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Nabeel A, Jadhav V, M DR, Sire C, Theraulaz G, Escobedo R, Iyer SK, Guttal V. Data-driven discovery of stochastic dynamical equations of collective motion. Phys Biol 2023; 20:056003. [PMID: 37369222 DOI: 10.1088/1478-3975/ace22d] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2023] [Accepted: 06/27/2023] [Indexed: 06/29/2023]
Abstract
Coarse-grained descriptions of collective motion of flocking systems are often derived for the macroscopic or the thermodynamic limit. However, the size of many real flocks falls within 'mesoscopic' scales (10 to 100 individuals), where stochasticity arising from the finite flock sizes is important. Previous studies on mesoscopic models have typically focused on non-spatial models. Developing mesoscopic scale equations, typically in the form of stochastic differential equations, can be challenging even for the simplest of the collective motion models that explicitly account for space. To address this gap, here, we take a novel data-driven equation learning approach to construct the stochastic mesoscopic descriptions of a simple, spatial, self-propelled particle (SPP) model of collective motion. In the spatial model, a focal individual can interact withkrandomly chosen neighbours within an interaction radius. We considerk = 1 (called stochastic pairwise interactions),k = 2 (stochastic ternary interactions), andkequalling all available neighbours within the interaction radius (equivalent to Vicsek-like local averaging). For the stochastic pairwise interaction model, the data-driven mesoscopic equations reveal that the collective order is driven by a multiplicative noise term (hence termed, noise-induced flocking). In contrast, for higher order interactions (k > 1), including Vicsek-like averaging interactions, models yield collective order driven by a combination of deterministic and stochastic forces. We find that the relation between the parameters of the mesoscopic equations describing the dynamics and the population size are sensitive to the density and to the interaction radius, exhibiting deviations from mean-field theoretical expectations. We provide semi-analytic arguments potentially explaining these observed deviations. In summary, our study emphasises the importance of mesoscopic descriptions of flocking systems and demonstrates the potential of the data-driven equation discovery methods for complex systems studies.
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Affiliation(s)
- Arshed Nabeel
- Center for Ecological Sciences, Indian Institute of Science, Bengaluru, India
- IISc Mathematics Initiative, Indian Institute of Science, Bengaluru, India
| | - Vivek Jadhav
- Center for Ecological Sciences, Indian Institute of Science, Bengaluru, India
| | - Danny Raj M
- Department of Chemical Engineering, Indian Institute of Science, Bengaluru, India
| | - Clément Sire
- Laboratoire de Physique Théorique, CNRS, Université de Toulouse-Paul Sabatier, Toulouse, France
| | - Guy Theraulaz
- Center for Ecological Sciences, Indian Institute of Science, Bengaluru, India
- Centre de Recherches sur la Cognition Animale, Centre de Biologie Intégrative, CNRS, Université de Toulouse-Paul Sabatier, Toulouse, France
| | - Ramón Escobedo
- Centre de Recherches sur la Cognition Animale, Centre de Biologie Intégrative, CNRS, Université de Toulouse-Paul Sabatier, Toulouse, France
| | - Srikanth K Iyer
- Department of Mathematics, Indian Institute of Science, Bengaluru, India
| | - Vishwesha Guttal
- Center for Ecological Sciences, Indian Institute of Science, Bengaluru, India
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Jhawar J, Guttal V. Noise-induced effects in collective dynamics and inferring local interactions from data. Philos Trans R Soc Lond B Biol Sci 2020; 375:20190381. [PMID: 32713307 DOI: 10.1098/rstb.2019.0381] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
In animal groups, individual decisions are best characterized by probabilistic rules. Furthermore, animals of many species live in small groups. Probabilistic interactions among small numbers of individuals lead to a so-called intrinsic noise at the group level. Theory predicts that the strength of intrinsic noise is not a constant but often depends on the collective state of the group; hence, it is also called a state-dependent noise or a multiplicative noise. Surprisingly, such noise may produce collective order. However, only a few empirical studies on collective behaviour have paid attention to such effects owing to the lack of methods that enable us to connect data with theory. Here, we demonstrate a method to characterize the role of stochasticity directly from high-resolution time-series data of collective dynamics. We do this by employing two well-studied individual-based toy models of collective behaviour. We argue that the group-level noise may encode important information about the underlying processes at the individual scale. In summary, we describe a method that enables us to establish connections between empirical data of animal (or cellular) collectives and the phenomenon of noise-induced states, a field that is otherwise largely limited to the theoretical literature. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.
