PRINCIPLES OF TRAFFIC ORGANIZATION IN ANT TRANSPORTATION
NETWORKS
by
JUSTIN KITTELL
A DISSERTATION
Presented to the Department of Physics
and the Division of Graduate Studies of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
September 2023
DISSERTATION APPROVAL PAGE
Student: Justin Kittell
Title: Principles of Traffic Organization in Ant Transportation Networks
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Physics by:
Ben McMorran Chair
Robert Schofield Advisor
Tristan Ursell Core Member
John Toner Core Member
Scott Hansen Institutional Representative
and
Krista Chronister Vice Provost of Graduate Studies
Original approval signatures are on file with the University of Oregon Division of
Graduate Studies.
Degree awarded September 2023
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© 2023 Justin Kittell
This work is licensed under a Creative Commons Attribution (United States)
License.
3
DISSERTATION ABSTRACT
Justin Kittell
Doctor of Philosophy
Department of Physics
September 2023
Title: Principles of Traffic Organization in Ant Transportation Networks
Collectively, a colony of ants can execute complex and highly organized
behaviors, not least of which is the formation of ant ‘paths’ – the steady bidirectional
flow of individuals and resources that provides the colony with nutrients. This
bidirectionality necessitates the organization of opposing fluxes, with the choice of
organizational scheme impacting the energetic efficiency of the colony. In this work,
we perform an experimental investigation into the organizational principles employed
by the leaf-cutting ant Atta cephalotes under varying levels of lateral confinement.
We first extract the statistical properties of the unconfined path via automated
imaging and analysis. This characterization is a critical first step in understanding
the steady state organization resulting from ant behavior alone. We then explore
how the behavior and resulting path properties evolve for the same path under
different levels of confinement. This analysis reveals direct quantitative evidence
of a three-lane structures, as well as simple examples of energetic optimization at
critical widths. Finally, we verify the origin of these structures through simulation of a
’null’ model for insect behavior, revealing that the organization demonstrated by Atta
cephalotes foragers under confinement results from changes in individual behavior, not
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solely from ant-wall or ant-ant interaction. This analysis provides a framework for
understanding the behavioral trends of a natural transportation system in terms of
energetic optimization, with potential impacts on the development of autonomous
networks in human engineered systems.
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CURRICULUM VITAE
NAME OF AUTHOR: Justin Kittell
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, Oregon, USA
Indiana University, Bloomington, IN, USA
DEGREES AWARDED:
Doctor of Philosophy in Physics, 2023, University of Oregon
Bachelor of Science in Physics, 2017, Indiana University
AREAS OF SPECIAL INTEREST:
Organizational biophysics, energetics, multidimensional spectroscopy
PROFESSIONAL EXPERIENCE:
Graduate Research Assistant, University of Oregon, 2018-2020, 2023
Graduate Teaching Assistant, University of Oregon, 2017-2018, 2020-2023
Physics Demo Room Facilitator, University of Oregon, 2022-2023
GRANTS, AWARDS AND HONORS:
Weiser Graduate Leadership Award, University of Oregon, 2023
PUBLICATIONS:
Heussman, D., Kittell, J., von Hippel, P. H., & Marcus, A. H.
(2022). Temperature-dependent local conformations and conformational
distributions of cyanine dimer labeled single-stranded–double-stranded
DNA junctions by 2D fluorescence spectroscopy. The Journal of Chemical
Physics, 156(4).
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Heussman, D., Kittell, J., Kringle, L., Tamimi, A., von Hippel, P. H., & Marcus,
A. H. (2019). Measuring local conformations and conformational disorder of (Cy3)
2 dimer labeled DNA fork junctions using absorbance, circular dichroism and two-
dimensional fluorescence spectroscopy. Faraday discussions, 216, 211-235.
7
ACKNOWLEDGEMENTS
I would like to thank my advisors, Tristan Ursell and Robert Schofield for their
support and guidance through this project. I would also like to acknowledge my
committee for their continued engagement in my research, as well as the physics
department at the University of Oregon for providing a space to explore new and
interesting scientific ideas.
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To my parents, for teaching me to question.
To my friends, for leading the way.
To the ants, for everything.
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TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 18
1.1. The Coordinated Motion of Ants . . . . . . . . . . . . . . . 18
1.2. Energetic Optimization in a Natural System . . . . . . . . . . 20
1.3. Models for Coordinated Motion and Extensions to Bidirectional
Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 27
II. STATISTICS OF STEADY-STATE PATH STRUCTURE IN THE
UNCONFINED LIMIT . . . . . . . . . . . . . . . . . . . . . . 30
2.1. The ’Unconfined’ Path . . . . . . . . . . . . . . . . . . . 30
2.2. The Natural Path Width . . . . . . . . . . . . . . . . . . 32
2.3. Lab Frame Directionality . . . . . . . . . . . . . . . . . . 35
2.4. The Fundamental Laning Diagram (FLD) and the Lateral Dependence
of the Flux Correlate (FC) . . . . . . . . . . . . . . . . . 35
2.5. Ant Frame Polar Density and Polar T-T Correlation . . . . . . 38
2.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 43
III. THE DYNAMICS OF PATH STRUCTURE AND ORGANIZATION UNDER
CYCLIC CONFINEMENT . . . . . . . . . . . . . . . . . . . . 44
3.1. Moderate Confinement . . . . . . . . . . . . . . . . . . . 44
3.1.1. Structure in the Lab Frame . . . . . . . . . . . . . . . 44
3.1.2. Structure in the Ant Frame . . . . . . . . . . . . . . . 48
3.2. Extreme Confinement . . . . . . . . . . . . . . . . . . . 54
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Chapter Page
3.2.1. Structure in the Lab Frame . . . . . . . . . . . . . . . 54
3.2.2. Structure in the Ant Frame . . . . . . . . . . . . . . . 57
3.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 58
IV. COMPARING EXPERIMENTAL OBSERVATIONS TO THE ’NULL’ MODEL
OF ATTA CEPHALOTES TRANSPORTATION . . . . . . . . . . . 63
4.1. The ’Null’ Model of A.ceph Behavior . . . . . . . . . . . . . 63
4.2. Null Statistics in the Unconfined Limit . . . . . . . . . . . . 64
4.3. Null Statistics under Moderate and Extreme Confinement . . . . 66
4.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 68
V. MATERIALS AND METHODS . . . . . . . . . . . . . . . . . . 73
5.1. Ant Colony Maintenance . . . . . . . . . . . . . . . . . . 73
5.2. Data Collection and Analysis . . . . . . . . . . . . . . . . 75
VI. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . 78
APPENDIX: APPENDIX A . . . . . . . . . . . . . . . . . . . . . . 82
A.1. Ant Number Statistics and Total Fluxes . . . . . . . . . . . 82
A.1.1. The Unconfined Limit . . . . . . . . . . . . . . . . . 82
A.1.2. Moderate and Extreme Confinement . . . . . . . . . . . 83
A.2. Flux Correlate Under Moderate Confinement: All Trials . . . . 84
A.3. Neural Net Error Rates . . . . . . . . . . . . . . . . . . . 86
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . 89
11
LIST OF FIGURES
Figure Page
1. Examples of coordinated motion in (A) a sheep herd, (B) a school of fish, (C) a
flock of birds, and (D) an ant colony. . . . . . . . . . . . . . . . . 18
2. Atta cephalotes foragers follow and reinforce stigmergic pheromone signals while
transporting organic material, producing a laterally confined band of high ant
density along the food-nest axis. . . . . . . . . . . . . . . . . . . 19
3. Ant path length minimization. An unconfined ant transport network (A)
approximates a minimum length network (B) between three spatially separated
ant colony entrances (B) (Latty et al., 2011). (C) Argentinian ants find a
minimum length solution to a graph-theoretic representation of the Towers of
Hanoi puzzle (Reid et al., 2011) . . . . . . . . . . . . . . . . . . 20
4. Schematic examples of spatial and temporal organization in ant traffic
(Fourcassié et al., 2010). (A) Lane structures observed in the paths of army
ants and leaf cutting ants with unique spatial organizations represented by
each. (B) Observed yielding behavior for platooning leaf-cutters under lateral
confinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5. Comparing measurement in the lab frame and the ant frame (A) θi gives the
angle of the ith ant with respect to the lab coordinate system. A lab frame
orientation of 0 or pi is directly perpendicular to the path (B) The same ants
represented from the perspective of a single ant, showing the distances and
orientations (r and ϕ, respectively) to other ants in the same frame. . . 27
6. (A) Lab-frame representation of ant position and orientation, marked by Xs and
arrows respectively. Positions are given in units of ℓ, the average ant length,
where ants with a lateral position of x = 0 fall along the minimum length path.
Ants from separate video frames are differentiated by color. (B) Top-down view
of the sampling region (gray) in context, with relevant length scales shown.
Figure not to scale . . . . . . . . . . . . . . . . . . . . . . . . 30
7. Spatial and Lateral Ant Density in the Lab Frame (A) A spatial heat-map
showing average ant number per square ant length (ℓ−2) per frame. Horizontal
and vertical positions are given in units of ℓ (B) Lateral density profile, showing
the average linear ant density vs. lateral position in ant lengths. The blue curve
shows a normal distribution with a mean and standard deviation matching that
of the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
12
Figure Page
8. Linear ant density vs. lateral position. (A), (B), and (C) show results from
3 unique data collection periods. The blue curve in each shows a normal
distribution with mean and standard deviation matching that of the observed
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
9. Ant Direction in the Lab Frame (A) Net directionality as a function of lab frame
position (x, y) in units of ℓ, provided for the same 3 trials as above. Bin color
represents the average projection of the orientation vectors in each bin onto the
food-nest axis (B) Flux correlate, the product of net directionality and average
ant density in the lab frame. This emphasizes the regions of highest net flux. 34
10. (A) Fundamental Laning Diagram, showing the angular density distribution
as function of lateral position. W and L refer to the confining width and
vertical imaging extent, respectively (B) Linear ant density (green) and net
directionality (red) as function of lateral position. Their product, referred to
as the flux correlate, is shown in blue. Error bars (dashed lines) are estimated
from uncertainty in the net directionality and density, as well as the number of
ants in the corresponding bin. . . . . . . . . . . . . . . . . . . . 36
11. The lateral dependence of the flux correlate for the same three trials (A), (B),
and (C) as above. Low ant density and large error bars in the unconfined limit
prevent the direct observation of lane structures. . . . . . . . . . . . 38
12. Ant Frame Polar Density. Heat map showing ant density ( n2 ) as a function ofℓ
polar coordinates r and ϕ in the ant frame (see figure 5B). r is reported in ant
lengths, and ϕ in radians. Polar density observed by (A) all, (B) nest-bound,
and (C) food-bound ants . . . . . . . . . . . . . . . . . . . . . 39
13. Ant Frame Tangent-Tangent Correlation, showing average relative orientation
< cos(θi − θj) > as a function of polar coordinates r and ϕ in the ant frame
(see figure 5B). r is reported in ant lengths, and ϕ in radians, with the average
ant oriented directly up the page. TT Correlation observed by (A) all, (B)
nest-bound, and (C) food-bound ants . . . . . . . . . . . . . . . . 41
14. FLD as function of width, moderate confinement. (A) 5.0 cm, (B) 3.3 cm, (C)
2.5 cm, all during compression. (D) 2.5 cm, (E) 3.3 cm, (F) 5.0 cm, all during
expansion. The width and length of the confining region are provided for each
plot in units of average ant length. Red dashed lines enclose regions of ant-wall
exclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
15. Geometric modeling of ant-wall steric exclusion. In the lab frame, ants can
be represented as vectors with lengths and orientations. Certain orientations
and distances from the confining wall are excluded by steric interactions (red)
while others are not (green). Minimum allowed ant-wall distance depends on
the projection of any A.ceph vector onto the lateral axis. . . . . . . . 46
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Figure Page
16. Flux Correlate as function of width, moderate confinement. (A) 5.0 cm, (B)
3.3 cm, (C) 2.5 cm, all during compression. (D) 2.5 cm, (E) 3.3 cm, (F)
5.0 cm, all during expansion. Black dashed lines show error bars calculated
from the number of ants associated with each data point and the corresponding
uncertainty on density and average orientation. The flux correlate shows a 3 lane
