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 Systems Biology
Inherent noise can facilitate coherence in collective swarm motion

Edited by Simon A. Levin, Princeton University, Princeton, NJ, and approved February 11, 2009 (received for review November 5, 2008)
Abstract
Among the most striking aspects of the movement of many animal groups are their sudden coherent changes in direction. Recent observations of locusts and starlings have shown that this directional switching is an intrinsic property of their motion. Similar direction switches are seen in selfpropelled particle and other models of group motion. Comprehending the factors that determine such switches is key to understanding the movement of these groups. Here, we adopt a coarsegrained approach to the study of directional switching in a selfpropelled particle model assuming an underlying onedimensional Fokker–Planck equation for the mean velocity of the particles. We continue with this assumption in analyzing experimental data on locusts and use a similar systematic Fokker–Planck equation coefficient estimation approach to extract the relevant information for the assumed Fokker–Planck equation underlying that experimental data. In the experiment itself the motion of groups of 5 to 100 locust nymphs was investigated in a homogeneous laboratory environment, helping us to establish the intrinsic dynamics of locust marching bands. We determine the mean time between direction switches as a function of group density for the experimental data and the selfpropelled particle model. This systematic approach allows us to identify key differences between the experimental data and the model, revealing that individual locusts appear to increase the randomness of their movements in response to a loss of alignment by the group. We give a quantitative description of how locusts use noise to maintain swarm alignment. We discuss further how properties of individual animal behavior, inferred by using the Fokker–Planck equation coefficient estimation approach, can be implemented in the selfpropelled particle model to replicate qualitatively the group level dynamics seen in the experimental data.
While recent years have seen an explosion in the number of simulation models of moving animal groups, there is little detailed comparison between these models and experimental data (1, 2). The models usually produce motion that “looks like” that of a swarm of locusts, a school of fish, or a flock of birds, but the similarities are difficult to quantify (3). Furthermore, the simulation models themselves are often difficult to understand from a mathematical viewpoint since, by their nature, they resist simple meanfield descriptions. These complications make it difficult to use models to predict, for example, the rate at which groups change direction of travel or how spatial patterns evolve through time (4, 5). We are left with a multitude of models, all of which seem to relate to the available experimental data, but none of which provide clear predictive power.
One approach to the problem of linking experimental data to model behavior is the detailed study of the local interactions between animals. This approach has yielded better understanding of the rules that govern the interaction of fish (6, 7) and birds (8, 9). However, establishing these rules is technically difficult because it requires automated tracking of individuals over long periods of time and quantification of often complicated interactions.
Coherent animal groups often make sudden changes in direction (1, 10–12). In some cases a switch in direction is a response to an external influence, such as the presence of a predator, but in other cases animal groups appear to switch direction spontaneously. Recently, experiments on various densities of desert locusts ( Schistocerca gregaria) in a homogeneous environment confirmed that directional switching can occur without changes in the external environment (10). In experiments with lower locust densities it was found that groups of locust nymphs were highly aligned and marched in one direction around a ringshaped arena for up to 2 or 3 h, before spontaneously switching direction in the space of only a few minutes and marching in the opposite direction, again for a number of hours. In experiments with higher densities, marching groups travelled in the same direction for the 8h duration of the experiment. The group property of average velocity/alignment (as it will variously be denoted throughout the rest of this article) of the locust experiments was previously modeled by a 1dimensional selfpropelled particle (SPP) model similar to that in ref. 13. In a manner analogous to the experiments in ref. 10 this model also exhibits spontaneous direction switching where “particles” rapidly change alignment.