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Affiliation(s)
- Jitesh Jhawar
- Centre for Ecological Sciences, Indian Institute of Science, Bengaluru 560012, India
| | - Vishwesha Guttal
- Centre for Ecological Sciences, Indian Institute of Science, Bengaluru 560012, India
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Mahault B, Jiang XC, Bertin E, Ma YQ, Patelli A, Shi XQ, Chaté H. Self-Propelled Particles with Velocity Reversals and Ferromagnetic Alignment: Active Matter Class with Second-Order Transition to Quasi-Long-Range Polar Order. PHYSICAL REVIEW LETTERS 2018; 120:258002. [PMID: 29979075 DOI: 10.1103/physrevlett.120.258002] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2018] [Indexed: 06/08/2023]
Abstract
We introduce and study in two dimensions a new class of dry, aligning active matter that exhibits a direct transition to orientational order, without the phase-separation phenomenology usually observed in this context. Characterized by self-propelled particles with velocity reversals and a ferromagnetic alignment of polarities, systems in this class display quasi-long-range polar order with continuously varying scaling exponents, yet a numerical study of the transition leads to conclude that it does not belong to the Berezinskii-Kosterlitz-Thouless universality class but is best described as a standard critical point with an algebraic divergence of correlations. We rationalize these findings by showing that the interplay between order and density changes the role of defects.
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Affiliation(s)
- B Mahault
- Service de Physique de l'Etat Condensé, CEA, CNRS, Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France
| | - X-C Jiang
- Center for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, China
| | - E Bertin
- LIPHY, Université Grenoble Alpes and CNRS, F-38000 Grenoble, France
| | - Y-Q Ma
- Center for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, China
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
| | - A Patelli
- Service de Physique de l'Etat Condensé, CEA, CNRS, Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France
| | - X-Q Shi
- Service de Physique de l'Etat Condensé, CEA, CNRS, Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France
- Center for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, China
| | - H Chaté
- Service de Physique de l'Etat Condensé, CEA, CNRS, Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France
- Beijing Computational Science Research Center, Beijing 100094, China
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, 75005 Paris, France
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4
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Binhi VN. A limit in the dynamic increase in the accuracy of group migration. Biosystems 2018; 166:19-25. [DOI: 10.1016/j.biosystems.2018.02.003] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2017] [Revised: 10/11/2017] [Accepted: 02/14/2018] [Indexed: 11/24/2022]
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Kinetic order-disorder transitions in a pause-and-go swarming model with memory. J Theor Biol 2017; 419:90-99. [PMID: 28189669 DOI: 10.1016/j.jtbi.2017.02.009] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2016] [Revised: 01/02/2017] [Accepted: 02/06/2017] [Indexed: 11/20/2022]
Abstract
A two dimensional model of self-propelled particles combining both a pause-and-go movement pattern and memory is studied in simulations. It is shown, that in contrast to previously studied agent based models in two-dimensions, order and disorder are metastable states that can co-exist at some parameter range. In particular, this implies that the formation and decay of global order in swarms may be kinetic rather than a phase transition. Our results explain metastability recently observed in swarming locust and fish.
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Seyed-Allaei H, Schimansky-Geier L, Ejtehadi MR. Gaussian theory for spatially distributed self-propelled particles. Phys Rev E 2017; 94:062603. [PMID: 28085336 DOI: 10.1103/physreve.94.062603] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2016] [Indexed: 11/06/2022]
Abstract
Obtaining a reduced description with particle and momentum flux densities outgoing from the microscopic equations of motion of the particles requires approximations. The usual method, we refer to as truncation method, is to zero Fourier modes of the orientation distribution starting from a given number. Here we propose another method to derive continuum equations for interacting self-propelled particles. The derivation is based on a Gaussian approximation (GA) of the distribution of the direction of particles. First, by means of simulation of the microscopic model, we justify that the distribution of individual directions fits well to a wrapped Gaussian distribution. Second, we numerically integrate the continuum equations derived in the GA in order to compare with results of simulations. We obtain that the global polarization in the GA exhibits a hysteresis in dependence on the noise intensity. It shows qualitatively the same behavior as we find in particles simulations. Moreover, both global polarizations agree perfectly for low noise intensities. The spatiotemporal structures of the GA are also in agreement with simulations. We conclude that the GA shows qualitative agreement for a wide range of noise intensities. In particular, for low noise intensities the agreement with simulations is better as other approximations, making the GA to an acceptable candidates of describing spatially distributed self-propelled particles.
