Modeling collective animal behavior with a cognitive perspective: a methodological framework

Abstract

The last decades have seen an increasing interest in modeling collective animal behavior. Some studies try to reproduce as accurately as possible the collective dynamics and patterns observed in several animal groups with biologically plausible, individual behavioral rules. The objective is then essentially to demonstrate that the observed collective features may be the result of self-organizing processes involving quite simple individual behaviors. Other studies concentrate on the objective of establishing or enriching links between collective behavior researches and cognitive or physiological ones, which then requires that each individual rule be carefully validated. Here we discuss the methodological consequences of this additional requirement. Using the example of corpse clustering in ants, we first illustrate that it may be impossible to discriminate among alternative individual rules by considering only observational data collected at the group level. Six individual behavioral models are described: They are clearly distinct in terms of individual behaviors, they all reproduce satisfactorily the collective dynamics and distribution patterns observed in experiments, and we show theoretically that it is strictly impossible to discriminate two of these models even in the limit of an infinite amount of data whatever the accuracy level. A set of methodological steps are then listed and discussed as practical ways to partially overcome this problem. They involve complementary experimental protocols specifically designed to address the behavioral rules successively, conserving group-level data for the overall model validation. In this context, we highlight the importance of maintaining a sharp distinction between model enunciation, with explicit references to validated biological concepts, and formal translation of these concepts in terms of quantitative state variables and fittable functional dependences. Illustrative examples are provided of the benefits expected during the often long and difficult process of refining a behavioral model, designing adapted experimental protocols and inversing model parameters.

Figure 1. Clustering experiments.

Objects are uniformly distributed along the border of a 50 cm diameter circular arena. Ants enter the arena spontaneously from below, mounting along a wood stick through the hole in the arena center. Two different initial one-dimensional densities are used: 127 and 255 objects per meter. The duration of each experiment is 50 hours. Fig. A and Fig. B correspond respectively to the beginning and the end of a high density experiment. Fig. C and Fig. D display the time series of the number of piles (mean s.d.) for the low and the high density experiments, respectively. Piles are defined as follows: Two neighboring objects are considered to belong to the same cluster if the distance between them is less than 1 mm. A cluster constitutes a pile if it contains at least 6 objects. In another circular arena with a 25 cm diameter was also used (with the same low and high initial densities). For the purpose of the methodological illustration in the first part of the results section only the large arena is used, but the model built in the second part of the results section is compatible with all observations, including the ones in small arenas (see Fig. 6).

Figure 2. Clustering dynamics predicted by the six different models.

Figs. A and B indicate the time series of the number of piles for models 1–6 compared to the experimental data (mean s.d.) in the low and high density settings, respectively. The predictions of all the models are compatible with the experimental observations. Moreover, the predictions of model 5 are rigorously identical to those of model 3, and the predictions of model 6 are identical to those of model 4.

Figure 3. Definition of the object perception stimulus.

The grey area represents the objects distributed along the arena border and the dashed square the perception range of an ant at perimetric abscissa and distance to the border of the arena. The dashed grey area represents the fractions of the objects perceived by the ant (which corresponds to in the text). When defining the perception stimulus at perimetric abscissa , without knowing the distance to the border (one-dimensional modeling), is defined as the average of all values of when is uniformly distributed between and the distance of the external side of the farthest clustered object.

Figure 4. Parametric inversions for the direction change and deposition frequencies in the empty arena.

A) The survival curve of the proportion of ants still not having changed their direction ( trajectories) is compatible with an exponential fit , validating the memoryless and instantaneous turn assumptions and leading to the direction change frequency (mean s.e.), , , . B) The survival curve of the proportion of ants still not having deposited the object they loaded at time in the empty arena ( trajectories) is also compatible with an exponential fit and leads to the deposition frequency , , , . The black line represents the exponential fit and the dashed lines the 95% confidence interval. The -values correspond to a chi-squared test for goodness of fit as described in (pp. 131–137).

Figure 5. Parametric inversion for the deposition frequency.

A) Functional dependence of the deposition frequency on the perception stimulus. B) Fit of the deposition frequency to the experimental deposition probabilities (mean s.e.) on clusters of several sizes. The plain curve in A corresponds to the formal dependence retained in (, with and , , , ). The dashed curve is a linear fit ( with , , , ), very close to the previous formal dependence for small values of the perception stimulus. C) An alternative functional dependence of the deposition frequency on the perception stimulus, in which the deposition frequency is very low for small values of the perception stimulus and constant for values higher than a threshold corresponding to two objects entirely in the perception area. D) Adjustment of C to the experimental deposition probabilities ( if and else, with , and , , , ). The grey circles correspond to the experimental data (mean s.e.) and the -values to a weighted least squares procedure.

Figure 6. Comparison with experimental data of the collective dynamics predicted by the finally retained model (that of the dashed curve in Figs. 5A and B).

Figs. A and B correspond to the experiments in the big arena, and Figs. C and D to those in the small arena, with low and high object densities, respectively (see Fig. 1). The grey + correspond to the experimental data (mean s.d.) and the plain curves to the model predictions. The fitted number of ants in the arena is 50.

Figure 7. Significance of inter-individual variabilities and temporal correlations.

The survival probabilities of models 5 or 6 (also to be interpreted as the population average of the survival probabilities of models 3 or 4) compared to an exponential fit corresponding to small values (short time depositions). The extinction coefficient of the exponential is .