Emergence of coherent motion in aggregates of motile coupled maps

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Abstract

In this paper we study the emergence of coherence in collective motion described by a system of interacting motiles endowed with an inner, adaptative, steering mechanism. By means of a nonlinear parametric coupling, the system elements are able to swing along the route to chaos. Thereby, each motile can display different types of behavior, i.e. from ordered to fully erratic motion, accordingly with its surrounding conditions. The appearance of patterns of collective motion is shown to be related to the emergence of interparticle synchronization and the degree of coherence of motion is quantified by means of a graph representation. The effects related to the density of particles and to interparticle distances are explored. It is shown that the higher degrees of coherence and group cohesion are attained when the system elements display a combination of ordered and chaotic behaviors, which emerges from a collective self-organization process.

Highlights

► A minimal model of motile particles with adjustable intrinsic steering is presented. ► Collective motion emerges due to self-adaptation of each particle’s intrinsic state. ► Adaptation is achieved by a map which behavior ranges from periodic to chaotic. ► Higher cohesion occurs in a balanced combination of ordered and chaotic motion. ► Exhibits an abrupt change in degree of coherence as a function of particle density.

Introduction

Complex motion modes of collectives as a result of their constituent interacting entities occurs almost ubiquitously in nature and over the last decades it has provided a common ground for cross-disciplinary investigations among Physics, Biology and Mathematics. The range of applications of such studies is indeed extensive [1], [2], [3], [4], [5], [6], [7]. As a matter of fact, coherent patterns of collective motion found in distinct families of biological species such as fish schools, flocks of birds, swarms of insects and even colonies of bacteria [6], [5], [8], [1], [4], [9], [10], [11] have also been detected in granular matter systems, self-propelled particles with inelastic collisions and active Brownian particles in autonomous-motor groups [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. Early in the study of such collectives, modeling and simulation have been recognized as playing a crucial role in gaining insight of the mechanisms underlying such an emergence of global features from a set of simple rules [23], [24], [25].

As the interest of the scientific community in addressing such kind of systems increases, minimal microscopic models have been recently introduced. Most of them consider systems of many interacting elements whose couplings, being in general nonlinear, can be either local or global. From a purely deterministic perspective, such systems can be represented as high dimensional dynamical systems, which can be either discrete or continuous in their time evolution. This approach has been mainly spearheaded by Smale and collaborators [26], [27], [18], [21]. On the other hand, providing understanding of the connection between macroscopic collective features and microscopic scale interactions in multi-particle systems is well within the principal aims of statistical physics. Therefore, naturally, models of self-propelled interacting particles have provided a fertile ground for study using concepts and tools of statistical physics. This kind of approach has been mainly adopted in models where randomness is introduced in the dynamics of the particles by means of a Langevin-type description [22], [21], [20], [19], [5], [17], [28], [29], [13], [30]. Finally, another line of investigation aims at providing descriptions and modeling in purely probabilistic terms where interactions obey probabilistic ‘rules of engagement’. Notably, effective Fokker–Plank equations have been recently proposed for coarse-grained observables of ‘agent systems’ (see for example [12], [1], [31] and references therein for a more detailed presentation).

One of the earliest theoretical stochastic models of self-propelled interacting particles was introduced by Vicsek and collaborators as early as in 1995 [29], [5], which still possesses seminal value because of its minimal character. In Vicsek’s model, point particles move at discrete time steps with fixed speed. At every time step, the different particles velocities are determined by the average of neighboring particles. In other words, Vicsek’s model is an XY model in which the ‘spins are actively moving’ (see [32]). Furthermore, similarly as in ferromagnetic spin systems, Vicsek’s model exhibits a phase transition as a function of both the particle density and the intensity of noise. For a detailed investigation on the nature of such phase transitions we refer the reader to [29], [15], [30]. Further variations of Vicsek’s model have been recently proposed to account for changing symmetries, adding cohesion or taking into account a surrounding fluid interacting with the particles [32].

Although the application of concepts stemming from statistical physics research has led to identify some universal properties existing in these classes of systems, such as spontaneous symmetry breaking, phase transitions and synchronizing modes [14], [22], [20], [28], [15], the role of the individuals’ internal dynamics still remains veiled.

Whilst the explicit consideration of inner control processes could increase the complexity of models of interacting motiles, it is a necessary conceptual step in developing further insights into the mechanisms underlying the emergence of coherence of motion in biological systems. Historically, models of group motion where particles adapt to their environment by means of an inner steering mechanism have been developed in the context of traffic modeling. For an overview of such models we refer to [33]. In the context of biological systems, inner states have been considered in order to model the emergence of coherent behavior in groups of fireflies [34], [35]. More closely related to the problem here addressed is the study of the response and adaptation of populations of motiles to the information carried by external ‘fields’. Recently, by assuming inner state dynamics, attempts in this direction have been reported in the study of bacterial chemotaxis [36], [37] and biologically inspired collective robotics [38], [31].

