Conceived and designed the experiments: AP SG GT VF. Performed the experiments: AP SG ML. Analyzed the data: AP SCN DJTS. Contributed reagents/materials/analysis tools: BG SCN. Wrote the paper: AP DJTS.
The authors have declared that no competing interests exist.
We studied the formation of trail patterns by Argentine ants exploring an empty arena. Using a novel imaging and analysis technique we estimated pheromone concentrations at all spatial positions in the experimental arena and at different times. Then we derived the response function of individual ants to pheromone concentrations by looking at correlations between concentrations and changes in speed or direction of the ants. Ants were found to turn in response to local pheromone concentrations, while their speed was largely unaffected by these concentrations. Ants did not integrate pheromone concentrations over time, with the concentration of pheromone in a 1 cm radius in front of the ant determining the turning angle. The response to pheromone was found to follow a Weber's Law, such that the difference between quantities of pheromone on the two sides of the ant divided by their sum determines the magnitude of the turning angle. This proportional response is in apparent contradiction with the well-established non-linear choice function used in the literature to model the results of binary bridge experiments in ant colonies (Deneubourg et al. 1990). However, agent based simulations implementing the Weber's Law response function led to the formation of trails and reproduced results reported in the literature. We show analytically that a sigmoidal response, analogous to that in the classical Deneubourg model for collective decision making, can be derived from the individual Weber-type response to pheromone concentrations that we have established in our experiments when directional noise around the preferred direction of movement of the ants is assumed.
Many ant species produce large dendritic networks of trails around their nest. These networks result from self-organized feedback mechanisms: ants leave small amounts of a chemical -a pheromone- as they move across space. In turn, they are attracted by this same pheromone so that eventually a trail is formed. In our study, we introduce a new image analysis technique to estimate the concentrations of pheromone directly on the trails. In this way, we can characterise the ingredients of the feedback loop that ultimately leads to the formation of trails. We show that the response to pheromone concentrations is linear: an ant will turn to the left with frequency proportional to the difference between the pheromone concentrations on its left and right sides. Such a linear individual response was rejected by previous literature, as it would be incompatible with the results of a large number of experiments: trails can only be reinforced if the ants have a disproportionally higher probability to select the trail with higher pheromone concentration. However, we show that the required non-linearity does not reside in the perceptual response of the ants, but in the noise associated with their movement.
Many ant species produce large dendritic patterns of trails around their nests (
Each picture is obtained by summing all the ants detected from arena-level snapshots during 5 minutes (300 snapshots). The contrast and gamma are adjusted to make single ants visible in the images.
In spite of having coherent and efficient organization on a large scale, ant trail formation can be explained as the result of a completely self-organized process. Simulations supported by experiments have shown that the trails are the result of an autocatalytic process: ants move in response to local concentrations of pheromone and in turn change these same concentrations by laying new pheromone where they go
The link between models and experiment is weak in one important respect: the exact nature of how individual ants move on the trail and respond to pheromone remains unknown. Important work in the direction of answering this question was done by Jean Louis Deneubourg and collaborators
Equation 1 (or variants of it) accounts well for the results of experiments in which ants face a branching point in their trail, usually in the form of a double bridge. For example, Deneubourg et al.
Double bridge experiments do not however capture individual ant behaviour and are instead fitted directly to the global outcome. Many different mechanisms in which individual preferences are amplified by positive feedback can all explain the selection of one single branch (see e.g.
How insects perceive and react to environmental stimuli has been studied in many contexts other than pheromone trails. For example, various studies have established that perceptual errors are proportional to the magnitude of the stimuli
Studying individual responses to the trail pheromone is non-trivial since pheromone density is not readily visible. Two possible approaches are to test the ants' response to synthetic pheromone
In our study we use novel imaging and analysis techniques to overcome these limitations and study trail following behaviour over a large number of tracking events. We assume that the concentration of pheromone at a particular position in the arena at a particular time is proportional to the number of ants that have previously passed that position. We use various assumptions about evaporation to test the robustness of our results. In this way, we can derive the response function of individual ants to pheromone concentrations directly on the trails produced by the ants themselves.
The ants explore the arena uniformly in all directions around the entrance in the beginning of the experiment, but soon start to form trails that persist for some time. Later in the experiment these trails are either abandoned or amplified (see
A. Number of ants in the arena over time. B. Number of ants along the arena border (i.e. less than 2.5 cm from the border) over time. For each plot the curve gives the mean and standard deviation over all trials.
