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The authors have declared that no competing interests exist.

Conceived and designed the experiments: AJWW. Performed the experiments: AJWW JEH-R. Analyzed the data: RPM AP DJTS DS RG AJWW. Contributed reagents/materials/analysis tools: RPM AP DS DJTS. Wrote the paper: RPM DJTS AP.

Inference of interaction rules of animals moving in groups usually relies on an analysis of large scale system behaviour. Models are tuned through repeated simulation until they match the observed behaviour. More recent work has used the fine scale motions of animals to validate and fit the rules of interaction of animals in groups. Here, we use a Bayesian methodology to compare a variety of models to the collective motion of glass prawns (

The collective movement of animals in a group is an impressive phenomenon whereby large scale spatio-temporal patterns emerge from simple interactions between individuals. Theoretically, much of our understanding of animal group motion comes from models inspired by statistical physics. In these models, animals are treated as moving (self-propelled) particles that interact with each other according to simple rules. Recently, researchers have shown greater interest in using experimental data to verify which rules are actually implemented by a particular animal species. In our study, we present a rigorous selection between alternative models inspired by the literature for a system of glass prawns. We find that the classic theoretical models can accurately capture either the fine-scale behaviour or the large-scale collective patterns of movement of the prawns. However, none are able to reproduce both levels of description at the same time. To resolve this conflict we introduce a new class of models wherein prawns ‘remember’, their previous interactions, integrating their experiences over time when deciding to change behaviour. These outperform the traditional models in predicting when individual prawns will change their direction of motion and restore consistency between the fine-scale rules of interaction and the global behaviour of the group.

The most striking features of the collective motion of animal groups are the large-scale patterns produced by flocks, schools and other groups. These patterns can extend over scales that exceed the interaction ranges of the individuals within the group

Each of the models in the literature is capable of reproducing key aspects of the large-scale behaviour of one or more biological systems of interest. Together these models help explain what aspects of inter-individual interactions are most important for creating emergent patterns of coherent group motion. With this proliferation of putative interaction rules has come the recognition that some patterns of group behaviour are common to many models, and that different models can have large areas of overlapping behaviour depending on the choice of parameters

In many contexts however the rules of interaction are of more interest than the group behaviour they lead to. For example, when comparing the evolution of social behavior across different species, it is important to know if the same rules evolved independently in multiple instances, or whether each species evolved a different solution to the problem of behaving coherently as a group

The most frequent approach to inferring these rules has been to find correlations between important measurable aspects of the behaviour of a focal individual and its neighbours. For example, Ballerini

In these studies the rules of interaction are presented non-parametrically and cannot be immediately translated into a specific self-propelled particle model. Nor are these models validated in terms of the global schooling patterns produced by the fish. An alternative model-based approach that does fit self-propelled particle and similar models to data is proposed by Eriksson

What all previous empirical studies have lacked is a simultaneous verification of a model at both the individual and collective level. Either fine scale individual-level behaviour is observed without explicit fitting of a model

In this paper we study the collective motion of small groups of the glass prawn,

We study the positions and directions of co-moving prawns in a confined annular arena (See Methods and Materials and

Prawns moving within an annulus of 200 mm external diameter and 70 mm internal diameter. Red coloured prawns indicate a clockwise orientation, blue prawns a counter-clockwise orientation. In this instance the total number of prawns

We then calculated the number of prawns travelling CW or anti-CW at each time step of each experiment involving three, six or twelve prawns. From this we calculated the average number of CW and anti-CW prawns at a given time across experiments.

(A) The proportion of six-prawn experiments (

At a group level we see that prawns tend to align over time, producing a polarised stable state, which is higher for larger group sizes. We define the reproduction of these global patterns as the

Next we investigated a series of interaction models as to their ability to reproduce the fine scale interactions of the prawns. We predict the probability,

Each model specifies the probability that a focal prawn will change its direction in the next time step conditioned on the relative positions and directions of the other individuals in the arena. We use a logistic mapping to ensure probabilities remain between zero and one, so each model uses the relevant variables to determine a latent ‘turning-intensity’,

The models are, in increasing degree of complexity, as follows. Firstly to consider models that do not include zones-of-interaction – non-spatial models. We establish a baseline with a

Secondly we consider a class of

Model | Interaction zone | KL/bits | Consistent? | |||||||

Null | None | −7.5 | N/A | N/A | N/A | N/A | N/A | 10.2 | −944 | No |

MF | Global | −7.5 | N/A | N/A | 0.51 | N/A | N/A | 0.6 | −863 | Yes |

T | K nearest-neighbours | −7.5 | N/A | 3.4 | 0.72 | N/A | N/A | 2.0 | −864 | No |

S1 | Spatial, symmetric | −7.5 | N/A | 1.42 | N/A | N/A | 1.3 | −847 | No | |

S2 | Spatial, forward | −7.5 | N/A | 1.71 | N/A | N/A | 7.2 | −834 | No | |

S3 | Spatial, symmetric | −7.5 | N/A | 1.80 | −0.01 | N/A | 2.0 | −856 | No | |

S4 | Spatial, forward | −7.5 | N/A | 1.74 | 0.20 | N/A | 2.3 | −859 | No | |

D1 | Spatial, symmetric | −7.5 | N/A | 1.62 | N/A | 0.47 | 0.7 | −827 | Yes | |

D2 | Spatial, forward | −7.5 | N/A | 1.50 | N/A | 0.46 | 1.5 | −830 | No | |

D3 | Spatial, symmetric | −7.5 | N/A | 1.44 | 0.23 | 0.87 | 0.4 | −828 | Yes | |

D4 | Spatial, forward | −7.5 | N/A | 1.44 | 0.37 | 0.82 | 0.8 | −830 | Yes |

Visual inspection of the movements of the prawns suggests that interactions often follow a particular pattern. Two prawns, travelling in the opposite directions, collide. After the prawns have passed each other one of the prawns may subsequently decide to change direction. Self-propelled particle and other models of collective motion do not capture this type interaction. Such interactions are non-Markovian,