pattern at all widths, with maximum flux shown at the highest confinement level
(C, D). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
17. Perspective-based Polar Density as a Function of Width, moderate confinement.
(A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during compression. (D) 2.5 cm, (E) 3.3
cm, (F) 5.0 cm, all during expansion. . . . . . . . . . . . . . . . . 49
18. Perspective-based Polar Density as a Function of Width, moderate confinement,
nest-bound. (A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during compression. (D)
2.5 cm, (E) 3.3 cm, (F) 5.0 cm, all during expansion. . . . . . . . . . 51
19. Perspective-based Polar Density as a Function of Width, moderate confinement,
food-bound. (A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during compression. (D)
2.5 cm, (E) 3.3 cm, (F) 5.0 cm, all during expansion. . . . . . . . . . 51
20. Perspective-based TT Correlation as a Function of Width, moderate
confinement, all ants. (A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during compression.
(D) 2.5 cm, (E) 3.3 cm, (F) 5.0 cm, all during expansion. . . . . . . . 52
21. Perspective-based TT Correlation as a Function of Width, moderate
confinement, food-bound. (A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during
compression. (D) 2.5 cm, (E) 3.3 cm, (F) 5.0 cm, all during expansion. . 53
22. Perspective-based TT Correlation as a Function of Width, moderate
confinement, nest-bound. (A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during
compression. (D) 2.5 cm, (E) 3.3 cm, (F) 5.0 cm, all during expansion. . 53
23. FLD as function of width, extreme confinement. (A) 2.5 cm, (B) 1.8 cm, (C)
1.0 cm, all during compression. (D) 1.0 cm, (E) 1.8 cm, (F) 2.5 cm, all during
expansion. The width and length of the confining region are provided for each
plot in units of average ant length. Red dashed lines enclose regions of ant-wall
exclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
24. FC as function of width: Extreme Confinement. (A) 2.5 cm, (B) 1.8 cm, (C)
1.0 cm, all during compression. (D) 1.0 cm, (E) 1.8 cm, (F) 2.5 cm, all during
expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
25. Perspective-based Polar Density as a Function of Width: Extreme Confinement,
all ants (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D) 1.0 cm,
(E) 1.8 cm, (F) 2.5 cm, all during expansion. . . . . . . . . . . . . . 57
14
Figure Page
26. Perspective-based Polar Density as a Function of Width: Extreme Confinement,
food-bound. (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D)
1.0 cm, (E) 1.8 cm, (F) 2.5 cm, all during expansion. . . . . . . . . . 59
27. Perspective-based Polar Density as a Function of Width: Extreme Confinement,
nest-bound. (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D)
1.0 cm, (E) 1.8 cm, (F) 2.5 cm, all during expansion. . . . . . . . . . 59
28. Perspective-based TT Correlation as a Function of Width, extreme confinement,
all ants. (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D) 1.0
cm, (E) 1.8 cm, (F) 2.5 cm, all during expansion. . . . . . . . . . . . 60
29. Perspective-based TT Correlation as a Function of Width, extreme confinement,
food-bound. (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D)
1.0 cm, (E) 1.8 cm, (F) 2.5 cm, all during expansion. . . . . . . . . . 61
30. Perspective-based TT Correlation as a Function of Width, extreme confinement,
nest-bound. (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D)
1.0 cm, (E) 1.8 cm, (F) 2.5 cm, all during expansion. . . . . . . . . . 61
31. Laning in the Null Model (A) FLD and FC for the A.ceph foragers in the
unconfined limit (B) FLD and FC from simulated ants under the same steric
confinements. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
32. Polar density in the null model (A) Plots of polar density in the ant frame for
the A.ceph foragers in the unconfined limit (B) Polar density of simulated ants
under the same steric confinement. Both (A) and (B) show all ants, food-bound
ants, and nest-bound ants from left to right. . . . . . . . . . . . . 65
33. Polar TT correlation in the null model (A) Plots of the TT Correlation in the
ant frame for the A.ceph foragers in the unconfined limit (B) FLD and FC from
simulated ants under the same steric confinement. Both (A) and (B) show all
ants, food-bound ants, and nest-bound ants from left to right. . . . . . 66
34. FLDs under the null model, moderate confinement. Comparisons shown between
data and simulation (left and right, respectively) for confining widths of (A) 5cm,
(B) 3.8 cm, and (C) 2.5 cm . . . . . . . . . . . . . . . . . . . . 69
35. Ant frame polar density under the null model, moderate confinement.
Comparisons shown between data and simulation (left and right, respectively)
for confining widths of (A) 5cm, (B) 3.8 cm, and (C) 2.5 cm . . . . . 70
36. Polar TT correlation under the null model, moderate confinement. Comparisons
shown between data and simulation (left and right, respectively) for confining
widths of (A) 5cm, (B) 3.8 cm, and (C) 2.5 cm . . . . . . . . . . . . 71
15
Figure Page
37. (A) FLD, (B) Polar density, and (C) TT correlation under maximum
confinement (1 cm). Comparisons shown between data and simulation (left
and right, respectively) . . . . . . . . . . . . . . . . . . . . . . 72
38. Block diagram of a typical colony layout and location of observation platform,
with relevant lengths provided. Diagram not to scale. . . . . . . . . . 73
39. Ant imaging (A) Initial bright-field illumination, visible light (B) Dark-field
illumination, IR light . . . . . . . . . . . . . . . . . . . . . . . 74
40. Ant identification and pose estimation via the SLEAP neural net. Yellow dots
indicate the positions of the head (bold), thorax, and gaster of each ant. . 76
41. Ant transportation. . . . . . . . . . . . . . . . . . . . . . . . . 81
A.42.Ant Number statistics in the unconfined limit. (A) Average ant number is given
both before and after filtering (see chapter 5). (B) Ant Flux per frame, showing
number of food-bound ants in orange, nest-bound ants in blue, frame-wise net
count in black, and net integrated in green. . . . . . . . . . . . . . 82
A.43.Representative Ant Fluxes Under Moderate Confinement (A) 5 cm (B) 2.5
cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A.44.Representative Ant Fluxes Under Extreme Confinement (A) 2.5 cm (B) 1.0
cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A.45.Flux Correlate at each confinement level: moderate confinement. Color
corresponds to lab frame directionality, with magenta being towards the nest,
and green being towards the food. . . . . . . . . . . . . . . . . . 85
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LIST OF TABLES
Table Page
A.1. Neural net error rates . . . . . . . . . . . . . . . . . . . . . . . 86
17
CHAPTER I
INTRODUCTION
1.1 The Coordinated Motion of Ants
Biological systems in the natural world
often consist of many semi-identical, polar,
self-propelled individuals interacting at close
range. This description can be applied
across many orders of magnitude, including
bacterial swarms, bird flocks, schools of fish,
and herds of wildebeest. Despite the vast
differences in scale, there is a tendency for
common group movement dynamics to emerge
from the instinctual decision making of the
many actors. In these examples, individuals
tend to align themselves locally with their
neighbors while moving forward, resulting in
large-scale average alignment and dramatic
density fluctuations across the group. This
frequently observed ‘coordinated motion’ (CM) Figure 1. Examples of coordinated
motion in (A) a sheep herd, (B) a
is a common emergent pattern in active matter school of fish, (C) a flock of birds, and
(D) an ant colony.
systems [1]. As the decision making of most
animals is guided by an evolved, instinctual ’rule-set’ for responding to local stimuli,
the prevalence of CM across disparate biological systems is an example of convergent
evolution; the globally reduced risk of predation these movement patterns provide
18
!t
Figure 2. Atta cephalotes foragers follow and reinforce stigmergic pheromone signals while
transporting organic material, producing a laterally confined band of high ant density along
the food-nest axis.
acts as an evolutionary impetus for species to adopt local behavioral strategies that
produce them [2].
Though commonly employed for predation avoidance, a unique class of
coordinated motion can also be observed in the transportation networks of foraging
ants, in which ant-ant interactions and individual decision making confer dramatic
global benefits to the colony. Ants, an ecologically dominant species in nearly every
terrestrial environment on earth, engage in cooperative behaviors both in and around
their nest, directed primarily by chemical signaling [3][4]. While not true for all
ants, typically when a foraging ant discovers food far from the colony, it deposits a
volatile recruitment pheromone along the ground while returning to the nest; this
chemical guides other foragers to the food source. On their return trip, the recruited
foragers can reinforce or modify the original path by depositing their own pheromone.
This chemotactic navigation strategy generates the characteristic ant ‘path’, shown
19
in figure 2, which is the dense, temporally persistent network of foragers frequently
observed outside an ant colony.
1.2 Energetic Optimization in a Natural System
While transporting materials under these crowded conditions, the global dynamic
evolution of the foraging network is guided entirely by the local interaction rules of
individual ants [5][6]. Without any global control, these networks are capable of
approximating minimum length paths through both free [7] and spatially confined
[8] regions in 2D, as shown in figure 3. As some ant species are known to be
energetically limited due to the high energetic cost of transporting and processing
materials [3], minimizing overall network length may improve the reproductive fitness
of the colony, selecting for instinctual rule-sets which minimize energetic expenditure
Figure 3. Ant path length minimization. An unconfined ant transport network (A)
approximates a minimum length network (B) between three spatially separated ant colony
entrances (B) (Latty et al., 2011). (C) Argentinian ants find a minimum length solution to
a graph-theoretic representation of the Towers of Hanoi puzzle (Reid et al., 2011)
20
while maintaining a level of path flexibility and exploration appropriate for the
changing conditions of the local environment. This emergent property has inspired
novel computer algorithms for minimizing network lengths, commonly referred to
as the traveling salesman problems [9]. Generally, these algorithms simulate the
movement of many ’ants’ through a defined network, while keeping track of recent
activity levels at each node. By updating the positions of each ant based on relative
’pheromone concentration’ in any direction and connecting nodes along paths of
peak activity, algorithms like these find approximate minimum length paths through
complex environments. This underscores the potential for other improvements in
human designed systems to come from the study of biological transport networks.
