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This is to certify that the

dissertation entitled
How do bees compute the position of the sun?

Alternative Representations
presented by
Jeffrey A. Dickinson

has been accepted towards fulﬁllment
of the requirements for

Ph 0 D 0 degree in ZOOlogy

 

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MS U is an Aﬂ'irmau've Action/Equal Opportunity Institution 0- 12771

 

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Michigan State
. University

 

 

 

7 v-'v ‘—

HOW DO BEES COMPUTE THE POSITION OF THE SUN?
ALTERNATIVE REPRESENTATIONS

By

Jeffrey A. Dickinson

A DISSERTATION

Submitted to

Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Zoology

1997

ABSTRACT

HOW DO BEES COMPUTE THE POSITION OF THE SUN?
ALTERNATIVE REPRESENTATIONS

By

Jeffrey A. Dickinson

To use the sun as a compass, animals must compensate for its apparent movement. This is
complicated by the fact that the rate of change of the horizontal component of the sun's
position, or azimuth, which provides the directional information, varies over the course of
the day. Additionally, the daily pattern of change varies with season and latitude. It has
long been known that bees learn the local pattern of movement, and that they can estimate
the position of the sun at times of day when they have never seen it. However, the
mechanism has remained a mystery. When bees are restricted to ﬂy only in the aﬁemoon,
thereby limiting the information available for learning, they nonetheless have a relatively
accurate estimate the position of the sun in the morning, indicating a position for the
azimuth in the morning about 180° from the position experienced in the afternoon. These
results contradicted the predictions of all previous computational models of sun-compass
learning, which all assumed that the insects' neural computations rely on linear estimates of
azimuthal rate. However, these results suggest that they include aspects of the
nonlinearity of the natural pattern in their computations. In this dissertation, I present two
new methods of modeling the sun-compass-learning process. Both classes of models

incorporate the nonlinear aspects of the problem; one class of models is symbolic (it

maintains a symbolic representation of azimuth position and time) while the other is
nonsymbolic (it does not maintain a symbolic representation of azimuth position and time).
The nonsymbolic models are connectionist neural networks. The symbolic model attempts
to take into account the varying rate of change of the azimuth over the course of the day.
It is based on the geometry of an ellipse. In the nonsymbolic connectionist model, the
connection weights of the network are assumed to be preconﬁgured in such a way that a
bee is able to quickly recognize or quickly learn the course of the sun. Neural processes
that roughly correspond to either of these broad classes of models could account for the

bees’ behavior.

T 0 Sarah, for understanding

iv

ACKNOWLEDGMENTS

This work represents the culmination of a near ten-year quest. The ﬁrst seeds of these
ideas were planted at Carleton College, with the guidance of Jerry Eberhard and Julie

Neiworth.

This work would not have been possible without Fred Dyer, my advisor, who convinced
me to come to Michigan State University to work on this problem. Many of these ideas
were developed in concert with him. The errors that remain are mine. Fred provided the

necessary pushes to get things done throughout my time at Michigan State.

My other committee members, Tom Getty, Jim Miller and Mark Rilling, provided helpful

discussions at various times.

Fellow graduate student, Elizabeth Capaldi, provided support at many levels. Numerous
undergraduates have participated in the ﬁeld work at various stages; Chelsea Kostrub was
instrumental in ﬁeld work presented here. Comments from Rudolf Jander precipitated the

symbolic model, and a discussion with Clarence Lehman led to the neural network

modeling. Discussions with Paul Rodriges and Whitney Tabor at the Santa Fe Institute '8

Complex Systems Summer School led to development of some neural network models.

None of this would have been possible without the loving support and understanding of

my wife, Sarah.

TABLE OF CONTENTS

LIST OF TABLES ................................................................................................. ix
LIST OF FIGURES ............................................................................................... x
CHAPTER 1
OVERVIEW .......................................................................................................... 1
CHAPTER 2
SUN COMPASS ORIENTATION ......................................................................... 13
2.1 The Solar Reference ............................................................................ 14
2.2 Discovery of the Sun Compass ............................................................ 22
2.3 Distribution in the Animal Kingdom .................................................... 24
2.4 Directional Information from the Sun .................................................. 36
2.4.1 The Sun as a Point Source .................................................... 36
2.4.2 The Sun Deﬁnes a Local Direction ........................................ 37
2.4.3 The Sun Deﬁnes a Global Direction ...................................... 38
2.4.4 The Sun Deﬁnes Any Flexible Direction ................................ 39
2.5 Honey Bees as a Model System ........................................................... 39
2.5.1 Long Distance Orientation .................................................... 40
2.5.2 Short Distance Orientation .................................................... 41
2.5.3 The Dance Language ............................................................ 46
2.6 Other Celestial Compass Mechanisms ................................................. 49
2.6.1 The Lunar Compass .............................................................. 49
2.6.2 The Stellar Compass ............................................................. 49
2.7 Summary ............................................................................................ 50
CHAPTER 3
THE REPRESENTATION OF THE SUN‘S COURSE .......................................... 51
3.1 Previous Studies ................................................................................. 52
3.2 Experimental Methods ........................................................................ 63
3.2.1 Colony 1 ............................................................................... 63
3.2.2 Colony 2 ............................................................................... 64
3.2.3 Colony 3 ............................................................................... 64
3.2.4 Colony 4 ............................................................................... 65

vii

3.3 Results and Discussion ........................................................................ 67
3.3.1 Colony 1 ............................................................................... 67
3.3.2 Colony 2 ............................................................................... 75
3.3.3 Colony 3 ............................................................................... 79
3.3.4 Colony 4 ............................................................................... 83
3 .4 Summary ............................................................................................ 86
CHAPTER 4
NONLINEAR SYMBOLIC MODELS ................................................................... 87
4.1 A Symbolic Model of Sun Compass Learning ...................................... 87
4.2 The Ellipse Rate Function ................................................................... 91
4.2.1 The Geometry of the Ellipse ................................................. 92
4.2.2 The Equation of the Ellipse ................................................... 100
4.3 The Ellipse Azimuth Function ............................................................. 102
4.4 Estimating the Parameters of the Ellipse .............................................. 109
4.5 Fit of Ellipse Azimuth Function to Experimental Data ......................... 110
4.6 Fit of Ellipse Azimuth Functions to Other Data Sets ............................ 116
4.7 Reducing the Number of Parameters ................................................... 118
4.8 Estimating k with Restricted Experience .............................................. 120
4.9 The Effect of the Time Interval ........................................................... 123
4.10 Summary ............................................................................................. 125
CHAPTER 5
CONNECTIONIST MODELS ............................................................................... 126
5.1 Connectionist Computations ............................................................... 127
5.2 Connectionist Characteristics .............................................................. 130
5.3 Connectionist Representations of the Sun's Course .............................. 131
5.4 A Binary Representation ..................................................................... 132
5.4.1 Pattern Matching .................................................................. 133
5.4.2 Learning Advantage .............................................................. 143
5.5 A Continuous Representation .............................................................. 155
5.6 Summary ............................................................................................. 162
CHAPTER 6
CONCLUSIONS .................................................................................................... 163
6.1 Innate Structure .................................................................................. 165
6.2 Templates ........................................................................................... 166
6.3 Symbolic versus Nonsymbolic ............................................................. 169
6.4 A False Dichotomy? ............................................................................ 173
6.5 Caveat ................................................................................................ 175
6.6 Representing the Regularities of the World .......................................... 175
REFERENCES ...................................................................................................... 176

viii

LIST OF TABLES

Table 2.1 Distribution of the Sun Compass in the Animal Kingdom ........................ 26
Table 3.1 Regression Analysis of Individual Bees' Azimuth Estimates (Colony l)...70
Table 3 .2 Regression Analysis of Individual Bees' Azimuth Estimates (Colony 2)...78

Table 3.3 Regression Analysis of Individual Bees' Azimuth Estimates (Colony 3)...81

Table 4.1 Fit of Models to Experimental Data (R2) ................................................ 115
Table 4.2 Expected and Observed Parameters of the Ellipse ................................... 115
Table 4.3 Pit of Ellipse Model to New & New (1962) Data (R2) ............................ 117

Table 4.4 Expected and Observed Parameters of the Ellipse (New & New, 1962).. 117

Table 4.5 Estimates of k ﬁom the Training Period of Colony 1 .............................. 122
Table 5.1 Hamming Distance between Azimuth Training Patterns .......................... 150
Table 6.1 Comparison of New Models ................................................................... 165

ix

LIST OF FIGURES

Figure 1.1 ............................................................................................................... 5

The Computational-Representational Approach. A. The traditional view.
B. The computational-representational view.

Figure 2.1 ............................................................................................................... 17

Pattern of solar movement for different seasons and latitudes. The ﬁgure
represents a projection of the sun onto the celestial hemisphere. The center
points correspond to the zenith, which is the point directly overhead of the
observer. The outer circles correspond to the horizon line. The sun's
hourly position is plotted by the dark circles. The distance from the
horizon line to the sun corresponds to the sun's elevation in the sky. The
distance from the zenith point to the sun is the zenith distance. Extending
an imaginary line ﬁom the zenith through the sun's position to the horizon
reveals the position of the sun's azimuth which provides the directional
information for sun compass orientation. See example for 40° N equinox.
The rate of change of the azimuth varies over the course of the day and
with season and latitude. This can be seen in the comparison of the change
in azimuth between 7:00 and 9:00 vs. 11:00 and 13:00 (drawn for 40° N).

Figure 2.2 ............................................................................................................... 18

Seasonal and latitudinal variation of the ephemeris ﬁrnction. A. Ephemeris
functions for 5°, 25° and 40° North latitude for the equinox. B. Ephemeris
functions for 5°, 25° and 40° North latitude for the solstice. Solar azimuth
is plotted against local sun time. The two panels show the considerable
variation in the rate of change of the azimuth over the course of the day
and with season and latitude.

Figure 2.3 ............................................................................................................... 45
Proportion of choices to each of four directions with respect to the training

direction. A. Sunny. B. Partly cloudy C. Completely overcast. D.
Delay. The probabilities presented are the binomial probabilities that the

pattern of choices occurred by chance (25% correct choice; 75%
incorrect).

Figure 2.4 ............................................................................................................... 48

The dance language of the honey bee. The communicative waggle dance
indicates the distance and direction to the food. A. The angle indicated
by the waggle run (or) with respect to vertical corresponds to the angle of
the food with respect to the sun's azimuth. B. The ﬂight path from the
hive (H) to a food source (F) relative to the sun's azimuth the angle or is the
same as in (A). Over the course of the day, the dance angle changes as the
solar azimuth moves along the horizon. Distance is indicated by the length
of the waggle run.

Figure 3.1 ............................................................................................................... 56

Solar azimuths inferred from the dances of the bees in New and New's
(1962) experiments. Data are from colony 1 in Trinidad (10°3 8') on (A)
April 15, (B) April 16, (C) April 17, (D) April 18, (E) April 20, and (F)
April 21. Data were obtained from New and New's (1962) Figure 3 and
replotted to correspond to solar azimuths rather than dance angles.

Figure 3.2 ............................................................................................................... 62

Predictions from previous computational models about how insects ﬁll
gaps in their experience with the sun's course. The shaded region
corresponds to the time period of experience with the sun's course. The
lines represent the predictions of the position of the azimuth at other times
of day based on the various models. A. Interpolation: a constant rate of
change between the end of the one training period and the beginning of the
next on the subsequent day. B. Forward extrapolation: the rate of change
observed at the end of the training period extended forward through the
night and into the morning of the subsequent day. C. Backward
extrapolation: the rate of change of the azimuth observed at the beginning
of the training period and extended backward into earlier portions of the
day.

Figure 3.3 ............................................................................................................... 69

Solar azimuths inferred ﬁ'om the dances of restricted experienced bees
from colony 1. The curved line corresponds to the actual path of the sun
on July 22. The straight lines are the predictions from the previous
computational models. A. Interpolation. B. Forward extrapolation. C.
Backward extrapolation. D. 180° step function. The open symbols
correspond to two bees that differed qualitatively ﬁom the rest.

Figure 3.4 ............................................................................................................... 73

Solar azimuths inferred from the dances of four individual bees from colony
1 that show gradual shifts during the transition period. A. Bee 32. B.
Bee 74. C. Bee 78. D. Bee 87.

Figure 3.5 ............................................................................................................... 77

Solar azimuths inferred ﬁ'om the dances of restricted experienced bees (A)
and ﬁilly experienced bees (B) of colony 2.

Figure 3.6 ............................................................................................................... 82

Solar azimuths inferred from the dances of restricted experienced bees of
colony 3. Solar azimuth curve is for August 10.

Figure 3.7 ............................................................................................................... 85

Solar azimuths inferred from the vanishing bearings of colony 4. The data
are means :1: standard error of signiﬁcantly oriented groups of bees (10 of
12) at two different release sites (squares and circle). Tests occurred
between August 11 and 26. The azimuths inferred for each release day are
plotted. The solar azimuth shown corresponds to August 18.

Figure 4.1 ............................................................................................................... 95

The ellipse approximation of the azimuth rate ﬁinction. The rate of change
of the azimuth varies systematically over the course of the day. A. The ﬁrll
24 hour azimuth function for 40° N at the equinox. B. The 24 hour
azimuth function for 10° N at the equinox. C. The azimuth rate ﬁJnction
for the azimuth curve in A, computed over 8 minute intervals. D. The
azimuth rate ﬁinction for the azimuth curve in B, computed over 8 minute
intervals. E. An ellipse based on the rates in C (semimajor axis = 23 °/hr;
semiminor axis = 10°/hr). F. An ellipse based on the rates in D (semimajor
axis = 85°/hr; semiminor axis = 3°/hr).

Figure 4.2 ............................................................................................................... 99

The true azimuth rate function plotted in polar coordinates. The values
plotted correspond to the numerically calculated rate of change of the
azimuth over 8 minute intervals. A. 40° N and 10° N for the equinox. B.
10° N for April 16 (declination 9°54'). C. 40° N for the summer solstice.
In all cases the curves are not truly elliptical. The shape varies considerably
with season and latitude.

xii

Figure 4.3 ............................................................................................................... 105

The scaled and unscaled ellipse azimuth ﬁinctions for different seasons and
latitudes. A. The true solar azimuth is plotted with two ellipse azimuth
functions for 40° north latitude at the equinox. The squares correspond to
the scaled ellipse azimuth function and the diamonds correspond to the
unscaled azimuth ﬁinction. The ellipse parameters for both ellipse azimuth
functions are a = 23, b = 10 (as in Figure 4.1 E). The scaling ﬁrnction is
described in the text. The true sun azimuth curve is depicted by the heavy
line. B. The solar azimuth function (heavy line) and two ellipse azimuth
estimates (scaled = squares; unscaled = diamonds) for 10° north latitude at
the equinox. Ellipse parameters are a = 85, b = 3. C. The solar azimuth
function (heavy line) and two ellipse azimuth estimates (scaled = squares;
unscaled = diamonds) for 40° north latitude during the summer solstice.
Ellipse parameters are a = 48, b = 9 (these values were determined ﬁom
the actual azimuth rate function numerically calculated over 8 minute
intervals as in Figure 4.1 C and D, and depicted in Figure 4.2 C. D. The
solar azimuth function (heavy line) and two ellipse azimuth estimates
(scaled = squares; unscaled = diamonds) for 10° north latitude on April 16
(declination 9°54'). Ellipse parameters are a = 662, b = 1.5. The values
for the ellipse parameters a and b were determined ﬁom the actual azimuth
rate function, numerically calculated over 8 minute intervals as in Figure
4.1 C and D, and depicted in Figure 4.2.

Figure 4.4 ............................................................................................................... 108

Ellipse azimuth functions for 40° N and 5° N at the northern summer
solstice. The parameters of the ellipse rate function were estimated from
numerical calculations of the actual azimuth rate function over 8 minute
intervals (as in Figures 4.1, 4.2, and 4.3) (40° N: a = 48, b = 9; 5° N: a =
-43, b = -1). The actual ephemeris functions are plotted for comparison.

Figure 4.5 ............................................................................................................... 112

A range of ellipse azimuth ﬁinctions corresponding to the data from colony
1 (see Chapter 3). A. Parameters corresponding to the best ﬁtting ellipse
function (a = 1076, b = 1). B. Intermediate parameters (a = 600, b = 3).
C. Intermediate parameters (a = 300, b = 6). Theoretical (expected)
parameters of the ellipse (numerically calculated from the true azimuth rate
ﬁmction as in Figure 4.1, 4.2, and 4.3) (a = 36, b = 10). The shaded region
corresponds to an ideal step function based on the data from colony one.
The region corresponds to the mean (270° or 90°) i one standard deviation
(~30°) of the data.

xiii

Figure 4.6 ............................................................................................................... 119

Range of the parameters of the ellipse that generate equivalent ellipse
azimuth functions (see footnote 6). All of the parameters yield azimuth
curves that ﬁt the data of colony 2 equally well (R2 = .94). The linear
relationship suggests that the two parameters necessary to describe the
ellipse azimuth ﬁinction may be reduced to a single ratio.

Figure 4.7 ............................................................................................................... 124

The effect of time interval on the ellipse azimuth estimate. For all of the
curves, k is 1076 as in the best-ﬁtting ellipse azimuth ﬁmction for colony 1.
Time intervals are (A) 1, (B) 2, (C) 5, (D) 10, (E) 20, and (F) 40 minute
intervals. The curve that is most like a step function is the curve for 40
minute intervals.

Figure 5.1 ............................................................................................................... 128

General form of a feed-forward connectionist network. 1: input units H:
hidden units. 0: output units. w: connection weights.

Figure 5.2 ............................................................................................................... 136

Idealized solar ephemeris ﬁmctions used as the training set for the
connectionist models. The patterns consist of a matrix of ones and zeros,
with the ones corresponding to the black squares and the zeros
corresponding to the white squares. Each point corresponds to a pairing of
azimuth position and time of day. These patterns correspond roughly to
the natural range of solar azimuth ﬁrnctions (See Figure 2.2).

Figure 5.3 ............................................................................................................... 137

Network architecture. The network has 100 input units and 100 output
units that correspond to azimuth-time coordinates. There are 25 hidden
units. See text for explanation of connection patterns. The connections
are not shown because of the number of them. Activation is shown by the
gray scale (white = zero; black = one).

Figure 5.4 ............................................................................................................... 142

Responses of preconﬁgured network to test patterns consisting of partial
ephemeris functions. The test patterns (A) consist of inputs arranged in an
array of ones and zeros as in Figure 5.2. The responses (B) for each test
pattern are presented immediately to the right. The responses consist of

xiv

the activation level of each of the output units. They range from zero to
one and are all on the same scale. See text for a description of the
network.

Figure 5.5 ............................................................................................................... 146

Comparison of Hamming and Pythagorean distances. In this diagram, the
standard linear (Pythagorean) distance ﬁom the step ﬁmction to each of the
others pictured varies, while the Hamming distance is constant. A.
Pythagorean distance = 2, Hamming distance = 4. B. Pythagorean distance
= 4, Hamming distance = 4. Hamming distance is a measure of overlap
between the patterns. In each of the cases, two points have changed,
which gives a Hamming distance of 4 (turning off the old point and turning
on the new point for each of the two changes). The linear distance
between the old points and the new points makes a difference for the
Pythagorean distance but not for the Hamming distance.

Figure 5.6 ............................................................................................................... 149

Binary retraining trials for azimuth network trained on one azimuth pattern
and retrained on another. Binary retraining percentage is the number
retraining trials to reach an error criterion of .05 as a percentage of the
training time for the original pattern. Data are the means i standard error
for 10 replicates for each of the 7 azimuth patterns retrained on the 6 other
patterns. They are plotted against the Hamming distance (number of binary
substitutions) between the two patterns. This is clearly the major
explanatory variable.

Figure 5.7 ............................................................................................................... 153

Average hamming distance from each azimuth curve to every other curve
The curves are to the nearest ten degrees. The average Hamming distances
are plotted with respect to their latitude-declination coordinates. Panel A
shows the situation when the curves are restricted to the tropics (between
20° N and 20° S). Panel B shows the situation when temperate curves are
included (between 40° N and 40° S).

Figure 5.8 ............................................................................................................... 154

Binary retraining trials for azimuth curves discussed in Figure 5.7. For the
range of Hamming distances that exist between the azimuth curves
represented in ﬁgure 5.7, a network trained on one azimuth pattern and
retrained on another. The network had 324 inputs and outputs and 81
hidden units. Binary retraining is the number retraining trials to reach an
error criterion of .05 as a proportion of the training time for the original

pattern. Data are the means :t standard error for 5 replicates for each
Hamming distance (ﬁlled circles). They are plotted against the relative
Hamming distance (number of binary substitutions)/(number of elements)
between the patterns. For comparison the data from Figure 5.6 are
included (open circles).

Figure 5.9 ............................................................................................................... 158

Diagram of simple connectionist network used for the interpolation
network. 1: input unit. H: hidden unit. 0: output unit. B: bias unit. w:
modiﬁable connection weights.

Figure 5.10 ............................................................................................................. 161

Response of interpolation network to three different training regimes for
learning the same speciﬁc azimuth function. The azimuth ﬁmction
corresponds to 25° N on April 22 (solar declination 12°). A. Response of
a network over 24 hours at half-hour intervals for a training set of hourly
azimuth angles between sunrise (~06200) and sunset (~18:00) (local solar
time). B. 24 hour response of a network trained only on hourly positions
in the late aftemoon and evening (15:00 to 18:00). C. Response of a
network trained on aﬁemoon positions in B but with additional constraints
(Azimuthnm = Azimuth”,00 - 90°; Azimuthmoo = Azimuth”,00 - 180°).

Chapter 1

OVERVIEW

The natural world is full of regularities. Life itself depends on the regularities of the
world, and has evolved to take advantage of them. Many of the regularities are
incorporated into the genetic code. Others are learned and remembered by animals. The
distribution of resources in the environment is one factor that has led organisms to adopt
both of these strategies. In the short run resources may be distributed unequally through
the environment, and this makes certain locations more important than others for the
survival of the animal. The environment also varies over generations as conditions
ﬂuctuate and populations migrate into new areas. This requires an animal to learn the
local conditions or evolve to take advantage of the new conditions. It may be
advantageous to move between regular locations in the environment. This in turn may

lead to the evolution of orientation mechanisms and memory for places.

Learning has evolved to help maximize the performance of an animal in a world of
unpredictable predictabilities. These are events that occur with regularity and can be
predicted but are not regular enough to engender a speciﬁc response. Events of this type

cannot be genetically hard-wired into the behavior of an animal because the exact nature

of the events varies, yet there is a regularity that allows predictions to be made based on

previous experience.