In this article we investigate directional switching in a SPP model and in our experimental data of the motion of locusts. A coarsegrained model characterizes the behavior of a system in terms of a single “coarse” variable (average locust velocity in our case) and does not take into account the “fine” details of the behavior of individual locusts. If such a model is explicitly available we can apply relevant mathematical techniques to study the properties of group switching as will be shown later for a simple toy model. In a more complicated scenario, where we are not able to write a coarsegrained model explicitly, we can often still assume that there exists a coarsegrained Fokker–Planck equation (FPE) describing the evolution of the average velocity of the particles in the SPP model. We use appropriately initialized computational experiments to estimate the (explicitly unavailable) drift and diffusion coefficients of such a FPE. The drift coefficient represents the mean rate of change of the average velocity of the locusts, while the diffusion coefficient quantifies the randomness of the evolution of this coarsegrained variable (precise mathematical definitions are given later in Eqs. 13 and 14). We also estimate the drift and diffusion coefficients of a coarsegrained FPE based on experimental data. Coarse graining enables efficient description and analysis of these data and assists the refinement of the individualbased model so as to simulate observed group properties more accurately.
Our studies suggest that an individual's response to a loss of alignment in the group is increased randomness of its motion, until an aligned state is again achieved. This alignmentdependent stochasticity, using randomness to keep the group ordered, appears counterintuitive. Although the constructive effects of noise at the level of an individual have been reported in other biological systems (14), here, we present an example where noise has been found to be a constructive force at the collective level in an ecological system. Furthermore, the lack of sources of environmental noise in the experiment under consideration indicates the internal character of these fluctuations. Noiseinduced alignment seems, in this case, to be an intrinsic characteristic of collective coherent motion.
The article is organized as follows. We start by providing an analytical result for mean switching times of the SPP model with global interactions and compare this with its coarsegrained approximation in the case of local interactions. After demonstrating that coarse graining works well for our model simulations we apply a similar FPE coefficient estimation approach to the experimental locust data to find the drift and diffusion coefficients of the assumed underlying FPE. This analysis illuminates similarities and important differences between the model and the experimental data and allows us to better understand directional switching in both cases. We conclude by proposing a modified model that reproduces the experimental observations more accurately than the previously used SPP models.
SelfPropelled Particle (SPP) Model
We consider a group of N “locusts.” Each locust adjusts its behavior according to the behavior of its neighbors, which can be found less than a distance R (the interaction radius) from it. A locust's behavior is described by its position, x_{i} ≡ x_{i}(t), and velocity, u_{i} ≡ u_{i}(t), i = 1,…,N, which evolve according to the model adapted from Czirók et al. (13):
where Δt is the time step between successive position and velocity updates. ΔQ is a random variable uniformly distributed in
In refs. 10 and 13 the function η is chosen to be unity. Similarly we will choose η ≡ 1 in our initial model and later revise this choice after analysis of the experimental data. Let us note that the model in ref. 13 is a special case of model 1–2 where β = Δt = 1 and η ≡ 1 throughout.
Toy Model with Globally Interacting Particles
To illustrate our coarsegraining technique we will first study a modification of the model 1–2 where all particles interact with each other. The advantage of this model is that we can obtain the coarsegrained equations explicitly: we know that the assumptions, made later to justify coarsegraining, hold exactly and hence we can validate the approach. Moreover, we can obtain analytical results for this model, as shown below. We thus replace the local average (Eq. 3) in formula (Eq. 2) by the global average, to obtain: for i = 1,2,…,N, where η ≡ 1 for simplicity and the marching group comprises all N individuals. Note that this toy model is the original model with the domain length, L, equal to 2R. Eqs. 5 are now only oneway coupled with Eq. 1 (i.e., the positions of the locusts do not affect the velocities, but the velocities do affect the positions). Assuming that the number of locusts in the marching group is fixed and equal to N, this toy locust problem can be described by the system of N Eqs. (Eq. 5).