In addition to minimum length paths, the collective decision making of many
individual ants can produce other network level phenomena like context dependent
flux organization. When laden nest-bound foragers meet their unladen food-bound
peers on the path, the potential for head-to-head collisions threatens transportation
efficiency and necessitates structural organization of the local ant flux perpendicular
(lateral) to the direction of traffic flow. The resulting organized ant flux has been
shown to display both spatial and temporal organization in bidirectional flow, as
shown in figure 4 [10]. One type of spatial organization is ‘laning’, which arises when
incoming and outgoing groups occupy well-defined and separated positions across the
path. Biologists have described the observation of 3 lane traffic patterns in some
species of ant (Atta colombica, Eciton burchelli), a structure attributed to differences
in the yielding rules between incoming and outgoing ants, exemplified in figure 4A
[11]. This investigation aims to present a quantitative measure of the degree of laning
exhibited by the ant species Atta cephalotes (A.ceph) under varying levels of lateral
confinement.
21
When sufficiently confined, interactions between the walls and other ants
(referred to here as ’steric’ interactions) necessitate temporal organization, or
‘platooning’, the alternating occupation of a point on the path by groups of ants
heading in opposite directions. Like the three-laned structure above, the details of
platooning statistics such as platoon size and duration also depend on the rules of
interactions between ants, especially yielding, with examples shown in figure 4B.
Which organizational strategy emerges in the foraging column explicitly depends on
the confined width of the path [10], and while previous experiments have studied
Figure 4. Schematic examples of spatial and temporal organization in ant traffic (Fourcassié
et al., 2010). (A) Lane structures observed in the paths of army ants and leaf cutting ants
with unique spatial organizations represented by each. (B) Observed yielding behavior for
platooning leaf-cutters under lateral confinement.
22
ant traffic under different confining widths [12][13], no studies have yet explored how
the organization of flux dynamically evolves as the confining width is varied. As
with the path length minimization above, it is likely that the organizational strategy
which emerges in response to these external pressure minimizes some fraction of the
colonies energetic expenditure. By smoothly varying the level of steric confinement,
we expect to observe dynamic transitions in the organizational strategies employed by
the A.ceph – an organizational strategy phase transition driven by energetic necessity.
1.3 Models for Coordinated Motion and Extensions to Bidirectional
Traffic
To effectively extract energetically optimized rule-sets from the A.ceph foraging
column, physical models with adjustable behavioral parameters can be tuned to
match observations like those made in this study. For coordinated motion in general,
two broad classes of models have been employed to describe behaviors observed in
biological systems. The Vicsek model is an agent-based simulation that quantitatively
connects specific individual rule-sets with observable emergent properties. Under this
model, actors have a position vector r(t) and a scalar orientation θ (t) that evolve
according to
ri(t+ δt) = ri(t) + v∆t[cos(θi), sin(θi)]
23
and
θi(t+ δt) =< θj(t) >|ri−rj |<r +ηi(t)min
Here, v is the constant speed of individual actors, < θj(t) >|ri−r |<r is thej min
average orientation of all neighbors within a radius rmin of the focal actor, and ηi(t)
is an orientational noise term often equal to zero [1]. These equations encapsulate a
simple interaction rule: individuals attempt to align themselves with their neighbors
as they move forward. Simulations of this model reproduce the local orientational
alignment, density fluctuations, and global movement characteristic of coordinated
motion in biological systems. Further, the success of the Vicsek model highlights
the ability of simple rule-sets to produce coordinated motion. Increased complexity
can be built into agent-based models like this, a strategy employed in other works to
explore the glass-like dynamics of ant traffic jams [12]. Their discrete lattice based
model contained two groups of actors moving in opposite directions, and included a
variable ’interaction time’ Tint, specifiying the duration of head-to-head interactions.
Simulations of this model accurately reproduced the jamming statistics of fire-ants
for only certain values of Tint, those which matched the measured interaction time
for the same species. This result highlights how simulating different behaviors and
comparing the structural outputs to observational data can reveal which behavior
is best represented in the group. With data sets that include spatial and angular
distributions of individuals, expanding models like this to include differences in
yielding behavior may allow us to extract principles of efficient traffic organization
from organisms like Atta cephalotes.
The extraction of these principles from agent-based modeling requires long
periods of simulation to provide an effective comparison. As an alternative approach,
24
encoding behavior into a continuum description can provide direct insight into which
behavioral trends result in which global system output. An example of this can be
found in the continuum based description of flocking introduced by Toner and Tu in
1998. In this model, the properties of a generalized “flock” are described by a coarse-
grained scalar density field ρ(x, t) and vector velocity field v⃗(x, t), which evolve in
time according to the Toner-Tu equations
δ 2tv⃗ + λ1(v⃗ · ∇⃗)v⃗ + λ2(∇⃗ · v⃗)v⃗ + λ3∇⃗(|v⃗| )
= αv⃗ − β|v⃗|2v⃗ − ∇⃗ ∑P (ρ) +DB∇⃗(∇⃗ · v⃗) +DT∇
2v⃗ +D2(v⃗ · ∇⃗)2v⃗ + f⃗ ,
∞
∂ρ
P (ρ) = σn(ρ− ρ )n0 , +∇ · (v⃗ρ) = 0
∂t
n=1
The λn terms are the hydrodynamic derivatives of the velocity field, the α
and β terms ensure a non-zero velocity in the ordered phase, and the diffusive Dn
terms encompass the coupling between neighboring members of the flock. P is the
effective pressure of the system, defined in term of density relative to ρ0, and f⃗
is a noise term. This equation is analogous to the Navier-Stokes equation but for
active (i.e. self-propelled) hydrodynamic systems [1]. Remarkably, by considering the
allowed solutions to this equation, Toner and Tu were able to mathematically predict
the continuous breaking of rotational symmetry that occurs when active matter
systems exhibit global motion. By encapsulating local rule-set interactions into a
continuum description of matter, the spontaneous emergence of a global property can
be predicted, bypassing the simulation necessary in agent-based approaches.
In order to predict the long term behavior of a bidirectional flow in a
human engineered system, including any lateral flux organization, a hydrodynamic
25
description of counter-propagating density fields may allow us to anticipate the
emergence of these trends from encoded behavioral inputs. Surprisingly, flux
organization can spontaneously emerge in bidirectional flows consisting of simple
molecules, complex plasmas, and driven colloids. The emergence of these structures
in flows not guided by behavioral rule-sets hints at a more fundamental mechanism
for lane formation, which recent work has explored in the hydrodynamic limit [14].
This work shows that spatial inhomogeneities and gradients in counter-propagating
density fields can drive the temporal evolution of these opposing fluxes. This model
predicts the instability of homogeneous bidirectional flow, providing a mathematical
basis for the nucleation of lanes in systems with and without behavioral rule-sets.
These results are derived for symmetric flows, where the yielding characteristics of
counter propagating individuals are assumed to be identical. While effective for many
pedestrian and automotive models, as well as non-sentient systems like plasmas and
colloids, the emergence of the traffic patterns observed in the foraging columns of the
A.ceph are believed to be a direct consequence of differences in behavior between nest-
bound and food-bound ants [11]. While finding a mathematical basis for the steady
state flux distributions observed within the ant path would require an extension of
this hydrodynamic model, successfully encoding the behavioral characteristics found
in this study into an accurate hydrodynamic description would provide a framework
for extracting the rule-set parameters evolutionarily encoded in the individual ants.
Data-sets like the one utilized in this study may provide the comparison necessary
for extracting these behavioral parameters from the simulation of such hydrodynamic
models.
26
1.4 Outline
In this work, we provide quantitative evidence of spatial organization in the
foraging column of the leaf-cutting ant Atta cephalotes and explore how this lane
structure evolves under varying levels of external confinement. We first sample the
positions and orientations of foraging ants over many days in the unconfined limit;
this allows us to characterize both the natural width of the ant path and observe any
steady state lane structure through plots of average directionality and flux correlate in
the ’lab frame’, a coordinate system defined in figure 5A. We then explore how these
measures of organization respond to cyclic changes in the degree of steric confinement.
Results are presented for A.ceph foragers under moderate (2.5 – 5.0 cm) and heavy
(1.0 – 3.5 cm) confinement, allowing us to identify organizational transition points
under both compression and expansion.
Figure 5. Comparing measurement in the lab frame and the ant frame (A) θi gives the angle
of the ith ant with respect to the lab coordinate system. A lab frame orientation of 0 or pi
is directly perpendicular to the path (B) The same ants represented from the perspective
of a single ant, showing the distances and orientations (r and ϕ, respectively) to other ants
in the same frame.
27
In addition, we compute the statistics of ant-ant interactions from the reference
frame of each individual to explore the perspective-based density and alignment fields
around the ‘average’ individual in our system, with results presented in the ’ant
frame’, described in figure 5B. Similar analysis techniques have been employed to
study the statistics of collective motion and group alignment in herds of migrating
caribou [15], allowing researchers to identify unique social interaction rules for the
two species under study. By sorting the A.ceph into 2 groups based on lab frame
directionality (nest-oriented vs food-oriented) and calculating the average agent-
based density and tangent-tangent pair correlations maps for each, we show a width
dependent contrast between the social interaction rules of the two groups. By
connecting the organizational changes observed in the lab frame with corresponding
changes in these agent based maps, this analysis aims to provide a link between
average local information and global outcomes in the ant column.
Finally, we compare the statistical lane structures we observe in our system with
those produced by a ’null’ model to fully explore the degree of self organization
the foraging column achieves. Under this simple hypothesis, the positions and
orientations of individual ants are selected from uniform distributions, only restricted
by steric interactions with physical barriers and other ants. By matching experimental
parameters like ant number, confining width, and ant size between our simulated and
collected data, we observe how strongly the emergent organizational structure in
the ant column differs from the simplest ‘null’ prediction. This modeling process
aims to better differentiate the organizational transitions arising due to changes in
ant behavior from those created by physical changes in the apparatus, verifying the
origin of any structural transitions as behavioral, not apparatus-dependent.
28
Ants are but one example of a self-organizing biological system where the
rule-sets guiding individual behavior can lead to global benefits. The rule-sets
and behavioral principles that shape these transportation networks, and their
global properties, may contain useful design principles for human systems with
similar spatial or energetic constraints. Although the chemotactic mechanisms that
underlie ant behavior have been explored in a biological context, many salient
questions about the ants’ organization strategies remain poorly understood. This
includes the dynamic evolution of the ants’ flux distribution, which, as a first goal,
this project seeks to illuminate through a combination of environmental control,
quantitative imaging, and perspective-based analysis techniques. Revelation of these
organizational strategies and their emergence from individual rule-sets will provide
key insights into the structure of biological transport networks and may provide
a conceptual framework for the design of robust autonomous networks in human
transportation systems.