A first characterisation of individual ant motion is provided by the measure of ant speed (
The graph in A is from all data, while the one in B is limited to “moving” ants (ants that move at least 0.4 cm in each of the two time intervals; see
We tested whether speed depends on pheromone concentrations experienced by the ants.
For each tracking event we get the position of the ant at time
Let us consider the angle changes made by ants in response to pheromone concentrations. For each level of total pheromone
Each graph is for a different range of values of total pheromone
Error bars associated with each data point are
Trial I.D. |
|
|
|
|
|
T01 |
|
|
|
|
|
T02 |
|
|
|
|
|
T03 |
|
|
|
|
|
T04 |
|
|
|
|
|
T05 |
|
|
|
|
|
T06 |
|
|
|
|
|
T07 |
|
|
|
|
|
T08 |
|
|
|
|
|
T09 |
|
|
|
|
|
T10 |
|
|
|
|
|
T11 |
|
|
|
|
|
T12 |
|
|
|
|
|
The table reports for each trial the values of a power law fit of the type
Trial I.D. |
|
|
|
|
|
T01 |
|
|
|
|
|
T02 |
|
|
|
|
|
T03 |
|
|
|
|
|
T04 |
|
|
|
|
|
T05 |
|
|
|
|
|
T06 |
|
|
|
|
|
T07 |
|
|
|
|
|
T08 |
|
|
|
|
|
T09 |
|
|
|
|
|
T10 |
|
|
|
|
|
T11 |
|
|
|
|
|
T12 |
|
|
|
|
|
The same as
In order to visualize how the slope
It is important to notice that this relation does not depend on the specific scale used to measure pheromone: the same relation holds if we multiply both
We also made the assumption that pheromone marking is reasonably constant in time. This implies that the values of
In our analyses, we assumed that ants respond to pheromone in front of them and up to a distance of one centimetre. We still have no real knowledge of what regions around their position ants actually use to detect pheromone. This raises several questions. For instance, we would like to know the perception radius over which ants can respond to pheromone, or at which points in front of the ant pheromone concentrations have the greatest effect on movement decisions. We would also like to check whether locations behind the ant are important. If this were the case, it would imply integration of pheromone concentrations over some time before deciding to change direction.
In order to explore these issues we recomputed the predicted turning angle, but this time instead of integrating over the entire regions
The colour associated with each point (
Negative correlations behind the ant are small, and can probably result from sensory adaptation at the level of the antennae: the left and right antenna would underestimate the amount of pheromone if they have just been exposed to high pheromone concentrations. The correlation becomes positive around the position of the ant, indicating a small temporal delay between perception and response (absence of integration). Given the measured ant speed of about 2 cm/s, if ants required a processing time of, for example, 0.5 seconds from the moment when they sense the pheromone until the moment when they change direction, then the correlation should become positive already
The response rules of individual ants to pheromone concentrations established here are, at first sight, different from the disproportional response (equation 1) found in the literature for large-scale binary choice experiments. To explain this discrepancy we ran an agent based simulation in a binary bridge setup where each agent (each ant) responds to pheromone according to the same rules identified from the experiments (see
To see why a proportional response inherent in Weber's law produces a disproportional outcome, consider the case of one ant approaching the branching point in which there is slightly more pheromone on the right than on the left and the total pheromone concentration is low (i.e.
When the simulation is run with the same parameters in a circular open arena, it leads to the formation of distinct trails (
Each image is obtained by summing 300 snapshots of the simulation taken at equal intervals of 1 second of simulation time (corresponding to 5 minutes of simulation) in a similar way to what had been done for the experimental data.
Weber's Law has previously been established in a wide range of animals and for different sensory stimuli
We have shown that the use of Weber's Law also explains the formation of ant trail networks. In the context of social interactions, the perceptual response is coupled with positive feedback to generate collective patterns. In our case, positive feedback is mediated through leaving pheromone, and the collective pattern is the trail network. We can imagine that other collective phenomena, such as group decision-making, could also be founded on coupling between Weber's Law and simple feedback mechanisms.
