The interaction between the space and phase dy-namics is captured by the term 1 + J cos( θ j − θ i ). If J >
0, “like attracts like”: swarmalators are prefer-entially attracted to other swarmalators with the samephase, while
J < ω i = ω ,and by a change of reference they set ω = 0.The swarmalator model exhibits five long-term col-lective states. Figure 1 showcases these states as scatterplots in the ( x, y ) plane, where swarmalators are repre-sented by dots and the color of each dot represents theswarmalator’s phase θ (color, recall, can be mapped to S and so can be used to represent swarmalators’ phases).The parameter dependence of these states are encapsu-lated in the phase diagram shown in Figure 1. The firstthree states, named the static sync , static async , and static phase wave , are – as their names suggest – static inthe sense that the individual swarmalators are stationaryin both position and phase. This stationarity allows thedensity of these swarmalators ρ ( x , θ ) in these states to beconstructed explicitly (the density ρ ( x , θ ) is interprettedin the Eulerian sense, so that ρ ( x + dx , θ + dθ ) gives thefraction of swarmalators with positions between x and x + d x and phases between θ and θ + dθ ). In the remaining splintered phase wave and active phase waves states swar-malators are no longer stationary. In the splintered phasewave state swarmalators execute small oscillation in bothspace and phase within each cluster. In the active phasewave the swarmalators split into counter-rotating groups– in both space and phase – so that (cid:104) ˙ θ i (cid:105) = (cid:104) ˙ φ i (cid:105) = 0,where φ i = arctan( y i /x i ) is the spatial angle of the i -thoscillator and angle brackets denote population averages.The conservation of these two quantities follows from thegoverning equations; the pairwise terms are odd and thuscancel under summation. Three-dimensional analoguesof the five collective states were also reported.The stability properties of the static async state areunusual. Via a linear stability analysis in density space,an integral equation for the eigenvalues λ was derived.Yet numerical solutions to this integral equation, in theparameter regime where the static async state should bestable, produced a leading eigenvalue so small in magni-tude that its sign could not be determined reliably. Thus,the stability of the state could not be analytically con-firmed. There is however a parameter value at whichthe magnitude of λ increases sharply, which was usedto derive a pseudo critical parameter value K c ≈ − . J marking the apparent (i.e. as observed in simulations)destabilization of the state. Nevertheless, the true sta-bility of the static async state remains a puzzle.Some extensions of the work in [48] have been car-ried out. One addresses the conspicuously empty up-per right quadrant in the ( J, K ) parameter plane in Fig-ure 2, where only the trivial static sync state appears.By adding phase noise to (4), Hong discovered [53] theactive phase wave exists for
K >
0. Unexpectedly, thesplintered phase wave was not observed. O’Keeffe, Ev-ers, and Kolokolnikov [52] have extended (3) by allowing - - - - - - - - FIG. 3:
Ring phase state.
Swarmalators arerepresented by colored dots in the ( x, y ) plane wherethe color of each dot represents its phase. The state isfound by numerically integrating the governingequations in [52] with ( J , J , K, N ) = (0 , . , , J cos( θ j − θ i )). Thisled to the emergence of ring states, an example of whichis depicted in Figure 3. They constructed and analyzedthe stability of these ring states explicitly for a popula-tion of given size N . Analytic results for arbitrary N arepotentially useful for robotic swarms, which presumablyare realized in the small N limit. Another offshoot of thisanalysis was a heuristic method to predict the number ofclusters which form in the splintered phase wave state;they viewed each cluster of synchronized swarmalatorsas one giant swarmalator which let them re-imagine thesplintered phase wave state as a ring, allowing them toleverage their analysis. A precise description of the num-ber of clusters formed is an open problem. Iwasa-Tanaka model . Iwasa and Tanaka pro-posed and studied a ‘swarm-oscillator’ model in a se-ries of papers [44, 54–57]. The inspiration for their workcomes from chemotactic oscillators, i.e., oscillators mov-ing around in a diffusing chemical which mediates theirinteractions. They began with the general model˙ X i ( t ) = f ( X i ) + kg ( S ( r i , t )) (5) m ¨ r i ( t ) = − γ ˙ r i − σ ( X i ) ∇ S (6) τ ∂ τ S ( r , t ) = − S + d ∇ S + (cid:88) i h ( X ) δ ( r − r i ) , (7)where X i represents the internal state (which will laterbe identified as a phase), r represents the position of the i -th oscillator, and S represents the concentration of thebackground chemical. By means of a center manifoldcalculation and a phase reduction technique they derivedthe simpler equations˙ ψ i ( t ) = (cid:88) j (cid:54) = i e −| R ji | sin(Ψ ji − α | R ji | − c ) (8)˙ r i ( t ) = c (cid:88) j (cid:54) = i ˆ R ji e −| R ji | sin(Ψ ji − α | R ji | − c ) (9)where R ji = R j − R i , Ψ ji = ψ j − ψ i , and ψ i is the i -th os-cillator’s phase. We call this Iwasa-Tanaka model. Noticethe space-phase coupling in this model is somewhat pe-culiar; in contrast to the swarmalator model given by (3)and (4) the relative position R ji and relative