Vortex formation by active agents as a model for Daphnia swarming

Dynamics and risk sharing in groups of selfish individuals

Understanding why animals organize in collective states is a central question of current research in, e.g., biology, physics, and psychology. More than 50 years ago, W.D. Hamilton postulated that the formation of animal herds may simply result from the individual's selfish motivation to minimize their predation risk. The latter is quantified by the domain of danger (DOD) which is given by the Voronoi area around each individual. In fact, simulations show that individuals aiming to reduce their DODs form compact groups similar to what is observed in many living systems. However, despite the apparent simplicity of this problem, it is not clear what motional strategy is required to find an optimal solution. Here, we use the framework of Multi Agent Reinforcement Learning (MARL) which gives the unbiased and optimal strategy of individuals to solve the selfish herd problem. We demonstrate that the motivation of individuals to reduce their predation risk naturally leads to pronounced collective behaviors including the formation of cohesive swirls. We reveal a previously unexplored rather complex intra-group motion which eventually leads to a evenly shared predation risk amongst selfish individuals.

Spontaneous vortex formation by microswimmers with retarded attractions

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.

Daphnia magna model in the toxicity assessment of pharmaceuticals: A review

Daphnia magna is one of the most commonly used model organism to assess toxicity of wide range of pharmaceuticals such as antibiotics, anticancer drugs, antidepressants, anti-inflammatory drugs, beta-blockers and lipid-regulating agents. Currently, daphnia toxicity tests based on immobilisation and lethality standardised by OECD, acute immobilisation test and reproduction test, are mainly used in toxicological studies. Detailed analysis of Daphnia biology allows distinguishing the swimming behaviour and physiological endpoints such as swimming speed, distance travelled, hopping frequency, heart rate, ingestion rate, feeding rate, oxygen consumption, thoracic limb activity which could be also useful in assessment of toxic effects. The advantage of behavioural and physiological parameters is the possibility to observe sublethal effects induced by lower concentrations of pharmaceuticals which would not be possible to notice by using OECD tests. Additionally, toxic effects of tested drugs could be assessed using enzymatic and non-enzymatic biomarkers of daphnia toxicity. This review presents scientific data considering characteristics of D. magna, analysis of immobilisation, lethality, reproductive, behavioural, physiological and biochemical parameters used in the toxicity assessment of pharmaceuticals. The aim of this paper is also to emphasize usefulness, advantages and disadvantages of these invertebrate model organisms to assess toxicity of different therapeutic classes of pharmaceuticals. Also, various examples of application of D. magna in studies on pharmaceutical toxicity are presented.

Analysis of aligning active local searchers orbiting around their common home position

We discuss effects of pairwise aligning interactions in an ensemble of central place foragers or of searchers that are connected to a common home. In a wider sense, we also consider self-moving entities that are attracted to a central place such as, for instance, the zooplankton Daphnia being attracted to a beam of light. Single foragers move with constant speed due to some propulsive mechanism. They explore at random loops the space around and return rhytmically to their home. In the ensemble, the direction of the velocity of a searcher is aligned to the motion of its neighbors. At first, we perform simulations of this ensemble and find a cooperative behavior of the entities. Above an overcritical interaction strength the trajectories of the searcher qualitatively changes and searchers start to move along circles around the home position. Thereby, all searchers rotate either clockwise or anticlockwise around the central home position as it was reported for the zooplankton Daphnia. At second, the computational findings are analytically explained by the formulation of transport equations outgoing from the nonlinear mean field Fokker-Planck equation of the considered situation. In the asymptotic stationary limit, we find expressions for the critical interaction strength, the mean radial and orbital velocities of the searchers and their velocity variances. We also obtain the marginal spatial and angular densities in the undercritical regime where the foragers behave like individuals as well as in the overcritical regime where they rotate collectively around the considered home. We additionally elaborate the overdamped Smoluchowski-limit for the ensemble.

Using a new high-throughput video-tracking platform to assess behavioural changes in Daphnia magna exposed to neuro-active drugs

Recent advances in imaging allow to monitor in real time the behaviour of individuals under a given stress. Light is a common stressor that alters the behaviour of fish larvae and many aquatic invertebrate species. The water flea Daphnia magna exhibits a vertical negative phototaxis, swimming against light trying to avoid fish predation. The aim of this study was to develop a high-throughput image analysis system to study changes in the vertical negative phototaxis of D. magna first reproductive adult females exposed to 0.1 and 1 μg/L of four neuro-active drugs: diazepam, fluoxetine, propranolol and carbamazepine. Experiments were conducted using a custom designed experimental chamber containing four independent arenas and infrared illumination. The apical-located visible light and the GigE camera located in front of the arenas were controlled by the Ethovision XT 11.5 sofware (Noldus Information Technology, Leesburg, VA). Total distance moved, time spent per zone (bottom vs upper zones) and distance among individuals were analyzed in dark and light conditions, and the effect of different intensities of the apical-located visible light was also investigated. Results indicated that light intensity increased the locomotor activity and low light intensities allowed to better discriminate individual responses to the studied drugs. The four tested drugs decreased the response of exposed organisms to light: individuals moved less, were closer to the bottom and at low light intensities were closer each other. At high light intensities, however, exposed individuals were less aggregated. Propranolol, carbamazepine and fluoxetine induced the most severe behavioural effects. The tested drugs at environmental relevant concentrations altered locomotor activity, geotaxis, phototaxis and aggregation in D. magna individuals in the lab. Therefore the new image analysis system presented here was proven to be sensitive and versatile enough to detect changes in diel vertical migration across light intensities and low concentration levels of neuro-active drugs.