The brain is the organ of learning. Recent theories in psychology have suggested that
there is a close correspondence between the regularities of the world and those of the
brain (Shepard, 1987). The brain is an immensely complicated computational system
(Churchland & Sejnowski, 1989) that evolved over millennia in the context of ﬂuctuating
yet regular environments. One approach to understanding the function of the brain
involves understanding the environmental pressures that lead to its evolution. A
fundamental view of the brain is that it consists of a number of modules with speciﬁc
computational characteristics (Gallistel, 1990). These may have evolved as solutions to
very speciﬁc problems presented by the environment. In order to understand the
ﬁinctioning of the brain as a whole it is important to understand the ﬁrnctioning of its
parts. The properties of the brain that seem uniﬁed in experience may in fact be a
haphazard conglomeration of modules that evolved to solve very speciﬁc yet different
environmental problems. The human brain is particularly complex and it is quite diﬁicult
to decipher individual computational modules with the exception of the those involved in
the most basic levels of sensory processing. Therefore, to understand the ﬁmctioning of
individual specialized modules it is fruitful to look at animals that have much smaller brains

and have very speciﬁc environmental problems to solve.

Gallistel (1990) has focused considerable attention on such animals as a means of
understanding the organization of learning and of the brain. Gallistel has championed a
computational-representational approach to animal learning. In this framework important
elements of the animal's external world are represented in the animal's internal world.
Gallistel maintains that these representations are ﬁrnctionally isomorphic with the entities
in the real world. He uses isomorphism in the formal mathematical sense, in which one set
of entities can be mapped onto another, such as the relationship between geometry and
algebra (Gallistel, 1990). Thus, for Gallistel, there is a one-to-one relationship maintained
between the parameters of the natural world and the parameters of the representations. In
Gallistel's view, computations can be performed on the representations to yield new forms
of the representations needed to guide behavior. He calls these derived representations.
For example, a representation of quantity can be combined with a representation of time to
yield a new representation of rate. This general scheme is diagrammed in Figure 1. For
comparison, the computational-representational approach contrasts with the traditional
behaviorist approach (Watson, 1925), which posits that only behavior can be studied
experimentally and not the internal structures that determine behavior. Although this
approach is essentially dead, it has left an indelible mark on American psychology and the
study of learning (e. g. Macphail, 1987). By contrast, the computational-representational
approach maintains that behavioral experiments can be used to distinguish between
alternative internal processing mechanisms. Thus it basically opens the black box of the
behaviorist approach. In addition, black boxes remain within the former boundaries of the

stimulus-response box. These new boxes contain speciﬁc neural computations. The goal

of research in this paradigm is to decipher the nature of the representations and the
computations involved in speciﬁc information-processing problems. A signiﬁcant part of
Gallistel's motivation is to guide neurophysiological research to the right questions based
on the contents of computational boxes. That is, what types of neural processes are

necessary to perform these speciﬁc computations.

Gallistel (1990) highlights navigation as a subject to explore the computational solutions
to speciﬁc environmental problems. The ability to remember locations in the environment
and to navigate between known locations is ﬁindamental for the competitive ability of
many animals. This ability is ubiquitous in the animal kingdom. To navigate accurately
the animal must be able to solve certain computational problems. Understanding the types
of problems that need to be solved and the ways that animals solve them sheds

considerable light on cognitive processes and the neural mechanisms that underlie them.

Navigation is one realm of learning that may have very speciﬁc information-processing
requirements. By understanding the speciﬁc requirements and the particular solutions to
navigational problems, we may gain insight into the range of information-processing
capabilities that animals have. Additionally, we can see how those capabilities may be

shaped by evolution to solve very speciﬁc problems.

 

A
Stimulus Response
B
Sensory Inputs Computations Behavioral Outputs

 

Figure 1.1 The Computational-Representational Approach A. The

AA.

traditional view B The r ., 'view.

In this dissertation, I will show how an environmental regularity has been encoded in the
brain of an animal to facilitate the rapid learning of a pattern that forms a critical reference
point for effective navigation. The environmental regularity I consider is the sun and the

animal is Apis mellifera, the common honey bee.

For aeons the earth has alternatively been shrouded in darkness and bathed in the light of
the sun. During the day, the sun appears to make a circuit across the sky. This forms a
regular pattern, yet the exact pattern depends on the vantage point on the earth and the

time of the season.

The sun is a dominant reference point for orientation in many species (reviewed in Able,
1980). As a compass mechanism it is pervasive in the animal kingdom (see Table 2.1).
Yet using the sun as a compass poses very speciﬁc information-processing problems that
must be solved. Since the sun appears to move over the course of a day, an animal that
uses it as a compass reference point must account for that movement. This is a signiﬁcant
problem that has attracted the attention of researchers for most of this century, but it has

yet to be explained.

The problem emerges because the directional reference of the sun compass, the azimuth,
changes at a varying rate over the course of the day. In addition the daily pattern of

movement varies with season and with latitude, so an animal must be able to use the

correct local pattern. A wide range of animals are able to deal with the special
requirements of using the sun as a compass. Throughout the dissertation, I will focus on

honey bees as a model system for understanding this problem.

In Chapter 2, I discuss the nature of the sun as a directional reference. In particular, its
apparent movement deﬁnes a complex pattern that has certain regular features but is not
completely predictable because the pattern varies with season and with latitude. I review
the distribution and use of the sun compass in the animal kingdom. I consider the
information that is available from the sun, and the ways in which animals use this

information.

The major portion of this dissertation is concerned with how insects learn about the course
of the sun for use as a sun compass. Chapter 3 describes the previous computational
models that have attempted to explain this process, and it introduces a series of
experiments that were designed to distinguish among the previous models. The results
were surprising and suggested that none of the previous models could account for the
behavior of the bees. In particular they suggested that the bees may have some sort of
innate template that is modiﬁed with experience. Speciﬁcally, their representation of the
sun's course includes aspects that they could not have observed. These include
nonlinearities in the natural pattern that were not incorporated into the previous

computational models.

In Chapters 4 and 5, I explore alternative ways of modeling the sun compass learning
problem. Chapter 4 extends the previous models to deal with the additional information
that the insects appear to possess. The previous models were of a symbolic algebraic
nature. The predictions that they produced emerged from the algebraic manipulation of
information available to the insects in their experience: the position of the sun at particular
times of day. The models assume that a neural process carries out the equivalent of the
algebraic manipulation of the experienced information about the sun. In the new model,
the symbolic nature of the process is retained. Information about solar position and time
are used to form estimates of sun's position at new times of day. Unlike the previous
models, there is additional information included in the process about the general pattern of
movement of the sun. In particular, whereas previous computational models assumed that
insects relied on linear estimates of the rate of change of the sun's azimuth, the new
symbolic model accounts for the new evidence that insects are informed of the non-linear
relationship between azimuth and time. This new model utilizes the geometry of an ellipse

to express the pattern of movement of the sun.

In Chapter 5, I present an alternative method of modeling the sun compass learning
problem. This is a connectionist neural network model (F eldman & Ballard, 1982). This
contrasts with all of the previous computational models that were of a symbolic nature,
including the new model presented in Chapter 4. A connectionist neural network is a
nonsymbolic model (Rumelhart & McClelland, 1986; Smolensky, 1988). In contrast with

the symbolic models, the representations in connectionist neural network models consist of

the distributed pattern of connections between neural elements. In the case of the sun
compass learning problem, a connectionist representation does not encode an explicit
algebraic equation describing the position of the sun at different times of day.
Connectionist neural networks are loosely inspired by the structure of real biological
neural networks (McCulloch & Pitts, 1943). They consist of a large number of simple
processing elements operating in parallel (Feldman & Ballard, 1982). A representation in
a connectionist neural network is distributed across the connection weights between the
neural elements (Hinton et al., 1986). This contrasts with the structure of a representation
in a symbolic model in which the representation consists of symbols manipulated by logical
operations (Feldman & Ballard, 1982). In the connectionist framework, the operations on
the neural elements are numerical and can be considered subsymbolic (Smolensky, 1988)
because they operate on a level below the traditional symbol. A symbol of the traditional
artiﬁcial intelligence ﬁamework may be distributed across hundreds of neural elements in
the connectionist framework. The elements independently perform numerical operations
on their component of the higher level symbol. Thus, the computations on the symbols
are not isomorphic to the higher level logical operations. For example, in a connectionist
model of stereopsis, there is no unit that computes depth (Boden, 1991). Rather the
computation of depth is distributed across numerous simple elements that each respond to
local inputs, computing some microfeature of the overall pattern. The representation of

depth emerges from the activity pattern of the network as a whole.

10

In the symbolic representation of the sun compass learning problem, input symbols
corresponding to azimuth position and time of day are manipulated by an explicit algebraic
function to yield output symbols also corresponding to azimuth position and time of day.
Since there is no explicit algebraic function that computes outputs ﬁ'om inputs in the
connectionist framework, the representation of the sun's course emerges from the activity
pattern of an entire network. It is computed from the numerous independent numerical
computations of the units in the network. This is a new way of looking at the sun

compass learning problem.

One problem with the approach advocated by Gallistel is that it focuses extensively on the
symbolic aspects of computation to the point of excluding a priori nonsymbolic solutions
to certain computational problems. Connectionist neural networks (Feldman & Ballard,
1982) may form just such a solution to some of the computational problems that Gallistel
(1990) presents. Gallistel (in press) has argued strongly that certain processes in insect
navigation cannot be implemented in a nonsymbolic (connectionist) ﬁ'amework. He
singles out the sun compass learning problem as one such process. This issue will

comprise a central focus of this dissertation.

Yet Gallistel (1990) deserves credit for focusing attention on the information-processing
problems of navigation. This presents clear computational problems that must be solved,

even in the brains of some of the smallest animals. Connectionism has often focused on

ll

much larger and more intractable problems such as language (e. g. Sejnowski &

Rosenberg, 1986).

The content of Chapters 4 and 5 sets up a dichotomy between symbolic and nonsymbolic
models. Some would argue that this is a false dichotomy (see Boden, 1991). Boden
argues that the two approaches are not as different as they might seem. In fact, both can
trace their heritage to the seminal work of McCulloch and Pitts (1943). The differences
between the approaches may reduce to differences in focus: while the symbolic approach
focuses on what is being computed, the connectionist approach focuses on how it is being
computed. It may in fact be more ﬁ'uitful to use Marr's (1982) framework, which divides
information-processing problems into three levels: computation, algorithm, and
implementation. The question of what is being computed lies at the ﬁrst level. The
second level consists of the speciﬁc algorithm used to perform the computation. There are
potentially multiple algorithms for any given computation. Finally, there is the level of the
hardware implementation. Likewise, the same algorithm may be implemented on different
hardware. The traditional symbolic approach is top-down, starting with the computational
level and proceeding to the algorithmic level. In contrast, the connectionist approach is
essentially bottom-up, although it does not start with a speciﬁc neural implementation but
a generalized one. Nevertheless, it is not easy to integrate the approaches, because they
meet on uneasy ground at the level of the algorithm and do not ﬂow smoothly together
(Clark, 1990). With this in mind, I will follow the distinction that has been maintained

from both sides of the issue (Smolensky, 1988; Gallistel, in press).

12

In Chapter 6, I present a general discussion of the issues raised in the previous ﬁve
chapters. I will focus on comparing the assumptions and predictions of the models
presented in Chapters 4 and 5. Although this entire project was based on behavioral
experiments and modeling, it attempts to describe something about the neural
computations that are occurring in the brain of the bee. From the data presented in
Chapter 3, it is clear that insects account for the nonlinear pattern of the movement of the
sun's azimuth in order to use it as a compass reference. It is still unclear exactly how they
do this. Through the construction of models presented in Chapters 4 and 5, I have
explored the domains in which this behavioral problem must be solved in the brain. There
is no question that this a computational problem that an animal with a relatively small brain
can solve. There is considerable debate about the nature of representations in the brain
and the types of computational process that exist (Smolensky, 1988; Gallistel, in press).
The symbolic-nonsymbolic dichotomy is a major question in this debate. The sun compass
problem may be able to shed some light on this issue. In Chapters 4 and 5, I show that the
neural processes necessary for sun-azimuth estimation could exist in either a symbolic or a
nonsymbolic form. Although there may be an inﬁnite number of computational models
that could approximate the behavior of the bees in sun compass learning, the modeling
effort yields testable predictions about the behavior that may be able to guide
neurophysiological research. Thus behavioral experiments combined with modeling can
be a powerful tool to help decipher the nature of neural computations on a complex

spatiotemporal representation.

Chapter 2

SUN COMPASS ORIENTATION

Compasses are mechanisms for determining direction. For most people, the most familiar
compass measures directions relative to the earth's magnetic ﬁeld. The needle on the
compass points to the geomagnetic pole. Many animals have a compass sense (reviewed
in Able, 1980). The most common of these are magnetic and celestial. I will focus on the
latter, and I will be speciﬁcally concerned with the use of the sun as a celestial compass.

A compass can be used in combination with other mechanisms to determine position, but a
compass itself only provides directional information. In this dissertation, I will be
concerned solely with the determination of direction. I will not be concerned about
determinations of position or distance, thought they are essential aspects of the navigation
process. These are interesting problems in their own right, but they are independent from
questions about the sun compass that I will be considering. Two animals could have
entirely different representations of position or distance, including none at all, and still use

the same compass mechanism for determining direction.

13

14

2.1 The Solar Reference

The sun is by far the most prominent object in the daytime sky, and it provides a useﬁil
compass reference for diurnal animals. However, unlike the geomagnetic poles, the sun is
not a ﬁxed geographic point. Rather the sun appears to move during the day because of
the rotation of the earth. Any animal that uses the sun as a compass must account for this

apparent movement.

The pattern of solar movement is consistent and predictable--it always rises in the eastern
part of the sky and sets in the western part of the sky--but the precise pattern of
movement varies with season and with latitude. During the northern summer, the sun rises
in the northeast and sets in the northwest, while during the northern winter, it rises in the
southeast and sets in the southwest. In the northern temperate regions, the sun always
passes to the south at local noon, while in the southern temperate regions it always passes
to the north at local noon. In the tropics at noon, it passes either north, south or directly

overhead depending on the season.

The general features of solar movement are conveyed in Figure 2.1 which shows the
pattern of solar movement for 40°, 25° and 5° north latitude at the equinox and at the
northern summer solstice. The ﬁgure represents a projection of the sun onto the celestial
hemisphere. The center points correspond to the zenith, which is the point directly
overhead of the observer. The outer circles correspond to the horizon line. The sun's

hourly position is plotted by the dark circles. The distance from the horizon line to the sun

15

corresponds to the sun's elevation in the sky. The distance from the zenith point to the

sun is the zenith distance.

The ﬁgure illustrates an important element of solar movement that complicates its use a
compass. This emerges from the fact that the directional information is given by the sun's
position along the horizon. In the ﬁgure, the angle on the polar plots corresponds to
direction, which is measured clockwise ﬁom north (0°). This angle is the sun's azimuth.
The pattern of movement of the azimuth is more complex than the pattern of movement of
the sun itself. The azimuth does not change at a constant rate throughout the day. During
an hour the sun moves 15° along its are, but the distance traveled along the horizon varies
with time of day, season and latitude. Extending arcs from the zenith through the sun's
hourly position to the horizon illustrates this point (see example for 40° N at the equinox
in Figure 2.1). The angular change in the sun's hourly position around the horizon is
relatively small during the early and late parts of the day as the sun is rising and setting,
but is much larger during the middle of the day. This pattern of variation is slightly
different for different seasons and latitudes. At high latitudes the variation in the rates of

change of the azimuth is much less than it is in low latitudes.

16

Figure 2.1 Pattern of solar movement for different seasons and latitudes.
The ﬁgure represents a projection of the sun onto the celestial hemisphere.
The center points correspond to the zenith, which is the point directly
overhead of the observer. The outer circles correspond to the horizon line.
The sun's hourly position is plotted by the dark circles. The distance from
the horizon line to the sun corresponds to the sun's elevation in the sky.
The distance from the zenith point to the sun is the zenith distance.
Extending an imaginary line from the zenith through the sun's position to
the horizon reveals the position of the sun's azimuth which provides the
directional information for sun compass orientation. See example for 40°
N equinox. The rate of change of the azimuth varies over the course of the
day and with season and latitude. This can be seen in the comparison of
the change in azimuth between 7 :00 and 9:00 vs. 11:00 and 13:00 (drawn
for 40° N).

l7

Equinox Solstice
N

40°NW E

1.6.00.0..S
14 12 10

 

 
 

16 8
SON W 0000030.... E W 14 11210
16141210 8

 

Figure 2.1

18

360 ~—
A - Equinox

270

Sun Azimuth (°)
8

 

90

 

 

 

 

 

0 J l
6 8 10 12 14 16 18
Local Sun Time
360 ——
B _ Solstice
270
C
:8
:3
S 180
<
S
(I)
90
O I
6 8 10 12 14 16 18
Local Sun Time

Figure 2.2 Seasonal and latitudinal variation of the ephemeris function. A.
Ephemeris functions for 5°, 25° and 40° North latitude for the equinox. B.
Ephemeris functions for 5°, 25° and 40° North latitude for the solstice.
Solar azimuth is plotted against local sun time. The two panels show the
considerable variation in the rate of change of the azimuth over the course
of the day and with season and latitude.

19

The variation in the rate of change of the azimuth, with respect to time of day, season and
latitude, is more easily analyzed when the azimuth angle is plotted against time as in
Figure 4.2. Figure 4.2 presents the "azimuth curves" for the same six season-latitude
combinations illustrated in Figure 4.1. In general, the graphs of azimuth position against
time take the form of S-shaped curves. The curves are steepest during the middle of the
day when the azimuth is changing most rapidly and ﬂatter in the early and late portions of
the day when the sun moves more slowly along the horizon. Azimuth curves ﬁom tropical
regions show a more extreme variation in slope (or rate of change of the azimuth) between
the middle and ends of the day than curves from temperate regions. A negative slope at
lower latitudes indicates that the azimuth is moving counter-clockwise around the horizon
and passing to the north of the observer at noon rather than to the south as is typical of

northern latitudes.

The shapes of the azimuth curves reveal that there are considerable nonlinearities in the
rate of change of the azimuth. To obtain directional information from the sun, an animal
must be able to account for the nonlinear relationship between the sun's azimuth and time
of day. Furthermore, since the pattern of change varies with season and latitude, the
animal must be able to utilize the appropriate local pattern of change. It has long been
known that animals are able to do this. Such an animal is said to have a time-compensated
sun compass. There are several possible mechanisms that could accomplish time

compensation.

20

The mechanisms can be grouped in two main categories. First, information about the
position of the sun's azimuth at a particular time of day could be extracted from a look-up
table (Churchland & Sejnowski, 1989) (analogous to an ephemeris table, e. g. Whiting,
1669). Such a look-up table could conceivably be genetically hard-wired. Second, the
azimuth could be computed ﬁ'om the relevant information, which could include time of
day, season and latitude. This second class of solutions corresponds to Gallistel's (1990)
computational-representational framework. Both these methods could be used to obtain
either the actual azimuth angle for a speciﬁc time of day or they could be used to obtain

approximate or estimated azimuth angles for speciﬁc times of day.

There are three further subdivisions based on the amount of information about the
movement patterns of the sun that is encoded in the animal's genes. First, the animal could
have a complete innate ephemeris ﬁrnction. This would be a universal emphemeris that
would be good any time of day and season, anywhere on the planet. To account for
seasonal and geographic variability, the animal would need to have some way of using the
appropriate innate ephemeris function. This is true regardless of whether the animal is
using a look-up table or computing the function. In order to access the true ephemeris
ﬁinction, the animal would have to assess time of day, time of year (solar declination, the
angle of latitude at which the sun is directly overhead at noon) and latitude. The true solar
ephemeris is a function of these three variables. To compute the position of the azimuth

with these parameters, the following equations are used (Brines, 1978):

21

cos(ZS) = sin(D)sin(L) + cos(D)cos(L)cos(T) (2_ 1)

sin (D) — cos(ZS)sin(L)
sin (ZS)cos (L)

 

sin(A) = (2-2)
where A is azimuth, ZS is solar zenith distance, D is solar declination, L is the latitude of
the observer, and T is time degrees from local noon. The solar zenith distance (ZS) can

be determined from equation 2.1 and substituted into equation 2.2 to obtain azimuth.

The main problem for an animal using these equations to compute the azimuth angle is in
obtaining information about the latitude and the declination of the sun. Both of these
variables are confounded in the zenith distance of the sun (equation 2.1), so it would be

difficult to assess them independently.

Second, the animal could have an innately determined ephemeris that is not universal, but
is adapted to the local geographic range of the animal. This eliminates the need to assess
latitude. This function could vary with season, or, if the animal is particularly short-lived,
and lives at a certain time of year, a single genetically determined ephemeris function may
be suitable for the animal's entire life. Again, this could be implemented with either a
look-up table or an explicit ﬁinction. In this case, some of the variables in equations 2.1

and 2.2 would be constants.

22

Third, the animal could learn the local pattern of movement with respect to local
geographic features. This could be done by ﬁlling a look-up table with time-linked
azimuth positions. This would be a form of associative learning. Alternatively, the animal
could use observable quantities such as time of day and position of the azimuth to
calculate the position of the azimuth. Such a calculation would not depend on the latitude
or declination of the sun. This calculation could be an approximation of the true azimuth
function. If the animal is long-lived and/or travels long distances latitudinally the animal

would have to update its representation of the sun's course.

Many experimental studies have implicated this third solution to the problem of accurately
compensating for the changing azimuth over the course of the day (see below for details).
Different animals may use different solutions to the problem, but the main emphasis of this
dissertation will be on the animals that rely on the third alternative. Of those animals, I

will be primarily concerned with how insects (particularly honey bees) solve this problem.

2.2 Discovery of the Sun Compass

Suggested as early as 1911 by Santschi (Santschi, 1911, discussed by Wehner, 1990), the
use of a time-compensated sun compass for orientation was ﬁrst demonstrated for both
birds (Kramer, 1950) and bees (Frisch, 1950) in 1950. The sun compass has subsequently

been demonstrated in a wide range of vertebrates and invertebrates.

23

In order to conclude that an animal is using a sun compass, its effect on the homing ability
of an animal must be dissociated from other possible compass mechanisms such as a
magnetic compass or other cues such as landmarks. If an animal orients accurately on a
sunny day, but is disoriented on a completely overcast day, that is strong evidence that the
sun is necessary for orientation. Note that even if the sun is obscured by clouds, but blue
sky is visible, a sun compass may be used, since the position of the sun can be determined
from the polarization patterns of skylight. Many animals effectively use these cues for
orientation (Adler & Phillips, 1985; Fent, 1986; Rossel & Wehner, 1986; Phillips &

Moore, 1992).

Evidence that an animal orients correctly on a completely overcast day does not rule out
the possibility that it can use the sun as a compass on a sunny day. Most orientation
mechanisms have multiple redundant systems (Able, 1980). The animal may simply use an
alternative compass mechanism or landmarks if the sky is cloudy. To distinguish between
the possible roles of different orientation cues, the cues need to be put in opposition.
Many animals will favor the sun compass when it is in opposition with other cues. One
method of accomplishing this is to clockshift the animal (Hoffmann, 1960), by putting it
on a light-dark cycle that is shifted in phase relative to the actual light-dark cycle. In these
circumstances an animal will orient in a predictable wrong direction, since the actual
position of the sun is different from the predicted position based on the animal's internal
sense of time. Another method of dissociating the direction indicated by the sun compass

from alternative compass mechanisms is to transport the study animal to a new longitude

24

where the animal's internal clock is out of phase with the local time (Papi, 1955; Renner,
1959). Again, if the animal is relying on the sun compass it will orient in a predictable
wrong direction. Longitudinal translocations also prove that the temporal information is

endogenous and independent ﬁom the position of the sun itself.