29
CHAPTER II
STATISTICS OF STEADY-STATE PATH STRUCTURE IN THE UNCONFINED
LIMIT
2.1 The ’Unconfined’ Path
Figure 6. (A) Lab-frame representation of ant position and orientation, marked by Xs and
arrows respectively. Positions are given in units of ℓ, the average ant length, where ants
with a lateral position of x = 0 fall along the minimum length path. Ants from separate
video frames are differentiated by color. (B) Top-down view of the sampling region (gray)
in context, with relevant length scales shown. Figure not to scale
The foraging column of the A.ceph is a dynamic system with network structure
and inter-ant properties which vary dramatically over both space and time. Figure
6A shows the positions and orientations of ants in three consecutive frames, collected
at 1 FPS, with positions given in units of the average ant length. These position
maps reveal dramatic changes in ant density and orientation on a frame-by-frame
basis. To explore the statistical properties of this foraging column in the steady
state, we must integrate the position and orientation information of individual ants
over many such frames. As the auto-chemotactic navigation strategy of the ants
30
results in a laterally localized band of high density like that exemplified in figure
7A, characterizing the width of this narrow spatial region will define a critical length
scale for the ant’s transportation network. In this investigation, the path ’width’
is taken to be a statistical quantity referring to the average lateral spread of ant
locations about their mean x position µx which likely corresponds with the location
of peak pheromone concentration. In order to quantify this local path property in
the steady state, we route the foraging column through a 2D region where the ant
path is ’unconfined’. This condition is satisfied when most ants remain far from steric
barriers, with the bulk of the ant traffic moving through free space. This prevents
steric interactions from impacting the lateral density profile, ensuring any observed
structure results from the decision making of the ants alone. Figure 6B shows a
schematic of the observation platform utilized in this study, with the sample region
indicated in gray. We define the confining width Wconf. as the minimum distance
between the steric barriers perpendicular to the food-nest axis. For Wconf. = 30cm,
the frequency of ant-wall steric interaction is extremely limited by the energetic cost of
the associated path length increase. Under these conditions, the navigation strategy
employed by the A.ceph foragers results in a mean ant position far from any confining
walls, allowing us to observe the unconfined path over long time scales. Exploring the
lateral distribution of ant positions under these conditions is necessary as the degree
of steric confinement we later impose with physical barriers can only be understood
in terms of the spatial distribution of ants across the natural, unconfined path. For
the trials presented below, each plot draws on frames collected consistently over 4
days at 1 FPS, averaging distributions from both high and low ant density. Data
collection parameters can be found in chapter 5, and a temporal breakdown of ant
flux can be found in appendix A.1.
31
Figure 7. Spatial and Lateral Ant Density in the Lab Frame (A) A spatial heat-map showing
average ant number per square ant length (ℓ−2) per frame. Horizontal and vertical positions
are given in units of ℓ (B) Lateral density profile, showing the average linear ant density vs.
lateral position in ant lengths. The blue curve shows a normal distribution with a mean
and standard deviation matching that of the data.
2.2 The Natural Path Width
In Figure 7A, we present the lab frame mean ant density of A.ceph foragers
for the sample region shown in figure 6B. A lateral position of x = 0 marks the
location of the straight-line path between the two entrances, with the x axis running
perpendicular to the presumed direction of traffic flow. This plot, referred to as a
’heat map’, shows the average ant density (N2 ) at a given position (x, y) inside theℓ
observation region, with higher densities represented by warmer colors. If we assume
any lateral structure we observe is independent of our sample location along the
ant path, we can collapse the 2-dimensional data represented by figure 7A along its
vertical axis, considering only the lateral (horizontal) position of each ant. Under this
assumption, we can produce a lateral density plot like the one shown in figure 7B.
This plot shows the average linear ant density (N ) observed at each lateral position
ℓ
x. We can calculate statistical properties of the resulting distribution to provide an
estimate of the ‘natural’ path width maintained by Atta cephalotes under laboratory
conditions. For the specific trial shown in figure 7, the lateral mean of the ants’
32
Figure 8. Linear ant density vs. lateral position. (A), (B), and (C) show results from 3
unique data collection periods. The blue curve in each shows a normal distribution with
mean and standard deviation matching that of the observed data.
x positions µx deviates from the shortest length path (x = 0) by 7.98 ant lengths,
with a standard deviation of σx = 4.89 ant lengths. We include a scaled Gaussian
distribution with the same mean and standard deviation to emphasize any deviation
from normalcy present in these lateral distributions. For this trial, the density profile
shows a pronounced negative skew.
For all trials, we find the mean standard deviation of ant position to be σx = 6.7±
2.2 ℓ, which implies that on average only 32% of ants stray further than 6.7± 2.2 ant
lengths from the path center. This sets an important length scale for our system, as for
all confining widthsWconf. >>> 2σ , we expect the density profile and organization to
be largely independent of width, while for Wconf. close to 2σx or below, we anticipate
that the organizational strategy employed by the ants will strongly depend on the
level of confinement. Over all trials, the average lateral deviation of the path center
(determined by µx) from the shortest, straight-line path at x = 0 is approximately
7.4 ± 1.4 ant lengths. While this lateral deviation is persistent over the timescales
explored in this study (96 hours) necessitating a small increase in the ant’s energy
expenditure compared to travel along the minimum-length path, the lateral deviation
is very small compared to the total distance between input ports, which for our system
is greater than 250 ℓ. Given the small change in total length caused by this lateral
33
Figure 9. Ant Direction in the Lab Frame (A) Net directionality as a function of lab frame
position (x, y) in units of ℓ, provided for the same 3 trials as above. Bin color represents the
average projection of the orientation vectors in each bin onto the food-nest axis (B) Flux
correlate, the product of net directionality and average ant density in the lab frame. This
emphasizes the regions of highest net flux.
deviation, it is unlikely that any minimization procedure which emerges from the
behavior of the A.ceph foragers would be sensitive to such a small deviation from
the shortest path. Surprisingly, the path established by the ants always maintained
a positive µx, with the ant path always drifting slightly to the right in the lab frame.
This reproducibility suggests such lateral drift may result from an innate behavioral
mechanism which further studies may illuminate.
34
2.3 Lab Frame Directionality
While the analysis so far has focused only on the lateral position of the
ants, including orientation information allows us to estimate the steady state flux
distribution of the unconfined foraging column. In our analysis, θ refers to the angle
of an ant in the lab frame as defined in figure 5A, with π > θ > −π and θ = π
2
corresponding to ants oriented away from the colony. In order to identify regions
in our observation window with a preferred direction of travel, we bin the A.ceph
in 2D based on their (x, y) positions and report the average projection of the ants’
orientation vector on the food-nest axis (< sin(θi) >) in each bin. Figure 9A shows
these plots of ’net directionality’ for the same three trials shown above, in which
we can identify regions which show a persistent tendency for food-bound or nest-
bound travel. Regions more likely to be occupied by nest-bound ants are magenta,
while regions more likely to be occupied by food-bound ants are green. In order to
emphasize the regions which contribute most to the total flux, we take the product of
the net-directionality with the spatial density, yielding a quantity referred to as the
’flux correlate’ (FC), shown in figure 9B. On these plots, temporally persistent lanes
appear as vertical bands of contiguous bright colors. While such regions appear in our
analysis, in the next section we will attempt to characterize the emergent organization
perpendicular to the direction of travel, which may have the largest impact on the
energetic efficiency of the colony.
.
2.4 The Fundamental Laning Diagram (FLD) and the Lateral Dependence
of the Flux Correlate (FC)
As described above, if we assume that over the vertical extent of our observation
region the distribution of orientations are constant, we can ignore spatial information
35
Figure 10. (A) Fundamental Laning Diagram, showing the angular density distribution as
function of lateral position. W and L refer to the confining width and vertical imaging
extent, respectively (B) Linear ant density (green) and net directionality (red) as function
of lateral position. Their product, referred to as the flux correlate, is shown in blue. Error
bars (dashed lines) are estimated from uncertainty in the net directionality and density, as
well as the number of ants in the corresponding bin.
36
along the longitudinal direction. To fully explore the relationship between lateral
position and lab-frame orientation, we can replace this vertical spatial axis with the
lab frame angle θ, allowing us to organize the ants in 2D based on their positions
and orientations (xi, θi). If we bin the ants found near each position and angle (x, θ),
we can measure the average density of ants per ant length per radian ( N ) in any
(rad)(ℓ)
particular frame. Figure 10A shows a heat map of this density field, with higher ant
density represented by warmer colors. We refer to this plot as the fundamental laning
diagram (FLD), as spatially and temporally persistent lanes which align with the
food-nest axis result in localized peaks in both lateral position and angular density.
Here, W refers to the lateral confining width, and L refers to the vertical extent
of the observation region. For the unconfined A.ceph column, the FLD shows two
regions of high ant density along the angular axis, one food-bound and one nest-
bound (π/2 and −π/2 in the lab frame, respectively). These two angular peaks are
nearly aligned along the lateral axis, emphasizing the bidirectional nature of the ants’
transportation network. In this unconfined limit, the FLD does not reveal any obvious
lane structures perpendicular to the direction of travel. This observation is reinforced
by the 1D analysis presented in figure 10B, which shows the lateral density in green,
lateral directionality in red, and corresponding flux correlate in blue. Error bars
(dashed lines) are estimated from uncertainty in the net directionality and density,
as well as the number of ants in the corresponding bin. Figure 11 shows the FCs
and associated error bars for the same unconfined trials presented above. Due to the
low density of A.ceph foragers and the high degree of mixed traffic in the unconfined
limit, the estimates of the FCs’ lateral dependence are statistically limited, with error
bars which largely encompass the vertical axis and obscure any direct evidence of a
37
Figure 11. The lateral dependence of the flux correlate for the same three trials (A), (B),
and (C) as above. Low ant density and large error bars in the unconfined limit prevent the
direct observation of lane structures.
preferred flux orientation, especially in figure 11C. While some trials have regions do
show preferred flux orientation, these structures are not consistent between trials.
2.5 Ant Frame Polar Density and Polar T-T Correlation
As previously described, the global structural output in an ant transportation
network is an emergent property resulting from many individuals making decisions
based on their unique local environments. Rather than analyze the dynamics of
interactions on an individual level, we will instead attempt to characterize the
perspective of the ‘average’ ant by calculating system properties like density and
relative alignment in the reference frame of every individual (see figure 5B) and then
averaging these results over all ants. We refer to these plots as ’perspective-based’ or
’ant-frame’, since the represent system parameters from the perspective of an average
ant. Figure 12A shows the average perspective-based ant probability density in the
unconfined foraging column averaged over all ants, and normalized over the circular
region shown. For an ant located at r = 0 and oriented directly up the page, this
plot shows the probability of finding a neighbor ant at each relative position and
orientation (r, ϕ), with warmer colors indicating higher probability densities. The
38
Figure 12. Ant Frame Polar Density. Heat map showing ant density ( n2 ) as a function ofℓ
polar coordinates r and ϕ in the ant frame (see figure 5B). r is reported in ant lengths, and
ϕ in radians. Polar density observed by (A) all, (B) nest-bound, and (C) food-bound ants
distribution reveals a peak neighbor density at a distance r between 1 and 3 ant
lengths over a wide range of relative angles. The decrease in relative density at larger
distances hints at the spatial inhomogeneity of ant density in the lab frame. If the
ants spread uniformly through the observation region, there would be no decrease
in ant density at higher distances from the average ant. Once again, the details of
the ants’ auto-chemotactic navigation result in specific signatures in our analysis. In
addition to the radial density peak, figure 12A also shows a depletion zone directly
to the rear of the average ant. This decreased ant density likely results from the
preference for forward motion exhibited by individual ants, as the region directly
39
behind an ant in motion can only be immediately occupied by another ant moving
in the same direction. In contrast, the region directly in front of any particular ant
is not limited by prior occupation, and can be occupied by a forward or backward
moving ant. This may result in the up-down asymmetry demonstrated in the polar
density plot for all ants shown in figure 12A.