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Affiliation(s)
- Hamid Seyed-Allaei
- Department of Physics, Sharif University of Technology, P. O. Box 11155-9161, Tehran, Iran
| | - Lutz Schimansky-Geier
- Department of Physics, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany
| | - Mohammad Reza Ejtehadi
- Department of Physics, Sharif University of Technology, P. O. Box 11155-9161, Tehran, Iran.,School of Nano Science, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran
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Yoshioka H. Mathematical analysis and validation of an exactly solvable model for upstream migration of fish schools in one-dimensional rivers. Math Biosci 2016; 281:139-148. [PMID: 27693303 DOI: 10.1016/j.mbs.2016.09.014] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2016] [Revised: 09/16/2016] [Accepted: 09/20/2016] [Indexed: 11/15/2022]
Abstract
Upstream migration of fish schools in 1-D rivers as an optimal control problem is formulated where their swimming velocity and the horizontal oblateness are taken as control variables. The objective function to be maximized through a migration process consists of the biological and ecological profit to be gained at the upstream-end of a river, energetic cost of swimming against the flow, and conceptual cost of forming a school. Under simplified conditions where the flow is uniform in both space and time and the profit to be gained at the goal of migration is sufficiently large, the optimal control variables are determined from a system of algebraic equations that can be solved in a cascading manner. Mathematical analysis of the system reveals that the optimal controls are uniquely found and the model is exactly solvable under certain conditions on the functions and parameters, which turn out to be realistic and actually satisfied in experimental fish migration. Identification results of the functional shapes of the functions and the parameters with experimentally observed data of swimming schools of Plecoglossus altivelis (Ayu) validate the present mathematical model from both qualitative and quantitative viewpoints. The present model thus turns out to be consistent with the reality, showing its potential applicability to assessing fish migration in applications.
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Affiliation(s)
- Hidekazu Yoshioka
- Faculty of Life and Environmental Science, Shimane University, Nishikawatsu-cho 1060, Matsue, Shimane 690-8504, Japan.
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Seyed-Allaei H, Ejtehadi MR. Vortex with fourfold defect lines in a simple model of self-propelled particles. Phys Rev E 2016; 93:032113. [PMID: 27078298 DOI: 10.1103/physreve.93.032113] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2015] [Indexed: 11/07/2022]
Abstract
We study the formation of a vortex with fourfold symmetry in a minimal model of self-propelled particles, confined inside a squared box, using computer simulations and also theoretical analysis. In addition to the vortex pattern, we observe five other regimes in the system: a homogeneous gaseous phase, band structures, moving clumps, moving clusters, and vibrating rings. All six regimes emerge from controlling the strength of noise and from the contribution of repulsion and alignment interactions. We study the shape of the vortex and its symmetry in detail. The pattern shows exponential defect lines where incoming and outgoing flows of particles collide. We show that alignment and repulsion interactions between particles are necessary to form such patterns. We derive hydrodynamical equations with an introduction of the "small deviation" technique to describe the vortex phase. The method is applicable to other systems as well. Finally, we compare the theory with the results of both computer simulations and an experiment using Quincke rotors. A good agreement between the three is observed.
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Affiliation(s)
- Hamid Seyed-Allaei
- Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran
| | - Mohammad Reza Ejtehadi
- Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran
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Caussin JB, Solon A, Peshkov A, Chaté H, Dauxois T, Tailleur J, Vitelli V, Bartolo D. Emergent spatial structures in flocking models: a dynamical system insight. PHYSICAL REVIEW LETTERS 2014; 112:148102. [PMID: 24766020 DOI: 10.1103/physrevlett.112.148102] [Citation(s) in RCA: 43] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2013] [Indexed: 06/03/2023]
Abstract
We show that hydrodynamic theories of polar active matter generically possess inhomogeneous traveling solutions. We introduce a unifying dynamical-system framework to establish the shape of these intrinsically nonlinear patterns, and show that they correspond to those hitherto observed in experiments and numerical simulation: periodic density waves, and solitonic bands, or polar-liquid droplets both cruising in isotropic phases. We elucidate their respective multiplicity and mutual relations, as well as their existence domain.