2.3 Distribution in the Animal Kingdom

A wide range of animals can use the sun for orientation. Table 2.1 provides a
compendium of the animals in which a sun compass has been strongly implicated. Time
compensation is suggested (though not deﬁnitively proved) in all of the cases in the table.
Historical precedence has been implemented throughout the table. In many cases, later
studies provided additional (sometimes critical) evidence, but they have not been included.

The list of references for each species is not exhaustive.

Much of the work on the sun compass in animals has focused on birds and arthropods.
Nevertheless, the assortment of work in other groups reveals the widespread use of the
sun compass. One is present for example in nearly all groups of extant vertebrates.
Intriguingly, our own group, the mammals, may be the least likely to use this mechanism.
This probably stems from the nocturnal activity patterns of most mammals (Bovet, 1992).

But there are nevertheless several examples of sun compass orientation in mammals.

Among the invertebrates, numerous arthropods have been shown to use the sun as a

compass. Other groups have been investigated much less, but the sun compass has been

25

implicated in mollusks (Warburton, 1973; Hamilton & Russel, 1982) and even in a
cnidarian (Scyphozoa) (Hamner et al., 1994). This latter fact is particularly intriguing

since jellyﬁsh do not have a centralized nervous system.

The variety of animals that exhibit sun compass orientation is impressive, and one could
potentially draw some conclusions regarding the evolution of the trait. But when
considering the evolutionary implications of this list, it is important to distinguish between
species that have not been tested and those that have been tested, and no evidence of
sun-compass orientation was found. It is much harder to isolate examples of this latter
type from those in which the sun compass has been veriﬁed. It is less clear which groups
speciﬁcally do not have the ability to use the sun as a navigational compass. This problem
could stem in part from non-publication of negative results, or even more signiﬁcantly,
from the lack of study of groups assumed not to have complex orientation mechanisms.
The context in which deﬁnitive negative results are likely to be published are in
comparative studies among groups of species, some of which use the sun as a compass
and some of which do not. Unfortunately, there are very few comparative studies of this
type. In the cases where this has been studied, some interesting patterns of the presence
and absence of a sun compass emerge. This is particularly true in the Coleoptera, where
some families have sun-compass orientation (e. g. Frantsevish, 1977) and others do not
(Scapini et al., 1993). A similar pattern appears to exist within the isopod genus T ylos
(Hamner et al., 1968; Ugolini et al., 1995). Further investigation of cases of this type

would be of considerable interest in an evolutionary analysis of the sun compass.

26

 

Table 2.1 Distribution of the Sun Compass in the Animal Kingdom

 

Phylum Class Genus species Reference
(Order: Family)

 

Cnidaria
Scyphozoa

Aurelia aurita Hamner et al., 1994
(Semaeostomae)

(jellyﬁsh)

 

Mollusca
Gastropoda

Nerita plicata Warburton, 1973
(Prosobranchia:

Neritidae)

(snail)

Aplysia brasiliana Hamilton & Russel, 1982
(Opisthobranchia:

Anaspidea)

(sea hare)

 

Arthropoda
Chelicerata

Arctosa perita Papi, 1955a
(Araneae: Lycosidae)
(wolf spider)

Lycosaﬂuviatilis Papi & Syrjamaki, 1963
(Araneae: Lycosidae)
(wolf spider)

 

(Crustacea)
Malacostraca

Amphipoda
T alitrus saltator Pardi & Papi, 1952
(Amphipoda:
Talitridae)
(sandhopper)
Talorchestia spp. Pardi & Grassi, 195 5
(Amphipoda:
Talitridae)
(sandhopper)

 

27

 

Table 2.1 (cont'd)

 

Orchestia
mediterranea
(Amphipoda:
Talitridae)
(sandhopper)
Orchestoidea spp.
(Amphipoda:
Talitridae)
(sandhopper)

Pardi, 1960

cited in Herrnkind, 1972

 

Decapoda

Goniopsis cruentata
(Decapoda:
Grapsidae)
(mangrove crab)

Ocypode
ceratophthalma
(Decapoda:
Ocypodidae)
(ghost crab)

Uca tangeri
(Decapoda:
Ocypodidae)
(ﬁddler crab)

Pagurus longicarpus
(Decapoda:
Paguridae)

(hermit crab)

Callinectes sapidus
(Decapoda:
Portunidae)

(blue crab)

Palaemonetes
antennarius
(Decapoda:
Palaemonidae)
(freshwater shrimp)

Schone,1963

Daumer et al., 1963

Altevogt & von Hagen,
1964

Rebach, 1978

Nishimoto & Hermkind,
l 982

Ugolini et al., 1989

 

28

 

Table 2.1 (cont'd)

 

Isopoda

T ylos latreilli Pardi, 1954

(Isopoda: Tylidae)

(littoral isopod)

Idotea baltica Ugolini & Messana, 1988
(Isopoda: Idoteidae)

(marine isopod)

 

Insecta

Collembola

Hypogastrura socialis Hagvar, 1992
(Collembola:

Hypogastruridae)

(springtail)

 

Orthoptera

Nemobius sylvestris Beugnon, 1983
(Orthoptera: Gryllidae)
(wood cricket)

Pteronemobius Beugnon, 1987
lineolatus

(Orthoptera: Gryllidae)

(swimming cricket)

Gryllotalpa Ugolini & Felicioni, 1991

gryllotalpa
(Orthoptera:
Gryllotalpidae)
(mole cricket)

 

Derrnaptera

Labidura riparia Ugolini & Chiussi, 1996
(Dermaptera:

Labiduridae)

(earwig)

 

Hemiptera

Velia currens Birukow, 1956
(Hemiptera: Veliidae) (but of Heran (1962) and
(waterstrider) Schmidt-Koenig (1975))

29

 

Table 2.1 (cont'd)

 

Coleoptera

Scarites terricola
(Coleoptera:
Carabidae)
(ground beetle)

Omophron Iimbatum
(Coleoptera:
Carabidae)

(ground beetle)

Dyschirus numidicus
(Coleoptera:
Carabidae)

(ground beetle)

Phaleria provincialis
(Coleoptera:
Tenebrionidae)
(darkling beetle)

Paederus
rubrothoracicus
(Coleoptera:
Staphylinidae)
(rove beetle)

S tenus bipunctatus
(Coleoptera:
Staphylinidae)
(rove beetle)
Lethrus spp.
(Coleoptera:
Scarabaeidae)

Eurynebria
complanata
(Coleoptera:

Carabidae)
(ground beetle)

Papi, 1955b

Papi, 1955b

Papi, 1955b

Pardi, 1956

Ercolini & Badino, 1961

Ercolini & Scapini, 1976

Frantsevish et al., 1977

Colombini et al., 1994

 

30

 

Table 2.1 (cont'd)

 

Hymenoptera
Apis mellifera
(Hymenoptera:
Apidae)
(honey bee)

Formica rufa
(Hymenoptera:
Formicidae)
(wood ant)
Lasius niger
(Hymenoptera:
Formicidae)
Cataglyphis spp.
(Hymenoptera:
Formicidae)
(desert ant)

Frisch, 1950

Jander, 1957

Jander, 1957

Wehner, 1972

 

Diptera

Diamesa spp.
(Diptera:
Chironomidae)
(Wingless glacier
nudge)

Kohshima, 1985

 

Chordata

(Pisces)
Osteichthyes

Lepomis spp.

(Perciforrnes:

Centrachidae)
(sunﬁsh)

Roccus (Alorone)
chrysops
(Perciforrnes:
Centrachidae)
(white bass)
Aequidens
portalegrensis
(Perciformes:
Cichlidae)
(port-cichlid)

Hasler et al., 1958

Hasler et al., 1958

Braemer, 1959

31

 

Table 2.1 (cont'd)

 

C ichlaurus severus
(Perciformes:
Cichlidae)

(South American
cichlid)

Uaru
amphiacanthoides
(Perciformes:
Cichlidae)

(South American
cichlid)

Scarus spp.
(Perciformes:
Scaridae)

(parrot ﬁsh)

Oncorhynchus nerka
(Salmoniforrnes:
Salrnonidae)
(sockeye salmon)

Anableps spp.
(Cyprinodontiformes:
Anablepidae)
(four-eyed ﬁsh)

Anguilla rostrata
(Anguilliforrnes:
Anguillidae)
(American eel)

Gambusia aﬂinis
(Cyprinodontiformes:
Poeciliidae)
(mosquito ﬁsh)

F undulus notti

(Cyprinodontiformes:
Poeciliidae)

(starhead topminnow)
Zenarchopterus dispar

(Cyprinodontiformes:
Hemiramphidae)
(halfbeak)

Schwassmann & Hasler,
1 964

Schwassmann & Hasler,
1 964

Winn et al. 1964

Groot, 1965

Schwassmann, 1967

Miles, 1968

Goodyear & Ferguson,

1969

Goodyear, 1970

Forward, et al., 1972

 

32

 

Table 2.1 (cont'd)

 

Micropterus salmoides Loyacano et al., 1977
(Perciformes:

Centrachidae)

(largemouth bass)

Cheirodon pulcher Levin et al., 1992
(Characifonnes:

Microcharacidae)

(tetra)

 

Amphibia

Acris spp. Ferguson 1963
(Anura: Hylidae)
(cricket frog)

Bufofowleri Ferguson & Landreth,
(Anura: Bufonidae) 1966
(Fowler's toad)

Pseudacris triseriata Landreth & Ferguson,

(Anura: Hylidae) 1966

(chorus ﬁ'og)

Ascaphus truei Landreth & Ferguson,
(Anura: Ascaphidae) 1967a

(tailed frog)

Taricha granulosa Landreth & Ferguson,
(Caudata: 1967b
Salamandridae)

(rough-skinned newt)

Rana catesbeiana Ferguson et al., 1968
(Anura: Ranidae)

(bullfrog)

Rana pipiens Jordan et al., 1968
(Anura: Ranidae)

(southern leopard frog)

Bufo boreas Gorman & Ferguson,
(Anura: Bufonidae) 1970

(western toad)

Ambystoma tigrinum Taylor, 1972
(Caudata:

Ambystomatidae)

(tiger salamander)

33

 

Table 2.1 (cont'd)

 

Reptilia

T errapene c. carolina Gould, 1957
(Testudinata:

Testudinidae)

(box turtle)

Chrysemyspicta Gould, 1959
(Testudinata:

Testudinidae)

(painted turtle)

Lacerta viridis Fisher & Birukow, 1960
(Squamata:

Lacertidae)

(emerald lizard)

Crotalus atrox Landreth, 1973
(Squamata: Viperidae)
(rattlesnake)

Natrix sipedon Newcomer et al., 1974
(water snake)

(Squamata:

Colubridae)

Regina septemvittata Newcomer et al., 1974
(water snake)

(Squamata:

Colubridae)

T rionyx spinifer DeRosa & Taylor, 1980
(Testudinata:

Trionychidae)

(softshell turtle)

Alligator Murphy, 1981
mississippiensis

(Crocodylia:

Alligatoridae)

(American alligator)

Uma notata Adler & Phillips, 1985
(Squamata: Iguanidae)
(fringe-toed lizard)

34

 

Table 2.1 (cont'd)

 

Sceloporusjarrovi Ellis-Quinn & Simon,
(Squamata: Iguanidae) 1991

(desert lizard)

Thamnophis spp. Lawson, 1994
(Squamata:

Colubridae)
(garter snake)

 

Aves

S turnus vulgaris Kramer, 1950
(Passeriformes:

Sturnidae)

(Starling)

Columba livia Matthews, 1953
(Columbiformes:

Columbidae)

(pigeon)

Lanius collurio von St. Paul, 1953
(Passeriformes:

Laniidae)

(red-backed shrike)

Sylvia nisoria von St. Paul, 1953
(Passeriformes:

Sylviidae)

(three barred warbler)

Sturnella neglecta von St. Paul, 1956
(Passeriformes:

Icteridae)

(western meadowlark)

Anas plaiyrhynchos Matthews, 1963
(Anseriforrnes:

Anatidae)

(mallard)

Pygoscelis adeliae Emlen & Penney, 1964
(Sphenisciformes:

Spheniscidae)

(Adelie penguin)

 

35

 

Table 2.1 (cont'd)

 

Zonotrichia albicollis Able & Dillon, 1977
(Passeriformes:

Emberizidae)

(white-throated

sparrow)

Aphelocoma Wiltschko & Balda, 1989
coerulescens

(Passeriformes:

Corvidae)

(scrub jay)

Erilhacus rubecula Helbig, 1991
(Passeriformes:

Turdidae)

(European robin)

 

Mammalia

Apodemus agrarius Li'iters & Birukow, 1963
(ﬁeld mouse)
(Rodentia: Muridae)

Delphinus delphis Pilleri & Knuckey, 1969
(Cetacea:

Delphinidae)

(common dolphin)

Microlus Fluharty et al., 1976
pennsylvanicus

(Rodentia: Cricetidae)

(meadow vole)

Spermophilus Haigh, 1979
tridecemlineatus
(Rodentia: Sciuridae)
(thirteen-lined ground
squirrel)
Eptesicusﬁiscus
(Chiroptera)

(big brown bat)

Buchler & Childs, 1982

 

36
2.4 Directional Information from the Sun
The sun can provide a directional reference for many different types of oriented activity.
These range from relatively simple directional movements to true navigation over long
distances. In this section, these categories are introduced. The next section provides a

detailed account of sun compass orientation in a particular animal.

2.4.1 The Sun as a Point Source

The simplest form of directional orientation takes the form of a taxis in which the
organism makes directed movements toward or away from a particular cue (see Schone,
1984). Phototaxis is a common type of this movement. Since the sun is the strongest
point source of light on the planet, many organisms make directed movements toward it.
This would not be classiﬁed as sun compass orientation, even though it can lead to
oriented behavior. Organisms that exhibit merely a phototactic response rather than

time-compensated sun compass orientation are not included in Table 2.1.

When an animal moves at a ﬁxed angle relative to a directional point source, it exhibits a
form of orientation termed menotaxis (see Schone, 1984). This mechanism can be relative
to the sun or other celestial bodies and involve time compensation: chronometric
astromenotaxis. For many animals that use this orientation mechanism, the sun deﬁnes a

locally important direction.

37

2.4.2 The Sun Deﬁnes a Local Direction

Many animals live in environments with important directions deﬁned only in one
dimension. This occurs for species that live along an ecotone, such as the shore of a
ocean, lake or river. Many of the species that live in such areas have orientation
mechanisms that allow them to detemrine the local direction perpendicular to the ecotone.
This form of orientation has been termed y-axis orientation (Ferguson, 1967) to
distinguish it from orientation along the x-axis: the shore or ecotone. This mechanism has
also been called zonal orientation (Jander, 1975) and it has received considerable attention
(see Hermkind, 1983; Pardi & Ercolini, 1986). The sun compass is a dominant orientation
reference that is used in these circumstances, although it is one of many possible
mechanisms (Hartwick, 1976). In these cases, the sun deﬁnes a locally important direction
for species that move efficiently to safety zones that are in one particular direction. The
directed movements occur irrespective of time of day, so this compass mechanism is time
compensated. An example of this type of movement is between the edge and the middle
of a lake. This mechanism is used by many ﬁshes (Goodyear, 1970; Goodyear & Bennett,
1979) and amphibians (Ferguson & Landreth, 1966; Ferguson, 1967; Landreth &
Ferguson, 1967; Ferguson et al., 1968; Jordan et al., 1968) in addition to some insects and
crustaceans. An analogous situation occurs on beaches where the y-axis direction is
perpendicular to the beach. Many littoral zone animals use this mechanism as an escape
response (reviewed in Hermkind, 1972, 1983). This orientation system has been

extensively studied in the amphipod crustacean T alilrus saltator, which has revealed

38

genetic and learned components of the behavior (Pardi & Papi, 1952; Pardi, 1960; Pardi &

Scapini, 1983; Ugolini & Macchi, 1988; Scapini & Fasinella, 1990).

2.4.3 The Sun Deﬁnes a Global Direction

Y-axis orientation can be thought of as a mechanism of local direction-ﬁnding.
Speciﬁcally, the local direction is the direction toward or away from a shoreline. The sun
can also deﬁne a global direction. Thus the local environment can be linked with a global
geocentric reference frame. This is particularly important for long distance migrants.
Migrating birds for example must ﬂy in a particular geocentric direction (e. g. north).
Many birds use the sun or the pattern of skylight polarization to determine their
migrational direction and/or calibrate their magnetic compass (reviewed in Wiltschko, &
Wiltschko, 1991). In some cases there is no evidence of time compensation (or it has not
been examined) (Moore, 1980) while in others there is (Helbig, 1991). Thus the sun or
skylight polarization patterns can be used to deﬁne a geocentric coordinate without
necessarily being used as a time-compensated compass. Since the sun always sets in the
western half of the sky if restricted to a particular time of year, the sunset deﬁnes a static
direction. Sunset calibration could thus occur without a true time-compensating sun
compass. In Table 2.1 I have attempted to restrict the cases to species in which time

compensation is likely.

39

2.4.4 The Sun Deﬁnes Any Flexible Direction

In many cases, the sun compass can be used to obtain directional information about any
ﬂexible local direction. An example of this type of orientation is seen in the decapod
crustacean Ocypode ceratophthalma (Daumer et al., 1963). This crab provides a contrast
to the many beach-dwelling crustaceans and insects that merely exhibit y-axis orientation.
The crab maintains a burrow in the sand, the entrance of which is not demarcated by
landmarks. After moving about the beach in search of food, if this crab encounters a
potential predator it will run directly to its burrow instead of adopting a stereotyped
response direction toward or away from the water. The escape run is oriented by the sun
and the skylight polarization patterns and its use of the sun compass is time-compensated.
This suggests a ﬂexible system that is capable of learning any direction with respect to the
sun. Such a system is constantly updated as the position of the animal and the position of
the sun change. This system, known as path integration, has been extensively studied in
desert ants (W ehner & Wehner, 1986; Muller & Wehner, 1988) and it appears to exist in
beetles (Frantsevish et al., 1977) and bees (Frisch, 1950, 1967) among other arthropods.
Many vertebrates have the ability to use the sun to deﬁne ﬂexible directions. It has been
particularly well studied in homing pigeons (e. g. Matthews, 1953; Keeton, 1969;

Wiltschko, et al., 1976, 1984; Wiltschko, & Wiltschko, 1981).

2.5 Honey Bees as a Model System

It is clear that numerous animals may use the sun as a compass. In this dissertation, I am

mainly concerned with how the sun compass is learned and how animals are able to

40

estimate the position of the sun at times of day when they have never seen it. (I will
describe this phenomenon in detail in the next chapter). The data that I rely most heavily
on come from insects, particularly honey bees. This section examines the role of the sun

compass in the life of the bee.

2.5.1 Long Distance Orientation

The sun compass provides directional information for orientation over long distances. It is
by no means the only source of such information. Other potential compass mechanisms
such as a magnetic compass could play a role, and landmarks play a prominent role in
orientation. It is unclear exactly what mechanisms freely ﬂying bees use most often to
navigate, but it is safe to assume that the sun compass and landmarks play the dominant
roles (Dyer, 1996). This information has been gained through displacement experiments.
In the most common type, bees are trained to obtain a sucrose solution from an artiﬁcial
feeding station a particular distance and direction from the hive in one landscape.
Subsequently, overnight, the hive is moved to a new landscape and when the hive is
opened during the day, the landings of bees are recorded at identical feeders placed at
various directions from the hive (F risch, 1967). Alternatively, bees are captured at the
feeding station and subsequently released in a novel location. The new environment in
either of these cases may have landmarks that resemble the site from which they originally
experienced, or the site may have no prominent landmarks or landmarks that do not

resemble the ones from the original feeding site. Hives can also be moved over much

41

greater distances longitudinally (Renner, 1959) or latitudinally (Lindauer, 195 7) to

distinguish the role of the celestial compass from a magnetic compass.

It is clear that when there are no landmarks present, the bees use the sun to navigate

(F risch, 1967). When landmarks are present that mimic the landmarks from the training
site, the bees generally use the landmarks (F risch & Lindauer, 1954). However, the
landmarks must be very prominent to overrule the sun compass. In some cases, bees will
ignore even very large landmarks and follow the direction indicated by the sun (Menzel et

31,1990)

2.5.2 Short Distance Orientation
In addition to determining a directional heading for long distance orientation, the sun
compass can play a role in short-distance orientation. In this context, the sun compass can

be used to resolve ambiguous landmarks (Dickinson, 1994).

It has long been known that insects use landmarks to guide their approach to a goal
(Tinbergen, 193 8). Recent studies have suggested that the insects move to a position to
match a stored retinal image of the landmarks (Wehner, 1972; Collett & Land, 1975;
Wehner & Raber, 1979; Cartwright & Collett, 1983; Dill et al., 1993). This poses a
problem for a ﬁ'ee ﬂying insect that might encounter a landmark from any number of
angles. Many potential landmarks such as trees and shrubs have radial symmetry which

would render them directionally ambiguous. Because they look the same ﬁom all

42

directions, an insect trying to match a retinal image would search in an annulus. To

resolve this ambiguity an external direction reference is required.

Lindauer (1960) ﬁrst suggested that the sun might provide such a reference. He trained
bees to obtain food ﬁom one comer of a symmetrical table. The rewarded comer was
consistently in a particular compass direction. Subsequently, he moved the hive to a new
location and placed the feeding table in a new direction. Nonetheless the bees
predominantly visited the original position on the feeder. Lindauer concluded that the
bees were using a time-compensated celestial compass mechanism in this task. However,
other potential directional references could not be ruled out, such as the earth's magnetic

ﬁeld.

To determine if bees were using the sun compass to resolve ambiguous landmarks on this
small of a scale, I trained bees to obtain food in an arena at one of four identical feeders
placed in the cardinal compass directions from a prominent cylindrical landmark
(Dickinson, 1994). Walls around the arena excluded panoramic landmarks and the
cylinder was symmetrical so any directional information the bees used would have to come
from an external reference. Bees were trained individually to ﬁnd food at one of the
feeders. In the test, there was no food present, and the number of landings on each feeder
(out of the ﬁrst 10) were scored. Bees learn this task very rapidly; on a sunny day they

search in the appropriate direction after only two visits to the correct feeder. Figure 2.3

43

(A) shows the proportion of visits to the correct feeder on their ﬁfth return to the arena.

Clearly they are able to resolve the ambiguous landmark.

To determine what directional reference the bees were using, I tested them under three
additional conditions. Figure 2.3 (B) shows the responses under partly cloudy sky with
the sun obscured but with polarization cues available; Figure 2.3 (C) shows the responses
of the bees on completely overcast days; and Figure 2.3 (D) shows the responses of bees
that were subjected to a delay of three to four hours (90° of azimuthal arc) between the
last training trial and the test trial. The bees were signiﬁcantly oriented under all
conditions except those of complete overcast. This suggests that they were using a
time-compensated celestial compass to resolve the ambiguity of the landmarks as Lindauer

(1960) originally suggested.