While useful for making observations about all actors in a foraging column, ant
frame maps also allow us to explore how individual perspectives depend on their
direction of travel. In figures 12B and 12C, we show the average perspective of
ants who are food-bound and nest-bound respectively. As the three lane traffic
pattern sometimes observed in ant foraging columns likely arises due to asymmetries
in yielding between these groups [11], maps like these may provide insights into
differences in decision making between the two classes, as demonstrated in other
studies [15]. In the unconfined limit explored here, however, no obvious distinction
can be drawn between these two groups of foragers. As the FC for these trials
failed to reveal characteristics of a 3 lane structure, it is possible that some degree of
confinement may be necessary for laning signatures to emerge in our analysis, with
accompanying differences in these ant frame measurements.
In analogy with the lab frame measurements presented above, we can include
orientation information in our ant frame measurements to explore the frequency of
aligned and anti-aligned interactions as a function of distance and relative orientation
from the average ant. As the orientation of each ant is given by the direction of its
tangent vector, a good measure of relative alignment is the dot product between two
unit vectors for a pair of ants, also known as the tangent-tangent (TT) correlation,
equivalent to cos(θi − θj) for lab frame orientations θi and θj. This measure varies
from 1 to −1 for aligned and anti-aligned pairs, respectively. As with the density plots
40
Figure 13. Ant Frame Tangent-Tangent Correlation, showing average relative orientation
< cos(θi− θj) > as a function of polar coordinates r and ϕ in the ant frame (see figure 5B).
r is reported in ant lengths, and ϕ in radians, with the average ant oriented directly up the
page. TT Correlation observed by (A) all, (B) nest-bound, and (C) food-bound ants
above, we calculate this observable in the reference frame of each ant, measuring the
TT correlation between the central ant and every other individual in the frame at a
particular r and ϕ before merging these observations into a single reference frame.
We then spatially bin the TT values in r and ϕ and average them to explore the
alignment field around the ’average’ ant. We refer to the resulting plot as the polar
TT correlation, shown in figure 13. When considering all ants as in figure 13A, we
observe that individuals frequently experience alignment with neighbors behind them,
while they are more likely to meet an anti-aligned partner directly in front of them.
41
These features can be explained by the ant’s practice of ‘antenating’, or stopping to
touch antennas [16]. The process can take less than a second, but this is enough to bias
our sampled statistics in the fore region towards anti-alignment. Another feature of
interest is the prominent anti-alignment behind and to either side of the average ant.
Like water in front of a boat, antenating ants disengage and pass on either side while
maintaining their proximity, resulting in the observed anti-alignment profile referred
to as a ’wake’. For the trial shown, these anti-aligned regions are slightly asymmetric,
suggesting the ants in our system show a slight preference for passing opposing traffic
on the right. As with the density plots above, we can group individuals based on their
global orientation to explore the differences in alignment observed by food-bound and
nest-bound ants, as shown in figures 13B and 13C. While the wake described above
is present from the perspective of both classes, food-bound ants experience a higher
degree of anti-alignment in their immediate vicinity compared with nest-bound ants
in the same observation region. This observation is in agreement with the observed
tendency for unladen, food-bound ants to yield preferentially to nest-bound traffic
[11], resulting in a lower frequency of head-to-head collisions and a reduced wake for
laden ants.
In addition to these proximal differences, we can also observe a large scale right-
left asymmetry in the relative orientation of more distant ants. Figure 13A suggests
an overall preference for alignment on the right and anti-alignment on the left. This
trend can also be seen in the conditional plots in figure 13B and 13C as well. While
this asymmetry does not seem to result in any obvious organization in the lateral flux
distribution, it is possible that under sufficient confinement this statistical tendency
may result in the sudden emergence of an asymmetric lane structure, highlighting the
potential for dynamic phase transitions to occur in active matter systems.
42
2.6 Conclusion
By characterizing the statistical properties of this foraging column in the
unconfined limit, we measured the natural width of the ant path, as well as the
details of its lateral flux distribution in the steady state. While we can easily extract
measures of mean position and lateral spread from this data set, the low ant density
and minimal directional segregation across the lateral extent of the path limits the
amount of information we can extract about the columns organization of flux. Despite
this limitation, observations reported in the ant frame show organizational signatures
that we speculate result from nematic alignment, spatial localization, antenating,
and a preference for forward motion. This underscores the value of these perspective
based plots for studying interactions between individuals, as well as the behavioral
characteristics of the average ant, even when lab frame measurements do not show
strong evidence of spatial structure. With the results shown in this chapter, we are
now prepared to study the structure and dynamic evolution of the foraging column
under different levels of lateral confinement.
43
CHAPTER III
THE DYNAMICS OF PATH STRUCTURE AND ORGANIZATION UNDER
CYCLIC CONFINEMENT
We now explore how the ant transportation network responds to changes in
lateral confinement by restricting the A.ceph foragers to a 2D plane of variable width.
In order to explore extreme levels of confinement for long periods without severely
limiting the colonies caloric intake, we cyclically compress and expand the confining
barriers and collect data under 10 different conditions: 5 widths under expansion,
and 5 under compression to complete a single cycle. This cycle is repeated 5 times
with the resulting ant observations grouped by condition before analysis. We present
the results of this experimental procedure from two separate width ranges. In both
regimes, physical barriers produced regions in the observation window with an average
ant density of zero. To ease visual comparison between these different widths, we
normalize the lateral representation of the data based on confinement level and report
all positions in ant lengths.
3.1 Moderate Confinement
3.1.1 Structure in the Lab Frame
Our results in section 2.2 suggest that the unconfined path has a natural width
of Wpath ≈ 2σx ≈ 13.4 ant lengths, approximately 5.1 cm. Our first investigation into
the statistical properties of the confined path explore it’s properties under moderate
confinement, here defined as confining widths from approximately 1.0 Wconf. to 0.5
Wconf., a range of 5.0 to 2.5 cm. Figure 14 shows a sample of 6 width-normalized
FLDs averaged over 5 compression-expansion cycles under moderate width restriction.
These plots reveal that the foraging column experiences peak ant density when
maximally confined. As the foragers travel at approximately constant speeds and
44
Figure 14. FLD as function of width, moderate confinement. (A) 5.0 cm, (B) 3.3 cm, (C)
2.5 cm, all during compression. (D) 2.5 cm, (E) 3.3 cm, (F) 5.0 cm, all during expansion.
The width and length of the confining region are provided for each plot in units of average
ant length. Red dashed lines enclose regions of ant-wall exclusion.
must collectively maintain some baseline level of caloric input, the survival of the
colony depends on the A.ceph’s ability to maintain different densities under different
degrees of lateral restriction. At high density, the energetic cost of individual collisions
can be mitigated by improving the lateral organization of ant flux through the creation
of lanes. Examples of this increased organization can be found in Fig 14C and 14D,
where ant traffic centered around the food-bound direction π/2 nucleates into two
distinct spatial peaks, and traffic centered around the nest-bound direction −π/2
becomes more concentrated near the path center. This trend suggests that the Atta
cephalotes (A.ceph) engage in a statistical three lane traffic pattern under moderate
confinement, where the outer sides of the path are preferentially occupied by food-
bound ants, with a central lane dominated by nest-bound individuals.
45
At these widths, steric interactions between the ants and move-able barriers can
create exclusion zones whose properties depend on the positions and orientations of
the barriers, as well as the instantaneous confining width. Figure 15 shows a schematic
representation of this steric exclusion process, highlighting combinations of positions
and orientations which are allowed and disallowed due to ant-wall interaction in green
and red, respectively.
To verify that the organizational
changes observed in Fig 14 are not a
consequence of these exclusion zones,
and instead arise from behavioral shifts
in the ant population, we calculate
the minimum distance to the wall
the average ant can attain based on
their vector orientation (see chapter
4.1) and record these values. We
then plot this minimum displacement
Figure 15. Geometric modeling of ant-wall
vertically on the FLD as a function of steric exclusion. In the lab frame, ants can
be represented as vectors with lengths and
orientation. In figure 14, this minimum orientations. Certain orientations and distances
from the confining wall are excluded by steric
displacement is indicated for both walls
interactions (red) while others are not (green).
with red dashed lines. These curves Minimum allowed ant-wall distance depends on
the projection of any A.ceph vector onto the
allow us to enclose the geometrically lateral axis.
excluded regions on the FLD and
compare the predicted depletion with that in the observational data. Under moderate
confinement (2.5 – 5.0 cm) these marked regions closely align with boundary of the
ant density profile, especially for ants closely aligned with ±π . Conveniently, the
2
46
high degree of nematic alignment exhibited by the A.ceph in this system keeps most
individuals near these orientations, where the width of the excluded region goes to
0, having a minimal impact on the lateral density profile. For ants oriented far from
the food-nest axis, the depletion in the density profile becomes spatially asymmetric,
showing a larger exclusion zone than predicted by the static geometric theory for some
combinations of position of orientation. This additional depletion can be observed
on the left side or the right side of the path for lab frame orientations of θ = 0 or
±π respectively, resulting in an ‘s’ shaped density profile. This shape likely arises
from a combination of the ant’s preference for forward motion and their limited
turn radius [17], since some combinations of orientation and ant-wall distance would
require individuals to ‘back-up’ to the confining walls. Further theoretical modeling
is required to fully include this observed trend in our hypothesized exclusion zones,
and fitting these parameters in future investigations may provide insight into system
parameters like minimum turn radius and average ant velocity.
To best characterize the lateral flux distribution as a function of width, figure
16 shows the width normalized FC for the same 6 trials presented in figure 14. The
large uncertainty near the right and left edges of these plots result from the decrease
in ant density closest to the confining walls. In this representation, the three lane
traffic pattern is readily apparent at all confinement levels, with regions of flux which
are definitively positive or negative, even with the inclusion of estimated uncertainty.
This analysis also allows us to explore any differences in system behavior due to
direction of barrier motion, a property referred to as hysteresis. At this level of
confinement, neither the FLDs nor the FCs show any significant structural hysteresis
due to direction of barrier motion, although the overall ant flux is slightly lower
during expansion compared to compression. In contrast with the unconfined path,
47
Figure 16. Flux Correlate as function of width, moderate confinement. (A) 5.0 cm, (B)
3.3 cm, (C) 2.5 cm, all during compression. (D) 2.5 cm, (E) 3.3 cm, (F) 5.0 cm, all during
expansion. Black dashed lines show error bars calculated from the number of ants associated
with each data point and the corresponding uncertainty on density and average orientation.
The flux correlate shows a 3 lane pattern at all widths, with maximum flux shown at the
highest confinement level (C, D).
a foraging column under moderate confinement shows a dramatic increase in lateral
organization, as seen in the lab frame.