phase Φ ji appear inside the sine terms in both the ˙ r i and ˙ ψ i equa-tions. Another difference between the two models is that˙ r i in (8) has no hardshell repulsion term, which means theoscillators can occupy the same position in space.The Iwasa-Tanaka model has rich collective states.An exhaustive catalogue of these states with respect tothe model’s four parameters is an ongoing effort [57].Highlights include a family of clustered states [44, 45, 56]in which swarmalators collect in synchronous groups.The spatial distributions of these groups depend on theirphase, similar to the splintered phase wave (Figure 1).The authors speculate this phase clustering is reminiscentof the chemotactic cell sorting during biological develop-ment [44]. The Iwasa-Tanaka model also produces ringstates [45], as well as an interesting ‘juggling’ state [54] inwhich the population forms a “rotating triangular struc-ture whose corers appear to ‘catch’ and ‘throw’ individualelements” – in other words, the population juggles the el-ements around the corners of a triangle (see Figure 1 in[54]). Aside from theoretical novelty, this juggling couldconceivably be exploited in robotic swarms, potentiallyallowing some form of relay between the elements. B. Alignment and synchronization
In our proposed taxonomy, systems that combinealignment and synchronization are characterized by aninternal phase θ and an orientation β without a dynamicspatial degree of freedom. In other words, the particles’position x might affect their θ and β dynamics, but x itself does not evolve in time. Although units character-ized by just a phase θ and orientation β might seem odd,Leon and Liverpool studied a system with units whichmeet these criteria: a class of soft active fluids whichconstitute a ‘new type of nonequilibrium soft matter –a space-time liquid crystal’ [58]. They developed a phe-nomenological theory of these soft fluids which allowedthem to derive dynamical equations for order parame-ters quantifying the orientation order, phase order, andorientation-phase order. These revealed collective stateswhich maximize each of these order parameters: alignedstates with orientational order but no phase order, sync’dstates with phase order but no orientational order, andstates with both phase and orientational order. They were able to partially analyze these states, and conjec-tured the states could be realized in protein-filamentsmixtures, such as cell cytoskeletons [59, 60] or tissue-forming cells [61]. C. Alignment, aggregation, and synchronization
Systems with units that align, aggregate, and syn-chronize – and therefore have dynamic state variables x , θ, β – are the least well studied. A small study wascarried out in [48], with the aim of checking if the swar-malator states reported are generic, i.e., robust to the in-clusion of alignment dynamics (it was found they were).Beyond this preliminary study, the space of possible be-haviors arising from alignment, aggregation, and syn-chronization is largely unexplored. D. Alignment and aggregation
While not strictly within our proposed taxonomy ofswarmalator systems, we give a brief review of studieson aggregation and alignment. We do this because align-ment can be viewed as a type of synchronization, whereinstead of units adjusting to a common phase θ , unitsinstead adjust a common orientation β . Or put anotherway, because an internal phase θ and orientation β areformally equivalent – both being circular variables – par-ticles aligning their orientations is analogous to oscilla-tors aligning their phases.Vicsek set the paradigm of this class of models witha beautiful and simple model: r i ( t + δt ) = r i ( t ) + ( νδt )ˆ n (10) β i ( t + δt ) = (cid:104) β j (cid:105) | r j − r i | To our knowledge, there are just two works which un-equivocally realize swarmalators in the real world – bythis we mean, precisely, a real-world system in which abidirectional space-phase coupling is unambiguously ex-hibited. These works are: Swarmalatorbots . Bettstetter et al first realizedthe collective states of the swarmalator model (Figure 1)in the lab using small robots [64]. They programmed‘swarmalatorbots’ whose governing equations were de-rived from the swarmalator model (3), (4). Magnetic domain walls . To our knowledge, thefirst realization of a natural swarmalator system wasfound by Hrabec et al when studying the magnetic do-main walls [65]. Ordinarily, domain walls are describedby a single spatial degree of freedom x . But as the au-thors note, the dynamics on the walls also depend ontheir internal structure. The authors minimally describethis internal state by a one dimensional polarization an-gle of the internal magnetic field θ . This allows the wallto be viewed as a point particle with position x and phase θ – to be viewed as a swarmalator [65]. An experimentalstudy of coupling between two domain walls – usually justone wall is studied – revealed the walls can synchronize,which in turn affects the walls’ velocity. Richer space-phase dynamics, such as families of Lissajous curves, arealso reported. ***Beyond swarmalatorbots and magnetic domain walls,there are many systems in which a bidirectional couplingbetween swarming and synchronization might exist. Welist these candidate swarmalator systems below: Myxobacteria . Myxobacteria are bacteria com-monly found in soil, and can produce interesting col-lective effects [66]. The phase variable of myxobacteriacharacterizes the internal growth cycle of the cell. Groupsof cells interact through cell-to-cell contact, theorized toprovide a channel through which swarming and synchro-nization can couple. Populations of myxobacteria can ex-hibit a ‘ripple phase’, in which complex patterns of wavespropagate through population [66]. In contrast to otherwave phenomena in biological systems, such as the well-studied Dictyostelium discoideum, these rippling wavesdo not annihilate on collision. In [66] a Fokker-Plancktype equation was used to analyze rippling waves. Re-alizing them in a microscopic swarmalator system is anopen problem. Biological microswimmers . ‘Microswimmers’ isan umbrella term for self-propelled microorganisms con-fined to fluids, such as celia, bacteria, and sperm [67].Groups of microswimmers show swarming behavior as aresult of cooperative goal seeking, such as searching forfood or light. They can also synchronize: Here the phasevariable is associated with the rhythmic beating of swim-mers’ tails which – through hydrodynamic interactions –can synchronize with the beatings of others swimmers’tails. Whether or not this hydrodynamics provides abidirectional coupling between sync and swarming – asrequired of swarmalators – is unclear, but to us seemsplausible. Researchers have developed models of spermunder this assumption [68] and found clusters of synchro-nized sperm consistent with real data [69]. Vortex arraysof sperm have also been reported [70]. Here, sperm self-organize into subgroups arranged in a lattice; within eachsubgroup, the sperm move in a vortex, wherein a corre-lation between the their angular velocity and their phasevelocity is realized, reminiscent of the splintered phasewave state (Figure 1). Simulations of realistic modelshave been carried out [47] which show other interest-ing phenomena. Japanese tree frogs . During mating season, maleJapanese tree frogs attract females by croaking rhythmi-cally. The croaking of neighbouring frogs tend to anti-synchronize due to a precedence effect: croaking shortlyafter a rival makes a frog look less dominant. Researchershave theorized that this competition leads to mutual in-teraction between frogs’ space and phase dynamics, cou-pling sync and swarming. Models based on these as-sumption produce ring-like states where frogs arrangethemselves on the borders of fields with interesting space-phase patterns, some of which are consistent with dataon real tree-frogs’ behavior collected in the wild. Magnetic colloids . Similar to magnetic domainwalls, a colloid’s constituent particles can synchronizethe orientation of their magnetic dipole vector when suf-ficiently close. When in solution, they are free to movearound in space, creating a feedback loop between theirspace and phase dynamics. In ferromagnetic colloidsconfined to liquid-liquid interfaces, Snezhko and Aran-son [71] found this interaction leads to the formation of‘asters’ – star-like arrangements of particles whose spa-tial angles correlate with the orientations of their mag-netic dipole vectors, equivalent to the static phase wave(Figure 1). Yan et al explored how synchronization isuseful in colloids of Janus particles [72]. Janus particlesare micrometer-sized spheres with one hemisphere cov-ered with nickle which gives them non-standard magneticproperties. In particular when subject to a precessingmagnetic field, they oscillate about their centers of mass.This oscillation creates a coupling between particles, giv-ing rise to ‘synchronization-selected’ self-assembly. Forexample, zig-zag chains of particles and microtubes ofsynchronized particles can be realized. IV. SWARMALATORS IN BIO-INSPIREDCOMPUTING The computer engineering community has been in-spired by coupled oscillators to develop new techniquesfor synchronization in communication and sensor net-works [21–24, 73, 74]. Here, instead of Kuramoto-typeoscillators (2), models of pulse-coupled oscillators havebeen borrowed. As the name suggests, pulse-coupled os-cillators communicate by exchanging short signals; thistime-discrete variant of coupling is a more natural fit forapplication in technology, where smooth, continuous cou-pling – as exemplified by Kuramoto oscillators – is costlyto achieve. The canonical model of pulse-coupled oscil-lators is the Peskin model [17], defined by˙ x i = S − γx i (14)with S , γ > x i is a voltage like state variable forthe i -th oscillator. When the oscillators’ voltage reacha threshold value they ( i ) fire a pulse which instan-taneously raises the voltage of all the other oscillators( ii ) reset their voltage to zero, along with any other os-cillator whose voltage was raised above the threshold onaccount of receiving a pulse. The synchronization prop-erties of the Peskin model are well-studied [75, 76]. Morerecently, estimates for the convergence speed have beenderived [77, 78].The simplicity, distributed nature, adaptability, andscalability of pulse-coupled oscillators make them attrac-tive from an engineering point of view, where tempo-ral coordination is often a goal. For example, synchro-nization is required for many tasks in different layers ofcomputing and communications systems [79] such as inthe alignment of transmission slots for efficient mediumaccess [80], scheduling of sleep cycles for energy effi-ciency [81], and coordination of sensor readings to cap-ture a