These results are somewhat puzzling in light of some recent results obtained by Collett
and Baron (1994). In a similar study to the one described above, they found that bees
adopt a stereotypic viewing angle of the landmarks near a goal. The viewing angle is
maintained with respect to the earth's magnetic ﬁeld. Adopting such a consistent viewing

angle is an alternative means of resolving the ambiguity of landmarks (Collett, 1992).

In additional studies using a large screen to exclude polarization patterns, I replicated my
original ﬁndings and found no evidence that the bees could use a magnetic compass to

choose the appropriate direction (Dickinson, unpublished data). However, it appears that

44

the bees may simply take much longer to learn to rely on magnetic cues in these
circumstances. In Collett and Baron's (1994) experiments, the bees had a much longer
training period, amounting to more than a day. This contrasts considerably with the ﬁve

visits that the bees had in my experiments.

Nevertheless, whether or not the bees are capable of using a magnetic compass under
certain circumstances, it is clear that bees are able to use the sun as a directional reference

for orientation over short distances.

45

75

A p = .0004 B p = .007

 

 

 

 

Figure 2.3 Proportion of choices to each of four directions with respect to
the training direction. A. Sunny. B. Partly cloudy C. Completely
overcast. D. Delay. The probabilities presented are the binomial
probabilities that the pattern of choices occurred by chance (25% correct
choice; 75% incorrect).

46

2.5.3 The Dance Language

In addition to using the sun as a navigational compass, the sun's azimuth plays a prominent
role in the communicative dance language of honey bees (Frisch, 1967). Honey bees
direct colony mates to proﬁtable food sources by means of a symbolic waggle dance. The
dance is so named because the prominent feature of the dance involves the bee vigorously
shaking her abdomen. In the process, she completes a number of circuits in which she
waggles in a straight line at a particular angle on the vertical comb. Between waggle runs,
the bee turns alternatively to the left or right and returns to the starting position where she

commences another waggle run.

The dance communicates the distance and direction to a food source. Distance is
determined by the duration of the waggle run. Direction is determined by the angle of the
waggle run with respect to vertical. The angle of deviation ﬁ'om vertical corresponds to
the angle of deviation of the direction of the food from the sun's azimuth. For example, a
bee dancing 75° to the left of vertical is indicating direction 75° to the left of the sun's

azimuth (see Figure 2.4).

Over the course of a day, a bee visiting the same food source will change her angle of
dance as the sun's azimuth changes position. On a cloudy day, when the bee cannot see
the sun, she orients her dance to a memory of the sun with respect to landmarks (Dyer &

Gould, 1981). Over an entire day, her dances changes orientation to reﬂect the changing

47

position of the sun (Dyer, 1987). This provides an essential tool for studying the bee's
representation of the sun's course (Lindauer, 1957; New & New, 1962; Wehner, 1972;

Wehner & Raber, 1979; Dyer, 1985, 1987; Dyer & Dickinson, 1994).

48

 

 

 

Figure 2.4 The dance language of the honey bee. The communicative
waggle dance indicates the distance and direction to the food. A. The
angle indicated by the waggle run (or) with respect to vertical corresponds
to the angle of the food with respect to the sun's azimuth. B. The ﬂight
path from the hive (H) to a food source (F) relative to the sun's azimuth the
angle or is the same as in (A). Over the course of the day, the dance angle
changes as the solar azimuth moves along the horizon. Distance is
indicated by the length of the waggle run.

49

2.6 Other Celestial Compass Mechanisms

Besides the sun, there are other celestial bodies that can contribute to orientation. In
principle they have similar requirements to the sun compass, and they may reveal similar

learning mechanisms.

2.6.1 The Lunar Compass

The lunar compass has been implicated in at least two species of amphipod crustaceans:
T aliirus saltator (Papi, 1960) and Orchestoidea corniculata (Enright, 1961).
Compensating for the change in lunar azimuth is considerably more difﬁcult than
compensating for the change in solar azimuth, because the pattern of lunar azimuthal

change differs nightly.

The Asian honey bee Apis dorsata ﬂies by the light of the moon at night (Dyer, 1985), but
it does not use the moon in place of the sun to orient its dances at night. Instead, it
continues to use the sun as a reference in the communicative waggle dance. It appears to
extrapolate the position of the sun after sunset (see Chapter 3). Diurnal insects may orient

to the moon if they are forced to be active at night, but they appear to use it as the sun

(Jander, 1957; Wehner, 1982).

2.6.2 The Stellar Compass
Just as ancient sailors did, animals that are active during the night have the opportunity to

use the stars as a navigational guide. For some birds that migrate at night, this seems to

50

be the case (Sauer & Sauer, 1960). In the case that has been most thoroughly studied, the
birds obtain global directional information from the stars. This information comes from
the overall pattern of rotation (Emlen, 1967a, b). The center point of the rotation deﬁnes
true geographic north (or south). This orientation mechanism is not affected by clock
shifting the birds (Emlen, 1967b), so with respect to time compensation the system is

distinct from the sun compass.

2.7 Summary

Numerous animals use the sun compass for orientation in many different circumstances.
Animals that use a time-compensated sun compass for orientation must account for a
nonlinear change in direction indicated by the sun's azimuth with respect to time of day.
Further complicating the matter is the fact that the nonlinear pattern of change varies with
season and latitude. Yet the animals that use the sun compass for orientation are able to
solve these problems. The mechanisms that could account for this ability are considered in

the subsequent chapters.

Chapter 3

THE REPRESENTATION OF THE SUN'S COURSE

The complex pattern of movement of the sun's azimuth (described and illustrated in the
previous chapter) presents a problem to all animals that use the sun as a compass. Since
the pattern of movement of the azimuth varies with time of day, season and latitude, the
animal must somehow be informed of the correct local pattern of movement. As
discussed in the previous chapter, there are several possible types of solutions to this
problem, which vary in the degree of learned versus innate information about the sun's
course included in the model. In this chapter I will discuss recent experiments with honey

bees that suggest that the sun compass is neither exclusively innate nor learned.

These studies suggest that honey bees employ a computational solution to the problem of
compensating for the sun's apparent movement. First I will describe the previous studies
that set the stage for a thorough investigation into the nature of the computations and
representations involved. In the main body of this chapter, I will present the data from
recent experiments by Dyer and Dickinson (1994) that provided critical new insights into
the nature of the computations and representations. In addition to presenting the original

data from Dyer and Dickinson (1994), I present two other data sets, and I provide new

51

52

analyses of all of the data to further illuminate the nature of the computations involved. In
Chapters 4 and 5, I present new models of the computational processes of sun compass

learning that may shed light on the underlying mechanisms.

3.1 Previous Studies

For honey bees it has long been known not only that bees learn the course of the sun, but
that they can estimate the position of the sun at times of day when they had never seen it
(Lindauer, 195 7, 1959). Behavioral experiments over the nearly 40 intervening years have
elucidated to a considerable degree the nature of the honey bee's representation of the

sun's course.

Some of the earliest evidence about the nature of the bee's representation came from the
observation of bees dancing within a colony during the night (Lindauer, 1954). These
"marathon" dances occur for considerable lengths of time without the dancing bee leaving
the colony. Over time the dance angles change in correspondence with the change in the
sun's azimuth. While "marathon" dances can occur during the day or the night, the
nocturnal dances are particularly striking because the bees would have never had an

opportunity to observe the position of the sun during the night.

The use of the sun compass during an animal's physiological night has been reported for
several other species including sandhoppers (Pardi, 1958b) spiders (Tongiorgi, 1959; Papi

& Syrjamaki, 1963), beetles (Pardi, 1958a), water striders (Birukow, 1956; Birukow &

53

Busch, 1957) (but cf. Heran (1962) and Schmidt-Koenig (1975)), ants (Wehner, 1982),
ﬁsh (Braemer, 1959) and birds (Hoffrnann, 1959; Schmidt-Koenig, 1961, 1963). These
studies have relied on three types of methods: shifting the phase of the animal's internal
clock (Birukow & Busch, 1957; Pardi, 1958b; Braemer, 1959; Tongiorgi, 1959;
Schmidt-Koenig, 1961; Papi & Syrjamaki, 1963), presenting an artiﬁcial light source
(Birukow, 1956; Pardi, 1958a; Braemer, 1959), and transporting the animal north of the
Arctic Circle to expose it to the midnight sun (I-Ioﬂinann, 1959; Papi & Syrjamaki, 1963;

Schmidt-Koenig, 1963).

Bees can also estimate the position of the sun during the day at times when they have
never seen it (Lindauer, 1959). When honey bees are restricted to view the path of the
sun during a portion of the whole day, they nonetheless can estimate the position of the
sun during other times of the day. In Lindauer's (1959) experiments, be restricted the
experience of bees by raising brood in an incubator and establishing a colony of which he
could completely control the experience. He only opened the colony and allowed the bees

to acquire information about the sun during a limited portion of the day.

Lindauer (195 7) trained the bees in the afternoon to ﬁnd food in the south. Subsequently,
the hive was moved to a new location (so the bees could not rely on landmarks), and the
bees were allowed to ﬂy for the ﬁrst time in the morning. Four feeding stations were set
up in the cardinal compass directions and the number of bees arriving were recorded. The

bees ﬂew predominantly to the south in search of food if they had seen the afternoon sun

54

for ﬁve days. With only three days of experience, the bees failed to choose the correct

feeding station.

This was the ﬁrst indication that the course of the sun was learned by the bees. Indeed, it
also showed that bees do not simply remember observed positions of the sun at various
times of the day, but that they can compute the position of the sun at times of day when
they had never seen it. The nocturnal dances and the behavior of Lindauer's (1959) bees
in the morning support the conclusion that bees can generalize their experience to other
times of day. In other words, the bees appear to form a representation of the sun's course
that consists of a complete ftmction, even though they may have experience with only a

portion of the ﬁinction.

Unfortunately, Lindauer's (195 9) data were not accurate enough to determine the
computational methods the bees used to ﬁll in the gaps in their experience with the sun.
This is because in the experiments in which be restricted the bees' experience, he only
measured their response to the nearest 90° (since the feeders were placed in four
directions). However, subsequent studies have provided potential answers to this question

and have revealed more of the details of the insect's representation of the sun's course.

Some of the most illuminating data come ﬁom studies conducted in the tropics. As
illustrated in the previous chapter, the pattern of change in the solar azimuth is most

variable in the tropics. The most extreme variation in the pattern of solar movement in the

55

tropics occurs when the azimuth abruptly switches from passing to the south at noon to
passing to the north (or vice versa). New and New (1962) studied this transition period in
colonies of bees in the tropics. They used the honey bees' dance language (F risch, 1967)
(see section 2.5.3) to investigate the bees' representation of the sun's course during the
transition time. They trained bees to an artiﬁcial food source and recorded the dance
angles of the bees visiting the food source as they changed from indicating the position of
the food relative to an eastern azimuth and indicating the position of the food relative to a
western azimuth. They found that the bees had consistently oriented dances even when
the sun passed very close to the zenith. However, the bees' representation of the sun
differed from the actual sun's course in systematic ways (see Figure 3.1). During the
transition phase, the rate of change of the sun's azimuth indicated in the bees' dances was
less rapid than the actual change in the sun's azimuth. Perhaps more striking is the fact
that the bees anticipated the transition ﬁ'om a southern to a northern noon position before
it actually occurred. Some individual bees (not shown in Figure 3.1) differed from each
other in the pattern of azimuthal movement they indicated. New and New (1962)
concluded that the bees were interpolating between known positions of the sun. That is,
the bees behaved as if they were shifting their dance angles at a constant rate between
memorized positions of the sun. New and New suggested that the interpolation occurred
during the noon transition because the actual changes in the azimuth were below the visual
acuity threshold of the bees. A similar mechanism could be used to ﬁll other gaps in
experience, such as during the night or during long periods of time during the day that the

bees do not experience the pattern of azimuthal movement.

>

5

0

(°) ,3

 

 

 

    

 

 

Em
C:
3
U)
90
O
O
o 1 I
10 12 14
LocalSunTime
360 —
0:
27o _ ﬁ
C
.grso —-
5
U.)
90 .e.
O
_ O
0 . 1 . 1
10 12 14
LocalSunTimc
E
360 — .
t .: e
l' O
270 —
6
§
.Erso _
2 .
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VJ
90
I o
0 . it 1

 

 

12
Local Sun Time

(°) N

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Sun Azimuth

 

\l
0

(°) N

5

Sun Azimuth

8

\l
o

(°)N

'55
c

Sun Azimuth

8

 

 

 

 

10 12 14

Local Sun Time

Figure 3.1 Solar azimuths inferred from the dances of the bees in New and
New's (1962) experiments. Data are from colony 1 in Trinidad (10°3 8') on
(A) April 15, (B) April 16, (C) April 17, (D) April 18, (E) April 20, and (F)
April 21. Data were obtained from New and New's (1962) Figure 3 and
replotted to correspond to solar azimuths rather than dance angles.

57

The interpolation hypothesis of New and New (1962) is one of four distinct
computational strategies that have been proposed to explain how insects can ﬁll in the
gaps of their experience with the sun's course. Three of the mechanisms assume that
observed positions of the sun at speciﬁc times of day are used to calculate the position of
the sun at other times of the day. The fourth mechanism assumes that the insects rely on
the average rate of movement of the sun (15° per hour). Once a rate is obtained, all of the
models assume that the position of the sun's azimuth could be determined by a neural

implementation of an equation equivalent to the following:
At+ngt+rt (3.1)

where Am is the unknown azimuth; A, is the observed azimuth; t is the time of
observation; t is the time interval between the observed and estimated azimuths; and r is
the rate of change of the azimuth. The computational models differ in how the rate of

change of the azimuth (r) is determined.

First, in the average rate hypothesis, r is simply 15° per hour. This is distinct from the
other three methods which imply that the bees measure or calculate a rate of compensation

but that they differ in the method of calculation.

Second, in the interpolation hypothesis, r is determined by computing the rate of change of
the azimuth between two observed positions of the sun. This assumes a neural

implementation of the standard rate equation:

58

_A2-A1

— 12;“ (3.2)

where Al and A2 are two observations of azimuth position at two speciﬁc times of day (t]

and 122).

Third, an alternative mechanism of computing r for use in equation 3.1 that has been
proposed relies on the extrapolation of a single observed rate of change of the azimuth
(Gould, 1980). This single value of r could be used to determine the position of the sun's

azimuth later in the day (forward extrapolation).

Fourth, in a variation of the extrapolation hypothesis, a single value of r could be used to

determine the position of the sun's azimuth earlier in the day (backward extrapolation)

(Dyer, 1985).

Until recently, the experimental results were ambiguous regarding which method of
computing the rate of change of the azimuth was used to ﬁll in the gaps of an insect's
experience with the pattern of change of the sun's azimuth. None of the data supports the
conclusion that the average rate of movement is used. Several studies with honey bees
(New & New, 1962; Dyer, 1987) and desert ants (Wehner & Lanfranconi, 1981; Wehner,
1982) support the interpolation hypothesis. On the other hand other studies with bees

have been more consistent with the forward (Gould, 1980; Dyer, 1985) or backward

59

(Dyer, 1985) extrapolation hypotheses. Lindauer's (1959) data are consistent with all of

these mechanisms because of the 90° separation between the test feeding stations.

Recent data (Wehner & Muller, 1993; Dyer & Dickinson, 1994) have suggested a new
way of thinking about how insects ﬁll the gaps in their experience. When bees and ants
were confronted with large gaps in their experience of solar movement they nevertheless
form a representation of the sun's course that approximated its actual course during the
entire day. None of the previous computational mechanisms could account for this
performance. The results of these experiments suggest that the insects have a certain
amount of innate structure in their representation about the sun's course. Presumably the
innate structure could be modiﬁed with experience. The previously hypothesized
computational mechanisms may still play some role in the process (Dyer, 1987), but they

are not used to estimate the position of the sun's azimuth during large gaps in experience.

In the subsequent sections of this chapter I will describe the experiments of Dyer and
Dickinson (1994) in further detail. In addition, data from two additional colonies are
presented, and new analysis are presented for all of the data. In the subsequent chapters,
new computational models are presented which may shed ﬁirther illumination on the
process leading to the patterns observed by Dyer and Dickinson (1994) and Wehner and

Muller (1993).

60

In the experiments with Fred Dyer, I utilized the technique developed by Lindauer (1959)
of establishing colony of bees naive to a portion of the sun's course in an attempt to
distinguish between the previous computational hypotheses. Instead of relying on feeding
stations, I utilized the communicative waggle dance of the honey bee (F risch, 1967) (see
section 2.5.3), which can provide a readout of the position of the sun. This occurs
because in the symbolic system, the distance and direction to the food is indicated with
respect to the sun. The angle of the dance on the vertical comb in the hive corresponds to
the angular deviation between the solar azimuth and the food source. This system is used
even on completely overcast days when bees do not have a view of the sun (Dyer &
Gould, 1981). During these times, bees rely on a memory of the sun's position with

respect to landmarks.

In these experiments, bees were allowed to ﬂy only in the late afternoon, hence they had
experience only with a small portion of the sun's course (~20%). How the bees estimated
the position of the sun in the morning was determined by their dances on a completely
overcast day. Figure 3.2 shows the predictions for this experiment based on the previous
computational models. With the large size of the gaps in the experience in the bees in
these experiments, the predictions for the previous computational models of interpolation,
and forward and backward extrapolation were very different. For interpolation (line A), a
linear rate of time compensation is expected between the last view of the sun in the
evening and the ﬁrst view of the sun in the afternoon of the following day. This prediction

overestimates the movement of the sun in the morning. For forward extrapolation (line

61

B), a linear rate of compensation based on the observed rate during the training period is
expected. This rate is used to extrapolate the position of the sun's azimuth forward,
through the night and into the morning of the following day. This prediction considerably
underestimates the movement of the sun in the intervening time period. For backward
extrapolation (line C), the same linear rate of compensation is used as in forward
extrapolation, only it is extended backward into an earlier portion of the day. This
prediction overestimates the position of the sun in the morning by a considerable margin.
It was our hope with this experiment that we would be able to decide among the three

hypothesized mechanisms.

62

Training Period

360
F

270 —

   
 

180 Sun Azimuth

Sun Azimuth (°)

90

 

 

Local Sun Time

Figure 3.2 Predictions from previous computational models about how
insects ﬁll gaps in their experience with the sun's course. The shaded
region corresponds to the time period of experience with the sun's course.
The lines represent the predictions of the position of the azimuth at other
times of day based on the various models. A. Interpolation: a constant
rate of change between the end of the one training period and the beginning
of the next on the subsequent day. B. Forward extrapolation: the rate of
change observed at the end of the training period extended forward
through the night and into the morning of the subsequent day. C.
Backward extrapolation: the rate of change of the azimuth observed at the
beginning of the training period and extended backward into earlier
portions of the day.

63
3.2 Experimental Methods
Combs of the European honey bee (Apis mellifera) were hatched out in an incubator and
used to establish colonies of bees whose experience was completely known. In the four
colonies, the bees were allowed to ﬂy only during a small fraction of the day. During this
time, the bees gained experience with the sun as it moved over a portion of its diurnal
course. Three of the four colonies were exposed to portions of the afternoon course of
the sun. The fourth colony experienced a portion of the sun's course in the morning. For
the ﬁrst three colonies the technique of measuring the dance angles of the bees on a cloudy
day was used (Dyer, 1987). For the fourth colony, the technique of measuring ﬂight

bearings on a sunny day was used (Meder, 1958).

3.2.1 Colony 1

The ﬁrst colony was established from incubator-reared bees in a two-frame observation
colony adjacent to an alfalfa ﬁeld on the farms of Michigan State University, East Lansing,
Michigan (43° 45' N). Bees were trained to ﬁnd food at an artiﬁcial feeder containing
sucrose solution 350 m to the south along a prominent line of trees. Bees were
individually labeled with numbered tags. The colony was open from 15:00 until dark
(~19200) (local sun time ). Each night after dark the colony was closed and removed from
the ﬁeld. Bees in the hive were exposed to diffuse light during daylight hours in an

attempt to maintain the colony's circadian cycle.

64

On a cloudy day when bees cannot see the sun, the colony was opened in the morning,
The dances were recorded to the nearest 5° with a protractor referenced to gravity. The
inferred solar azimuths were calculated from the dance angles (azimuth = 180° - dance

angle). This colony corresponds to colony 1 in Dyer and Dickinson (1994).

3.2.2 Colony 2

The experiment was repeated with similar conditions for the second colony. The colony
was established in the same location and the bees received similar regime of training.
Again, the colony experienced the sun for about the last four hours of daylight. In
contrast to colony 1, the hive contained two groups of bees that received different
treatments. One group was allowed to ﬂy only during the late afternoon. The second
group, however, was able to ﬂy during the entire day. To accomplish this, the
restricted-experience bees had small pieces of plastic glued to their thoraxes (G. E.
Robinson, personal communication). A grating over the entrance of the hive precluded
these bees from leaving the colony during certain times. The other bees were able to
crawl through the small openings of the grating. This colony corresponds to colony 2 in

Dyer and Dickinson (1994).

3.2.3 Colony 3
Colony 3 received a different amount of experience with the sun's course than colony 1

and colony 2. Colony 3 was established in the same location as colonies 1 and 2.

65

This colony was open from noon (local sun time) onward, so they experienced a
considerably larger portion of the sun's course than either colony 1 or colony 2. This
amounted to 50% of the diurnal course of the sun. The bees were trained as in colonies 1
and 2, but they were only trained 150 m south along the treeline. (To separate the
potential ambiguity between an accurate representation of the sun's course and the ability
of the bees to see the sun or polarized light through the clouds, these bees were originally
intended to be tested in an alternative location, but no bees reached the feeder in several

attempts).

3.2.4 Colony 4

For this manipulation, the ﬂight bearings of bees captured at the feeder and released in a
novel environment after a holding period were used to determined the bees' estimates of
the sun's course. This technique has been successfully used in the past as a measure of the
honey bee's representation of the sun's course (Meder, 1958). Under these circumstances,
bees orient their ﬂights with respect to the sun compass, compensating for the change

during captivity.

Colony 4 was established in a ten-frame hive at the Inland Lakes Research Center on the
Michigan State University campus. This colony experienced the sun only during the
morning hours from dawn until 10:00 local sun time. The colony was closed at that time
by simulating rain with a garden sprinkler (G. E. Robinson, personal communication).

This stops the majority of bees from leaving the colony. After a steady "rain" of about 30

66

minutes, the colony was sealed with a screen and the entrances shaded from a view of the

sky.

The bees were trained to an artiﬁcial sucrose source 400 m west through an open ﬁeld of
grass and wild ﬂowers. A small lake was located 30 m north of the ﬂight route and a few

scattered trees and an interstate highway was located 30 m south of the ﬂight route.

Bees leaving the colony were automatically dusted with a bright ﬂuorescent pigment
(Day-Glo Color, Cleveland, Ohio). This allowed bees visiting the feeder from other
colonies to be identiﬁed (by their lack of pigment), so that only restricted-experience bees
were used in the releases. Additionally, bees were painted at the feeder and were given a
unique color for each day they were at the feeder. After several days of foraging at the
feeder, the bees were captured individually in vials and held in a dark place before their

release time.