In the present work we introduce a purely deterministic model where, in analogy to the Vicsek’s class of models, particles can display phenomenological random-like motion and exhibit a sharp change of coherence of motion as a function of the particle density. Furthermore, in contradistinction to Vicsek’s class of models, no boundary conditions are considered, since a feature of the group’s cohesion is that it is built by the collective dynamics ‘per se’. Even in the absence of explicit interparticle attraction, it is the coordinated effectively synchronized collective motion that keeps the group together. In our model every motile is endowed with an inner ‘steering’ variable that evolves according to an heuristic discrete-time equation. For the sake of simplicity, such an evolution law has the structure of the logistic map. The latter provides a suitable, well understood, combination of chaotic and ordered behavior to account for the coherence and novelty aspects observed in real collectives of motiles. Communication between a particle and its environment occurs via a control parameter that tunes its value according to the external states of the surrounding particles. At the microscopic level the features of motiles are summarized by the following conditions, along the lines of [19]:

  • (α)

    Each element has a time dependent internal state and spatial position.

  • (β)

    Each element is ‘active’ in the sense that its internal state can exhibit chaotic behavior, both in presence and absence of interactions with other particles.

  • (γ)

    The dynamics of the internal state of a given element is determined by local, short range interactions, effectuated within a neighborhood of a characteristic radius.

  • (δ)

    The interparticle interaction depends on the internal states of the participating particles.

These general rules have been found to give rise to nontrivial emergent behavior which cannot be readily deduced from the microscopic parameters of the system.

At the collective or ‘macroscopic’ level, the characteristic emergent phenomena observed in such kind of locally coupled systems are mainly described by the notion of ‘clustering’, either in real or in state space. As it has been reported in [19], distinct classes of clustering behavior accompany, in a generic way, such a coupling:

  • (i)

    Elements forming a cluster merge in and out of the cluster.

  • (ii)

    Elements can remain separated from neighboring clusters but they form a bridge between distinct clusters facilitating information flow, exchange of elements between clusters and adding cohesion.

  • (iii)

    Presence of independent clusters separated by distances larger than the interaction, with elements rarely merging in and out of the clusters amidst them.

  • (iv)

    Cluster - cluster interactions such as aggregation, segregation and competitive growth between various sized clusters.

In this work, we shall focus on the description and quantification of emergent collective motion based on a clustering index that accounts simultaneously for both, the degree of spatial clustering and the degree of interparticle alignment of velocities.

The paper is organized as follows: In Section 2, we introduce a general formulation of the model. Next, in Section 3 we address the possible types of stationary behavior in the motion of an individual particle, as well as the basic phase-locking synchronization process that results from local interparticle pair-interactions. Section 4 presents the case of the many particle system. Its typical evolution patterns, regimes of motion, degree of synchronization and the dependency on both, density of particles and interparticle distance, are investigated. Finally, Section 5 concludes the present work with a brief summary, discussion of the results and possible further extentions.

Section snippets

Formulation of the model

We consider N particles, labeled through an index i = 1, 2,  , N, whose positions at time t are denoted by the vectors {rti}. They evolve on a plane (two dimensional motion) where their positions change simultaneously at discrete time steps Δt, according tort+Δti=rti+vtiΔt.Similarly as in Vicsek’s original model [29], [5], we assume here that at every time step the speeds of all particles are equal to a common constant values=vti.Changes in the particles’ velocities occur via an inherent

Local dynamics

We proceed now to the more ‘microscopic’ local level in order to elucidate the underlying dynamics of particle–particle interactions. It is instructive to consider one particle in isolation and subsequently a single pair of particles and their interaction. In a sense this is analogous to the statistical mechanical treatment of ‘dilute gas’ where the particles’ collisions, being rare, are described very accurately by their binary collisions. Since, at very low densities, binary interactions are

Emergent global dynamics and patterns of motion

Having seen the role of spontaneous synchronization in binary interactions we can now consider the full collective motion problem. As we shall demonstrate further in this section, synchronization is an underlying mechanism for coherent and cohesive motion. Another feature of this intrinsic and coherent steering, which keeps a large fraction of the group together, is that the role of boundary conditions becomes really secondary. Furthermore, we emphasize the fact that the aggregates of motiles

Conclusions

In this work we have introduced a minimal model of motile particles which takes into account an intrinsic steering mechanism. Its main feature is that it exhibits the emergence of patterns of coherent collective motion. Since each particle is endowed with an intrinsic mechanism, it allows it to adjust its trajectory according to the surrounding conditions. With this extra feature the model can be of utility as an augmented inert particle model. The adaptation is achieved by changes which are

Acknowledgments

The authors thank G. Nicolis, J - L. Deneubourg and T. Bountis for offering their encouragement, insightful comments and most fruitful criticism during the preparation of the present paper. We would, also, like to thank E. Toffin and M. Lefebre for motivating this work and for suggesting relevant references. The work of A. G. C. R. was supported by the ‘Communauté Francaise de Belgique’ (contract ‘Actions de Recherche Concertées’ No. 04/09-312) and by the Federal Ministry of Education and

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