Ant tracking experiments

We performed ant tracking experiments using Anoplolepis gracilipes (“yellow crazy ant” or “long-legged ant” [39]) worker specimens (approx. 5mm long) collected from around the campus of the Okinawa Institute of Science and Technology Graduate University, in Okinawa, Japan. After being collected manually, specimens were placed in a small container (approx. 5x5x10cm) potentially with conspecifics from the same area, for up to a few hours. One-by-one, specimens were placed alone in an acrylic, rectangular arena of size 15x21x16.5cm (Fig 1a), where they were recorded using a Nikon D800E camera at 60 Hz for roughly 20 minutes. This arena size is typical of works interested in the confined behavior of insects on the scale of A. gracilipes [15]. Experiments were performed over several weeks between May 2024 and July 2024. Each ant was used only once.

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Fig 1. Overview of experiment to extract ant trajectories in a confined arena.

Specimens are placed alone in an acrylic arena and recorded for 20 minutes (a). Ant position is tracked from videos using a neural network tracking approach (b), giving spatial trajectories (c).

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Control trials were performed to confirm that being placed with conspecifics prior to the tracking experiments did not have a significant effect on ant’s motion. Ants were either isolated after collection, or placed with conspecifics, but in all cases the tracking experiments were always performed with only a single ant present.

Control trials were also performed in which the enclosure was sanitized — washed with soap and water, then cleaned with 70% ethanol [38] — between each experiment, or left as is. Long-legged ants are known to produce several types of pheromones for navigation and orientation [10,39], so these trials were intended to determine if there were stimuli induced from previous ants in the otherwise static environment.

Finally, all trials were performed during either the morning (08:00–12:00) or afternoon (12:00–17:00). It has been suggested that these ants tend to be more active in the morning [16], so we looked to compare if the time of day of the experiment affected the statistics.

We saw no significant differences in any of the trajectory statistics presented in this work across all of these control-test pairs (S1 Fig, compare to Fig 5). This is not to say that these factors had no effect at all on the trajectories, but just that the quantities of interest were robust to these changes in experimental protocol. As such, the figures in the main text of this work include data from both the control and test groups for the above three comparisons.

A limited set of experiments was performed with a different camera (Phantom v641 high-speed camera) that is capable of shooting at a much higher framerate and resolution (120 and 300 Hz, 2560x1600 pixels), in order to characterize the scale of temporal and spatial discretization artifacts in the trajectories. This dataset generally shows similar statistics to the main dataset, suggesting that the principal statistical properties are not discretization artifacts. While one can intuitively predict the scale of such artifacts a priori in other cases, the neural network tracking approach (and its uncertainty) makes this more difficult. No analysis of this dataset is included here, though it is available, alongside the rest of the data and code, in the online repository [40].

Other factors of collection and treatment were kept constant where possible. Temperature and humidity were kept roughly constant around 22° C and 50% relative humidity by the climate control in the laboratory for the duration of experiments. This is towards the lower end of, though still within, the temperature range for which these ants forage [16,39,41]. Overhead fluorescent lights were kept on for all experiments, though lighting is suspected not to affect these ants’ activity [16].

Trajectory analysis

Following experiments, the video data was consolidated and used to train a deep convolutional neural network to track the posture of the ants using SLEAP [42]. A multi-animal (top down) model is used (Fig 1b), composed of two U-net neural networks [43], one which identifies the rough location of the ant (“centroid model”), and the second which identifies specific keypoints in a limited region around the identified location (“centered instance”). This latter network tracks three keypoints on the ants: the tip of the head, the center of the thorax, and the rear tip of the abdomen. This structure is preferred to a single-animal model since the ant is so small compared to overall arena size. The parameters of these networks are described in S1 Table.

The positions of the keypoints are extracted to give the spatial trajectory of each body part through time. After training on 280 manually annotated frames, we observe an average error per keypoint of mm, which is estimated to be of similar magnitude to the error in the annotations of these features. All keypoints within a single frame are averaged to find the center-of-mass position of the ant; we don’t consider the relative motion of the ant’s body in this work. To further reduce the effect of artifacts, we perform windowed averaging (smoothing) on the center-of-mass position of the ants with a window size of 10 (representing a time window of about 0.15 seconds, chosen empirically).

