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.

Department of Macromolecular Physics Faculty of Mathematics and Physics Charles University 18000 Prague Czech Republic

Institute for Theoretical Physics Leipzig University Postfach 100 920 04009 Leipzig Germany

Peter Debye Institute for Soft Matter Physics Leipzig University 04103 Leipzig Germany

Zobrazit více v PubMed

Kauffman, S. The Origins of Order: Self-organization and Selection in Evolution (Oxford Univ. Press, 1993).

Vicsek T, Zafeiris A. Collective motion. Phys. Rep. 2012;517:71–140. doi: 10.1016/j.physrep.2012.03.004. DOI

Delcourt J, Bode NWF, Denoël M. Collective vortex behaviors: diversity, proximate, and ultimate causes of circular animal group movements. Q. Rev. Biol. 2016;91:1–24. doi: 10.1086/685301. PubMed DOI

Strandburg-Peshkin A, et al. Visual sensory networks and effective information transfer in animal groups. Curr. Biol. 2013;23:R709–R711. doi: 10.1016/j.cub.2013.07.059. PubMed DOI PMC

Pearce DJG, Miller AM, Rowlands G, Turner MS. Role of projection in the control of bird flocks. Proc. Natl. Acad. Sci. USA. 2014;111:10422–10426. doi: 10.1073/pnas.1402202111. PubMed DOI PMC

Cremer J, et al. Chemotaxis as a navigation strategy to boost range expansion. Nature. 2019;575:658–663. doi: 10.1038/s41586-019-1733-y. PubMed DOI PMC

Couzin, I. D. & Krause, J. Self-organization and collective behavior in vertebrates. Adv. Study Behav. 32, 1–75 (2003).

Ioannou CC, Guttal V, Couzin ID. Predatory fish select for coordinated collective motion in virtual Prey. Science. 2012;337:1212–1215. doi: 10.1126/science.1218919. PubMed DOI

Berdahl AM, et al. Collective animal navigation and migratory culture: from theoretical models to empirical evidence. Philos. Trans. R. Soc. B Biol. Sci. 2018;373:20170009. doi: 10.1098/rstb.2017.0009. PubMed DOI PMC

Vicsek T, Czirók A, Ben-Jacob E, Cohen I, Shochet O. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 1995;75:1226–1229. doi: 10.1103/PhysRevLett.75.1226. PubMed DOI

Hemelrijk CK, Hildenbrandt H. Some causes of the variable shape of flocks of birds. PLoS ONE. 2011;6:e22479. doi: 10.1371/journal.pone.0022479. PubMed DOI PMC

Costanzo A, Hemelrijk CK. Spontaneous emergence of milling (vortex state) in a Vicsek-like model. J. Phys. D Appl. Phys. 2018;51:134004. doi: 10.1088/1361-6463/aab0d4. DOI

Cavagna A, et al. Scale-free correlations in starling flocks. Proc. Natl. Acad. Sci. USA. 2010;107:11865–11870. doi: 10.1073/pnas.1005766107. PubMed DOI PMC

Mora T, Bialek W. Are biological systems poised at criticality? J. Stat. Phys. 2011;144:268–302. doi: 10.1007/s10955-011-0229-4. DOI

Muñoz MA. Colloquium: criticality and dynamical scaling in living systems. Rev. Mod. Phys. 2018;90:031001. doi: 10.1103/RevModPhys.90.031001. DOI

Cavagna A, et al. Dynamic scaling in natural swarms. Nat. Phys. 2017;13:914–918. doi: 10.1038/nphys4153. DOI

Kim, D. W., Hong, H. & Kim, J. K. Systematic inference identifies a major source of heterogeneity in cell signaling dynamics: the rate-limiting step number. Sci. Adv. 8, eabl4598 (2022). PubMed PMC

Zhang J, Zhou T. Markovian approaches to modeling intracellular reaction processes with molecular memory. Proc. Natl. Acad. Sci. USA. 2019;116:23542–23550. doi: 10.1073/pnas.1913926116. PubMed DOI PMC

More HL, Donelan JM. Scaling of sensorimotor delays in terrestrial mammals. Proc. R. Soc. B Biol. 2018;285:20180613. doi: 10.1098/rspb.2018.0613. PubMed DOI PMC

Mijalkov M, McDaniel A, Wehr J, Volpe G. Engineering sensorial delay to control phototaxis and emergent collective behaviors. Phys. Rev. X. 2016;6:1—16.