Daphnia swimming behaviour as a biomarker in toxicity assessment: A review

Daphnia is a motile common model organism widely used in ecotoxicological testing. Although mortality and immobilisation are the main endpoints used for determination of toxicity, detection of subtle alterations induced by some chemicals particularly at lower levels may require more sensitive biomarkers. As a number of studies indicated that swimming behaviour may be altered by pesticides, nanoparticles, bacterial products or other chemicals, analysis of its various parameters is considered as a novel methodological approach for toxicity assessment and monitoring of water quality. This paper presents the current state of knowledge on the effects induced by various chemical compounds on the parameters of swimming behaviour of Daphnia and systems developed for its analysis. Advantages and limitations of swimming behaviour as a tool in toxicological studies are also discussed.

COLLECTIVE VORTEX BEHAVIORS: DIVERSITY, PROXIMATE, AND ULTIMATE CAUSES OF CIRCULAR ANIMAL GROUP MOVEMENTS

Ant mill, caterpillar circle, bat doughnut, amphibian vortex, duck swirl, and fish torus are different names for rotating circular animal formations, where individuals turn around a common center. These "collective vortex behaviors" occur at different group sizes from pairs to several million individuals and have been reported in a large number of organisms, from bacteria to vertebrates, including humans. However, to date, no comprehensive review and synthesis of the literature on vortex behaviors has been conducted. Here, we review the state of the art of the proximate and ultimate causes of vortex behaviors. The ubiquity of this behavioral phenomenon could suggest common causes or fundamental underlying principles across contexts. However, we find that a variety of proximate mechanisms give rise to vortex behaviors. We highlight the potential benefits of collective vortex behaviors to individuals involved in them. For example, in some species, vortices increase feeding efficiency and could give protection against predators. It has also been argued that vortices could improve collective decision-making and information transfer. We highlight gaps in our understanding of these ubiquitous behavioral phenomena and discuss future directions for research in vortex studies.

Circularly confined microswimmers exhibit multiple global patterns

Geometric confinement plays an important role in the dynamics of natural and synthetic microswimmers from bacterial cells to self-propelled particles in high-throughput microfluidic devices. However, little is known about the effects of geometric confinement on the emergent global patterns in such self-propelled systems. Recent experiments on bacterial cells report that, depending on the cell concentration, cells either spontaneously organize into vortical motion in thin cylindrical and spherical droplets or aggregate at the inner boundary of the droplets. Our goal in this paper is to investigate, in the context of an idealized physical model, the interplay between geometric confinement and level of flagellar activity on the emergent collective patterns. We show that decreasing flagellar activity induces a hydrodynamically triggered transition in confined microswimmers from swirling to global circulation (vortex) to boundary aggregation and clustering. These results highlight that the complex interplay between confinement, flagellar activity, and hydrodynamic flows in concentrated suspensions of microswimmers could lead to a plethora of global patterns that are difficult to predict from geometric consideration alone.

Self-induced polar order of active Brownian particles in a harmonic trap

Hydrodynamically interacting active particles in an external harmonic potential form a self-assembled fluid pump at large enough Péclet numbers. Here, we give a quantitative criterion for the formation of the pump and show that particle orientations align in the self-induced flow field in surprising analogy to ferromagnetic order where the active Péclet number plays the role of inverse temperature. The particle orientations follow a Boltzmann distribution Φ(p) ∼ exp(Ap(z)) where the ordering mean field A scales with the active Péclet number and polar order parameter. The mean flow field in which the particles' swimming directions align corresponds to a regularized Stokeslet with strength proportional to swimming speed. Analytic mean-field results are compared with results from Brownian dynamics simulations with hydrodynamic interactions included and are found to capture the self-induced alignment very well.

Macroscopic traveling packet and soliton states of quasi-one-dimensional flocks

Using a continuum model for inhomogeneous flocks, we show that a finite but arbitrarily large moving "packet" of active particles (e.g., moving creatures) can form in a background of a lower density disordered phase of these particles, like a liquid drop surrounded by vapor. The "vapor density" of the disordered background can be made arbitrarily low. We find three basic types of quasi-one-dimensional states: "longitudinal", "transverse", and "oblique" states, with their internal velocity fields, respectively, parallel, perpendicular, and oblique to the interface. The transitions between these states are also studied.