Subsequently, the bees were released at a new location 3.2 km from the training location.
The bees were released between an alfalfa ﬁeld and an abandoned ﬁeld on the Michigan

State University Farms.

67
3.3 Results and Discussion
3.3.1 Colony 1
An unexpected pattern was revealed in the bees' estimates of the position of the solar
azimuth in the morning. Figure 3.3 shows the solar azimuths inferred from the dances of
the bees in the ﬁrst experiment. In the morning, the bees danced as if they estimated the
sun to be approximately 180° from the position in the middle of the afternoon training
period (M = 89.5°, SD = 290°, n = 133). The colony shifted from indicating a morning
direction to indicating an afternoon position around noon. For this colony the shift
appears to occur at 10:50 rather than noon. The descriptive statistics presented here
employ the assumption that time of day distinguishes the morning and afternoon groups.
The means and standard deviations were computed using the appropriate methods for the

analysis of circularly distributed data (Batschelet, 1981).

Over the entire morning the bee's estimation of the sun's position did not change. A linear
regression of the morning data was not signiﬁcant, (T (two-tailed) = 1.77, df = 134, p >
.025) meaning that during the morning they were not compensating for a changing solar
azimuth (i.e. the slope of the regression line was not different from zero) (see Table 3.1).
For all of the regression analyses reported, the groups were split by angle rather than time
(i.e. morning is less than 180° and afternoon is greater than 180°). I did this because I
wanted to examine the behavior of individual bees, and I assumed that individual bees

would have slightly different circadian rhythms. The individual would nonetheless shift

68

from a morning angle (< 180°) to an afternoon angle (> 180°) at her individual

representation of noon.

Around noon the bees abruptly shifted as a group to indicate solar positions consistent
with the middle of the training period (M = 276.77°, SD = 240°, n = 404). By contrast,
the afternoon group did have a signiﬁcant regression slope, but it was very small (1 .4°/hr,
R2 = .02, T = 2.94, df = 399, p < .005) (see also Table 3.1). The actual rate of change in

the azimuth during the middle of the training period was 9.5°/hr.

69

Training Period

Sun Azimuth (°)

 

 

 

0 l E l 1 1 l l l
6 s 10 12 14 16 1s 20
Local Sun Time

Figure 3.3 Solar azimuths inferred from the dances of restricted
experienced bees from colony 1. The curved line corresponds to the actual
path of the sun on July 22. The straight lines are the predictions from the
previous computational models. A. Interpolation. B. Forward
extrapolation. C. Backward extrapolation. D. 180° step ﬁinction. The
open symbols correspond to two bees that differed qualitatively from the
rest.

 

7O

 

Table 3.1 Regression Analysis of Individual Bees' Azimuth Estimates (Colony 1)

 

 

 

 

 

Bee Morning Afternoon
Slope R2 prob. n Slope R2 prob. n
11 3.64 0.22 0.06 12
20 7.3 0.09 0.17 12 -2.79 0.53 0.03 7
22 2.76 0.12 0.15 11
23 7.23 0.56 0.04 6
32 2.19 0.04 0.18 25
48 2.46 0.06 0.29 8 -4.24 0.4 0.01 13
49 2.44 0.12 0.17 12 -3.37 0.14 0.06 12
51 -1.37 0.06 0.28 8
52 3.18 0.04 0.24 16 1.43 0.05 0.18 18
55 5.08 0.33 0.02 14
61 2.36 0.17 0.02 25
64 6.13 0.77 < 0.005 8
74 15.99 0.6 < 0.005 10 7.86 0.69 < 0.001 25
77 -4.71 0.63 0.02 7
78 -8.16 0.63 < 0.001 16
85 -1.83 0.16 0.13 10
87 13.4 0.46 < 0.01 12 3.3 0.17 0.04 19
88 -3.46 0.23 0.11 8
90 4.23 0.16 0.08 14 1.06 0.06 0.2 14
93 6.36 0.35 0.02 12
98 -0.85 0 0.43 14 -l.53 0.04 0.2 21
104 0.9 0.01 0.37 15
106 -14.06 0.86 < 0.005 6
108 -4.98 0.25 0.02 19
114 3.79 0.34 0.02 13
115 4.93 0.54 < 0.005 15
191 -0.06 < 0.001 0.49 19

 

 

Total 3.3 0.02 0.04 136 1.41 0.02 <0.005 401

 

71

The bees' estimate of the position of the sun in the morning is almost exactly 180° from
the sun's mean position in the afternoon during the training period and the mean azimuth
indicated in the afternoon by the bees. Two bees indicated positions of the sun throughout
the day that were qualitatively different from the rest of the bees. These bees are indicated
by the open symbols in Figure 3.3 and they are not included in the calculation of the
statistics. These two bees estimate a pattern of solar movement during the day that differs
from the remainder of the bees in a systematic way. These bees indicate an abrupt
transition the position of the azimuth around noon as well, but indicate a transition from

approximately 270° to 90° instead of from 90° to 270°.

Overall, the behavior of the bees is well described by a 180° step ﬁinction. This model
assumes that the morning angle is exactly 180° from the azimuth experienced at the middle
of the training period (270°). At midday (10:50 for this colony), the afternoon angle
abruptly replaces the angle assumed in the morning. This model explains a high
proportion of the variance in the data (R2 = .85) and the ﬁt of the data to the 180° step
function is signiﬁcantly better than the ﬁt of the data to the actual ephemeris function (R2
= .59) (F3 = 2.01, df= 536,536, p < 0.001). We tested this by comparing the variances
described by each model ﬁinction. It is used throughout, as a means of comparing the ﬁt
of data to nonlinear curves. This is of course a post hoc test, but it describes the data

well, and forms a standard and a prediction for future comparisons.

72

The 180° step function raises questions about the mechanisms underlying the sun
compass. It is clear from previous data (Dyer, 1987), that the estimates of the sun's
position throughout the day appear to be based on a continuous function for bees with
complete experience with the sun. It is difficult to imagine what type of mechanism would
lead to a transition ﬁom a discrete ﬁinction to a continuous ﬁinction (this will be explored
extensively in the subsequent chapters). Preliminary evidence suggests, however, that
individual bees from colony 1 may indeed represent the sun's course as a continuous
ﬁinction. Although the group of bees shifts abruptly at noon, bees that performed dances
during the transition period seemed to change their dance angles gradually. Figure 3.4
shows the solar azimuths inferred from the dances of four individual bees that danced
during the transition period. Several of the bees appear to indicate solar transitions that
are considerably less abrupt than the 180° ﬁmction would suggest. One of the bees
indicates a solar transition to the north at noon rather than the south. This is particularly
striking since the sun never passes to the north at noon in the temperate zone. This
behavior bears a striking resemblance to the data that New and New (1962) obtained (see

Figure 3.1).

73

>
w

 

 

 

 

 

 

 

 

 

270 — 270 — ‘
S A: h
a .. o-o >-
3 :3
5180 — 5180 ——
G G
:3 '- 3 ..
w -5 «1:90 43/

F O _

O
OilAlililililil oiliLilililil.l
6 8 10 l2 l4 l6 18 20 6 8 10 12 14 16 18 20
Local Sun Time Local Sun Time

0
U

 

 

 

 

 

 

 

 

36o — 360 —
. v -
a‘
r w ' /
V
270 — V / 270
6" // 6"
V " v
.5 . ‘5
3 D
5180 —— .3180
s . s
VJ /’ U)
90 —/ 90
,_ V
0 l l 11 l I l l l I L I ll 0
6 8 10 12 14 16 18 20 6 8 10 12 l4 l6 18 20
Local Sun Time Local Sun Time

Figure 3.4 Solar azimuths inferred from the dances of four individual bees
from colony 1 that show gradual shifts during the transition period. A.
Bee 32. B. Bee 74. C. Bee 78. D. Bee 87.

74

An alternative and more quantitative way of analyzing the nature of the representation of
individual bees is by a linear regression analysis of the solar azimuths indicated in the
morning and afternoon. Although a linear model clearly cannot explain the underlying
mechanism for the entire day, by examining portions of the overall curve, it may be
possible to discern whether the underlying function is truly a 180° step function, or a more

gradual function.

The results of this analysis for colony 1 are presented in Table 3.1. Data were analyzed
for bees that had more than ﬁve dances in either the morning or the afternoon. Whether
solar azimuths were grouped with the morning data or the afternoon data depended on the
azimuth angle rather than the time. Two-tailed T tests were used to assess whether the
slopes of the regressions were signiﬁcantly diﬁ‘erent from zero. Data with signiﬁcant

slopes are presented in bold typeface.

In the afternoon, there were 11 signiﬁcant regression slopes out of 26 tested. Six of the
signiﬁcant slopes were positive while ﬁve were negative. Some of the slopes are relatively
large compared to the colony as a whole. In the morning, there were fewer bees to
compare; three of nine had signiﬁcant slopes. All three of the signiﬁcant regressions had
positive slopes. The fact that there are a variety of slopes present in the data sorted by
individual bees suggest that there is more to the overall pattern than a simple step

ﬁinction. This is one of the primary concerns of the subsequent chapters.

75

The clearest conclusion from the data of colony 1 is that none of the previous
computational models are supported by the results. For all of the previous computational
models, a single constant rate of time compensation was expected. This is clearly not
what occurred. By indicating the position of the sun in the morning to be approximately
180° ﬁ'om the afternoon position, the bees incorporated some of the nonlinearities of the

rate of change of the azimuth into their estimates.

3.3.2 Colony 2

The data ﬁom colony 2 support and extend the main conclusions from colony 1. Figure
3.5 shows the inferred estimates of the sun from the dances of the restricted-experience
and the ﬁilly experienced bees of colony 2. Although not as extreme of a pattern is
produced, the restricted-experience bees of colony 2 estimated positions of the sun in the
morning (M = 97.8°, SD = 194°, n = 95) that were approximately 180° from the position

indicated in the afternoon (M = 257.6°, SD = 20.3, n = 85).

The step ﬁmction provides a signiﬁcantly better ﬁt than the actual ephemeris function (R2
= .91, F3 = 2.76, df = 179, 179, p < .001). The experienced bees, by contrast, tracked the
su n's course more accurately. For these bees, the actual ephemeris ﬁinction provides a

better ﬁt than the step ﬁinction (R2 = .85, F3 = 2.64, df= 59, 59, p < .001).

Unlike colony 1, the solar azimuths indicated in the morning and the afternoon appear to

change over time at a rate that roughly matches the rate of change of azimuth that the bees

76

experienced in the afternoon. Both the morning data and the afternoon data have
signiﬁcant regression slopes. In the morning the slope is 6.9° per hour (R2 = .24, T =
5.31, df= 91, p < .001) while in the afternoon, the slope is 122° per hour (R2 = .47, T =
8.63, df = 85, p < .001). These are both quite close to the actual pattern of movement of
the azimuth in the middle of the afternoon training pattern, which was 10.8°/hr. The
regression slopes for the moming and the afternoon are signiﬁcantly different from each
other (T = -2.74, df = 176, p < .005); however, for the two bees that had signiﬁcant
regression slopes for both the moming and the afternoon, the slopes did not differ
signiﬁcantly from each other. All of the individual bees in colony 2 that had signiﬁcant

linear regressions had positive slopes (see Table 3.2).

These results from restricted-experience bees in colony 2 contrast markedly with those
from colony 1. In particular, the representation formed by the bees in colony 2, though
still approximate, more closely matched the local emphemeris ﬁmction than did the
representation formed by bees in colony 1. The bees in colony 2 had a more accurate
representation of the rate of change of the azimuth in the morning and the afternoon.
From the positive slopes of the regression lines, one could infer that the bees also have a
more accurate representation of the midday transition (the fact that the sun passes to the
south at noon at this latitude) than colony 1. Unfortunately, because of a period of heavy

rain, there are too few dances during the transition to test this conclusion.

77

A Training Period

Sun Azimuth (°)

 

 

Local Sun Time

Sun Azimuth (°)

 

 

6 8 10 12 14 16 18
Local Sun Time

Figure 3.5 Solar azimuths inferred from the dances of restricted
experienced bees (A) and ﬁilly experienced bees (B) of colony 2.

78

 

Table 3.2 Regression Analysis of Individual Bees' Azimuth Estimates (Colony 2)

 

 

 

 

Bee Morning Afternoon

Slope R2 prob. n Slope R2 prob. n
210 1.79 0.03 0.28 16 8.18 0.81 < 0.001 12
233 8.97 0.59 < 0.005 11 17.84 0.87 < 0.005 6
235 5.57 0.33 < 0.01 18
265 31.04 0.96 < 0.001 6
270 11.34 0.74 < 0.001 11
291 5.33 0.45 0.02 9 13.86 0.78 < 0.005
311 10.68 0.32 0.07 8
3 16 6.68 0. 19 0.2
373 7.2 0.59 0.01 8 14.33 0.33 0.07 8

Total 6.94 0.24 < 0.001 93 12.22 0.47 < 0.001 87

 

79

3.3.3 Colony 3

Since colony 3 had three additional hours of experience with the sun's course in the early
afternoon, the results were expected to differ qualitatively from those of colony 1 and
colony 2. This is because they had experience with a much larger portion of the sun's
course, including portions with relatively fast and relatively slow rates of change of the
azimuth. However, the data presented in ﬁgure 3.6 suggest that this is not the case. The
data for both the aﬁernoon (M = 223.7°, SD = 14.0°, n = 67) and the morning (M =
655°, SD = 166°, n = 25) are consistently below the expectations from the actual
ephemeris function. The afternoon data are puzzling because in both colony 1 and colony
2 the solar azimuths indicated in the aftemoon initially fell above the curve of the true
solar azimuth for the early portion of the afternoon. Like colony 1, however, the rate of
change of the azimuth in the aﬁemoon is lower than the expected rate of change (slope =
3.7°/hr, R2 = .16, T = 3.49, df = 65, p < .001). The afternoon slope is considerably less
than the morning slope (20.6°lhr, T = 3.35, df= 23, p = .001) (see Table 3.3). The two

slopes are signiﬁcantly different (T = 2.71, df = 88, p < .005).

A 180° step ﬁmction through the middle of the training time (azimuth = 255°) explains a
high proportion of the variance (R2 = .82) but the ﬁt is not signiﬁcantly better than the ﬁt
with the true ephemeris ﬁmction (R2 = .76) (F, = 1.11, df= 91, 91, p > .05). This result is
somewhat troublesome in light of the results from colonies 1 and 2. Colony 3 had a

different training regime than the two other colonies. The bees of this colony experienced

80

a much larger proportion of the sun's daily course. The original prediction for the
emphemeris ﬁmction estimated by the bees of this colony was not a step function. With
the increased experience, one of the predictions was that the bees would have a more
accurate representation of the sun's course for the entire day. In fact, the prediction
corresponded to actual emphemeris function. This presented a potential problem of
interpretation because if the bees indicated the accurate azimuth function, it would be
impossible to distinguish between whether they were relying on an accurate representation
of the sun's course or a view of the sun through the clouds. Because of this, the
experimental plan included transporting this colony to a nearby location with a similar
array of landmarks in a different orientation (this technique had been successfully used in
the past (Dyer, 1987)). Several attempts were made to complete this manipulation, but
the bees failed to visit the feeder. After an unsuccessﬁil attempt to get the bees to ﬂy in
the test location in the morning, the colony was opened in the training location in the
aﬁernoon and the bees immediately began to visit the feeder and dance in the hive. These
data were subsequently recorded knowing the potential ambiguities that could result. The
overcast consisted of a dense fog at times, so it seems unlikely that the bees could directly
detect the sun. Systematic errors of the type evident in the data of colony 3 are not
uncommon in the sun compass literature (e. g. Wehner and Lanfranconi (1981)). It is this
type of systematic error, and the error evident in the step ﬁinction of colonies l and 2 that
will allow different potential mechanisms to be distinguished. The following chapters will

explore the nature of the potential mechanisms.

81

 

Table 3.3 Regression Analysis of Individual Bees' Azimuth Estimates (Colony 3)

 

 

 

 

Bee Morning Afternoon
Slope R2 prob. n Slope R2 prob. n
41 -1.99 0.16 0.22 6
59 6.29 0.37 0.07 7 5.11 0.71 < 0.001 22

Total 20.58 0.33 <0.001 25 3.66 0.16 <0.001 67

 

82

Training Period

360 —

Sun Azimuth (°)

 

 

Local Sun Time

Figure 3.6 Solar azimuths inferred from the dances of restricted
experienced bees of colony 3. Solar azimuth curve is for August 10.

83

3.3.4 Colony 4

The procedures used for colony 4 differed from the other three colonies in several
respects. This colony was restricted to ﬂy only in the morning instead of the afternoon,
and I used an alternative method to determine these bees' estimates of the solar positions
during the unexperienced portion of the day. Instead of using the dances of the bees to
infer their representation of the sun's position, I used their ﬂight bearings in unfamiliar
territory on a sunny day. Bees were captured at the feeder and released in a novel location
alter a holding period. Under these circumstances, bees normally use the sun compass to
set a homeward course (even though in these circumstances, the homeward direction takes
them in the wrong direction) and they compensate for the apparent movement of the sun

during their captivity (Meder, 1958).

Results of these manipulations are presented in Figure 3.7. The results show considerably
more variance than method of using the dance language. Therefore, I have plotted the
means :1: standard errors for the 10 signiﬁcantly oriented groups of bees (out of 12) in
Figure 3 .7. Nine of these were from a single release site and one was from another release
site. Most of the data corresponds roughly to the true ephemeris ﬁinction. With the high
error variance, the ﬁt to the azimuth ﬁmction was low (R2= .25), but the ﬁt to a step

ﬁinction was considerably worse (R2: -.61).

84

Several factors could have contributed to these results. First, the colony was set up in an
area with numerous other large hives, so it is possible that experienced bees could have
driﬁed to the restricted experience colony. Second, the bees could have used landmarks
for orientation. Both the training site and the test site had a row of trees that bore some
resemblance. They were of different densities and of different distances, but they may
have contributed to the orientation. This conclusion is supported (albeit weakly) by the
fact that one set of releases from an alternative site has a mean that falls a considerable
distance from the sun azimuth curve (see the circle in Figure 3 .7). The mean for these
data is almost exactly 270°, and it is further from the sun azimuth curves than any of the
other sets. Unfortunately, there was not time to make further releases ﬁ'om this site before

the colony was compromised by an insufﬁcient closing.

Although the data from ﬂight bearings is considerably more variable than the data from
dance angles, it is somewhat easier to collect because of the required weather conditions.
This suggests that although colony 4 does not contribute much to the results presented

here, it may nonetheless serve as a useful guide to future investigations.

85

360 ~

270 ~

- I ,////
~ ﬁg

Sun Azimuth (°)

90

 

 

Local Sun Time

Figure 3.7 Solar azimuths inferred from the vanishing bearings of colony
4. The data are means i standard error of signiﬁcantly oriented groups of
bees (10 of 12) at two different release sites (squares and circle). Tests
occurred between August 11 and 26. The azimuths inferred for each
release day are plotted. The solar azimuth shown corresponds to August
1 8.

86
3.4 Summary
The data presented here for honey bees and the data obtained from similar experiments
with ants (W ehner & Muller, 1993) suggest that the previous computational models of
linear interpolation and extrapolation are inadequate in their explanation of the
mechanisms by which insects (at least hymenopterans) compute the position of the sun to
ﬁll gaps in their experience. New models are required to understand the underlying
computational mechanisms that results in the ability of ants and bees toﬁll large gaps in
their experience and incorporate aspects of the nonlinear pattern of the movement of the

azimuth into their estimates.

The new models must satisfy several conditions that the experiments with experience-
restricted bees revealed. First, the bees indicate a position for the morning sun that is 180°
from the position of the sun in the afternoon. Second, the transition between these
positions occurs at about noon. Third, the bees seem to rely on a continuous function to
estimate the position of the sun. Fourth, the representation of the sun's course is more

accurate with more experience.

Dyer and Dickinson (1994) suggested that the bees may have something analogous to a
template as described for bird song (Marler, 1976, 1984) that would account for their
behavior. It may be fruitﬁJl to think about the models in the subsequent chapters as

templates for learning about the sun's course.

Chapter 4

NONLINEAR SYMBOLIC MODELS

It is clear from the data presented in Chapter 3 that the previous computational models of
sun compass learning cannot adequately account for the behavior of the insects in the most
recent experiments. New models are needed to ﬁll these gaps. In this chapter and in the
subsequent chapter, several new models will be presented. The models are based on
contrasting approaches to modeling cognition and they differ considerably in their
underlying representational structure (i.e. how they represent the sun's course). In this
chapter I will consider a nonlinear symbolic model, while in the next chapter I will
consider a nonlinear nonsymbolic (or connectionist neural network) model. My goal will
be to examine how each of these approaches might be applied to the sun compass learning

problem.

4.1 A Symbolic Model of Sun Compass Learning

In the previous models of sun compass learning, measurements of the rate of change of the
azimuth at an observed time were used to compute the position of the sun at a new time.
The symbolic quantities of azimuth position and time are maintained throughout the

computation. The input symbols are manipulated mathematically to yield output symbols.

87

88

The neural implementation of this computation would involve neural symbols that
represent azimuth position and time. Gallistel has speciﬁcally argued that the sun
compass learning problem can only be solved symbolically (Gallistel, in press). This
conclusion will be critically examined in the following chapter with the development of a
nonsymbolic model, but the ﬁrst priority is to determine if there is a symbolic process that

the insects could use to ﬁll gaps in their experience with the sun.

The equations of the true solar azimuth ﬁinction are symbolic (equations 2.1 and 2.2).
These equations have three input variables (latitude, solar declination, and time of day)
that could be used to calculate the output variable of azimuth position. As indicated in
Chapter 2, two of the input variables (latitude and declination) would be difﬁcult for an
animal to assess. Both of these variables are confounded in the zenith distance of the sun.
In order to use the zenith distance as a means of determining the azimuth function, the
animal would have to have an independent measure of either declination or latitude. This
is not out of the range of possibility. Latitude for instance could be determined ﬁ'om the
inclination of the earth's magnetic ﬁeld (Wiltschko & Wiltschko, 1995). However, it does
not appear necessary to postulate a mechanism for the independent assessment of latitude
and declination. Experiments suggest that the zenith distance of the sun does not play a
role in the sun compass of many animals. Recall that longitudinal translocations indicate
that the animals respond to the sun's azimuth and their internal sense of time without
regard for the sun's zenith distance (Papi, 1955; Renner, 1959). The equivalent effect has

been shown in animals with a phase-shifted circadian clock (Hoffmann, 1960). In

89

addition, experiments with artiﬁcial suns have shown that the zenith distance does not play

a role in the determination of direction (St. Paul, 1953; Brines & Gould, 1979).

