Kolbjørn Tunstrøm

Inferring the structure and dynamics of interactions in schooling fish

Determining individual-level interactions that govern highly coordinated motion in animal groups or cellular aggregates has been a long-standing challenge, central to understanding the mechanisms and evolution of collective behavior. Numerous models have been proposed, many of which display realistic-looking dynamics, but nonetheless rely on untested assumptions about how individuals integrate information to guide movement. Here we infer behavioral rules directly from experimental data. We begin by analyzing trajectories of golden shiners (Notemigonus crysoleucas) swimming in two-fish and three-fish shoals to map the mean effective forces as a function of fish positions and velocities. Speeding and turning responses are dynamically modulated and clearly delineated. Speed regulation is a dominant component of how fish interact, and changes in speed are transmitted to those both behind and ahead. Alignment emerges from attraction and repulsion, and fish tend to copy directional changes made by those ahead. We find no evidence for explicit matching of body orientation. By comparing data from two-fish and three-fish shoals, we challenge the standard assumption, ubiquitous in physics-inspired models of collective behavior, that individual motion results from averaging responses to each neighbor considered separately; three-body interactions make a substantial contribution to fish dynamics. However, pairwise interactions qualitatively capture the correct spatial interaction structure in small groups, and this structure persists in larger groups of 10 and 30 fish. The interactions revealed here may help account for the rapid changes in speed and direction that enable real animal groups to stay cohesive and amplify important social information.

Collective States, Multistability and Transitional Behavior in Schooling Fish

The spontaneous emergence of pattern formation is ubiquitous in nature, often arising as a collective phenomenon from interactions among a large number of individual constituents or sub-systems. Understanding, and controlling, collective behavior is dependent on determining the low-level dynamical principles from which spatial and temporal patterns emerge; a key question is whether different group-level patterns result from all components of a system responding to the same external factor, individual components changing behavior but in a distributed self-organized way, or whether multiple collective states co-exist for the same individual behaviors. Using schooling fish (golden shiners, in groups of 30 to 300 fish) as a model system, we demonstrate that collective motion can be effectively mapped onto a set of order parameters describing the macroscopic group structure, revealing the existence of at least three dynamically-stable collective states; swarm, milling and polarized groups. Swarms are characterized by slow individual motion and a relatively dense, disordered structure. Increasing swim speed is associated with a transition to one of two locally-ordered states, milling or highly-mobile polarized groups. The stability of the discrete collective behaviors exhibited by a group depends on the number of group members. Transitions between states are influenced by both external (boundary-driven) and internal (changing motion of group members) factors. Whereas transitions between locally-disordered and locally-ordered group states are speed dependent, analysis of local and global properties of groups suggests that, congruent with theory, milling and polarized states co-exist in a bistable regime with transitions largely driven by perturbations. Our study allows us to relate theoretical and empirical understanding of animal group behavior and emphasizes dynamic changes in the structure of such groups.

Determining interaction rules in animal swarms

In this paper, we introduce a method for determining local interaction rules in animal swarms. The method is based on the assumption that the behavior of individuals in a swarm can be treated as a set of mechanistic rules.The principal idea behind the technique is to vary parameters that define a set of hypothetical interactions, as for example, a rule for aligning with neighbors. The parameter values are optimized so that the deviation between the observed movements in an animal swarm and the movements predicted by the assumed rule set is minimal. We demonstrate the method by reconstructing the interaction rules from the trajectories produced by a computer simulation. Copyright 2010, Oxford University Press.

Collective dynamics of self-propelled particles with variable speed

Understanding the organization of collective motion in biological systems is an ongoing challenge. In this paper we consider a minimal model of self-propelled particles with variable speed. Inspired by experimental data from schooling fish, we introduce a power-law dependency of the speed of each particle on the degree of polarization order in its neighborhood. We derive analytically a coarse-grained continuous approximation for this model and find that, while the specific variable speed rule used does not change the details of the ordering transition leading to collective motion, it induces an inverse power-law correlation between the speed or the local polarization order and the local density. Using numerical simulations, we verify the range of validity of this continuous description and explore regimes beyond it. We discover, in disordered states close to the transition, a phase-segregated regime where most particles cluster into almost static groups surrounded by isolated high-speed particles. We argue that the mechanism responsible for this regime could be present in a wide range of collective motion dynamics.

Effective thermostat induced by coarse graining of simple point charge water

We investigate how the transport properties of a united atom fluid with a dissipative particle dynamics thermostat depend on the functional form and magnitude of both the conservative and the stochastic interactions. We demonstrate how the thermostat strongly affects the hydrodynamics, especially diffusion, viscosity, and local escape times. As model system we use simple point charge (SPC) water, from which projected trajectories are used to determine the effective interactions in the united atom model. The simulation results support our argument that the thermostat should be viewed as an integral part of the coarse-grained dynamics rather than a tool for approaching thermal equilibrium. As our main result we show that the united atom model with the adjusted effective interactions approximately reproduces the diffusion constant and the viscosity of the underlying detailed SPC water model.