In contexts where groups of animals are faced with choosing between multiple options, such as the shortest path to food or the best direction to move
While our observations are consistent with double bridge experiments, we do not have a complete explanation for the formation of trail networks. Comparing
Antennal contacts can also help an ant head toward the nest, or away from it, by sensing whether or not colony mates travelling on the same trail have recently come out of the nest. Another aspect that is not reproduced in our model but is often seen in ant trails is branching. Surprisingly, however, trail branching is not particularly prevalent in arena-level observations. The branching that does occur maybe only be explainable in terms of trails “colliding” instead of an active formation of bifurcations. We can speculate that crowding and ant interactions on the trail are the main factors that determine branching
To answer these remaining questions more detailed arena-level observations of trail formation are needed, combined with detailed observations of how the ants interact with each other on the trails. We believe that the computer-automated approach we have used here can be further refined to produce such analyses.
We used colonies of the Argentine ant
Argentine ants leave pheromone both when moving out of the nest in search for food and when going back to the nest in food recruitment
Under these conditions, Argentine ants start exploring the arena homogeneously around the entrance, but soon end up forming trails, some of which are then abandoned after some time and some of which are reinforced (
Two sets of data were collected for each trial. Snapshots of the whole arena were collected every 1 s with a digital photo camera (Canon EOS 20D) and stored as 3504×2336 pixels RGB colour images. Image quality was sufficient to clearly see all the ants in the arena. These images were used for all arena-level observations of trail formation. At the same time, a smaller portion of the arena
The positions of all the ants were detected from each frame (and from each camera snapshot) with standard image analysis techniques (see e.g.
Pheromone levels cannot be measured directly in our experimental setup. Instead, as a proxy for the concentration of “pheromone” at each particular point in the arena we use the number of passages of ants over that point. Here we used only data from the individual-level film camera. An empty “pheromone map”, corresponding to the field of view of the camcorder was initialized at the beginning of each trial. Then, at each frame, the “pheromone” of all the pixels that were covered by one ant was incremented by a fixed amount
In order to characterise the movement of ants we automatically tracked all the ants in the individual level videos over short time periods. This involves about 50,000 tracking events for each experimental trial, for a total of
We looked at the total pheromone within a one centimetre radius of the ant. In particular, we define L to be the total pheromone in a 90 degrees front-left sector relative to the ants position (see
In order to test the ability of the individual response rules to the pheromone concentration to explain trail formation at a larger scale, we set up an agent-based model of Argentine ant arena exploration using the NetLogo 4.1.1 modelling environment. The NetLogo world consists of square patches each with its own pheromone concentration. We run the simulation both in an open arena setup (diameter = 1000 patches) and in a “binary bridge” setup (total length of the setup 440 patches, all other proportions as in
1000 ants are initialized inside the nest at the beginning of the simulation; each ant enters the arena with a probability of 1/1000 per time step. Once in the arena, ants move at a constant speed (2 patches per time step, equal to the average speed of ants measured in the experiment). Their movement is not bound to the dimensions of the patches (ants move off-lattice), while pheromone concentrations are updated at the grain size of the grid, as pheromone is a property of the patches themselves. The ants move at every time step and update their direction of movement every four time steps (equivalent to 0.4 s, the same interval used when analysing the data). The new direction of movement is determined by the concentration of pheromone within two circular sectors oriented 45 degrees to the left
All the simulation parameters are chosen to match as closely as possible the experimental measures on real ant behaviour. We do not have experimental data to characterise the behaviour of ants along the arena border. In the simulation, ants heading against the border align with it, pointing to the direction that involves the minimum change from previous direction. In the binary bridge setup, ants heading against the border align with the border pointing away from the latest visited site (either nest or food source).
Ants that have been out in the arena for a long period can be marked with a special “back-to-nest” label. If such a labelled ant happens to be within two centimetres of the nest, then it will set its heading towards the nest and go directly there. In the open arena simulation, the ants are given a back-to-nest label randomly, with probability of 1 every 10 minutes of simulation time; in the binary bridge setup, the label is given to all the ants that have reached the food source at the distal extremity of the bridge (the dark blue region in
The simulations implement a very simplistic model of ant behaviour: ants respond only to local concentrations of pheromone, with no memory of past position and direction of movement. There are no direct ant-ant interactions. As such their purpose is to test to what degree the observed pheromone trail patterns are explainable simply in terms of their reaction to pheromone concentrations using equation 10.
In the introduction, we stated that equation 2 is similar to equation 1 with
In the derivation of equation 8 for the double bridge experiment the question is not about expected position, but rather the probability of ants arriving at either side of a branching point on the bridge. Consider ants which move up a single bridge toward a branching point. We can model the time evolution of the probability density
The initial condition is a delta function at zero,
Let
At the stationary state, putting
(MOV)
(MOV)
Simon Garnier is grateful to Iain Couzin for fruitful and inspiring discussion.