Instead of attempting to disentangle the potential inputs of zenith distance and latitude, I
will focus on the directly observable quantities. This is what the previous computational
models have done. The variables of azimuth angle and time are directly measurable by the
animal. Speciﬁc instances of these variables could be used to estimate a parameter that
describes the relationship between the variables. In the case of the previous computational
models, the parameter is the rate of change of the azimuth. The parameter is a constant
that can be used to compute speciﬁc values for the output variables of azimuth position
and time, given speciﬁc input values. The data suggest that more is needed. None of the
previous models were supported by the experimental results of Wehner and Muller (1993)
and Dyer and Dickinson (1994). They found that insects somehow account for the
varying rate of change of the azimuth over the day. Thus the rate of change of the
azimuth (r) is not a constant, but is itself a variable. The goal of the present model is to
determine a ﬁmction that could describe the varying rate of change of the azimuth over the

course of the day.

The pattern of the curves in Figure 2.2 indicates that the rate of change of the azimuth
systematically varies over the course of the day. Thus r is also a function of time. It is the
azimuth rate function. But what form does this ﬁinction take? In nature this equation is

the ﬁrst derivative of the azimuth ﬁrnction (equations 2.1 and 2.2) with respect to time.

90

But if it is reasonable to assume that the animals do not have access to the true azimuth
ﬁmction (since they would need to independently assess latitude and declination), it is
reasonable to assume that they do not have access to its ﬁrst derivative. These two
suppositions are further substantiated by the fact that bees that have restricted experience
do not have completely accurate representations of the sun's course (Dyer & Dickinson,
1994). Therefore, the azimuth rate ﬁrnction that the bees used is assumed to be an
approximation of the true azimuth rate function. Furthermore, this approximation should
be based on the readily observable quantities of azimuth angle and time of day. These

quantities would allow the parameters of the function to be estimated.

One function that meets these qualiﬁcations is based on the geometry of an ellipse. This
model was originally suggested in qualitative terms by Rudolf J ander (personal
communication). I have formalized this model and will show that it describes the
experimental data well. This model allows the generation of complete azimuth ﬂmctions
from experience with very small portions of the actual azimuth ﬁinction. The azimuth
ﬁmctions generated from the model are approximations of the true azimuth function, but
they are approximations that would allow a high degree of accuracy with a relatively small
investment in time to learn the pattern. The ﬁt of the model functions to previous data

sets is very good.

91

4.2 The Ellipse Rate Function

In this new symbolic model, the true azimuth rate function is assumed to be approximated
by an ellipse plotted in polar coordinates. In this form, the angle of the plot corresponds
to time, which cycles over 24 hours, and the length of the radius vector corresponds to the
rate of change of the azimuth. Ellipses of different shapes would correspond to azimuth
ﬁrnctions for different seasons and latitudes. Speciﬁcally, rounded ellipses correspond to
temperate latitudes and squashed ellipses correspond to tropical latitudes. This provides a
means of visualizing the relationship between the rate of change of the azimuth and the

time of day.

The geometry of an ellipse intuitively seems to provide a good description of the
variations in the rate of change of the azimuth over the course of the day. The rate
(radius) varies systematically with time of day (angle). As the radius sweeps around the
ellipse (like the hands of a distorted clock) it increases to a maximum at the semimajor axis
and decreases to a minimum at the semiminor axis. Thus the equation of the ellipse in
polar coordinates may be a ﬁmction that closely approximates the rate ﬁinction of the

actual azimuth.

In the following sections, I will ﬁrst bolster the case that the geometry of the ellipse
provides a good approximation for the true rate ﬁmction. I will subsequently introduce
the equation of the ellipse in polar coordinates as an approximation for the true rate

ﬁmction. The ﬁrst part of my argument is purely geometrical.

92

4.2.1 The Geometry of the Ellipse

From Figure 2.2 (see also Figure 4.1 below), it is clear that the rate of change of the
azimuth varies systematically over the course of the day. It reaches a maximum at midday
and symmetrically increases prior to midday and decreases after midday. The true rate
function and its hypothesized relationship with an ellipse function is portrayed in Figure
4.1. The ﬁrst panel of Figure 4.1 (A and B) shows the full 24 hr azimuth ﬁmction for a
temperate latitude (40° N) and a tropical latitude (10° N) for the equinox. In both azimuth
functions, the rate of change of the azimuth varies systematically over the course of the
day. The rate of change of the azimuth for a given time of day corresponds to the slope of
the azimuth curve at that point. In both curves the slope starts out high and decreases. It
subsequently increases to a maximum at noon before decreasing again. The systematic
variation in the slope is what needs to be explained by a rate function. The pattern of the

variation of the slope over the course of the day deﬁnes the azimuth rate ﬁmction.

In the second set of graphs (Figure 4.1 C and D), the rate of change of the azimuth is
plotted against time. This plot shows the systematic variation in the rate of change of the
azimuth with time of day that was described in reference to the curves in Figure 4.1 A and
B. The rates plotted in the second panel were calculated numerically over intervals of 8
minutes. The plots for both latitudes reveal a similar pattern, although the range between
the maximal and minimal rates is more extreme in the tropical curve. Ifthe x-axis were

stretched out to include several days, the pattern of change in the rate would repeat itself;

93

therefore, the true rate function is a cyclic ﬁmction with alternating increasing and
decreasing rates of change of the azimuth. Polar coordinates are well suited to the
representation of cyclical functions. The idea of the ellipse function is that the pattern of
increasing and decreasing rates of change of the azimuth would correspond to an ellipse if
they were plotted in polar coordinates. Thus, the equation of the ellipse would form an
approximation of the true rate function. In the third pair of graphs in Figure 4.1 (E and
F), a hypothetical pair of ellipses are plotted. Time is plotted in degrees with the following
relationships: 00:00 = 0°; 06:00 = 90°; 12:00 = 180°; 18:00 = 270°. The lengths of the
semimajor and semiminor axes of the ellipses correspond to the maximal and minimal rates
of change in the azimuth plotted in C and D of the ﬁgure. The resulting pattern is clear.
An ellipse has the potential to describe a wide variety of emphemeris ﬁrnctions. Relatively
rounded ellipses correspond to the ephemeris functions of temperate latitudes, while

relatively squashed ellipses correspond to the ephemeris ﬁmctions of tropical latitudes.

94

Figure 4.1 The ellipse approximation of the azimuth rate function. The
rate of change of the azimuth varies systematically over the course of the
day. A. The full 24 hour azimuth function for 40° N at the equinox. B.
The 24 hour azimuth ﬁinction for 10° N at the equinox. C. The azimuth
rate function for the azimuth curve in A, computed over 8 minute intervals.
D. The azimuth rate function for the azimuth curve in B, computed over 8
minute intervals. E. An ellipse based on the rates in C (semimajor axis =
23°/hr; semiminor axis = 10°/hr). F. An ellipse based on the rates in D
(semimajor axis = 85°/hr; semiminor axis = 3°/hr).

>

270

Sun Azi_r_nuth (°)
8

8

 

 

01411111111111111111411L1
0 3 6 9121518 2124

Local Sun Time

('3

1'3
:5
1

I

8
I ﬁT

8
T

Azimuth Rate Function (°/hr)
8
1

lllLllth111LLLllll1111
O 3 6 9 121518 2124

Local Sun Time

 

C

 

180

Figure 4.1

 

 

 

 

 

 

360

270

6‘

V

-5

3

5180

g

w

90
0lllllLllllllllllllLllllJ
0 3 691215182124

LocalSunTime

120——

g C

2,90_

8

E

“n60

8

5

E30
0L1 11111 111
0 3 6 9 121518 2124

LocalSunTime
0
270 90
180

96

The hypothetical correspondence between the shape of an ellipse and the latitude of an
azimuth function is intuitively satisfying. Whether it actually approximates the true rate
function is another question. The actual values of the rate of change of the azimuth
plotted in Figure 4.1 C and D can be plotted in polar coordinates to check their
correspondence of the ellipse functions plotted in E and F of Figure 4.1. Figure 4.2 (A)
provides a comparison of the true rate functions plotted in polar coordinates with the
ellipses generated in Figure 4.1 E and F. In this case, the curves for the different latitudes
are plotted in the same ﬁgure (the scale is the same as Figure 4.1 E and F). The ellipse is a
very good description of the azimuth rate curve for 40° north latitude. The azimuth rate
curve is nearly identical to the ellipse when plotted in polar coordinates. In contrast, the
azimuth rate curve varies slightly from the form of the ellipse for the 10° N curve.

However, the ellipse may still provide a good approximation to this 10° N rate function.

Figure 4.2 (A) still shows the special case of the equinox. As the declination of the sun
changes, the pattern of the polar plot of the true rate ﬁrnction changes considerably, and
the shape diverges signiﬁcantly ﬁ'om that of a true ellipse. Panels B and C of Figure 4.2
show this effect. In panel B, the true rate function is plotted for 10° N on April 16
(declination 9°54') when the sun passes nearly directly overhead at noon. The scale of the
plot has been changed 100 fold to reveal the change in the pattern. Since the azimuth is
shiﬁing abruptly from east to west, there is a spike in the rate around noon. Note that the

pattern is now asymmetrical, since there is no corresponding spike at midnight. The

97

pattern is more clearly evident in panel C of Figure 4.2 which shows the true rate function

for 40° north latitude at the equinox.

Although the pattern of the true rate function plotted in polar coordinates is clearly
asymmetrical and not elliptical, the ellipse may nonetheless provide a good approximation
of the true rate ﬁrnction, particularly during the day. The pattern of the function between
06:00 (90°) and 18:00 (270°) is symmetrical around noon and well approximated by half
an ellipse. This corresponds to the majority of the hours of daylight, particularly in the
tropics. The fact that the nocturnal values would not be well approximated by the daytime
ellipse is not a serious problem. This would in fact be an interesting prediction of the
ellipse model. This would suggest that the solar positions estimated at night by animals
relying on a representation of the sun's movement based on an ellipse would correspond to
the pattern of movement of the sun during the day, even if the true nocturnal pattern
differed considerably. For temperate latitudes, the nocturnal values that would be

observed do not deviate considerably ﬁom an ellipse estimated from the daytime values.

98

Figure 4.2 The true azimuth rate ﬁmction plotted in polar coordinates.
The values plotted correspond to the numerically calculated rate of change
of the azimuth over 8 minute intervals. A. 40° N and 10° N for the
equinox. B. 10° N for April 16 (declination 9°54'). C. 40° N for the
summer solstice. In all cases the curves are not truly elliptical. The shape
varies considerably with season and latitude.

 

 

 

180

Figure 4.2

100

4.2.2 The Equation of the Ellipse

Since the ellipse ﬁinction appears to be a relatively good candidate as an approximation to
the true rate ﬁinction, the equation of the ellipse in polar coordinates can be used to
describe the ﬁinctional relationship between time of day (1:) and the rate of change of the

azimuth (r). Thus:

ab

Jaz sin2(t) + b2c0S2(‘C)

 

r

(4.1)

 

where r is the rate of change of the azimuth; ‘C is the time angle (with 15° per hour); and a
and b are the parameters of the ellipse. These parameters are equivalent to the parameters
a and b in the familiar Cartesian equation of the ellipse which correspond to length of the

semimajor axis and the semiminor axis respectively:

— + — = (4.2)

The values of the parameters a and b determine the shape of the ellipse. When a and b
are roughly equal, the ellipse is nearly circular. When a is very large with respect to b,
the ellipse has a much more squashed shape. In terms of the rate of change of the
azimuth, the greater the variation between the maximal and minimal rates, the more

tropical the azimuth ﬁinction is (see Figure 4.1).

Equation 4.1 can be used in combination with equation 3.1 to determine the position of

the azimuth for a particular time of day. Equation 4.1 yields a series of values for the rate

101

of change of the azimuth (r) for different times of day. These values can substituted into

equation 3.1 at the appropriate times of day in the calculation of the azimuth.

Using these two equations to calculate azimuth, one of which is a discrete difference
equation, bypasses an obvious alternative approach. Since the rate of change of the
azimuth (r) is a function of time, and it is theoretically the ﬁrst derivative of the azimuth
ﬁmction, if the antiderivative of the rate function can be found, an explicit ﬁinction of the
azimuth in terms of time would exist and could be used to directly calculate azimuth
positions by time of day. However, no such explicit ﬁrnction appears to exist for equation
4.1. In addition, using the difference equation (3.1) with the ellipse rate function (4.1)
provides a direct parallel between the previous linear models of sun compass learning and
this nonlinear model, since all of the previous models can be expressed in terms of

equation 3.1 (see Chapter 3).

One byproduct of using a difference equation is that shape of the resulting ﬁmction varies
with the number of time steps used, as in any difference equation. This technically adds
another parameter to the azimuth estimation ﬁinction, although it is not a parameter that
the bees would have to estimate. However, this effect of the time steps may help describe
some of the results of previous experiments (Dyer & Dickinson, 1994), because it turns
out that using a function with fewer time intervals increases the likelihood of generating an

azimuth estimate that corresponds to a step function (see section 4.9 below).

102

In the next section, I will consider the range of functions that result ﬁ'om a use of a
combination of the ellipse rate ﬁrnction (equation 4.1) and the azimuth difference
equation (3.1). The values of the ellipse parameters a and b required to produce these
curves will be generated, but I will leave it to the following section to discuss how the

parameters can be estimated by the insects.

4.3 The Ellipse Azimuth Function

Figure 4.3 shows the azimuth ﬁmctions generated using equations 3.1 and 4.1 for the
same seasons and latitudes as discussed in the previous section (and Figures 4.1 and 4.2).
Panel A of Figure 4.3 shows the true azimuth ﬁrnction for 10° north latitude at the
equinox and two azimuth ﬁmctions generated from the model equations. Clearly, the
model does a very good job of approximating the true azimuth function, since one of the
lines completely overlaps the true function. The azimuth functions produced from the
ellipse rate ﬁinction in Figure 4.3 (A) both used an ellipse with the same parameters (a =
23, b = 10). These are the same parameters that were derived ﬁom the true rate function
in Figure 4.1. The two curves differ slightly because one has been scaled to 360°
(squares) and the other has not (diamonds). Looking at Figure 4.3 (B) it is apparent that
scaling is necessary to accurately approximate the true azimuth function. This shows the
actual azimuth function and two ellipse azimuth estimates for 10° north latitude at the
equinox. The ellipse parameters for this ellipse azimuth function were again taken from the
true rate function for 10° north latitude as in Figure 4.1 (D) (a = 85, b = 3). The scaled

ellipse azimuth ﬁmction again provides an extremely close correspondence to the actual

103

azimuth ﬁmction. In this case, however, the unsealed ellipse azimuth function provides an
extremely poor approximation. This probably results from the fact that the true rate
function does not correspond to an ellipse in this case (see Figure 4.2 A). Scaling the
ellipse azimuth function forces equation 3.1 to sum to the appropriate level (360°).
Without scaling, the accumulation of errors can lead to results that deviate substantially

from the true pattern.

The third and fourth panels of Figure 4.3 (C and D) show the effects of seasonal variation
on the approximation of the ellipse azimuth ﬁmction to the actual azimuth ﬁmction. Again
the curves are for 10° and 40° north latitude. The declinations of the sun correspond to
those considered in Figure 4.2 The summer solstice curve is plotted for the 10° latitude
line and the April 16 curve is plotted for the 10° north latitude. The scaled and unscaled
ellipse azimuth ﬁmctions are plotted as in A and B above. Again, the need for scaling the
resulting functions is apparent, particularly in D. Additionally, even the approximation of
the scaled ellipse azimuth ﬂinction is not as good as in the functions plotted for the
equinox. Because of the symmetry of the ellipse, the ellipse azimuth ﬁinctions are
constrained to pass through 90° at 06:00 solar time and 270° at 18:00 solar time. Thus
the correspondence to the actual azimuth ﬁmctions is not as good, since the true azimuth
ﬁrnctions do not have this constraint. This is one of the most important predictions of the

ellipse model.

104

The necessity of scaling the output is probably not a serious problem for this model. Since
it is a neural model (the symbolic calculations are assumed to be implemented in
isomorphic neural processes) the outputs would not be scaled identically to the numerical
values presented here. The output could easily be a ratio based on another value in the

calculation. The scaling function I used relies on this simple manipulation:

A: = i4£360 (4.3)
AN

where the scaled azimuth at time I (AI) is equal to the ratio of the unscaled azimuth at
time I (A) and the unscaled azimuth for the last time interval (A N) multiplied by 360°. This
implies that the azimuth at midnight is 360°. The ellipse azimuth function could just as
easily be scaled to 180° at noon. Both of these alternatives force the ellipse function to
take on values that may not necessarily exist for a given latitude and season (i.e. if the sun
actually passes to the north at noon). However, the ellipse azimuth function could also be
scaled to pass through 90° at 06:00 or 270° at 18:00. All actual azimuth functions pass

close to these values and all scaled ellipse azimuth ﬁmctions pass through them.

Sun Aa'lnum (°)

 

 

 

 

0 11111111111111411111111 1
0 3 6 9121518 2124 0 3 6 9121518 2124

Local Sun Time Local Sun Time

 

 

 

 

 

D
360— I
270-
6‘: 1
.5180
5
U)
90
0 1111111111LLL111111144_1 o
0 3 6 9121518 2124 0 3 6 9121518 2124
IncalSmrTimc LocalSunTimc

Figure 4.3 The scaled and unscaled ellipse azimuth functions for different
seasons and latitudes. A. The true solar azimuth is plotted with two ellipse
azimuth ﬁmctions for 40° north latitude at the equinox. The squares
correspond to the scaled ellipse azimuth ﬁrnction and the diamonds
correspond to the unscaled azimuth ﬁmction. The ellipse parameters for
both ellipse azimuth ﬁmctions are a = 23, b = 10 (as in Figure 4.1 E). The
scaling function is described in the text. The true sun azimuth curve is
depicted by the heavy line. B. The solar azimuth function (heavy line) and
two ellipse azimuth estimates (scaled = squares; unscaled = diamonds) for
10° north latitude at the equinox. Ellipse parameters are a = 85, b = 3. C.
The solar azimuth ﬁmction (heavy line) and two ellipse azimuth estimates
(scaled = squares; unscaled = diamonds) for 40° north latitude during the
summer solstice. Ellipse parameters are a = 48, b = 9 (these values were
determined from the actual azimuth rate function numerically calculated
over 8 minute intervals as in Figure 4.1 C and D, and depicted in Figure 4.2
C. D. The solar azimuth ﬁinction (heavy line) and two ellipse azimuth
estimates (scaled = squares; unsealed = diamonds) for 10° north latitude on
April 16 (declination 9°54'). Ellipse parameters are a = 662, b = 1.5. The
values for the ellipse parameters a and b were determined ﬁ'om the actual
azimuth rate ﬁinction, numerically calculated over 8 minute intervals as in
Figure 4.1 C and D, and depicted in Figure 4.2.

106

All of the ellipse azimuth functions presented up to this point concern azimuth ﬁmctions
that pass to the south at noon: those typically viewed from the northern latitudes. But
during the northern summer, from most vantage points on the earth, the sun passes to the
north at noon. At the northern summer solstice this occurs ﬁom viewing positions from
the tropic of Cancer (23°26' N) southward. In these cases, the sun appears to move
counterclockwise around the northern horizon. By convention, the rate of change of the
azimuth in these cases would be negative. The rate of change is still maximal around
noon. The ﬁmctional relationship between the rates of different times of day is
maintained; therefore, it would seem that they too can be described by an ellipse function.
However, by inspection of equations 4.1 and 4.2 (the equations of the ellipse), it is clear
that the ellipse will have a positive radius (r, the rate of change of the azimuth) even if the
parameters a and b are negative. The easiest way around this problem is to assume that
information about direction of movement is maintained in a separate channel of
information ﬂow in the nervous system. This information can be implemented by changing
the sign in equation 3.1 and subtracting changes in azimuth ﬁ'om previous values

throughout the day.

Figure 4.4 shows the results of a case of counterclockwise movement. In this ﬁgure, the
azimuth functions for 5° N and 40° N on the summer solstice are plotted with the
corresponding scaled ellipse ﬁinctions (40° N: a = 48, b = 9; 5° N: a = -43, b = -1).
Although it is technically inaccurate, for simplicity, I have conveyed the information about

the direction of movement in the signs of the parameters a and b. I will use this

107

convention henceforth. The values for a and b are theoretical estimates, derived from a
numerical calculation of the true azimuth rate ﬁrnctions for the two ephemeris functions
(as in Figures 4.1, 4.2, and 4.3). Additionally, the actual theoretical value for b is 1,
instead of -1. This occurs because in the tropics the azimuth's movement along the
horizon can reverse directions. But for the model, I will assume that b has the same sign

as a.

From Figures 4.3 and 4.4 it is apparent that the combined use of equations 4.1 and 3.1
(the ellipse azimuth function) with appropriate scaling functions, can yield sun azimuth
positions that closely approximate the true sun azimuth ﬁrnction for any particular season

and latitude.

108

 

 

 

360 —
_ Solstice
27o —
Q
J: — o
.5 / 40 N
E
5‘ 180 —
a /
:5 _
U)
90 .
5°N
O 1 1 1 1 1 v 1 1 l 1 1 1
6 8 10 12 14 16 18

Local Sun Time

Figure 4.4 Ellipse azimuth ﬁmctions for 40° N and 5° N at the northern
summer solstice. The parameters of the ellipse rate function were
estimated ﬁ'om numerical calculations of the actual azimuth rate ﬁinction
over 8 minute intervals (as in Figures 4.1, 4.2, and 4.3) (40° N: a = 48, b =
9; 5° N: a = -43, b = -1). The actual ephemeris ﬁinctions are plotted for
comparison.

109

4.4 Estimating the Parameters of the Ellipse

The ellipse rate ﬁrnction (equation 4.1) has two parameters that determine its shape (a and
b). These parameters can be estimated from the readily observable quantities that have
been used in the previous symbolic models of this process: azimuth position and time of
day. For a bee to estimate the position of the sun using computations equivalent to
equations 3.1 and 4.1, she would have to estimate the parameters of the ellipse at a time of
day when the rate of azimuthal movement has been observed. With two observations of
the rate of change of the azimuth and time (this could involve four observations of azimuth
position and time, see equation 3.2), the parameters a and b can be estimated with the

following two equations:

 

- 2 - 2
'81!) T] —SlIl ‘Czl

 

a=r1r2 (4.4)

. 2 2 - 2
lrfsrn “cl -r2srn tzl

 

- 2 - 2
Isrn Tl-Sm Tzl
2

 

b=r1r2 (4.5)

lrfcos 11 —r§cos212|

in which a and b are the estimated parameters of the ellipse and r,, r2, 1,, and 132 are
speciﬁc observations of the rate of change of the azimuth at speciﬁc times of day.
Equations 4.3 and 4.4 were generated by solving equation 4.1 for a and b and substituting
each equation into the other. I have added the absolute value operators to prevent
imaginary numbers from resulting. They have no effect on the magnitude of a or b, only

whether they are real or imaginary. This manipulation is required because not all rates

110

and times that could be observed by a bee necessarily fall on a true ellipse since the ellipse
is an approximation of the true function. If equations 4.4 and 4.5 were not constrained in

this way, the resulting ﬁmction would not necessarily be an ellipse.

Note that once again the effects of observed negative rates (counterclockwise movement
of the azimuth) are eliminated in equations 4.4 and 4.5 (assuming the two observations of
rate have the same sign). This occurs because the rates are multiplied together and
squared in equations 4.4 and 4.5, eliminating negative sign of the rates. This substantiates

the need for an independent channel to carry information about the direction of movement.