A quantity of interest is the mean switching time between different directions of motion of the group (left or right) for which, in this special case of global interaction, we can find an explicit formula. Adding Eqs. 5 and dividing through by N and invoking the central limit theorem where ΔW is the normal random variable with mean 0 and variance Δt (i.e., the standard Brownian process). Assuming the approximation in Eq. 6 to be exact, let f_{N}(U,t) be the probability distribution function of the random variable U. Given initial condition f_{N}(U,0), the distribution f_{N}(U,t) can be computed (as in ref. 15) as a solution of the FPE, Notice that, apart from the small approximation due to the application of the central limit theorem, this is an exact equation for the coarsegrained observable U. The steady solution of Eq. 7 is given by where C is a normalization constant and the potential, ϕ_{N}(U), can be computed as The potential is symmetric with respect to U = 0. It has two global minima at U_{±} = ±1. SI Appendix, Fig. S1A shows the stationary probability distribution (SPD), f_{Nst}(U), given by Eq. 8, and SI Appendix, Fig. S1B shows the potential, ϕ_{N}(U), given by Eq. 9, both plotted as solid lines. The mean switching time can be computed (as in ref. 15) as a function of N, The function T_{1}(N) is plotted (full line) in SI Appendix, Fig. S1C with approximations to the variation of the mean switching time with N (squares) given by stochastically simulating the model. There is quantitative agreement between the simulations and the theoretical formula (Eq. 10). There is no doubt, in the case of global interaction, that the dependence of mean switching time on the number of particles in the system is exponential. It can be approximated (see SI Appendix, section S1) as
ComputerAssisted Methodology
We now return to the original model 1–2 with finite interaction radius. Unlike in the case of the toy global interaction model, we are unable to derive an explicit, closedform equation for the coarsegrained observable U, i.e., the analogue of Eq. 7. However, we hypothesize that such a closed equation, exists, where D(U) and F(U) are the diffusion and drift coefficients, respectively. Using an equationfree approach (16, 17) we are still able to approximate quantities such as the mean switching time by designing and performing short computational experiments to estimate, on demand, the drift and diffusion coefficients of the unavailable, coarsegrained FPE (Eq. 12) for the mean velocity, U, of the locusts. Following refs. 16 and 18, we approximate where 〈·〉 represents an ensemble average over several, consistently initialized, short replica simulations with the detailed model 1–2. δt is a small number of time steps of the process (typically between one and five) dictated by the timescale on which the coarsegrained Eq. 12 becomes valid. We can then use the drift and diffusion coefficients to approximate, via quadrature, the potential: and hence the SPD of the underlying FPE (using Eq. 8).
The equationfree approach involves initializing the particles consistently [which is called “lifting” in the equationfree terminology (17)], so that their velocities give a specific value of alignment, U. We perform a large number of replica simulations (typically 1,000) allowing each one to evolve for a short time period δt. This provides representative averages for the drift and diffusion coefficients of FPE (Eq. 12) using formulae 13 and 14, respectively. After we have found the values of these desired quantities at the first alignment we repeat the process on an evenly spaced grid of possible alignments. Applying this technique in the case of global interaction gives an excellent fit to the analytical solutions, as expected (see SI Appendix, Fig. S1A and S1B).
In the more biologically realistic ‘local’ model, 1–2, we found that slightly smoother results were obtained by initializing particles with both a prescribed alignment and a velocity variance, which, in long simulations, was observed to be consistent with the particular alignment value. The diffusion coefficient of a specific case (N = 30) of the local model is shown in Fig. 1A to be approximately constant, while the drift coefficient has a characteristic antisymmetric cubic shape, indicative of the symmetric doublewelled potential (see SI Appendix, Fig. S2). Such an effective potential is consistent with most particle velocities being aligned (in one direction or in the other) most of the time, with occasional switches between directions whose frequency depends on the height of the potential barrier between the wells relative to their depth. This is corroborated by the approximations to the SPDs given in Fig. 1B. The histogram represents the proportion of time the mean velocity of the particles spends at each allowed value bin during a long time simulation, while the curve is the equationfree approximation to the SPD.
For a range of N we used the shortburst, equationfree derived potential to estimate the mean switching time by using a modified version of Eq. 10: where U_{−} and U_{+} are the mean velocities at which the two minima of the potential occur. We also established the relationship between mean switching time and N by counting the switches during a long simulation for the same range of N. The two methods give similar, apparently exponential, relationships (see Fig. 1C).