This tracking model was used to extract the trajectories of all ants across all videos. In some cases, the trajectory is split into separate parts because either the ant climbs the walls of the arena — and therefore leaves the camera view — or the tracking algorithm fails. The latter case mostly consists of very short interruptions, which can be fixed by linearly interpolating the trajectory for the missing points. We perform this interpolation up to a maximum of 30 missing data points (chosen empirically, 0.5 seconds in real-time). If there is a gap larger than 30 frames, we don’t interpolate the trajectory as it could introduce artifacts. Most trials had a few long segments representing the majority of the motion, with some shorter segments punctuating these longer ones. Except for computing the occupation maps, these short segments are excluded from analysis by taking only segments longer than a cutoff time, t_min = 5 seconds (chosen empirically).

We mark 15 trials as outliers due to various experimental differences to the rest of the data, which are summarized in S2 Table. Excluding these trials results in an ensemble of ant segments representing more than 24 hours of footage capturing the motion of nearly 100 individual specimens. Some example segments are shown in Fig 1c. All data is available in the online repository [40].

Ant motion as a run-and-tumble process

We propose that the run-and-tumble [44–46] framework qualitatively matches the typical movement patterns of these ants. This stochastic process involves short time-scale ballistic motion, punctuated by discrete reorientation events. Given an ant (or particle) with position and orientation , the most basic Langevin equation for this process in two dimensions looks like:

The first equation represents the translational motion of the ant, implementing the “run” evolution, with speed v and direction vector :

We assume that the ant moves with a constant speed v to simplify the model; we find that this gives a reasonable characterization of the locomotion behavior, though we discuss the ramifications of this assumption later on.

The second equation describes the tumbling behavior of the ant, where the orientation is updated at discrete times by some amount . In principle these could be drawn from any arbitrary distribution, though common choices for these two are:

(1)

(2)

That is, the reorientation kernel K is a zero-centered von Mises distribution with width parameter (I₀ is the modified bessel function of the first kind of order zero), and the time between tumbles (run time) is drawn from an exponential distribution with average . This is equivalent to defining a constant tumbling rate . As we will see, these choices are well suited for the ant data. On top of this basic model, we consider three additions to more realistically model the ant motion: boundary conditions, rotational diffusion, and finite tumbling times.

Other studies have proposed piecewise models that evolve according to different governing equations in the bulk and near the walls [38]. Here, we propose to model our data with a more unified approach: using the same fundamental dynamics across the entire space, though with a local boundary force/torque at the edge of the arena. The fundamental difference is that the piecewise approach introduces non-local effects of the boundaries, invoking some assumptions about the sensory system of the specimen (i.e., how does the insect “know” that it is at the boundary?). Such assumptions can be very helpful, though our approach provides a more minimal (and therefore, hopefully more widely applicable) model, which might be expanded based on the anatomy and behavior of a specific organism.

Including rotational diffusion allows the model to capture some of the roughness and curvature that real ants demonstrate when walking. This introduces a qualitative separation between small angle changes that result in curvature along an otherwise straight path, and large angle changes that constitute reorientation behavior.

Finally, we introduce waiting times at each tumble, a phenomenon that is repeatedly observed both in our data and in that for other insects [18]. This is implemented via a “waiting function“ H(t) that is zero during waiting times of length , and 1 otherwise,

(3)

where is the Heaviside function. For now, we make no assumptions about the form of the distribution from which the are drawn.

Altogether, we have the governing equations:

(4)

(5)

The translational boundary force is taken to be simply a rigid body force along the boundary normal, :

(6)

This is equivalent to defining an infinite well potential that the process evolves within, and invokes the assumption that the ant does not sense the boundary prior to coming into contact with it. The rotational boundary condition is taken to be a torque that aligns the particle with the closer tangent of the wall:

(7)

Both of these choices of boundary terms are relatively simple, though seem to recreate the observed motion quite well. We discuss further, biologically-inspired extensions one could make to these terms towards the end of this work. Finally, is a delta-correlated random variable with unit variance.