4.5 Fit of Ellipse Azimuth Function to Experimental Data

The ellipse azimuth ﬁinction is ﬂexible enough to represent a wide range of azimuth
functions. By determining the values of a few parameters, an animal can have a
representation of a full 24-hour ﬁmction which closely approximates the course of the sun.
In the data presented in Chapter 3, it was established that honey bees can relatively
accurately estimate the position of the sun in the morning, even if they have previously
seen it only during the late aftemoon. However, the bees' representation of the sun's
course was not completely accurate. Instead, the data suggested that a 180° step ﬁinction
describes the mechanism they were using. Figure 4.5 shows that the ellipse azimuth
ﬁmction can produce curves that range from 180° step ﬁrnctions to relatively ﬂat curves
that correspond more closely to a sun azimuth curve from a temperate region. The

parameters (a and b) for the ellipse azimuth functions range from the theoretically

111

expected values (given the actual maximum and minimum rates of change of the azimuth
for those dates) to the values that best ﬁt the experimental data. The set of parameters of
the ellipse function that produced the best ﬁtting azimuth curves to the data was actually
one set of out of many sets of parameters that ﬁt the data equally well. Because of the
discrete nature of one of the equations (3.1) used to generate the ellipse azimuth function,
the best-ﬁtting parameters had to be determined iteratively rather than analytically. I used
a hill climbing algorithm to make small adjustments to a and b in order to ﬁnd the values
of the parameters that maximized R2. The algorithm stopped repeatedly at local maximum
that produced R2 values very close to each other, often differing only in the ﬁfth decimal
place. A systematic exploration of the space revealed a long ridge of roughly equal height.
For the values of the parameters reported, I used the values closest to the theoretically

predicted values that resulted from 50 runs from different randomly chosen starting points.

112

r (a= 1076; b= 1)

 

 

 

 

Local Sun Time Local Sun Time

 

 

 

0 1 1 1 l 1 1 L 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 l 1
6 9 12 15 18 6 9 12 15 18

Local Sun Time Local Sun Time

 

Figure 4.5 A range of ellipse azimuth functions corresponding to the data
from colony 1 (see Chapter 3). A. Parameters corresponding to the best
ﬁtting ellipse ﬁmction (a = 1076, b = 1). B. Intermediate parameters (a =
600, b = 3). C. Intermediate parameters (a = 300, b = 6). Theoretical
(expected) parameters of the ellipse (numerically calculated from the true
azimuth rate ﬁmction as in Figure 4.1, 4.2, and 4.3) (a = 36, b = 10). The
shaded region corresponds to an ideal step function based on the data from
colony one. The region corresponds to the mean (270° or 90°) i one
standard deviation (~3 0°) of the data.

113

An azimuth ﬁinction generated with a combination of equations 3.1 and 4.1 ﬁts the data
from colony 1 well (R2 = .87; a = 1076, b = 1). This is 2% better than the ﬁt of the 180°
step function, but it is a statistically signiﬁcant difference (Fs = 1.16, df = 536,536, p <
.05). This difference probably results from the fact that the ellipse function explains the
variance of the bees' estimates during the midday transition period better than a step
ﬁmction. See Table 4.1 for a comparison of goodness-of-ﬁt with the actual azimuth
function and the 180° step ﬁmction. The parameters of the ellipse ﬁmction that produce
the azimuth curve that best ﬁts the data are considerably different from the theoretically
expected parameters (see Figure 4.5 and Table 4.2). The theoretically expected
parameters were derived from numerical calculations of the true rate function. The value
of a in the best-ﬁtting ellipse was 1,076° per hour which was nearly 30 times the 36° per
hour expected maximal rate around midday. It would seem that if the bees are using this
type of mechanism to estimate the unknown azimuth, they are using an ellipse that is
considerably more squashed than the most appropriate ellipse for this season and latitude.

This could be a default mechanism.

For colony 2 an azimuth ﬁrnction generated from the ellipse rate model explains a

considerable proportion of the data (R2 = .94; a = 421, b = 3) of the experience-restricted
bees. The ﬁt of the data to the ellipse model is signiﬁcantly better than the ﬁt of the data
to the actual ephemeris ﬁmction (Fs = 4.23, df = 179,179, p< .001), and to the 180° step

function (Fs = 1.53, df = 179,179, p < .005). Again, the observed parameters of the ellipse

114

that produced the best ﬁtting azimuth curve were different from the theoretically expected

values (see Table 4.2).

The ellipse model, with different parameters of shape, also explains a considerable fraction
of the data for the fully experienced bees (R2 = .87; a = 36, b = 3). This is signiﬁcantly
better than the 180° step ﬁmction 073 = 2.85, df = 59, 59, p < .001). The ellipse function
appears to explain slightly more of the data than the actual azimuth function, but the
difference is not statistically signiﬁcant (F, = 1.29, df = 59, 59, p = .17). Thus the fully
experienced bees appear to be using an ellipse azimuth function based on a more accurate

estimate of a and b.

For colony 3, the ellipse azimuth function ﬁts the data as well as the 180° step ﬁinction

(R2 = .82; a = 13; b = 9); there is no statistical difference between them.

For colony 4, the ellipse azimuth function ﬁts the data about as well as the true azimuth
ﬁmction (R2 = .27; a = 31; b =16). The ﬁt of the data to the ellipse azimuth was not
statistically different from the ﬁt to the true sun azimuth, although both were substantially
better than the step function. There is considerably more error variance for this colony,
which relied on ﬂight bearings rather than dance angles to determine their estimates of the

solar azimuth.

115

 

Table 4.1 Fit of Models to Experimental Data (R’)

 

 

 

 

 

Data Set Azimuth Step-Function Ellipse

Colony 1 0.59 0.85 0.87
Colony 2 (RE) 0.73 0.91 0.94
Colony 2 (FE) 0.85 0.65 0.87
Colony 3 0.76 0.82 0.82
Colony 4 0.25 -0.61 0.27

 

RE: restricted experience; FE: full experience

 

Table 4.2 Expected and Observed Parameters of the Ellipse

 

 

 

 

 

 

Data Set Expected Observed

a b a b
Colony 1 36 10 1,076 1
Colony 2 (RE) 27 10 421 3
Colony 2 (FE) 27 10 36 3
Colony 3 32 10 13 9
Colony 4 30 10 31 16

 

RE: restricted experience; FE: full experience

116

4.6 Fit of Ellipse Azimuth Functions to Other Data Sets

In addition to ﬁtting the experimental data from Chapter 3 well, the ellipse model may be
applied to other data sets with similar success. The data from New and New (1962)
provide an example of the generality of this model as a description of sun compensation.
Recall that New and New (1962) observed the dance of bees in the tropics as the sun
passed very close to the zenith at noon. They found that bees compensated during midday
at a very rapid rate over the approximately 180° separating the morning and aﬁemoon
azimuths of the sun. The data (see Figure 3.3) have long been interpreted as evidence that
bees interpolated linearly between the moming and aﬁemoon positions of the sun. The
alternative explored here is that the bees were using a continuous function based on an
ellipse. With suitable parameters, the ellipse azimuth ﬁmction describes the data very well
(see Table 4.3). In this case, the observed parameters of the best ﬁtting ellipse azimuth

functions are quite close to the expected parameters (see Table 4.4).

One striking result of New and New (1962) was the fact that the bees anticipated the sun
passing to the north at noon several days before this occurred. With the ellipse model, this
would suggest that the bees were using the parameters of a single ellipse throughout the
days during the transition period. The pooled data (for colony 1) ﬁ’om New and New
(1962) for April 15 to 21 are well described by a single ellipse azimuth function (R2 = .91,

a=-251,b=-2).

117

 

Table 4.3 Fit of Ellipse Model to New & New (1962) Data (R2)

 

Data Set

 

April 15
April 16
April 17
April 18
April 20
April 21

Azimuth

180° Step-Function

 

 

-0.42
0.14
0.48
0.36
0.72
0.74

0.51

0.5
0.63
0.19
0.33
0.31

Ellipse

 

0.92
0.92

0.9
0.97
0.89
0.91

 

Data from New & New (1962): Colony 1, Trinidad (10°38' N)

 

Table 4.4 Expected and Observed Parameters of the Ellipse (New & New, 1962)

 

 

 

 

 

 

Data Set Expected Observed
a b a b

April 15 240 3 -229 -1
April 16 254 3 -359 -2
April 17 263 3 -415 -2
April 18 -262 -3 -l66 -2
April 20 -23 8 -3 -218 -2
April 21 -227 -3 -192 -2

R2

0.92
0.92

0.9
0.97
0.89
0.91

 

Data from New & New (1962): Colony 1, Trinidad (10°38' N)

118

4.7 Reducing the Number of Parameters

As 1 indicated above, in the search for the parameters that yielded the ellipse azimuth
functions that best ﬁt the data, I encountered multiple pairs of parameters that produced
curves that ﬁt the data equally well. In retrospect, this is to be expected since the effect
of the scaling function is to render the particular values assumed by the parameters a and b
somewhat irrelevant. This is because it is only the shape of the ellipse that is important in
determining the values that the ellipse azimuth function assumes. This suggests that it is
the relative magnitude of the parameters a and b that is important. Figure 4.6 further
substantiates this inference. Figure 4.6 shows the range of values of the parameters of the
ellipse (a and b) that yield ellipse azimuth functions that ﬁt the data equally well (or very
close to it). These data are for the restricted experienced bees of colony 2. The linear
relationship between the values of a and b indicates that it is the relative magnitude that is
important. This same pattern appears for almost all of the other data sets examined. The
data for colony 1 provide an apparent exception to this rule, which is probably a special
case where the slope of the line is zero. This probably results ﬁom the fact that all
parameters for the best-ﬁtting ellipses for colony 1 yield ellipse azimuth functions that
approximate step ﬁmctions. Thus, all of these functions explain the variance in colony 1
equally well. In other words, once the step-function threshold is surpassed, there is a

wider range of parameters that yield equivalent azimuth curves.

119

16—
14—

12—

10L

 

0 1.11...1...1...1..4111.1...1...1
0 200 400 600 800 1,000 1,200 1,400 1,600

a

Figure 4.6 Range of the parameters of the ellipse that generate equivalent
ellipse azimuth ﬁinctions (see footnote 6). All of the parameters yield
azimuth curves that ﬁt the data of colony 2 equally well (R2 = .94). The
linear relationship suggests that the two parameters necessary to describe
the ellipse azimuth function may be reduced to a single ratio.

120

This relationship between the parameters a and b suggests that the number of parameters
that need to be estimated to generate an ellipse azimuth ﬁmction may be reduced to one.
A single parameter would be powerﬁll enough to generate approximations to all of the
azimuth functions on the planet. I will deﬁne this parameter as k which is simply the
ratio of a to b. From equations 4.4 and 4.5, a means of estimating k directly can be

generated:

 

 

Irjcosztl —r§cos2121
k: 2.2 2.2 (4.6)
Irlsrn 11 —r2srn 12'

This parameter can be used with the following equation to produce an ellipse of

standardized size:

r = k (4.7)

sz 811121 + 00821

 

 

The semiminor axis of this ellipse is always one and the semimajor axis is k which
corresponds to the ratio of a and b. This ellipse requires the use of a scaling function to

produce the correct range of values.

4.8 Estimating k with Restricted Experience
So far I have shown that ellipses can be used to generate a range of azimuth functions and
that with the right parameters, these ellipse azimuth ﬁinctions provide a good ﬁt to the

experimental data. With equations 4.4, 4.5 and 4.6, I have shown how the bees could

121

theoretically estimate the parameters of the ellipse. There is a considerable gap, however,
between this theory and the practice of actually estimating the parameters given the
restricted experience of the bees. I conducted numerical simulations to see if this is at
least possible. I randomly picked four times of day falling within the training period of
colony 1. These deﬁned two time periods over which I estimated the rate of change of the
azimuth, using the azimuth for the last day of training. For 1000 simulation runs, the
results suggest that accurate estimates of k could be obtained with the restricted
experience (see Table 4.5). The mean k was 4.1 which is very closed to the predicted 3.6,
but very far from the observed 1,076 of the best ﬁtting ellipse azimuth ﬁmction. The

maximum k from this run was only 160.7.

This result raises the question of how the ellipse model can account for the data. This
question can be allayed somewhat by introducing error into the bee's ability to estimate the
position of the azimuth. New and New (1962) ﬁrst suggested this as one of the potential
reasons behind the data they observed. Introducing error, or limits on visual acuity into
the estimate greatly increases the range of parameters estimated from the restricted
experience period. Even small errors of 1° can have large effects. New and New (1962)
suggested that a 3° acuity level best described their data. Table 4.5 shows the estimates of

k from 1°, 3° and 5° limits on acuity.

122

 

Table 4.5 Estimates of k from the Training Period of Colony 1

 

 

 

Acuity Standard

Threshold Mean k Maximum k Median k Deviation of k
0° 4.1 106.7 1.9 10.4
1° 41.3 16,700 2.1 602.1
3° 51 13,000 2.2 561

5° 32.3 2,698 2.3 160.1

 

123

4.9 The Effect of the Time Interval

Since a discrete difference equation (3.1) is used in the generation of the ellipse azimuth
function, the length of the time interval over which it is summed has an effect on the
resulting function. I suggested above that this essentially added a parameter to the
equations. Presumably the bees (or other animals) have a speciﬁc time interval or average
interval that they would use in this process. It makes sense that it is a ﬁnite interval, since
the predicted change in azimuth should be above the perceptual threshold of the animal.
In the determination of the best ﬁtting ellipse parameters, a single time interval was used
for all of the data sets: 40 minutes. Initially, I did not hold this parameter constant. The
result of numerous runs suggested that the best-ﬁtting parameter was between 20 and 60
minutes. I assumed that the interval size should not vary considerably between individuals
or colonies. The 40 minute size is also corresponds to some empirical data that suggests
that bees update their information about the sun's position about every 40 minutes (Gould,

1984)

One of the effects of using this relatively large time interval (with respect to instantaneous
updates) is that error gets incorporated into the function that is generated. This is
equivalent to sloppy integration. It appears that this error increases the range of ﬁlnctions
that can be produced, particularly those resembling step functions, which correspond to

the experimental ﬁndings (Dyer & Dickinson, 1994). Figure 4.7 shows this effect.

124

 

 

 

360 —
L
270 —
ABCDEF
180 ~
90
6 12 18

Figure 4.7. The effect of time interval on the ellipse azimuth estimate. For
all of the curves, k is 1076 as in the best-ﬁtting ellipse azimuth function for

colony 1. Time intervals are (A) 1, (B) 2, (C) 5, (D) 10, (E) 20, and (F) 40
minute intervals. The curve that is most like a step function is the curve for
40 minute intervals.

125

4.10 Summary

In this chapter, I have shown that it is possible to derive a symbolic model that could
account for the recent experimental results regarding the sun-azimuth functions used by
ants (Muller & Wehner, 1993) and bees (Dyer & Dickinson, 1994). The critical departure
from previous models of sun compass learning is in the fact that this new model
incorporates the variation of the rate of change of the azimuth throughout the day.
Although other ﬁJnctions may perform as well, this model relied on the intuitively

satisfying model of a rate function based on the geometry of an ellipse.

Chapter 5

CONNECTIONIST MODELS

In this chapter I present a set of models that provides a contrast to the symbolic model
described in Chapter 4. In all of the models of sun compass learning presented in Chapters
3 and 4 the assumption was that insects rely on an explicit ﬁinction to calculate azimuth
position using the time of day. Regardless of whether the models relied on linear or
nonlinear rates of change of the azimuth in their calculations, the processes involved the
manipulation of symbols that corresponded to the observable quantities of azimuth

position and time.

In this chapter, I will present a set of connectionist models (F eldman & Ballard, 1982) that
contrast with all of the previous models of sun compass learning, including the model
presented in Chapter 4. In these models there is no explicit function computing azimuth
angle from inputs of time of day. There is no mechanism manipulating symbols that
correspond to azimuth position and time of day. These models will include nonlinearities
(Grossberg, 1988) that may allow them to account for the nonlinearities of the pattern of
solar movement that appear to be incorporated into the bees' (Dyer & Dickinson, 1994)

and ants' (W ehner & Muller, 1993) estimates of the sun's position at times of day they

126

127

have never seen it. These models could therefore be considered nonlinear nonsymbolic

models in contrast with the nonlinear symbolic model of Chapter 4.

These connectionist models of sun compass learning illustrate the types of behavior that
could result from a network of interconnected simple computing units (i.e. neurons) in
which there no explicit coding of an azimuth function. Such a network could form the
basis of an innate template that is modiﬁed with experience. This connectionist template
may allow insects to quickly recognize the local ephemeris function or quickly learn the

local ephemeris ﬁmction or both.

5.1 Connectionist Computations

Connectionist neural networks generally consist of a large number of interconnected
processing units (Feldman & Ballard, 1982; Rumelhart et al., 1986a). For the networks I
will consider, the neural elements are arranged in layers with connections between the
elements of each layer, but not within the layer. The information ﬂows in one direction
from the input layer (sensory) to the output layer, making them feed-forward networks or
perceptrons (Rosenblatt, 1958; Rumelhart et al., 1986b). The power of a network to learn
a complex representation comes from the number of layers in the network and the number

of units in the layers. Figure 5.1 shows a diagram of a simple feedforward network.

128

“ w a W e
04$ (9»
o W 6

Figure 5.1 General form of a feed-forward connectionist network. 1: input
units H: hidden units. 0: output units. w: connection weights.

129

Each individual unit essentially computes a weighted sum of its inputs to determine its
output. The sum is weighted by the connection strengths (or weights) between the units.

This weighted sum is described by the following equation:

x]: ZYI‘WJ‘I' (5-1)

in which the total input (x) of the jth neuron is the sum of the outputs 0),.) of the of the ith
layer and the connection weights between the layers (wﬁ) (Rumelhart et al., 1986c). The
output of the unit is a nonlinear ﬁrnction of the weighted sum of the inputs. Generally,

this ﬁmction is a sigmoid:

__ 1
y} _ 1 +e‘xf (5-2)

 

where yj is the output of the j th and x}. is the weighted sum of the inputs to the j th neuron

as in equation 5.1 (Rumelhart et al., 1986c).

Networks composed of these simple units that are arranged in at least three layers
(Rumelhart et al., 1986c) can learn a wide range of arbitrary mappings between inputs and
outputs. In fact, they are universal approximators (Hornik et al., 1989). This means that
given the appropriate network size and training time, they can learn to approximate any
functional relationship between a set of inputs and a set of desired outputs. Simple
perceptrons (Rosenblatt, 195 8) are limited in their learning abilities (Minsky & Papert,
1969), but the inclusion of the internal layer of "hidden units" greatly increases their

capacity to form a wide range of representations (Rumelhart et al., 1986c).

130

For a network to learn a complex representation it must have a mechanism for adjusting
the connection weights (wﬁ) between the layers of neural elements. Connection weights
could be set by hand, but the complexity of such networks usually requires a learning
algorithm that adjusts the connection weights progressively to reduce the error between
the outputs of a network and the desired outputs. One of the most common and most
powerful algorithms is the method of backpropagation of error (Rumelhart et al., 1986c).
This is a method for ﬁnding the global error minimum in multidimensional weight space.
The backpropagation algorithm adjusts each weight such that it achieves the steepest
descent in error. Thus errors in the output of the network as a whole can be used to

modify the connection weights within the network.

This learning method lacks biological realism in the exact mechanism of learning, the
modiﬁcation of the connection weights. Nonetheless, the major advantage of
backpropagation is that it allows the behavior of large networks of simple units to be
investigated. These are good models for the behavior of distributed representations that

can provide insights into brain mechanisms (McClelland et al., 1995).

5.2 Connectionist Characteristics
Aside from their ﬂexible learning abilities, perhaps the property of connectionist networks
that has attracted the most attention is their ability to generalize the learned relationship

between the input and output sets. This allows networks to classify novel inputs and to

131

complete partial patterns. The ability of connectionist networks to recognize and
complete patterns has long been recognized (Rosenblatt, 1958; Minsky & Papert, 1969),
but the widespread application to this area awaited a sufﬁcient learning algorithm

(Rumelhart et al., 1986b).

An example of this is seen in the face-recognition network of Cottrell and Metcalfe
(1991). This network, which had an architecture analogous to that in Figure 5.1 but with
different numbers of units, was trained on a set of images to classify human faces by name
and gender. Subsequently, the network could recognize different images of the same faces
in the training set and it could classify novel faces by gender. Additionally, when
presented with partial views of the faces, it could reconstruct the entire face. This was
done without any explicit coding of the geometrical properties of faces. The same
network would undoubtedly have been able to learn to recognize images of different bird
species, or the leaves of different tree species, and to classify partial images of these

natural shapes.

5.3 Connectionist Representations of the Sun's Course

The ability of connectionist networks to complete partial patterns seems to parallel the
ability of insects to ﬁll gaps in their experience with the sun's course. In this construction,
the sun compass learning problem is a pattern completion problem or a perceptual

recognition problem. This is based on the assumption that completing the pattern of solar

132

movement is analogous to the other types of patterns that connectionist neural networks

are able to complete.

I will consider both the learning capabilities and the pattern completion capabilities of
connectionist networks as I examine their relevance to the sun compass problem.
Additionally, I will consider two alternative ways of representing the sun's course in a

connectionist framework. The ﬁrst is a binary mapping and the second is a continuous

mapping.

5.4 A Binary Representation

It is clear from the experimental data presented in Chapter 3 that bees and ants are able to
ﬁll gaps in their experience with the sun to such an extent that they relatively accurately
infer the position of the sun during large portions of the day or night. To model this with
a connectionist architecture, we must consider that the network that the bees bring to the
problem must be innately conﬁgured so that it can estimate unknown solar positions.
Thus, the training of this network could not have involved the experience with these
portions of the sun's course. We are therefore dealing with a network trained during
evolutionary history. In principle, however, standard connectionist training principles
might apply. In this case, the errors generated by the network would be minimized

through the action of natural selection.

133

A preconﬁgured network could have several forms, each with different potential
advantages to a small-brained, short-lived animal. Such a network may allow an animal to
quickly access the appropriate solar ephemeris function for its foraging lifetime. Honey
bees forage outside the colony for only about 10 days before they die, and they make
relatively few ﬂights before they start foraging. A preconﬁgured network may allow them
either to quickly recognize the correct local ephemeris pattern or to quickly learn the

correct local ephemeris pattern or both. I will consider these alternatives in turn.

5.4.1 Pattern Matching

Like the face recognition problem outlined above, a network conﬁgured to recognize a
solar ephemeris function would be trained (over evolutionary time) on a range of azimuth
ﬁinctions. The strategy employed in this model was to train a network on a range of
azimuth functions and subsequently test the network with partial azimuth ﬁrnctions. This
is analogous to bees with an innately-conﬁgured network receiving restricted experience

with the sun's course (Dyer & Dickinson, 1994).