3.1.2 Structure in the Ant Frame
We now implement the perspective based analysis techniques presented in section
2.5 to explore how the perspective of the average ant depends on the instantaneous
confinement level. Figure 17 shows the polar density plots corresponding to the
same confining widths presented in figure 14. In contrast with the unconfined path
represented in section 2.5, here the radial distance to the peak density shows a strong
dependence on the relative angle; when looking to the left or right, the average ant
sees a dramatic drop-off in neighbor density compared to the vertical, aligned axis.
48
Figure 17. Perspective-based Polar Density as a Function of Width, moderate confinement.
(A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during compression. (D) 2.5 cm, (E) 3.3 cm, (F)
5.0 cm, all during expansion.
At the higher levels of confinement represented in fig 12C and 12D, this drop-off is
even more pronounced. This change in the density profile results from a combination
of steric confinement and the ant’s nematic alignment along the food nest axis. If
the ants were just as likely to be found oriented perpendicular to the path as along
it, then half of the sampled regions of steric depletion would fall in front of and
behind individual ants, resulting again in a radial density profile with no dependence
on relative direction. As emphasized in the FLDs presented in figure 14, the ants’
tendency to nematically align with the food-nest axis results in the observed depletion
zones.
While the wake observed in the unconfined limit is still present under this degree
of confinement, at the narrower widths represented by figures 17C and 17D the front-
49
back asymmetry described in 2.5 becomes less pronounced. This suggests that under
higher confinement levels, the statistical tendency for ants to engage in head on
collisions may be reduced by an increased degree of laning, balancing the ant density
found in front-of and behind the average ant, resulting in a higher degree of front-back
symmetry in plots of polar density.
Figures 18 and 19 present the ant-frame density plots for nest-bound and food-
bound ants respectively. Even at this confinement level, there is still no strong
distinction between the relative density observed by the two groups. These conditional
plots show similar degrees of width dependent asymmetry as that shown when
considering all ants. While not observed here, higher degrees of confinement may
cause orientation-dependent behavioral changes in the confined ants, resulting in
visible differences in this perspective based representation.
As first exemplified in the unconfined limit, we now include orientation
information in our perspective based to plots to study how the alignment field
around the average ant responds to changing confinement levels. The Polar TT
correlation introduced in section 2.5 is shown in figure 20 for the same trial and
confinement levels as above. These plots allow us to explore how the degree of relative
alignment experienced by the average ant in our system depends on the degree of steric
confinement.
In agreement with the TT Correlation for the unconfined foraging column (see
figure 13), at these widths the previously described front-back asymmetry and anti-
aligned wake are both present in the confined column over the full range of lateral
compression. These plots also reveal a slight right-left asymmetry, with the same
preference shown for passing oncoming traffic on the right. The persistence of this
asymmetry over such a wide range of width restriction suggests that at sufficient
50
Figure 18. Perspective-based Polar Density as a Function of Width, moderate confinement,
nest-bound. (A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during compression. (D) 2.5 cm, (E)
3.3 cm, (F) 5.0 cm, all during expansion.
Figure 19. Perspective-based Polar Density as a Function of Width, moderate confinement,
food-bound. (A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during compression. (D) 2.5 cm, (E)
3.3 cm, (F) 5.0 cm, all during expansion.
confinement, there will be a statistical preference for one organizational chirality,
resulting in a discernible 2 lane traffic pattern.
51
Figure 20. Perspective-based TT Correlation as a Function of Width, moderate
confinement, all ants. (A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during compression. (D)
2.5 cm, (E) 3.3 cm, (F) 5.0 cm, all during expansion.
In contrast with the polar density plots, the inclusion of orientation information
provides some evidence for hysteresis between compression and expansion in the ant
column, especially when the ants are grouped by their direction of travel. When
comparing the plots in figure 21B and 21E for all ants, there is a discernible depletion
in anti-alignment on the right side present under expansion, resulting in higher degree
of right-left asymmetry than that shown under compression. Of the two groupings
represented in figure 21 and 22, this trend is strongly mirrored by the food-bound
individuals, while the TT Correlation for nest-bound ants shows no strong changes
in right-left symmetry at any confinement level. This observation suggests that food
bound ants react more strongly to environmental changes. As the 3 lane traffic pattern
shown for Atta cephalotes likely results from an higher level of spatial reactivity in food
52
bound ants, the distinct asymmetry shown in these perspective-based plots reinforces
the potential for directionally dependent behavior among the A.ceph foragers.
Figure 21. Perspective-based TT Correlation as a Function of Width, moderate
confinement, food-bound. (A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during compression.
(D) 2.5 cm, (E) 3.3 cm, (F) 5.0 cm, all during expansion.
Figure 22. Perspective-based TT Correlation as a Function of Width, moderate
confinement, nest-bound. (A) 5.0 cm, (B) 3.3 cm, (C) 2.5 cm, all during compression.
(D) 2.5 cm, (E) 3.3 cm, (F) 5.0 cm, all during expansion.
53
3.2 Extreme Confinement
After quantifying the organizational characteristics of the path near its natural
width, we continue our investigation into the dynamic evolution of these structures
for widths much smaller than that imposed by the ants’ auto-chemotaxis. Here,
extreme confinement is defined as confining widths from approximately 0.7 Wpath to
0.2 Wpath, a range of 3.5 to 1.0 cm. As the average ant length in our system is
approximately ℓ = 3.8 mm, the maximum confinement level explored in experiment
is less than 3ℓ, and likely below the minimum width required to maintain a steady
3-lane traffic pattern. Such a narrow path will likely prompt significant changes
in the organizational structure when compared with the results under moderate
confinement.
3.2.1 Structure in the Lab Frame
To accommodate the vertical extent of the foragers leaf fragments, the move-
able walls separating the floor and ceiling of the of the observation region have a
height of 2.5 cm. To restrict foragers to the 2D plane below, we apply talc to these
vertical surfaces before each experiment, but over the long integration times ( 6
days) employed in these experiments, persistent attempts at climbing by the ants can
eventually remove the talc layer and provide alternative, out-of-plane paths. As our
current ant identification model can not discriminate between ants on the barriers
and ants on the observation platform, our presented results include the statistics of
these wall climbing individuals, especially at the minimum path width.
Figure 23 shows the FLD as a function of the provided lateral width restriction.
At these widths, a distinct aliasing pattern results from a mismatch in resolution
between the neural net’s localization and the histogram bin size. Under compression
and expansion, a dramatic change in the lateral distribution of both nest-bound and
54
Figure 23. FLD as function of width, extreme confinement. (A) 2.5 cm, (B) 1.8 cm, (C)
1.0 cm, all during compression. (D) 1.0 cm, (E) 1.8 cm, (F) 2.5 cm, all during expansion.
The width and length of the confining region are provided for each plot in units of average
ant length. Red dashed lines enclose regions of ant-wall exclusion.
food-bound ants can be observed between the widths of 5.1ℓ and 2.5ℓ. At a path
width of 2.5 ant lengths, food-bound ants are still likely to be found on either side of
the path, though with a slightly narrower spatial profile on the left side of the plot.
In contrast, nest-bound ants show a shift from a peak in density near the path center
to a strongly asymmetric density profile. If restricted to a 2D surface, this density
distribution would result in a high rate of head to head collisions and an overall
reduced foraging efficiency. Rather, the changes in the food-bound distribution show
the effects of ants not restricted to the plane. Viewed from below, these ants all appear
at the same x position, concentrating the density distribution on the side of the path
shown in figure 23, and producing the sharp spike seen in the width dependent FC
in figure 24 for the same trial.
55
Figure 24. FC as function of width: Extreme Confinement. (A) 2.5 cm, (B) 1.8 cm, (C)
1.0 cm, all during compression. (D) 1.0 cm, (E) 1.8 cm, (F) 2.5 cm, all during expansion.
The side of the path impacted by out-of-plane ants appears to be consistent
across the 5 trials integrated for this analysis. While this consistency may represent
a quantifiable preference for wall climbing on one side of the path, it may also result
from a stochastic process in the first compression-expansion cycle, with climbing
in consecutive trials influenced by which side of the path has a modified talc
barrier, in a simple example of energy minimization. Travel along a vertical surface
has a higher metabolic cost, explaining why the number of wall climbers and the
corresponding sharpness of the left-most peak in the FC change so dramatically over
such a small change in width. As maintaining the 3 lane traffic pattern shown at
lower confinement levels becomes increasingly difficult, it eventually outweighs the
inefficiencies associated with travel along a vertical surface. When the column is
allowed to expand, many of these wall climbers return to the observation platform.
56
Figure 25. Perspective-based Polar Density as a Function of Width: Extreme Confinement,
all ants (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D) 1.0 cm, (E) 1.8
cm, (F) 2.5 cm, all during expansion.
This width dependent behavioral switch is reproducible in our system, highlighting
additional ways in which the foraging column can optimize its opposing flux.
3.2.2 Structure in the Ant Frame
In that ant frame, we first present the width dependent polar density in figure
25. for the extremely confined path. Once again, the lateral width restriction coupled
with the average nematic alignment results in a higher pair density in front of and
behind the average ant in the system. In addition, the previously described wake
disappears at peak confinement, in agreement with the results from the moderately
confined path.
As before, we separate the ants into groups depending on their global orientation
to distinguish between the polar densities observed by individuals traveling in different
57
directions. Figure 26 shows the polar density from the perspective of the food-bound
ants, while figure 27 shows it from that of the nest-bound ants. There is a slight
distinction between the angular distribution of ant density at the narrowest width;
nest-bound ants observe an angled density profile with respect to their heading. As
the ants returning to the colony show the largest degree of spatial asymmetry in
lab frame density, we anticipate the emergence of these asymmetric features in their
perspective based plots as well, even if influenced by the inclusion of out-of-plane
ants.
Finally, we present the total agent-based TT Correlation for Atta cephalotes
under extreme confinement, as well as the same plots for individuals grouped by
global orientation in figures 28, 29 and 30.
In agreement with the TT Correlation for the unconfined foraging column (see
figure 13), at these widths the front-back asymmetry and anti-aligned wake are both
present in the confined column over the full range of lateral compression. These plots
also reveal a right-left asymmetry which may result from the out-of-plane individuals.
This possibility is reinforced by the results in figures 29 and 30, which show the
same plot for ants separated by global orientation. While unique width-dependent
asymmetries are present for both groups, extracting rules for interaction from these
plots will require alternative containment methods to prevent out-of-plane individuals
from influencing the interpretations.
3.3 Conclusion
Under confinement, the behavior of the A.ceph foragers results in an obvious
traffic pattern and corresponding flux organization with a direct dependence on
the imposed width. Under moderate confinement near the natural path width, the
58
Figure 26. Perspective-based Polar Density as a Function of Width: Extreme Confinement,
food-bound. (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D) 1.0 cm, (E)
1.8 cm, (F) 2.5 cm, all during expansion.
Figure 27. Perspective-based Polar Density as a Function of Width: Extreme Confinement,
nest-bound. (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D) 1.0 cm, (E)
1.8 cm, (F) 2.5 cm, all during expansion.
results presented above suggest that Atta cephalotes engage in the same 3 lane traffic
previously identified in other species, especially Eciton burchelli and Atta colombica.