Equations 4 and 5 can be used to simulate individual trajectories, allowing us to create an ensemble of simulated data. This simulation is done by defining a discrete timestep dt and evolving from a random initial condition for the desired amount of time. While the waiting function H(t) could be implemented using Heaviside functions as defined above, it is easier to use programmatic conditional statements to ignore the appropriate terms during each waiting time. The full simulation code in Python, along with all analysis methods used in this study, can be found in the data repository [40] as well as on Github (https://github.com/Jfeatherstone/tracker).

On top of the Langevin formulation above, we can also write a Fokker-Planck equation to describe how the probability of being in a particular state evolves [30,32]. This is particularly useful in deriving the expectation values of various quantities, for example, the persistence time (full derivation in S1 Appendix),

(8)

where I₁ is the modified bessel function of the first kind of order one. This does not take the finite size of the arena into account, and thus is only valid if we consider the motion in the center of the arena. The typical way to compute the persistence time of an arbitrary trajectory (i.e., imposing no model assumptions), which we compare to the above expression, is to consider the autocorrelation of the orientation through time. For a stationary, persistent process, we have:

(9)

By fitting this function with an exponential form, we can extract the persistence time . In subsequent sections, this is referred to as the “direct fit” method, whereas the formula derived from the discretized model is referred to as the “discrete fit” method.

We can make further use of this Fokker-Planck formulation by considering the run-and-tumble model as an active Brownian process with stochastic resetting [47]. This allows us to calculate the typical velocity from the short time scale mean squared displacement (MSD,see S1 Appendix for more details),

(10)

as well as the rotational diffusion coefficient from the persistence time during runs,

(11)

As above, computing the velocity by fitting the MSD is referred to as the “discrete fit” (since it invokes the model), whereas computing the median velocity from the derivative of the trajectory is referred to as the “direct fit”.

Trajectory discretization

In order to compare the experimental data with the stochastic model described in the previous section, we have to identify discrete run events, as well as the tumbling events that punctuate them (Fig 2) in this trajectory data. The idea of discretizing a continuous trajectory has been both applied and discussed at length in the movement ecology literature [8,48–50]. Such an analysis can reveal important information about the structure of movement [51], or the typical transitions made while moving [52]. The technical application of this method is usually relatively simple, and involves breaking up the trajectory at points when the animal turns more than some critical angle, and/or the speed is less than some threshold, . That being said, the justification for selection of parameter values for this technique is often not straightforward, and, in particular, the inherent subjectivity of this analysis is often criticized [53]: it is often the case that many different conclusions can be drawn from the same dataset by choosing different discretization parameter(s), all of which might seem reasonable at first glance.

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Fig 2. Trajectory analysis workflow.

Original sampled trajectories are used to compute trajectory statistics like occupation maps and mean squared displacement, as well as to identify points when the ant changes direction and/or stops moving, generating a discretized trajectory. This allows for the extraction of the times spent moving in a straight line, , the angle changes between these segments, , the time spent waiting before each reorientation, , and the angle drift during each straight run, . These values are used to derive statistics about the original trajectory using a Fokker-Planck equation, and to simulate new ant-like trajectories using a Langevin equation.

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This isn’t an issue when one can choose the discretization parameters a priori based on properties of the system being investigated, or when there is no ambiguity as to the different phases of behavior. For example when the behavior of the animal is strongly intermittent and/or when the motion is considered in only one dimension, it becomes very obvious when the organism is “running” and when the animal is “tumbling” [18,54]. Unfortunately, most animal behavior doesn’t demonstrate such a dichotomy, and we often have to determine the scale at which we discretize a posteriori. In this case, we may still apply this technique to a trajectory, though we should confirm several pieces of information to minimize the amount of subjectivity in the results. First, that it is reasonable to assume that there even exists a scale at which discretizing the trajectory gives relevant information about the underlying continuous process. In other words, does it, both qualitatively and quantitatively, make sense to model this process as having discrete phases? Second, that the discretization process is able to reliably extract the desired statistics from trajectories with known properties. And third, that the results are not especially sensitive to the choice of discretization parameter(s).