Forgoston E, Schwartz IB. Delay-induced instabilities in self-propelling swarms. Phys. Rev. E. 2008;77:035203. doi: 10.1103/PhysRevE.77.035203. PubMed DOI PMC

Piwowarczyk R, Selin M, Ihle T, Volpe G. Influence of sensorial delay on clustering and swarming. Phys. Rev. E. 2019;100:012607. doi: 10.1103/PhysRevE.100.012607. PubMed DOI

Khadka U, Holubec V, Yang H, Cichos F. Active particles bound by information flows. Nat. Commun. 2018;9:3864. doi: 10.1038/s41467-018-06445-1. PubMed DOI PMC

Holubec V, Geiss D, Loos SAM, Kroy K, Cichos F. Finite-size scaling at the edge of disorder in a time-delay vicsek model. Phys. Rev. Lett. 2021;127:258001. doi: 10.1103/PhysRevLett.127.258001. PubMed DOI

Attanasi A, et al. Information transfer and behavioural inertia in starling flocks. Nat. Phys. 2014;10:615–698. doi: 10.1038/nphys3035. PubMed DOI PMC

Qian B, Montiel D, Bregulla A, Cichos F, Yang H. Harnessing thermal fluctuations for purposeful activities: The manipulation of single micro-swimmers by adaptive photon nudging. Chem. Sci. 2013;4:1420—1429. doi: 10.1039/c2sc21263c. DOI

Bregulla AP, Yang H, Cichos F. Stochastic localization of microswimmers by photon nudging. ACS Nano. 2014;8:6542–6550. doi: 10.1021/nn501568e. PubMed DOI

Söker NA, Auschra S, Holubec V, Kroy K, Cichos F. How activity landscapes polarize microswimmers without alignment forces. Phys. Rev. Lett. 2021;126:228001. doi: 10.1103/PhysRevLett.126.228001. PubMed DOI

Baeuerle T, Loeffler RC, Bechinger C. Formation of stable and responsive collective states in suspensions of active colloids. Nat. Commun. 2020;11:2547. doi: 10.1038/s41467-020-16161-4. PubMed DOI PMC

Loeffler RC, Baeuerle T, Kardar M, Rohwer CM, Bechinger C. Behavior-dependent critical dynamics in collective states of active particles. EPL. 2021;134:64001. doi: 10.1209/0295-5075/ac0c68. DOI

Liebchen B, Löwen H. Which interactions dominate in active colloids? Chem. Phys. 2019;150:061102. PubMed

Stark H. Artificial chemotaxis of self-phoretic active colloids: collective behavior. Acc. Chem. Res. 2018;51:2681–2688. doi: 10.1021/acs.accounts.8b00259. PubMed DOI

Auschra S, Bregulla A, Kroy K, Cichos F. Thermotaxis of Janus particles. Eur. Phys. J. E. 2021;44:90. doi: 10.1140/epje/s10189-021-00090-1. PubMed DOI PMC

Fränzl M, Muinos-Landin S, Holubec V, Cichos F. Fully steerable symmetric thermoplasmonic microswimmers. ACS Nano. 2021;15:3434–3440. doi: 10.1021/acsnano.0c10598. PubMed DOI

Selmke M, Khadka U, Bregulla AP, Cichos F, Yang H. Theory for controlling individual self-propelled micro-swimmers by photon nudging I: directed transport. Phys. Chem. Chem. Phys. 2018;20:10502–10520. doi: 10.1039/C7CP06559K. PubMed DOI

Selmke M, Khadka U, Bregulla AP, Cichos F, Yang H. Theory for controlling individual self-propelled micro-swimmers by photon nudging II: confinement. Phys. Chem. Chem. Phys. 2018;20:10521–10532. doi: 10.1039/C7CP06560D. PubMed DOI

Tunstrøm K, et al. Collective states, multistability and transitional behavior in schooling fish. PLoS Comput. Biol. 2013;9:e1002915. doi: 10.1371/journal.pcbi.1002915. PubMed DOI PMC

Insperger, T. On the approximation of delayed systems by Taylor series expansion. J. Comput. Nonlinear Dyn.10, 024503 (2015).

Muinos-Landin S, Fischer A, Holubec V, Cichos F. Reinforcement learning with artificial microswimmers. Sci. Robot. 2021;6:eabd9285. doi: 10.1126/scirobotics.abd9285. PubMed DOI

Strogatz, S. H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering (Perseus Books, 1994).