Experimental Data
Locusts exhibit dynamic directional switching over a range of different densities. In previously reported experiments (10) we recorded the directional alignment of groups of between 5 and 100 locusts for 8 h. The ringshaped arena [an annulus with outer radius 40 cm and inner radius 17.5 cm (see SI Appendix, section S4)] in which the locusts were placed is analogous to the 1dimensional domain with periodic boundary conditions used in the SPP model. We can thus use similar techniques to analyze the experimental data as we did with the longterm simulations. However, the equationfree approach (involving many consistent initializations of the velocity—and even possibly of the variance of the velocity—of live locusts) is not practical in an experimental setting. Instead, we use a similar systematic FPE coefficient estimation approach from observations of the velocities of the locusts over a long period of time (an “equilibrium run”). During the first 2 h of the experiments the activity of the locusts changed significantly, but in the latter 6 h activity tended to be relatively consistent. We thus amalgamated, for each number of locusts up to 40, the observations of the locusts' velocities over several experiments after the first 2 h of each experiment. We collected instances of the same alignment, U(t) and the alignment, U(t + δt), a short time, δt, later and used these in Eqs. 13 and 14 to estimate the drift and diffusion coefficients of the underlying FPE. Although experiments were carried out for numbers of locusts >40 there were too few switches in the 6h interval to provide a meaningful estimate of drift and diffusion coefficients.
Fig. 2A and B show the estimated drift and diffusion coefficients of the alignment for the experiment with 30 locusts. The drift coefficient in Fig. 2B, although noisy, still has a roughly cubic shape consistent with that estimated for the SPP model. Unlike the SPP model, however, the diffusion coefficient appears to have a quadratic shape, with its maximum at zero alignment. The potential (see SI Appendix, Fig. S3), although not perfectly symmetric, is still doublewelled, indicating the tendency of the particles to exhibit collective motion in one direction or the other.
We used Eq. 16 to calculate, for different N, the mean switching times from the potentials constructed by systematic estimation. We also found the mean switching time directly by taking the total time of the experiment and dividing it by the number of switches made. Both methods give an approximately exponential relationship between mean switching time and N (see Fig. 2C). This implies that the more locusts there are, the less often they switch and, as the number of locusts becomes sufficiently large, there are effectively no switches over the duration of a day's marching (∼8 h). There are quantitative differences between the results obtained by direct estimation of number of switches and the result of the coarsegrained approach. Two key sources may contribute to this: (i) The assumption that there exists a coarsegrained FPE for the average velocity, U, is valid only approximately. Such a discrepancy was already observed when we substituted the toy model (Eq. 5) by model 1–2 (see Fig. 1C). In the case of the toy model (Eq. 5) we know that there exists a closed FPE (Eq. 7) and the results are exact, (see SI Appendix, Fig. S1C ). On the other hand, in the original model 1–2 the coarsegrained equation was not readily available and our computations revealed that it is valid only approximately. (ii) For the experiments there is also a second source of error since, unlike the computational model, we cannot obtain unlimited time series data. The length of the time series is limited by experimental restrictions. Binning the available time series as a histogram does not yield a symmetric SPD, which suggests that these time series are not long enough.
The fact that the diffusion coefficient increases when group alignment is low indicates that the locusts might respond to low group alignment by increasing the noisiness of their motion. To test this hypothesis we refined the SPP model 1–2. Instead of taking the function η(
In Fig. 3A the quadratic nature of the effective diffusion coefficient is recovered by the refined model. Qualitatively, the potential of the new model compares favorably with the potential of the actual experimental locust data (see SI Appendix, Figs. S3 and S4). A further useful validation of the revised model is that the mean switching time is significantly increased for all values of N in comparison with the original model [compare the simulationderived mean switching times (squares with the full best fit lines) in Fig. 1C with Fig. 3C]. A higher diffusion coefficient at lower alignments suggests that the locusts “prefer” to be in a highly aligned state: when the locusts leave this state they increase the randomness of their movements and consequently a new aligned state is arrived at more quickly. The effect of this altered diffusion coefficient on the evolution of the coarse variable, U, is evidenced in Fig. 4. The transitions between ordered states in the original model are very sharp (see Fig. 4A), whereas the transitions for the experimental data are relatively noisy (see Fig. 4B). The noisiness of the transitions appears to be replicated well by the revised model (see Fig. 4C). In general, it can be seen that the marching band switches direction more frequently in the original model than the revised model.