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Affiliation(s)
- Jean-Baptiste Caussin
- Laboratoire de Physique de l'Ecole Normale Supérieure de Lyon, Université de Lyon, CNRS, 46, Allée d'Italie, 69007 Lyon, France
| | - Alexandre Solon
- Université Paris Diderot, Sorbonne Paris Cité, MSC, CNRS, 75205 Paris, France
| | - Anton Peshkov
- LPTMC, CNRS, Université Pierre et Marie Curie, 75252 Paris, France and Service de Physique de l'Etat Condensé, CEA-Saclay, CNRS, 91191 Gif-sur-Yvettes, France and Max Planck Institute for Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Hugues Chaté
- LPTMC, CNRS, Université Pierre et Marie Curie, 75252 Paris, France and Service de Physique de l'Etat Condensé, CEA-Saclay, CNRS, 91191 Gif-sur-Yvettes, France and Max Planck Institute for Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Thierry Dauxois
- Laboratoire de Physique de l'Ecole Normale Supérieure de Lyon, Université de Lyon, CNRS, 46, Allée d'Italie, 69007 Lyon, France
| | - Julien Tailleur
- Université Paris Diderot, Sorbonne Paris Cité, MSC, CNRS, 75205 Paris, France
| | - Vincenzo Vitelli
- Instituut-Lorenz for Theoretical Physics, Universiteit Leiden, 2300 RA Leiden, Netherlands
| | - Denis Bartolo
- Laboratoire de Physique de l'Ecole Normale Supérieure de Lyon, Université de Lyon, CNRS, 46, Allée d'Italie, 69007 Lyon, France
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Romanczuk P, Schimansky-Geier L. Swarming and pattern formation due to selective attraction and repulsion. Interface Focus 2012; 2:746-56. [PMID: 24312728 DOI: 10.1098/rsfs.2012.0030] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2012] [Accepted: 08/30/2012] [Indexed: 11/12/2022] Open
Abstract
We discuss the collective dynamics of self-propelled particles with selective attraction and repulsion interactions. Each particle, or individual, may respond differently to its neighbours depending on the sign of their relative velocity. Thus, it is able to distinguish approaching (coming closer) and retreating (moving away) individuals. This differentiation of the social response is motivated by the response to looming visual stimuli and may be seen as a generalization of the previously proposed escape and pursuit interactions motivated by empirical evidence for cannibalism as a driving force of collective migration in locusts and Mormon crickets. The model can account for different types of behaviour such as pure attraction, pure repulsion or escape and pursuit, depending on the values (signs) of the different response strengths. It provides, in the light of recent experimental results, an interesting alternative to previously proposed models of collective motion with an explicit velocity-alignment interaction. We discuss the derivation of a coarse-grained description of the system dynamics, which allows us to derive analytically the necessary condition for emergence of collective motion. Furthermore, we analyse systematically the onset of collective motion and clustering in numerical simulations of the model for varying interaction strengths. We show that collective motion arises only in a subregion of the parameter space, which is consistent with the analytical prediction and corresponds to an effective escape and/or pursuit response.
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Affiliation(s)
- Pawel Romanczuk
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzerstrasse 38, 01187 Dresden, Germany
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Chou YL, Wolfe R, Ihle T. Kinetic theory for systems of self-propelled particles with metric-free interactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:021120. [PMID: 23005735 DOI: 10.1103/physreve.86.021120] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2012] [Indexed: 06/01/2023]
Abstract
A model of self-driven particles similar to the Vicsek model [Phys. Rev. Lett. 75, 1226 (1995)] but with metric-free interactions is studied by means of a novel Enskog-type kinetic theory. In this model, N particles of constant speed v(0) try to align their travel directions with the average direction of a fixed number of closest neighbors. At strong alignment a global flocking state forms. The alignment is defined by a stochastic rule, not by a Hamiltonian. The corresponding interactions are of genuine multibody nature. The theory is based on a Master equation in 3N-dimensional phase space, which is made tractable by means of the molecular chaos approximation. The phase diagram for the transition to collective motion is calculated and compared to direct numerical simulations. A linear stability analysis of a homogeneous ordered state is performed using the kinetic but not the hydrodynamic equations in order to achieve high accuracy. In contrast to the regular metric Vicsek-model no instabilities occur. This confirms previous direct simulations that, for Vicsek-like models with metric-free interactions, there is no formation of density bands and that the flocking transition is continuous.
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Affiliation(s)
- Yen-Liang Chou
- Department of Physics, North Dakota State University, Fargo, North Dakota 58108-6050, USA
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Morozov A, Poggiale JC. From spatially explicit ecological models to mean-field dynamics: The state of the art and perspectives. ECOLOGICAL COMPLEXITY 2012. [DOI: 10.1016/j.ecocom.2012.04.001] [Citation(s) in RCA: 36] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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