Using force covariance to derive effective stochastic interactions in dissipative particle dynamics

There exist methods for determining effective conservative interactions in coarse-grained particle-based mesoscopic simulations. The resulting models can be used to capture thermal equilibrium behavior, but the model system we study does not correctly represent transport properties. We suggest the use of force covariance to determine the full functional form of dissipative and stochastic interactions. We show that a combination of the RDF and a force covariance function can be used to determine all interactions in dissipative particle dynamics (DPD). Furthermore, we use the method to test whether the effective interactions in DPD can be adjusted to produce a force covariance consistent with the projection of a microscopic Lennard-Jones simulation. The results indicate that the DPD ansatz may not be consistent with the underlying microscopic dynamics. We discuss how this result relates to theoretical studies reported in the literature.

A method for estimating the interactions in dissipative particle dynamics from particle trajectories

We introduce a method for determining the functional form of the stochastic and dissipative interactions in a dissipative particle dynamics (DPD) model from projected phase space trajectories. The DPD model is viewed as a coarse graining of a detailed dynamics that displays a clear timescale separation. Based on the Mori-Zwanzig projection operator method we derive a consistency equation for the stochastic interaction in DPD. The consistency equation can be solved by an iterative bootstrapping procedure. Combined with standard techniques for estimating the conservative interaction, our method makes it possible to reconstruct all the forces in a coarse-grained DPD model. We demonstrate how the method works by recreating the interactions in a DPD model from its phase space trajectory. Furthermore, we discuss how our method can be used in realistic systems with finite timescale separation.

On the microscopic foundation of dissipative particle dynamics

Mesoscopic-particle-based fluid models, such as dissipative particle dynamics, are usually assumed to be coarse-grained representations of an underlying microscopic fluid. A fundamental question is whether there exists a map from microscopic particles in these systems to the corresponding coarse-grained particles, such that the coarse-grained system has the same bulk and transport properties as the underlying system. In this letter, we investigate the coarse-graining of microscopic fluids using a Voronoi-type projection that has been suggested in several studies. The simulations show that the projection fails in defining coarse-grained particles that have a physically meaningful connection to the microscopic fluid. In particular, the Voronoi projection produces identical coarse-grained equilibrium properties when applied to systems with different microscopic interactions and different bulk properties.

Modelling the dynamics of a minimal protocell container

This paper is a discussion on how reaction kinetics and three-dimensional (3D) lattice simulations can be used to elucidate the dynamical properties of micelles as a possible minimal protocell container. We start with a general discussion on the role of molecular self-assembly in prebiotic and contemporary biological systems. A simple reaction kinetic model of a micellation process of amphiphilic molecules in water is then presented and solved analytically. Amphiphilic molecules are polymers with hydrophobic (water-fearing), e.g. hydrocarbon tail groups, and hydrophilic (water-loving) head groups, e.g. fatty acids. By making a few simplifying assumptions an analytical expression for the size distribution of the resulting micelles can be derived. The main part of the paper presents and discusses a lattice gas technique for a more detailed 3D simulation of molecular self-assembly of amphiphilic polymers in aqueous environments. Water molecules, hydrocarbon tail groups and hydrophilic head groups are explicitly represented on a three-dimensional discrete lattice. Molecules move on the lattice proportional to their continuous momentum. Collision rules preserve momentum and kinetic energy. Potential energy from molecular interactions are also included explicitly. The non-trivial thermodynamics of large-scale and long-time dynamics are studied. In this paper we specifically demonstrate how, from a random initial distribution, micelles are formed and grow until they destabilize and can divide. Eventually a steady state of growing and dividing micelles is formed. Towards the end of the paper we discuss the relevance of the presented results to the design of a minimal artificial protocell.

Impact of anticipation in dynamical systems

Many animals, including humans, have predictive capabilities and, presumably, base their behavioral decisions-at least partially-upon an anticipated state of their environment. We explore a minimal version of this idea in the context of particles that interact according to a pairwise potential. Anticipation enters the picture by calculating the interparticle forces from linear extrapolations of the particle positions some time τ in the future. Simulations show that for intermediate values of τ, compared to a transient time scale defined by the potential and the initial conditions, the particles form rotating clusters in which the particles are arranged in a hexagonal pattern. Analysis of the system shows that anticipation induces energy dissipation and we show that the kinetic energy asymptotically decays as 1/t. Furthermore, we show that the angular momentum is not necessarily conserved for τ>0, and that asymmetries in the initial condition therefore can cause rotational movement. These results suggest that anticipation could play an important role in collective behavior, since it may induce pattern formation and stabilizes the dynamics of the system.