Discussion
Our analysis of selfpropelled particle models and of animal movement data has revealed a number of unique features. First, we established that the mean switching time increases exponentially with the number of particles/locusts. As locust density increases the turning rate of the group rapidly decreases. This observation has implications regarding attempts to control the locusts' motion: at high densities it becomes increasingly difficult to influence a group's direction. Second, we used the systematic Fokker–Planck equation coefficient estimation approach on experimental data. Our results indicated that the individuals move more randomly in locust groups with low alignment. This appears to enable the group to find (and remain in) a highly aligned state more easily.
One of the most useful aspects of our agentbased approach is that it enhances our ability to speculate on the behavior of individual locusts from grouplevel information: we alter individual behavioral rules and use selfpropelled particle models to test the effect that these alterations have on the coarse variable, U. This enables us to verify the validity of these individualscale alterations. We thus used our experimental observations to guide our modeling, changing the noise term for the individual locusts so that it becomes larger when the alignment is smaller. It should be noted that the randomness in the model does not necessarily mean that the locusts are making random choices. It could be that there exist underlying smallscale individual interactions that manifest themselves as noise in the individual motion of the locusts at the level of our experimental observation. We found the relationship between group number and mean switching time to again be exponential, but with longer mean switching times than in our previous model.
It would be interesting to consider whether recent findings about cannibalistic interactions between marching locusts (19) can provide a rationalization for the observation of apparently increased individual randomness in response to a loss of alignment at the group level: given the risk of exposing the rear of the abdomen to oncoming insects (19), there may be selection pressure on an individual to minimize the time spent in the disordered phase. A longer interswitch time might also be selected for in an evolutionary scenario since it allows the locusts to remain in a coherent group for longer periods, potentially increasing harvesting efficiency and reducing predation (7, 20–22). Noise in response to lost alignment may be an example of a general property of organization of collective motion. Another example is found in trafficjam models where one way of avoiding “phantom traffic jams” is to introduce additional noise to traffic motion (23–25).
We have provided evidence that our revised model is more biologically justifiable than our original model; unique functional forms such as Eq. 17 may be useful in the formulation of other selfpropelled particle models characterizing collective animal behavior. Our findings provide strong evidence for the seemingly unexpected phenomenon of randomness contributing to the creation of coherent behavior at the collective level.
Acknowledgments
This work was supported by Engineering and Physical Sciences Research Council and Biotechnology and Biological Sciences Research Council (C.A.Y.), the Ministry of Education and Science (Spain, Project FIS200501729) (C.E.), the Air Force Office of Scientific Research (I.G.K.), a Searle Scholar Award and Defense Advanced Research Planning Agency Grant HR0010510057 to Princeton University (I.D.C), St. John's College, Linacre College and Somerville College, Oxford (R.E.). This publication is based partially on work supported by King Abdullah University of Science and Technology Award No. KUKC101304 (R.E.). P.K.M. was partially supported by a Royal SocietyWolfson Merit Award. This work was partially supported by the OxfordPrinceton Partnership grant (P.K.M. and I.G.K.).
Footnotes
 ^{1}To whom correspondence should be addressed. Email: yatesc{at}maths.ox.ac.uk

Author contributions: C.A.Y., R.E., C.E., I.D.C., J.B., I.G.K., P.K.M., and D.J.T.S. designed research; C.A.Y., R.E., and C.E. performed research; C.A.Y. and J.B. analyzed data; and C.A.Y., R.E., C.E., I.D.C., J.B., I.G.K., P.K.M., and D.J.T.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0811195106/DCSupplemental.
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