We use an accumulator discretization method (sometimes called “non-local” [49]) in which the moving average of the current run direction is computed, and when the angle between that average direction and the next position is greater than some threshold value , a new run is defined. We also segment the trajectory by times when the velocity falls below a threshold , even if the ant continues moving in the same direction after resting. This combination allows for detection of runs punctuated either by changes in direction or waiting times (Fig 2). From this discretized trajectory, we extract the distribution of turn angles , run times , and waiting times . As discussed in the previous section, we also compute the rotational diffusion coefficient and velocity v using expressions derived from the model in the short time limit. All of these analyses are tested on synthetic run-and-tumble data generated using Eqs 4, 5, shown in S2 Fig; in general, the technique does a reasonable job of estimating the underlying parameters, including for data generated in the exact geometry used in the experiments.

All fitting is performed with either the powerlaw [55] or scipy [56] packages in Python. When a centrality measure is needed to characterize statistics below, the median is preferred over the mean, since some of the analyses that involve curve fitting can yield unrealistic values when there are few data, which heavily skew the mean.

Locomotion statistics

One of the most prominent features of the locomotion data, as one might expect, is the impact of the finite arena and its geometry. We observe that the most densely visited locations are those on the boundaries, visualized in Fig 3a, through the average spatial occupation map. We can also observe how the specimens align their motion with the boundary by examining various cuts of the phase space (spatial position and orientation) map, shown in Fig 3b. When near the left and right (speaking from a birds-eye view, as in the figure) walls, the ants overwhelmingly move up or down, and vice-versa for the top and bottom walls. Beyond this stereotyped behavior around the boundaries, we are next interested in the statistics of motion in the center of the arena. Based on these occupation maps, we define the center of the arena by cropping data within 1.5 cm of the walls.

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Fig 3. Spatial and phase space occupation maps for ant motion.

Average amount of time spent in each region of the experimental arena (a) and average amount of time spent in each region with a particular orientation (b). For phase space map, orange coloring represents time spent moving either up or down, and blue represents moving either left or right.

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One of the most classic objects of interest in trajectory analysis is the mean squared displacement (MSD), which is often used to identify whether a process is sub- or super-diffusive. Applied to the (uncropped) ant trajectories, this metric identifies three distinct regimes of motion, shown in Fig 4. At short time scales (blue shaded region), the ant moves ballistically, such that the displacement scales linearly with time, , and the trajectory consists primarily of straight segments. At longer time scales (orange shaded region), we see roughly Brownian scaling of the MSD, , indicating that the ant typically changes its orientation after the short time scale ballistic motion. Finally (gray shaded region), the finite size of the arena limits the distance the ant can displace from its previous positions, giving an MSD that no longer scales with time.

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Fig 4. Mean squared displacement (MSD) for (uncropped) ant motion.

Blue, orange, and grey shading indicates regions where the displacement scales linearly with time (ballistic motion), with the square root of time (Brownian motion), and independently of time (finite-size effects). Regions are drawn as rough guides, not based on specific quantifications. Faint curves are the MSD for each individual trial; solid black line represents the pooled statistics for all trials, with standard deviations shown as error bars.

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The location of the transition point between the ballistic and Brownian regimes is a (rough) quantification of the persistence time of the ant motion, i.e., the typical time after which the ant reorients herself. As we will see, indeed the typical persistence time computed from both the direct and discrete methods below is near the transition point, s. This transition point (and thus the persistence time) varies from experiment to experiment, which can be seen by looking at the MSD curves for each individual trial (faint curves in Fig 4). This clear separation in straight running behavior and reorientation behavior also provides quantitative evidence that indeed this process is well-described by a run-and-tumble model, justifying the application of the discretization procedure.

Discretized statistics

We now apply the previously-described discretization procedure to the trajectory data, allowing us to better understand the contribution of different elements to the persistence. As previously discussed, the choice of parameters for this discretization process, and , is important. That being said, we find that nearly all reasonable values of these parameters lead to the same trends in our data, with only minor quantitative differences (see S3 Fig); as such, we show each result for only a single, representative, choice of discretization parameters, though we mention the range of typical values resulting from other parametrizations.