The higher ant density exhibited by the A.ceph under this confinement gives our FC
59
Figure 28. Perspective-based TT Correlation as a Function of Width, extreme confinement,
all ants. (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D) 1.0 cm, (E) 1.8
cm, (F) 2.5 cm, all during expansion.
plots the statistical significance necessary to claim the definitive observation of lane
structures. In the ant frame, the effects of steric confinement can be easily observed
in plots of polar density, while the TT correlation emphasizes behavioral changes
under different widths for both nest-bound and food-bound individuals. These plots
provide an avenue for understanding how the behavior and spatial perspective of the
average individual react to dynamic environments.
Under extreme confinement, the A.ceph demonstrate a simple form of energetic
minimization by utilizing metabolically expensive vertical pathways only when the
cost of maintaining the steady-state 3 lane pattern becomes prohibitive due to the
level of confinement. This analysis suggests the critical width for the observed
transition is between 1.8 and 1.0 cm, or 5.1 and 2.5 ℓ. In order to identify a similar
60
Figure 29. Perspective-based TT Correlation as a Function of Width, extreme confinement,
food-bound. (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D) 1.0 cm, (E)
1.8 cm, (F) 2.5 cm, all during expansion.
Figure 30. Perspective-based TT Correlation as a Function of Width, extreme confinement,
nest-bound. (A) 2.5 cm, (B) 1.8 cm, (C) 1.0 cm, all during compression. (D) 1.0 cm, (E)
1.8 cm, (F) 2.5 cm, all during expansion.
transition point between a 3 and 2 laned path structure, alternative containment
methods will be necessary to better restrict the A.ceph to the 2D plane, or similar
61
analyses may be carried out on an ant species not dependent on a vertical clearance
like the leaf-cutters studied here.
While the analysis of our experimental data shows structures and trends which
support our understanding of ant behavior, it may be necessary to observe the same
outcomes for data which is completely unstructured. As a comparison, we now prepare
to explore the simplest model of A.ceph behavior, where structures and trends result
only from steric interactions, and not from behavioral changes in the ants.
62
CHAPTER IV
COMPARING EXPERIMENTAL OBSERVATIONS TO THE ’NULL’ MODEL OF
ATTA CEPHALOTES TRANSPORTATION
4.1 The ’Null’ Model of A.ceph Behavior
While the data presented so far has aimed to showcase the organizational
properties of the ant path, for both lab frame measurements and ant frame maps
the structure in these plots results from a combination of steric constraints in the
system, and the decision making of individual ants. To reliably extract information
about this decision making, we first need to understand what portion of the structure
in these plots comes from sterics alone. We achieve this by generating an artificial
data set under the assumption that average lab frame density is a constant, while the
frame-by-frame ant number number follows a Poisson Distribution with a fixed mean
number of ants. The position and orientation for each ant is chosen at ‘random’,
assumed initially to be uniform in both x, y, and θ, but restricted by the three basic
processes of occlusion: image bounds, ant-wall interactions, and ant-ant interactions.
In simulating the positions and orientations of the A.ceph, we restrict the foragers to
the same vertical extent as that sampled by our imaging system. Ant-wall occlusion,
first described in chapter 2 and presented schematically in figure 15, treats the A.ceph
foragers as orientable vectors in the lab frame. For ants of a given length ℓ, with a
lab frame orientation of θ, we model the minimum lateral distance dlat an in-plane
ant must maintain from a nest-food aligned barrier as
ℓ
dlat(θ) = cos(θ)
2
Conditional expressions in our simulation implement these two physical
restrictions on simulated ants before adding them to the data set. Finally, we prevent
63
ant-ant occlusion during the placement process by checking for overlap between the
line segment representations of the placed individual and those already in the frame.
If any ants overlap, all of the intersecting individuals are removed from the frame.
At very high confinement levels or extremely high densities this placement technique
can cause a dramatic drop off in total ant count. Since this effect only occurs as
a consequence of overlapping ant vectors, physical restrictions prevent A.ceph from
maintaining densities which would cause such dramatic percolation in simulation.
Overall, including the effects of ant-ant steric exclusion in our simulation results in a
slightly lower average ant number in each frame.
4.2 Null Statistics in the Unconfined Limit
Figure 31. Laning in the Null Model (A) FLD and FC for the A.ceph foragers in the
unconfined limit (B) FLD and FC from simulated ants under the same steric confinements.
We first explore the results of this null simulation for the unconfined foraging
column. Figure 31 compares the results presented in chapter 2 for the FLD, lateral
64
Figure 32. Polar density in the null model (A) Plots of polar density in the ant frame for
the A.ceph foragers in the unconfined limit (B) Polar density of simulated ants under the
same steric confinement. Both (A) and (B) show all ants, food-bound ants, and nest-bound
ants from left to right.
density and directionality, and corresponding FC with those predicted by the null
model of A.ceph behavior. In 31B, both appear completely flat, in alignment with
uniform sampling in both position and orientation used to generate the simulated
distribution. This comparison emphasizes the amount of organizational structure
present in the foraging column, especially along the orientation axis.
We also compare the ant frame polar density and TT correlation in figures. 32
and 33. The polar plots show a dramatic difference between the perspective-based
ant densities, as the auto-chemotactic navigation strategy represented by figure 32A
results in a peak local ant density which exceeds that of null prediction by an order
of magnitude. This observation holds when averaging over all ants or when grouping
by orientation.
For the TT Correlation, the results of the null simulation shown in figure 33B
contain a distinct lack of structure in the relative orientation of ants, especially when
65
Figure 33. Polar TT correlation in the null model (A) Plots of the TT Correlation in the
ant frame for the A.ceph foragers in the unconfined limit (B) FLD and FC from simulated
ants under the same steric confinement. Both (A) and (B) show all ants, food-bound ants,
and nest-bound ants from left to right.
directly compared with the same plots for the A.ceph foragers. This contrast serves
to emphasize the large amount of structural information which can be extracted from
this observable, potentially providing insights into the interaction rules and behavioral
rules employed by the A.ceph foragers.
4.3 Null Statistics under Moderate and Extreme Confinement
We continue our investigation into the null hypothesis by implementing the same
simulation and analysis techniques under varying levels of steric confinement. For this
section, we compare these results to those produced by the A.ceph under three distinct
widths, all representing behavior under compression (increasing confinement). While
we do not show results for expansion, the lack of structural hysteresis demonstrated in
chapter 3.1.1 suggests that our system operates near the adiabatic limit, with results
largely independent of the direction of barrier travel.
66
In figure 34, The FLDs for the null model and the collected data are presented
side by side for 3 confinement levels. For the simulated data, the effects of ant-wall
exclusion are apparent, with a high degree of correlation between the edge of the
depletion zone and its theoretical boundary. Without the internal behavioral rule-
sets present in the A.ceph, ants in the null model fully sample the available space,
yielding an FLD more sensitive to the degree of confinement at orientations far from
the nest-food axis. Aside from overall density increase at smaller widths, the lack
of structural change in these lab-frame plots further reinforces how the organization
exhibited by the true foragers results from individual behavior, and not from system
sterics. While not shown here, the FCs corresponding to these FLDs show the same
lack of organization, suggesting a net flux of zero at all lateral positions across the
path.
Figures 35 and 36 compare the ant frame measurements of polar density and
TT correlation between the simulated and collected data over the same 3 widths as
above.
Under extreme confinement, the comparisons are equally dramatic. Figure 37
gives the FLD, polar density, and TT correlation for the highest confinement level
explored in this study for both the observational data and the null simulation. While
largely in agreement with the moderately confined column, at the minimum width
the spatial regions impacted by ant wall sterics begin to dominate the statistics
of ant distribution, limiting the spread of individual orientation in the lab frame,
and causing a slight directional dependence on the radial distribution of ant density
observed by the average ant in our system, as shown in figure 37B. Despite this steric
structuring present in ant frame measurements, the polar density shown by the A.ceph
67
foragers still reveals organization far beyond that imposed by ant wall interactions,
and continues to emphasize the role of behavior in the results presented in chapter 3.
4.4 Conclusion
Primarily, this chapter serves to contrast the complexity of behavior exhibited
by the Atta cephalotes foragers with the simplest, null model of behavior where
individuals simply diffuse through the observation region. Through this method,
we can emphasize the degree of nematic alignment present in our foraging column,
and showcase its consequences in both the lab frame and the ant frame. We also
highlight the emergence of lanes, and show by comparison that the laning signatures
described in this study do not emerge as a consequence of ant-wall interactions, but
from individual decision making. As described in chapter 3, the models implemented
here for ant-wall exclusion do not consider the forward motion and limited turn radius
of individual ants when predicting depletion zones. Increasing the complexity of our
model from these simplest assumptions will allow us to better fit these depletion
zones, and continue to zoom in on how individual decision making directly influences
the global organization and structure of the foraging column.
68
Figure 34. FLDs under the null model, moderate confinement. Comparisons shown between
data and simulation (left and right, respectively) for confining widths of (A) 5cm, (B) 3.8
cm, and (C) 2.5 cm
69
Figure 35. Ant frame polar density under the null model, moderate confinement.
Comparisons shown between data and simulation (left and right, respectively) for confining
widths of (A) 5cm, (B) 3.8 cm, and (C) 2.5 cm
70
Figure 36. Polar TT correlation under the null model, moderate confinement. Comparisons
shown between data and simulation (left and right, respectively) for confining widths of (A)
5cm, (B) 3.8 cm, and (C) 2.5 cm
71
Figure 37. (A) FLD, (B) Polar density, and (C) TT correlation under maximum confinement
(1 cm). Comparisons shown between data and simulation (left and right, respectively)
72
CHAPTER V
MATERIALS AND METHODS
5.1 Ant Colony Maintenance
Figure 38. Block diagram of a typical colony layout and location of observation platform,
with relevant lengths provided. Diagram not to scale.
Several colonies of the leaf-cutting ant Atta cephalotes are maintained for
biophysical research at the University of Oregon. The block diagram in figure 38 shows
the layout of a typical enclosure. The workers in these highly specialized colonies
cut and collect fresh foliage from an isolated feeding platform before transporting it
through clear plastic tubing to their symbiotic fungal gardens, as shown schematically
in figure 38A. Throughout the experimental period, the colony under study was
provided a daily feeding of approximately 100 grams of Rubus armeniacus, also known
as the Armenian blackberry, collected daily by researchers in our lab. In order to best
study the spatial distribution of the ants during these regular foraging periods, the
foraging column is routed through an automated observation platform, consisting of
73
two 2’X4’ sheets of clear 1/4” acrylic separated by reposition-able 1 inch talc-coated
barriers (shown in figure 38B), with the entire structure supported horizontally above
an imaging apparatus. The width of the laterally confined region changes remotely via
computer using 4 threaded-shaft stepper motors, allowing accurate and reproducible
steric confinement on the time scale of hours to days. On either end of the lower panel,
holes act as input and output ports, while a coating of talc and alcohol on the interior
vertical surfaces prevents the A.ceph from moving vertically up the barriers, effectively
restricting foragers to a 2D plane. In practice, long collection times result in a steady
wear on the talc layer, and the need to frequently re-apply this preventative must
be balanced with minimizing outside interaction during data capture. We minimize
variation due to stagnant air, non-uniform pheromone evaporation, and humidity
by maintaining a steady rate of air-flow ( 3 cfm) through the observation chamber.