scene from different perspectives [82]. It is the rel-ative synchrony of the networked units, not necessarilythe absolute time, that is relevant here. The synchroniza-tion precision required is determined by the specific task.The achievable precision depends on the environment andhardware, and is influenced by deviations in delays andphase rates. Experiments in laboratory environmentsshow that the precision is in the order of hundred mi-croseconds with low-cost programmable sensor platforms[21, 83] and a few microseconds with field-programmablegate array (FPGA)-based radio boards [84]. Further usecases can be found in acoustics (synchronizing multipleloudspeakers) and energy systems (synchronizing decen-tralized grids [85, 86]), to give two examples.Synchronization is also important in robotics in orderto perform coordinated movements – and this is wheretemporal and spatial coordination unite. As mentioned,Bettstetter and colleagues [64] extended the swarmala-tor model [48] so that it could be applied to mobilerobots. They implemented the extended model in theRobot Operating System 2 (ROS 2) and experimentallydemonstrated that the space-time patterns achieved in theory (Figure 1) can be reproduced in the real world.Beyond realizing these specific states, we conjecture thatswarmalator-type models will enable novel self-assemblyprocedures in other robotics groups, in turn enabling col-laborative actions in monitoring, exploration, and ma-nipulation. As envisioned in [64], underwater robotscould be designed, which – in imitation of biological mi-croswimmers – could both swim in formation and syn-chronize their fin movements, thereby enhancing theirfunctional capabilities.More speculatively, the swarmalator concept could beuseful in the design of autonomous transport systems; forexample, when multiple vehicles driving in a convoy haveto avoid collisions with other convoys due to crossings andfor the purpose of overtaking. The model could also en-able self-configuring distributed antenna arrays (or loud-speakers) in which multiple antenna elements (or soundsources) automatically arrange their positions and orien-tations to create specific radiation patterns used to sendradio (audio) signals in a synchronous way. Anotherpromising application field is the planning and replan-ning of processes in factories, where products and ma-chines must follow a certain space-time order and wherestandard optimization methods reach their limits due tothe high system complexity. A final, more playful, ap-plication is in art: Artistic aerial light shows created bydrone swarms running the swarmalator model producecharming visual displays. V. DISCUSSION The implicit strategy in the theoretical works we re-viewed is: given a model, which spatiotemporal patternsemerge? Future work could consider the inverse strat-egy: given a desired spatiotemporal pattern, which formshould the model take on? – a question often consid-ered in engineering contexts. From a more general per-spective, one has to design the local rules and interac-tions that guide a self-organizing system toward a desiredglobal state [87]. Von Brecht et al have studied this in-verse question for swarming systems [88]; perhaps theirtools could be extended to swarmalators.Future work could also extend the reviewed models,which – recall – were designed to be as minimal as possi-ble. A wealth of synchronization phenomena have beenfound by adorning the Kuramoto model with new fea-tures like mixed-sign coupling [89–92], non-local coupling[11, 12], and delayed interactions [93]. Equipping swar-malators with these features would likely cure the povertyof phenomena in the first, third, and fourth quadrants ofthe ( J, K ) plane (Figure 2) and produce new dynamics inIswasa-tanaka and Vicesek type models, too. Phase mod-els more sophisticated than the Kuramoto model couldalso be explored; the Winfree model [94] or the newlyintroduced Janus oscillators [95] would be exciting to ex-periment with. Swarmalators with discrete and spatiallylocal coupling also merit study and are useful in com-puter and communication systems. Stitching the Peskinmodel (14) to the aggregation equation (1) or the Vicsekmodel (11) seems the natural way to do this. Continuingto study finite systems of swarmalators [52] is also perti-nent, since many robotic systems lie in the low N regime.And finally, beyond richer phase models, the spatial dy-namics of swarmalators could also be generalized. Evenwithin the framework of the aggregation equation (1), amenagerie of spatial patterns have been catalogued [96].Swarmalator counterparts to these states would be excit-ing to explore.We hope to have outlined that swarmalators havegreat potential in computing and other fields of tech-nology. Here we see swarmalators only as a case studyin a broader class of systems with large technologicalutility: systems whose units have both spatial andinternal degrees of freedom. The one-dimensional phase θ of a swarmalator is perhaps the simplest instance ofan internal degree of freedom, which can more generally be represented by a feature vector f . This vector coulddescribe a particle’s (three dimensional) magnetic orelectric field; a person’s political affiliation, mood,or health state; a bacteria’s phase in a non-circulardevelopmental cycle; and the mode of a robot, machine,or vehicle. 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