Each marginal-likelihood is calculated by importance sampling. The figure shows the mean and standard error from 10 instances, each of 5000 samples. Grey markers indicate models that are consistent with the observed large scale behaviour of the system, black markers indicate those that are not. Consistency is determined by alignment of the prawns towards CW or anti-CW movement in simulations.

The Null model, in which prawns do not interact, performs significantly worse than the mean-field model.

(A) Proportion of six-prawn simulations (

Are local spatial interactions important in reproducing observed direction changes? We note first that a topological interaction zone, where the focal prawn interacts with its

The empirical frequency of direction changing as a function of the distance to the nearest opposite facing prawn (grey markers) and the probability of changing direction when interacting with one (solid red line) or two (dashed red line) opposite facing prawns according to the optimal model (D1). The empirical data clearly shows the spatially localised interaction, which is confined to within approximately

This observation is further reflected in the marginal likelihood of the spatial models (S1–S4) in

However, simulations of the spatial models using the inferred interaction parameters (mean

(A) Proportion of six-prawn simulations (

The models incorporating a non-Markovian delayed response together with a spatial interaction zone (models D1–D4) outperformed the Markovian spatial models (

(A) Proportion of six-prawn simulations (

The difference in marginal likelihood between model D3 and model D1 is within the error of the sampling method, and therefore D3 should be considered as an alternative optimal model. Moreover, model D3 is globally more consistent with experiments when simulated.

(A) Proportion of six-prawn simulations (

For each model we report a measure of large-scale consistency with the experimental results, in terms of the final distribution of the proportion of CW-moving prawns. We use the Kullback-Leibler (KL) divergence

A number of physical

The true value of this study, however, is found not in the addition of one more species to this growing list, but in demonstrating a rigorous methodology for selecting an optimal and multi-scale consistent model for the interactions between individuals in a group. We have used a combination of techniques to identify the optimal model for our experiments: Bayesian model selection and validation against global properties. We applied Bayesian model selection to identify the model that best predicts the fine-scale interactions between prawns. This approach allows us to perform model selection in the presence of many competing hypotheses of varying complexity, while avoiding over fitting

The other approach we have employed in validating our model is consistency with large-scale dynamics. Reproduction of the large-scale dynamics is frequently used to validate mathematical models of biological systems, but presents only a necessary and not a sufficient condition for model validation. Indeed, all of the models we have assessed in this work can, with the appropriate parameters, generate aligned motion consistent with experiment. The fact that our mean-field model reproduces global dynamics, but fails at a fine scale level is not particularly surprising. Mean-field models are not designed to reproduce spatially local dynamics

To identify a better model we first visually inspected the interactions between the prawns. These observations suggested a ‘memory effect’, whereby a prawn would remain influenced by individuals beyond the moment of interaction. The resulting models, D1–D4, are both consistent with the polarisation condition and superior at predicting the fine-scale interactions, providing strong evidence for non-Markovian dynamics within this system. More generally, we would expect other examples of animal motion to be non-Markovian, with individuals taking time to react to others, to complete their own actions and also potentially reacting through memory of past situations. In this context, it is important to consider the limitations of recent studies identifying rules of interaction of fish

In what circumstances can we expect non-Markovian effects to play an important role in collective behaviour? Inference based on a Markovian model must account for behavioural changes of a focal individual in terms of their current environment. As such the crucial factor is how much the local environment changes between when the animal receives information and when it responds. Large changes in the local environment can be caused by long response times or by rapid movements of other animals relative to the focal individual. Where behavioural changes are strongly discontinuous, such as the binary one-dimensional movement in this study, non-Markovian effects may become especially important. This is because the focal individual may have to execute a number of small changes (such as stopping and turning through a several small angles) in order to register as having changed its direction of motion. Over the course of making many adjustments the environment can change dramatically from the moment that the change was initiated.

We have used qualitative replication of the large scale motion as a necessary condition for the correct model, and assigned zero probability to inconsistent models. A more subtle approach would be to give a weighting to global consistency. For example, D1 and D3 are both consistent at a global level and indistinguishable according to marginal-likelihood. As such, they should then be considered as equally viable alternative models for the real behaviour of the prawns. However, a visual inspection of global consistency favours D3 over D1 (see

Glass prawns (

The frame-by-frame movements of the prawns are imperfect representations of the true orientation, since a prawn will often stop or even drift slightly backwards without physically turning around. A Hidden Markov Model (HMM) allows the underlying orientation of the prawns to be discovered from the noisy frame-by-frame movements by demanding a higher degree of ‘evidence’ for a direction change, in essence only identifying direction changes when the prawn makes a sustained movement in the new direction. This gives a better estimate of the true orientation than given by the instantaneous velocity alone.

We constructed a two-state HMM

At any point in time the prawn is in a state of either CW or anti-CW orientation. The precise state is hidden but we make observations

A given model,

Johannes Alneberg provided assistance with figure creation. Three anonymous reviewers gave valuable advice to improve the manuscript.