Goldenfeld, N. Lectures on Phase Transitions and the Renormalization Group 1st edn (CRC Press, 1992).

Vollmer J, Vegh AG, Lange C, Eckhardt B. Vortex formation by active agents as a model for Daphnia swarming. Phys. Rev. E. 2006;73:061924. doi: 10.1103/PhysRevE.73.061924. PubMed DOI

Bregulla AP, Würger A, Günther K, Mertig M, Cichos F. Thermo-osmotic flow in thin films. Phys. Rev. Lett. 2016;116:188303. doi: 10.1103/PhysRevLett.116.188303. PubMed DOI

Popescu MN, Uspal WE, Dietrich S. Chemically active colloids near osmotic-responsive walls with surface-chemistry gradients. J. Phys. Condens. Matter. 2017;29:134001. doi: 10.1088/1361-648X/aa5bf1. PubMed DOI

Spagnolie SE, Lauga E. Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations. J. Fluid Mech. 2012;700:105–147. doi: 10.1017/jfm.2012.101. DOI

Lintuvuori JS, Würger A, Stratford K. Hydrodynamics defines the stable swimming direction of spherical squirmers in a nematic liquid crystal. Phys. Rev. Lett. 2017;119:068001. doi: 10.1103/PhysRevLett.119.068001. PubMed DOI

Wioland H, Woodhouse FG, Dunkel J, Goldstein RE. Ferromagnetic and antiferromagnetic order in bacterial vortex lattices. Nat. Phys. 2016;12:341–345. doi: 10.1038/nphys3607. PubMed DOI PMC

Nishiguchi D, Aranson IS, Snezhko A, Sokolov A. Engineering bacterial vortex lattice via direct laser lithography. Nat. Commun. 2018;9:4486. doi: 10.1038/s41467-018-06842-6. PubMed DOI PMC

Fletcher G. A mechanical analog of first- and second-order phase transitions. Am. J. Phys. 1997;65:74–81. doi: 10.1119/1.18522. DOI

Kuramoto Y. International symposium on mathematical problems in theoretical physics. Lect. Notes Phys. 1975;30:420. doi: 10.1007/BFb0013365. DOI

O’Keeffe KP, Hong H, Strogatz SH. Oscillators that sync and swarm. Nat. Commun. 2017;8:1504. doi: 10.1038/s41467-017-01190-3. PubMed DOI PMC

Fruchart M, Hanai R, Littlewood PB, Vitelli V. Non-reciprocal phase transitions. Nature. 2021;592:363–369. doi: 10.1038/s41586-021-03375-9. PubMed DOI

Loos SAM, Klapp SHL. Fokker–planck equations for time-delayed systems via markovian embedding. J. Stat. Phys. 2019;177:95–118. doi: 10.1007/s10955-019-02359-4. DOI

Geiss D, Kroy K, Holubec V. Brownian molecules formed by delayed harmonic interactions. New J. Phys. 2019;21:093014. doi: 10.1088/1367-2630/ab3d76. DOI

Drescher K, Dunkel J, Cisneros LH, Ganguly S, Goldstein RE. Fluid dynamics and noise in bacterial cell-cell and cell-surface scattering. Proc. Natl. Acad. Sci. USA. 2011;108:10940–10945. doi: 10.1073/pnas.1019079108. PubMed DOI PMC

Lauder GV, Drucker EG. Forces, fishes, and fluids: hydrodynamic mechanisms of aquatic locomotion. Physiology. 2002;17:235–240. doi: 10.1152/nips.01398.2002. PubMed DOI

Verma S, Novati G, Koumoutsakos P. Efficient collective swimming by harnessing vortices through deep reinforcement learning. Proc. Natl. Acad. Sci. USA. 2018;115:5849–5854. doi: 10.1073/pnas.1800923115. PubMed DOI PMC

Morin A, Caussin J-B, Eloy C, Bartolo D. Collective motion with anticipation: flocking, spinning, and swarming. Phys. Rev. E. 2015;91:012134. doi: 10.1103/PhysRevE.91.012134. PubMed DOI

Palmer SE, Marre O, Berry MJ, Bialek W. Predictive information in a sensory population. Proc. Natl. Acad. Sci. USA. 2015;112:6908–6913. doi: 10.1073/pnas.1506855112. PubMed DOI PMC