Deformable self-propelled particles with a global coupling

We have proposed a model of deformable self-propelled particles in which the time-evolution equations are given in terms of the center-of-mass velocity and a nematic order parameter representing the motion-induced deformation [T. Ohta and T. Ohkuma, Phys. Rev. Lett. 102, 154101 (2009)]. We investigate its many-body problem applying a global orientational coupling. Depending on the strength of the interaction, the self-propelled particles exhibit various kinds of collective dynamics and chaotic behavior: a ballistic procession state, a scattered state, a coherently phase synchronized state, two types of in-phase synchronized state, and an anomalously diffusive state. The phase reduction method for the weak coupling regime reveals the bifurcations between the secular collective motions. The phase boundary among the chaos regime and the synchronized regimes is determined by the linear stability analysis of the synchronized states.

Collective motion due to individual escape and pursuit response

Recent studies suggest that noncooperative behavior such as cannibalism may be a driving mechanism of collective motion. Motivated by these novel results we introduce a simple model of Brownian particles interacting by biologically motivated pursuit and escape interactions. We show the onset of collective motion for both interaction types and analyze their impact on the global dynamics. We demonstrate a strong dependence of experimentally accessible macroscopic observables on the relative strength of escape and pursuit and determine the scaling of the migration speed with model parameters.

Finite-size scaling analysis and dynamic study of the critical behavior of a model for the collective displacement of self-driven individuals

The Vicsek model (VM) [T. Vicsek, Phys. Rev. Lett. 75, 1226 (1995)], for the description of the collective behavior of self-driven individuals, assumes that each of them adopts the average direction of movement of its neighbors, perturbed by an external noise. A second-order transition between a state of ordered collective displacement (low-noise limit) and a disordered regime (high-noise limit) was found early on. However, this scenario has recently been challenged by Grégory and Chaté [G. Grégory and H. Chaté, Phys. Rev. Lett. 92, 025702 (2004)] who claim that the transition of the VM may be of first order. By performing extensive simulations of the VM, which are analyzed by means of both finite-size scaling methods and a dynamic scaling approach, we unambiguously demonstrate the critical nature of the transition. Furthermore, the complete set of critical exponents of the VM, in d=2 dimensions, is determined. By means of independent methods--i.e., stationary and dynamic measurements--we provide two tests showing that the standard hyperscaling relationship dnu-2beta=gamma holds, where beta, nu, and gamma are the order parameter, correlation length, and "susceptibility" critical exponents, respectively. Furthermore, we established that at criticality, the correlation length grows according to xi-t1z, with z approximately = 1.27(3) , independently of the degree of order of the initial configuration, in marked contrast with the behavior of the XY model.

Optimal foraging by zooplankton within patches: the case of Daphnia

The motions of many physical particles as well as living creatures are mediated by random influences or 'noise'. One might expect that over evolutionary time scales internal random processes found in living systems display characteristics that maximize fitness. Here we focus on animal random search strategies [G.M. Viswanathan, S.V. Buldyrev, S. Havlin, M.G.E. Da Luz, E.P. Raposo, H.E. Stanley, Optimizing the success of random searches, Nature 401 (1999) 911-914; F. Bartumeus, J. Catalan, U.L. Fulco, M.L. Lyra, G.M. Viswanathan, Optimizing the encounter rate in biological interactions: Lévy versus Brownian stratagies, Phys. Rev. Lett. 88 (2002) 097901 and 89 (2002) 109902], and we describe experiments with the following Daphnia species: D. magna, D. galeata, D. lumholtzi, D. pulicaria, and D. pulex. We observe that the animals, while foraging for food, choose turning angles from distributions that can be described by exponential functions with a range of widths. This observation leads us to speculate and test the notion that this characteristic distribution of turning angles evolved in order to enhance survival. In the case of theoretical agents, some form of randomness is often introduced into search algorithms, especially when information regarding the sought object(s) is incomplete or even misleading. In the case of living animals, many studies have focused on search strategies that involve randomness [H.C. Berg, Random Walks in Biology, Princeton University, Princeton, New Jersey, 1993; A. Okubo, S.A. Levin (Eds.), Diffusion and Ecological Problems: Modern Perspectives, second ed., Springer, New York, 2001]. A simple theory based on stochastic differential equations of the motion backed up by a simulation shows that the collection of material (information, energy, food, supplies, etc.) by an agent executing Brownian-type hopping motions is optimized while foraging for a finite time in a supply patch of limited spatial size if the agent chooses turning angles taken from an exponential distribution with a specific stochastic intensity or 'noise width'. Search strategies that lead to optimization is a topic of high current interest across many disciplines [D. Wolpert, W. MacReady, No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation 1 (1997) 67].