This also serves to prevent the A.ceph foragers from utilizing empty space within the
observational platform for colony structures like fungus gardens or waste chambers.
In initial experiments, foraging columns
clearly avoided the high intensity overhead light
utilized for bright-field imaging like that shown
in figure 39A. This resulted in a light-induced
path bias, especially for experiments in the
unconfined limit where the foraging column
would slowly drift out of frame. To remove
this bias, we switched to dark-field illumination,
as shown in figure 39B, and imaged ants using
an IR LED strip source (LightingWill). Under
Figure 39. Ant imaging (A) Initial
these conditions, the path drift observed under bright-field illumination, visible light
(B) Dark-field illumination, IR light
74
visible light did not occur. This reinforces
literature results showing a reduced sensitivity to longer wavelength light in many
ant species [18][19], as well as the demonstrated use of optical cues in the navigation
strategies of Atta cephalotes [6].
5.2 Data Collection and Analysis
We imaged ant positions and orientations in the foraging column with a Logitech
Brio webcam, modified with a near-infrared (LP830) filter and c-mount lens adapter
(KuroKesu Inc.) with fixed focus and imaging parameters selected to maximize
contrast. Data communication between camera and PC is provided by the OpenCV
module in Python. For the results presented in chapter 2 for the unconfined column,
each trial consisted of 50 videos collected over 4 days, with each video consisting of
5100 frames collected at 1 FPS, with a spatial resolution of 1920p x 1080p. Between
trials the system was removed from the foraging column and cleaned thoroughly
with alcohol to prepare for additional data collection. Following each cleaning the
ant path was reintroduced to the system with the barriers at a confining width of
approximately 5 cm. Once the path had reconnected, the barriers were withdrawn
and the path was given approximately 24 hours to stabilize. We include 3 such trials
in our analysis. For variable width confinement, one video consisting of 4500 frames
is collected at each width during a compression and expansion cycle. This results
in 10 videos per cycle (5 expansion and 5 compression), and 50 videos per trial at
the same frame-rate and resolution. We clean the system between each collection
period as described The results presented in chapter 3 utilize 3 consecutive trials of
this type. In total, these results for the unconfined column are extracted from over
750,000 individual frames, while those for the confined column utilize over 2.3 million.
In post analysis, each frame is converted to 8-bit monochrome, fish-eye corrected,
75
and passed through a background removal filter to flatten the image and eliminate
temporally persistent (i.e. stationary) features like dropped leaves or barriers, while
preserving pixel information about moving ants.
Figure 40. Ant identification and pose estimation via the SLEAP neural net. Yellow dots
indicate the positions of the head (bold), thorax, and gaster of each ant.
After image capture and pre-processing, we use the open-source neural net (NN)
software SLEAP to quickly and accurately identify ants in individual frames [20]. In
order to extract the position and orientation of each ant, our model identifies and
reports the position of the head, thorax, and gaster of each ant, as shown in figure 40.
From this, we determine the orientation of the vector from gaster to head, and take
its midpoint to represent the ants location. We trained this algorithm with a ground-
truth human-labeled data set containing 2̃500 ants. We estimated the error rate of our
current best model based on a sample of 4̃00 ants, and find that on average SLEAP
is able to correctly identify A.ceph foragers 94.6% of the time, with false positive and
false negative rates to be 2.2% and 5.8% respectively. This model is also insensitive
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to global orientation, with separated error rates for nest-bound and food-bound ants
of 94.7% and 94.1%, respectively (see Appendix A). Position and orientation data
were analyzed and distilled into lab-frame and ant-frame graphical data via custom
Python and Matlab (Mathworks, Inc) scripts. Sample raw data, pre-processing
scripts, trained models, sample position data, and python graphical analysis scripts
are freely available on FigShare (Kittell, Justin (2023). PRINCIPLES OF TRAFFIC
ORGANIZATION IN ANT TRANSPORTATIONNETWORKS. figshare. Dataset.
https://doi.org/10.6084/m9.figshare.23808960.v1)
Following the SLEAP identification, the data set is trimmed to remove ants based
on bend angles and immobility. Ants found at the same location and orientation
across many frames are removed, as well as any found with a head-thorax-gaster
angle of < 90◦.
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CHAPTER VI
CONCLUSIONS
Overall, the goal of this project is to increase our understanding of the
organizational structures employed by ants like Atta cephalotes in hopes that
understanding the behavioral rule-sets which tend to produce these structures may
help minimize energetic costs in human-engineered systems. To that end, we explored
global structure in the lab frame as well as the local structure observed by individual
ants over a wide range of confinement levels.
First, our measurements of the mean path width in the unconfined limit yielded
a critical length scale for the A.ceph foragers. This result provides context for not
only our studies of ants under confinement, but also establishes a standard non-
dimmensionalized technique for understanding path width in terms of another critical
length scale for ant transport networks, ℓ, the average ant length. While the actual
width exemplified by any other ant species may differ dramatically from that shown
in this work, the similarity in pheremone deposition, transportation, and ant-ant
interactions may result in similar paths widths when expressed in the system specific
units of ℓ. In future investigations, we will explore this possibility by comparing the
organization and fluxes demonstrated by the A.ceph with other species and sizes of
ants. Other species may be better suited to such investigations as well, given the
complexities and challenges associated with leaf-cutter maintenance and observation.
If the path widths observed in alternative model systems show path widths near
the non-dimmensionalized Wpath = 6.7 ± 2.2, this length scale may prove to be
fundamental for bidirectional auto-chemotactic traffic in the unconfined limit.
Once the statistical properties of the natural path were extracted from forager
behavior, we then explore how these measures of organization respond to cyclic
78
changes in the degree of steric confinement. Under moderate (2.5 – 5.0 cm) and
heavy (1.0 – 3.5 cm) confinement, plots of lab frame density, FLDs, FCs, average
agent-based densities and tangent-tangent pair correlations all reveal dramatic
organizational shifts in response to the decreased lateral extent of the observation
region. While not obvious for the unconfined column, the moderately confined path
showed clear signatures of a steady state 3 lane traffic pattern. This observation
of Atta cephalotes behavior stands in contrast with literature results indicating a
lack of any organization observed along their foraging routes [10]. This contrast
emphasizes the utility of our statistical approach when studying ant traffic, since
visually observing structural trends along sparsely populated foraging networks is
challenging due to local deviations and low density. We encourage the implementation
of similar techniques in future studies of ant traffic. especially when concerned with
the emergence and persistence of lanes.
While our investigation into the extremely confined path was hampered by
containment limits, our results still displayed a structural transition at a critical
width, likely driven by energetic considerations on an individual level. We also observe
sharp changes in the symmetry and structure of polar density and TT correlation at
these widths resulting from a dynamic transition in behavior triggered by the changing
local environmental around each individual. While extracting the details of this
behavioral change is still challenging, employing this analysis on a larger scale and over
a wider range of environmental restriction may provide a clearer picture of individual
behavior and continue to inform our understanding of energetically optimum behavior.
Connecting the organizational changes with corresponding changes in these agent
based maps provides a link between average local information and global outcomes
79
in the ant column, and may provide a useful analysis framework for human designed
systems experiencing traffic.
Finally, we verified the origin of the observed structural transitions as behavioral,
not steric by comparing the statistical lane structures we see in our system with
those produced by a ’null’ model. Through this method, we emphasize the degree
of nematic alignment present in our foraging column, and showcase its consequences
in both the lab frame and the ant frame. We also highlight the emergence of lanes
in the A.ceph foraging column, and show by comparison that the laning signatures
described in this study do not emerge as a consequence of ant-wall interactions, but
from individual decision making. As described in chapter 3, the models implemented
here for ant-wall exclusion do not consider the forward motion and limited turn radius
of individual ants when predicting depletion zones. Increasing the complexity of our
model from these simplest assumptions will allow us to better fit these depletion
zones, and continue to zoom in on how individual decision directly influences the
global organization and structure of the foraging column.
Through the data collection protocols and analysis techniques presented herein,
we have provided an insight into the organizational strategies and their emergence
from individual rule-sets for the ant Atta cephalotes, providing key insights into the
structure of a biological transport network, as well as a conceptual framework for the
design of robust autonomous networks in human transportation systems.
80
Figure 41. Ant transportation.
81
APPENDIX
APPENDIX A
A.1 Ant Number Statistics and Total Fluxes
In this section we provide representative plots of average ant number per frame
and the frame-wise flux for the experiments described in this work. We assume
outgoing food-bound ants count as positive flux, while returning nest-bound ants are
counted as negative flux. Each ant’s directionality is determined from its lab frame
orientation.
A.1.1 The Unconfined Limit
Figure A.42. Ant Number statistics in the unconfined limit. (A) Average ant number is
given both before and after filtering (see chapter 5). (B) Ant Flux per frame, showing
number of food-bound ants in orange, nest-bound ants in blue, frame-wise net count in
black, and net integrated in green.
Figure A.42A shows the normalized probability density for ant number per frame
for data collected in the unconfined limit, along with an accompanying Poissonian
distribution with the same mean. This comparison reveals that the ant number
distribution is not strictly Poissonian under our experimental conditions. Figure
A.42B gives the frame-by-frame breakdown of ant flux, as well as the net integrated
82
ant count. The 250,000 frames represented span 4 days and 4 feeding cycles, evidence
of which can seen in variation of net ant number, as pheremone recruitment results in
large increases in both positive and negative ant flux. While the net integrated flux
for this trial reveals 40,000 more nest bound ants than food bound ants, this is less
than 1% of the total number of ants identified.
A.1.2 Moderate and Extreme Confinement
Figure A.43 shows the frame-by-frame ant flux at the minimum and maximum
widths under moderate confinement. The average post-filtering ant number for figures
A.45A and A.45B are 8.2 and 9.6 respectively. Here, the well defined feeding cycles
seen for the unconfined case are not present at either width in figure A.45, since the
data set is a composite of samples from many compression and expansion cycles.
Figure A.43. Representative Ant Fluxes Under Moderate Confinement (A) 5 cm (B) 2.5
cm
Figure A.44 shows the frame-by-frame ant flux at the minimum and maximum
widths for the extremely confined case, with representative widths of 2.5 cm and 1.0
cm show in A and B respectively. The average post-filtering ant numbers are 12.0
and 11.3 respectively.
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Figure A.44. Representative Ant Fluxes Under Extreme Confinement (A) 2.5 cm (B) 1.0
cm
A.2 Flux Correlate Under Moderate Confinement: All Trials
In figure A.45 we present the complete set of spatial flux correlates (defined in
chapter 2) at each moderate confinement level, from 5.0 cm to 2.5 cm. This shows
the stability of the observed 3 lane traffic pattern across a wide range of confinement.
84
85
Figure A.45. Flux Correlate at each confinement level: moderate confinement. Color corresponds to lab frame directionality, with
magenta being towards the nest, and green being towards the food.
A.3 Neural Net Error Rates
In table A.1 below we show the frame-by-frame breakdown of our SLEAP models
neural net error rates. The analysis includes 3 separate videos at different densities,
and separates ants by global orientation.
Table A.1. Neural net error rates
(see following)
86
87
Table A.1.
88
Table A.1. (cont.)
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