Shown in Fig 5 are the turn angle, run time, and wait time distributions for the discretized trajectories. We find that the ant shows very stereotyped behavior around the edges and corners of the arena, leading to a “crown”-shaped turn angle distribution (Fig 5, left, blue squares) with peaks around 0°, °, and °. When we consider only the central part of the arena, this turn angle distribution reduces to a unimodal distribution (Fig 5, left, green circles and dashed line). We fit this latter distribution with a zero-centered von Mises functional form, giving a width parameter in the range [0.6, 1.3] ( for shown parameterization). Fixing the center of the distribution at zero assumes that the turn angle distribution is symmetric; this seems to be well justified, as allowing the center to vary as a fit parameter gives similar results, though with more noise.

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Fig 5. Turn angle, run time, and wait time distributions for discretized ant motion.

Probability distribution functions (PDFs) are created using pooled data from all ant trials. A single, representative choice of discretization parameters (°, mm/s) is shown here; for comparison with other parameter choices, see S3 Fig. Green dashed lines represent the median fit — von Mises, exponential, and power law, respectively — across the individual ant trials (in contrast to the pooled data shown in the curves with markers). Dashed vertical lines indicate the minimum value including in fitting.

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The run time distribution has an exponential form for both the uncropped and cropped data, with decay constant (average run time) in the range [1, 2.1] seconds (s for shown parametrization). The wait time distribution has a power law form for both the uncropped and cropped data with an exponent in the range [1.5, 1.7] ( for shown parametrization).

We estimate the value of the diffusion coefficient by fitting the persistence time within each discretized run, as discussed previously. This, together with the values for and , allows us to calculate the predicted persistence length of the process using Eq 8. This is shown for each individual ant trial in Fig 6a, labeled as “discrete fit”. We can compare this to the persistence time directly calculated from the trajectories using Eq 9, also shown in Fig 6a (“direct fit”). Both methods give similar distributions, with median values of s for the discrete fit and s for the direct fit.

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Fig 6. Persistence time and velocity for individual ants.

“Direct fit” distributions represent calculations of the persistence time (a) and velocity (b) without any model assumptions, while “discrete fit” distributions use an analytic expression derived assuming the proposed model. Both methods of calculating the velocity use the cropped trajectories (1.5 cm from the edge of the arena).

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We do a similar comparison for the typical velocity, computing this value either through expressions derived from the model (from Eq 10, “discrete fit”) or by taking the derivative of the trajectory (“direct fit”), shown in Fig 6b. As with the persistence, these two independent techniques gives very similar results, demonstrating the applicability of the proposed model.

Beyond guiding our analysis of the experimental trajectory data, one of the principle advantages of proposing a mathematical model of motion is that it allows us to simulate trajectories, as shown in Fig 7a. These simulated trajectories closely reproduce the statistics of the experimental data. For example, even though the simulation is derived from statistics in the center of the arena, the relatively simple boundary conditions implemented in the model qualitatively reproduce the multi-modal turn angle distribution throughout the entire arena (Fig 7b).

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Fig 7. Simulated ant trajectories and their statistics.

A simulated trajectory using the Langevin equation (Eqs 4, 5) with parameter values based on fits from experimental ant trajectories is compared to a segment from an experimental trajectory (a). The turn angle distribution for this simulated data qualitatively recreates the crown-shaped distribution from experimental data (b). The MSD of an ensemble of simulated trajectories (c) gives the same behavior as the experimental data (compare to Fig 4), with the added benefit of being able to explore behavior in an unbounded space.

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The simulations also allow us to explore how the behavior of these ants may change in different geometries. As an example, we show the mean squared displacement of simulated ants in the exact experimental geometry compared with an unbounded domain in Fig 7c. The fact that the finite size effects appear around the same time scale as the Brownian behavior might explain why the Brownian regime for the experimental trajectories (see Fig 4) is so limited.