The training patterns for this network (Figure 5.2) were idealizations of a range of solar
ephemeris functions that would normally occur at different seasons and latitudes (see
Figure 2.2 for a range of real azimuth functions). A binary mapping was used to represent
the ephemeris ﬁmctions (in Figure 5.1, black = one, white = zero). The pattern of ones
and zeros indicates the presence or absence of a particular combination of azimuth angle

and time of day. Some preprocessing would be required to get observations of azimuth

134

angles at times of day into this format. Each 10 by 10 array corresponds to a
coarse-grained depiction of a particular solar azimuth ﬁrnction. The speciﬁc size of the
array was originally constrained by the maximum size of the input array in the computer
simulation package that I initially used (Caudill & Butler, 1992). Seven patterns were
generated to correspond to a range of ephemeris functions: from a 180° step function
approximating a tropical azimuth curve to gradual curves approximating the typical
ephemeris ﬁrnctions of the northern and southern temperate regions. Given the coarseness
of the binary mapping, seven patterns provided the complete range between a step

function and a gradual function.

135

Figure 5.2 Idealized solar ephemeris ﬁlnctions used as the training set for
the connectionist models. The patterns consist of a matrix of ones and
zeros, with the ones corresponding to the black squares and the zeros
corresponding to the white squares. Each point corresponds to a pairing of
azimuth position and time of day. These patterns correspond roughly to
the natural range of solar azimuth ﬁmctions (See Figure 2.2).

DEBS}
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136

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Figure 5.2

137

Network Architecture

     
 

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Time Time
Input Layer Hidden Layer Output Layer

Figure 5.3 Network architecture. The network has 100 input units and
100 output units that correspond to azimuth-time coordinates. There are
25 hidden units. See text for explanation of connection patterns. The
connections are not shown because of the number of them. Activation is
shown by the gray scale (white = zero; black = one).

138

The network was trained to recognize each of the seven training patterns. The network
had 100 input units and 100 output units. Between the input and output layers was a
single hidden layer of 25 units. Each of the input units was ﬁilly connected with each of
the hidden units and each of the hidden units was ﬁilly connected with each of the output
units. The original ﬁndings were obtained with a commercially available simulation
package (Caudill & Butler, 1992), but they have been replicated and extended with
models run on Mathmalica® and Mathcad®. The initial connection weights were
pseudorandom numbers ranging from -0.3 to 0.3. The learning constant (11) was set at
0.5. (This determines the size of the weight changes). The networks were trained through
repeated exposure of the network to the patterns until the mean squared error between the
desired and the actual outputs of the network was reduced to 5%. The backpropagation
algorithm was used to adjust the connection weights during training (Rumelhart et al.,

1986c)

After the networks reached the criterion, they were presented with test patterns consisting
of fragments of ephemeris ﬁmctions (see Figure 5.4 A). These test patterns were
analogous to the partial ephemeris functions experienced by the bees in the experiments
presented in Chapter 3. The responses of a trained network to each of the test patterns
are presented in Figure 5.4 B. These responses consist of the activity levels (the result of
equations 5.1 and 5.2) for each of the output units. The values range from zero to one
because of the effect of the sigmoid activation ﬁinction (equation 5.2) of each of the units,

which asymptotically approaches zero and one. Although the patterns presented in Figure

139

5.4 (B) correspond to a single network conﬁguration produced through training, the
results of other replicates were consistent. The results are presented in this way to show
the subtleties that exist in the representation of the pattern in a single network. The results
presented in Figure 5.4 correspond to a network with 25 hidden units; nearly identical

effects are seen in networks with only 10 hidden units.

In response to the test patterns, the network ﬁlled the gaps in the partial azimuth curves.
When the test pattern consisted only of solar positions in the late aﬁemoon (analogous to
what afternoon-experienced honey bees faced (Dyer & Dickinson, 1994) or in the early
morning (analogous to what moring-experinced desert ants faced (Wehner & Muller,
1993)), the network produced a pattern of activation resembling the 180° step function
that the experience-restricted bees and ants develop. For several of the test patterns with
late aﬁemoon azimuth fragments, the network produced a step-function that also
exhibited variability during the midday transition. This mimics the behavior of some of the

bees from colony 1 (Figure 3.3 and 3.4) and of the bees in New and New's (1962) study.

In contrast with the patterns that produce a step ﬁmction, several of the test patterns,
corresponding to midday time-azimuth positions, resulted in gradual functions. This
suggests that experience at different times of day might lead to different representations of

the sun's course.

140

One of the test patterns was unnatural and indicated time-azimuth positions that would
never occur on the earth (e. g., bottom panel of Figure 5.4). The preconﬁgured network

ignores this type of spurious information.

These results demonstrate that certain aspects of the sun compass learning problem in
insects can be mimicked by a neural network that does not explicitly encode a
mathematical expression corresponding to the relationship between azimuth angle and
time. With experience over the entire day, the network may be able to recognize the
appropriate local ephemeris function. But with partial experience, the overlap among the
possibilities causes the network to partially activate several of the alternatives, leading to

the highest activation of the intermediate function: the 180° step function.

141

Figure 5.4 Responses of preconﬁgured network to test patterns consisting
of partial ephemeris functions. The test patterns (A) consist of inputs
arranged in an array of ones and zeros as in Figure 5.2. The responses (B)
for each test pattern are presented immediately to the right. The responses
consist of the activation level of each of the output units. They range ﬁom
zero to one and are all on the same scale. See text for a description of the
network.

142

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143

5.4.2 Learning Advantage

An alternative potential advantage of a preconﬁgured neural network may be that it allows
the insect to rapidly learn the appropriate local pattern. The question in this case is
whether a network that has learned one representation of the sun's course can quickly
replace that representation with one that corresponds to the actual pattern of movement

observed.

The 180° degree step function provides a logical starting place for this question. An insect
using a 180° step function during any season or at any latitude would experience the least
average error in its estimation of the sun's position than one using any other azimuth
function. This is because the 180° step function is the average of all of the solar ephemeris
ﬁrnctions encountered on the earth (i.e. all ephemeris ﬁmctions at all latitudes for all days
of the year) (Dyer & Dickinson, 1996). This suggests that a step ﬁrnction would be a
good template. By using the 180° step function as a template, an insect would on average
make fewer errors before the current local pattern is learned. This is not a learning
advantage per se but it is an advantage during the learning process. In addition, a step
function might also allow a more rapid learning of the local pattern than an unconﬁgured

network would. There are several reasons this might be true.

First, it may simply be easier to learn a pattern corresponding to an ephemeris function

once any other ephemeris function has been learned. Ifthis were the case, all ephemeris

144

ﬁrnctions would work equally well, although the step function would still have the

advantage of making fewer errors while learning.

Second, since the 180° step ﬁinction is the average of all ephemeris functions encountered
on the earth, it may be a good starting place for learning the other curves. This may
occur because the step function is intermediate between all of the other possible curves.
Thus the average amount the curve would have to change would be minimized (Dyer &

Dickinson, 1996).

Third, the step function may share more points in common with other ephemeris curves
than any of the other curves would. Because of this overlap, fewer substitutions (of
time-azimuth coordinates) would have to occur on average to move from a starting point
of a step ﬁinction to any other curve than from any other starting curve to another curve.
The number of substitutions between the patterns can be measured by the Hamming
distance (Hamming, 1986). Hamming distance is an information-theoretic concept that
describes the amount of overlap between binary strings. It is expressed as the number
positions in two strings that do not overlap, therefore, the Hamming distance between

{1,1,1} and {1,1,0} is one, and the distance between {1,1,1} and {0,1,0} is two.

This third alternative uses a different metric for measuring distance than the second
alternative does. The third alternative measures distance between points in the azimuth

patterns in Hamming distance rather than conventional linear distance, which is implied in

145

the second hypothesis above. Standard linear distance, or Pythagorean distance is

calculated with the following formula:

aﬂ = Z(x-y)2 (5.3)
while Hamming distance is calculated:

d = Z Ix -y| (5.4)

(see Hamming, 1986). Figure 5.5 illustrates these distance metrics with two time-azimuth
patterns. The ﬁgure depicts the transitions between a step ﬁrnction and two other azimuth
curves. The Hamming distance between the step function and each of the other curves is

the same (4), but the Pythagorean distance is different (2 and 4).

146

 

Figure 5.5 Comparison of Hamming and Pythagorean distances. In this
diagram, the standard linear (Pythagorean) distance from the step ﬁrnction
to each of the others pictured varies, while the Hamming distance is
constant. A. Pythagorean distance = 2, Hamming distance = 4. B.
Pythagorean distance = 4, Hamming distance = 4. Hamming distance is a
measure of overlap between the patterns. In each of the cases, two points
have changed, which gives a Hamming distance of 4 (turning off the old
point and turning on the new point for each of the two changes). The
linear distance between the old points and the new points makes a
difference for the Pythagorean distance but not for the Hamming distance.

147

1 investigated these alternative advantages of a preconﬁgured network with a series of
retraining experiments. In each case a network was trained to learn one of the ephemeris
ﬁinctions depicted in Figure 5.2 and was subsequently retrained on each of the other
curves (the ending weights for the ﬁrst training were used as the starting weights for each
of the retrainings). The network characteristics (e. g. layer size, learning constant,

criterion) were the same as described above.

Figure 5.6 shows the results of these experiments. In all of the cases, the retraining time
was considerably less than the initial training time. The retraining times are represented as
a percentage of the original training time, and they are all well under 100 percent. This
suggests that there is an advantage to starting with a preconﬁgured network no matter
what its form. It would be advantageous to start with a step function since the fewest
errors would occur during the learning process, but are there any ﬁrrther advantages of a
step ﬁmction, corresponding to the second or third possibilities listed above? This too is
apparent from the results of the retraining experiment (Figure 5.6). The results are
extremely consistent and fall into a clear pattern. There are three distinct groups of points.
The overlapping points in each group all are the same Hamming distance (Hamming,
1986) from the original training pattern. This corresponds to the third of the alternatives
listed above. The retraining time appears to be a function of the Hamming distance
between the patterns and not the linear distance (as implied by the second hypothesis listed

above). This makes sense in light of the fact that the networks considered are ﬁilly

148

interconnected; this means there is actually no geometry to the grids depicted in Figure

5.2. Each of the points in the grid is equally close to each of the other points.

The Hamming distance between each of the azimuth curves is shown in Table 5.1 (The
letters correspond to the letters in Figure 5.2). The step function (A) is among the curves
with equally short average Hamming distances to each of the other curves. In this case,
however, the step ﬁmction is only one of ﬁve curves that is closest to all of the other
curves in the number of binary substitutions that would have to occur during retraining.
This does not alone give it an advantage over the other curves as starting conﬁgurations
for an innate template of the sun compass. This may not be true for binary representations
of ephemeris curves that have greater detail. It is possible that with a smaller grid size the
step function would share more overlapping regions with more of the other curves. But it
is also the case that with greater detail will come a greater average Hamming distance

between the curves.

149

 

 

100 —
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ad
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0 h 1 1 1 L 1 l
2 4 6 8 10 12 14

Hamming Distance

Figure 5.6 Binary retraining trials for azimuth network trained on one
azimuth pattern and retrained on another. Binary retraining percentage is
the number retraining trials to reach an error criterion of .05 as a
percentage of the training time for the original pattern. Data are the means
i standard error for 10 replicates for each of the 7 azimuth patterns
retrained on the 6 other patterns. They are plotted against the Hamming
distance (number of binary substitutions) between the two patterns. This is
clearly the major explanatory variable.

150

 

Table 5.1 Hamming Distance between Azimuth Training Patterns

 

 

 

A(Step) B C D E F G
A (Step) 0 4 4 8 8 12 12
B 4 0 4 8 8 12 12
C 4 4 0 8 8 12 12
D 8 8 8 0 8 4 12
E 8 8 8 8 0 12 4
F 12 12 12 4 12 0 12
G 12 12 12 12 4 12 0
Mean 8 8 8 8 8 10.67 10.67

 

To examine the effect of grid size on the Hamming distance between the curves, 1
produced a set of azimuth curves to the nearest ten degrees. Like the curves in Figure
5.2, these azimuth curves were restricted to the daytime and consequently consisted of 18
by 18 grids. I subsequently looked at the average Hamming distance from each curve to
every other curve. If the search is conﬁned to the tropics, the step ﬁmction indeed has the
lowest average Hamming distance to every other curve (see Figure 5.7 A). However,
when the temperate latitudes are included, this changes (see Figure 5.7 B). Figure 5.7 A
shows the average Hamming distance between azimuth curves corresponding to each set
of latitude-declination coordinates to each of the others. Curves were produced for ten
degree increments of latitude and of declination from 20° N latitude and declination to 20°
S latitude and declination. The pit in the center of Figure 5.7 A shows that the lowest

average Hamming distance to all other curves corresponds to the azimuth curve for 0°

151

latitude and 0° declination, which is a 180° step function. However, when the temperate
latitudes are included, the picture changes, as Figure 5.7 B indicates. Figure 5.7 B shows
the average Hamming distance between all curves ﬁom 40° N latitude and 20° N
declination to 40° S latitude and 20° S declination. In this case, the curve with the lowest
average Hamming distance is a more temperate curve. The step ﬁinction no longer has
the lowest average Hamming distance to all other curves. This effect is somewhat
surprising, although in retrospect it is understandable. With higher latitudes come ﬂatter
azimuth curves; therefore, the more curves from high latitudes that are considered, the
more ﬂat curves there are. These curves subsequently have a high degree of overlap with

each other, and hence they have a lower average Hamming distance.

There is still a considerable retraining advantage, however, for the curves discussed in
Figure 5.7 B, even for the curves that are farthest apart. Figure 5.8 shows the results of a
retraining experiment involving these azimuth curves. Since Hamming distance is the
relevant factor, I restricted the experiment to a sampling of each of the Hamming distances
between the curves in Figure 5.7 B, instead of looking at all of the pairwise comparisons.
The Hamming distances range from 0 to 36. Each of the points in Figure 5.8 represents a
set of ﬁve replicates for each Hamming distance. Although the Hamming distances are
larger than those represented in the previous retraining experiment (Figure 5.6), the
retraining times are similar. This suggests that it is not Hamming distance per se that is
important. Since all of the azimuth curves have the same proportion of ones and zeros, it

might be reasonable to assume that it is the relative Hamming distance that is important.

152

For this reason, I plotted the results in terms of relative Hamming distance and have
included the results from the previous experiment for comparison. I have deﬁned relative
Hamming distance as the Hamming distance divided by the number of elements in the

pattern.

In summary, a preconﬁgured network would seem to confer a learning advantage. Once a
network has been trained on one curve, it can more quickly learn another curve. If the
preconﬁgured curve were in the form of a step ﬁinction, ﬁrrther advantages would accrue.
A preconﬁgured network that resulted in a step ﬁinction would give the insect an
advantage in the early stages of learning, because a step ﬁrnction leads to the smallest
average error for all possible ephemeris ﬁanctions observable on Earth. This is the
reduction of errors during the learning process that is mentioned above. Additionally, the
step function would be among the curves with the lowest average Hamming distance to all
other curves, particularly in the tropics. These three factors could lead to the evolution of
this mechanism. For a temperate animal it would make sense to have a preconﬁgured
network in the form of a ﬂat curve rather than a step ﬁmction; however, this is only the

case if the animal is conﬁned to either the northern or the southern temperate regions.

153

 

   

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Figure 5.7 Average hamming distance from each azimuth curve to every
other curve The curves are to the nearest ten degrees. The average
Hamming distances are plotted with respect to their latitude-declination
coordinates. Panel A shows the situation when the curves are restricted to
the tropics (between 20° N and 20° S). Panel B shows the situation when
temperate curves are included (between 40° N and 40° S).

       
  

154

 

 

 

 

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Figure 5.8 Binary retraining trials for azimuth curves discussed in Figure
5.7. For the range of Hamming distances that exist between the azimuth
curves represented in ﬁgure 5.7, a network trained on one azimuth pattern
and retrained on another. The network had 324 inputs and outputs and 81
hidden units. Binary retraining is the number retraining trials to reach an
error criterion of .05 as a proportion of the training time for the original
pattern. Data are the means i standard error for 5 replicates for each
Hamming distance (ﬁlled circles). They are plotted against the relative
Hamming distance (number of binary substitutions)/(number of elements)
between the patterns. For comparison the data from Figure 5.6 are
included (open circles).

155

5.5 A Continuous Representation

The models described in the previous section have a drawback that could limit their
applicability. Real insects can estimate the position of the sun at night. However, this part
of the sun's course falls outside of the range of the time-azimuth combinations in the
training patterns for the preconﬁgured networks. Since the networks are assumed to be
trained over evolutionary time, the responses of the ancestors of the bees to these times
would have to be incorporated into the training sets, yet no ancestor could have seen the
solar position at night. Additionally, in the binary mapping, not only can points outside
the range of the training pattern not be estimated, but also points within the range and
below the resolution of the network cannot be estimated, because of the discrete nature of
the binary mapping. For example, if the time dimension were coded at a resolution of one
point per hour, it would be impossible to estimate azimuth angles at intermediate points on
the half hour. This problem, and the problem of the range of the training set, may be
eliminated with an alternative mapping. Such an alternative representation would be to

use a continuous mapping rather than binary mapping.

To explore the effect of a continuous mapping, I turned to simple network with only three
neural elements and an input unit (see Figure 5.9). This network has one input unit and
one output unit. In addition there is one hidden unit and one bias unit. A bias unit is a
unit that whose activation is always one. It is connected to the other units via connections

with variable weights (See Figure 5.9). Bias units can increase the range of functions a

156

network can learn and decrease the convergence time (Rumelhart et al., 1986c). The bias
unit appears to be necessary for convergence in this case. The input corresponds to time
of day and the output corresponds to azimuth angle. Because of the sigmoid activation
function used, these numbers for both time and azimuth angle ranged ﬁ'om zero to one.
This network can be trained to learn particular time-azimuth correspondences. Because of
the continuous mappings of the inputs and outputs, their is no constraint on the values that
the test inputs can assume. A network trained on time-azimuth correspondences can be
tested with time inputs from the night. It can also be tested with inputs that are
intermediate to the training values. Both of these were impossible in the preconﬁgured

networks with binary mappings.

Figure 5.10 shows the responses of several networks with different training regimes. The
networks can be trained on a set of time-azimuth input/outputs and tested on a novel set

of time inputs.

In Figure 5.10 (A), the network has been trained on the azimuth angles at two hour
intervals between sunrise and sunset. The curve shows the responses of the network when
tested at half hour intervals throughout the 24-hour period. During the day, at values
intermediate to the training set, the network does a very good job of ﬁlling the gaps.
However, outside the range of the training set, the network performs much more poorly.

This is expected from the general behavior of neural networks, and it has been

157

demonstrated elsewhere (Johnson & Picton, 1996). The nocturnal azimuth estimates of

this network vary considerably from the true azimuth patterns.

The architecture and the mapping of this network allow it to interpolate between points in
the training set and to extrapolate forward and backward to points outside the range of the
points in the training set. This is a very important feature of the network that is parallel to
the mechanisms implicated in the sun compass learning problem (New & New, 1962;
Gould, 1980; Wehner & Lanfranconi, 1981; Wehner, 1982; Dyer, 1985, 1987) (and see
Chapter 3). Because of these distinctive features of this network, I will call it the

interpolation network.

The architecture of this network also allows it to be trained on a small portion of the sun's
daily course and to be tested on another part of it. This is analogous to the experiments
presented in Chapter 3 (Wehner & Muller, 1993; Dyer & Dickinson, 1994). In this case
(see Figure 5.10 B), the network does a very poor job of estimating the position of the

azimuth in the unexperienced portion of the day.

158

 

 

 

Figure 5.9. Diagram of simple connectionist network used for the
interpolation network. 1: input unit. H: hidden unit. 0: output unit. B:
bias unit. w: modiﬁable connection weights.

159

Although this network does a good job generalizing to azimuth positions temporally close
to those they have experienced, it does not estimate azimuths as well as insects at times
several hours from the training period. However, this model could be rescued if there
were a way to incorporate some structure in the model that is analogous to the type of

innate structure that appears to be present in bees (Dyer & Dickinson, 1994).

There are several mechanisms that could be used. Like the preconﬁgured networks in
sections 5.3 and 5.4, this network could have preconﬁgured connection weights.
Alternatively, there may be a different way of imposing structure on the learning process.
This could be achieved by incorporating some of the universal features of solar movement
into the training set. The 180° step function (Dyer & Dickinson, 1994) observed in the

behavior of experience-restricted bees can be used to accomplish this goal.

To do this, I trained a network with the same training set as in Figure 5.10 (B), but I also
included two additional points in the training set. These points were for time inputs of
06:00 and 12:00 (solar time). The azimuth angles for these times were derived from the
azimuth angle observe at 18:00. For the azimuth angle for 06:00, 180° is subtracted from
the azimuth angle at 18:00. Likewise for the angle at noon, it is obtained by subtracting

90° from the azimuth angle at 18:00.

By including these additional angles in the training set, the network produces a much

closer approximation to the true position of the sun when tested during the new time of

160

the day (see Figure 5.10 (C)). This suggests that this simple connectionist network is
capable of broadly simulating the behavior revealed in experiments. This seriously
challenges Gallistel's assertions about the ability of nonsymbolic models to solve the sun
compass learning problem. This last model might verge on symbolic elements, such as
deriving the additional training patterns through subtraction; but the essential
computations of the network are subsymbolic and they are not isomorphic to any logical

mechanism of computing azimuth angle from time.

161

Sun Azimuth (°)

 

 

Local Sun Time

Figure 5.10 Response of interpolation network to three different training
regimes for learning the same speciﬁc azimuth ﬁrnction. The azimuth
function corresponds to 25° N on April 22 (solar declination 12°). A.
Response of a network over 24 hours at half-hour intervals for a training
set of hourly azimuth angles between sunrise (~06:00) and sunset (~18:00)
(local solar time). B. 24 hour response of a network trained only on
hourly positions in the late aﬁemoon and evening (15:00 to 18:00). C.
Response of a network trained on aﬁemoon positions in B but with
additional constraints (Azimuthmo = Azimuthm00 - 90°; Azimuth“00 =
Azimuth“,00 — 180°).

162

5.6 Summary

In this chapter I have shown that connectionist models could account for several of the
characteristics of sun compass learning that have been revealed in recent experiments. In
contrast to the suggestion of Gallistel (in press), these types of models may provide a
viable alternative to a symbolic model as a means of explaining the sun compass learning
problem. To explain the phenomenon, however, a connectionist model must include a
certain amount of "innate" information about the sun's course. The results presented here
suggest that some kind of innately conﬁgured network could explain the behavior of the

animals in the experiments.

Chapter 6

CONCLUSIONS

The sun compass has proved to be an excellent avenue into gaining a more complete
understanding of the nature of a representation in the brain of an animal. Behavioral
experiments alone have gone a long way towards delineating computational mechanisms
by which the sun compass learning problem must be solved. The problem consists of the
ability of small-brained animals such as bees and ants to estimate positions of the sun at
times of day (and night) when they have never seen it. The actual solution is still a
mystery, but new experiments have shown that the problem may be solved by an innate
template that is modiﬁed with experience (Chapter 3), and I have shown how two broad

classes of computational mechanisms could solve the problem (Chapters 4 and 5).

I have focused on the dichotomy of symbolic and nonsymbolic processing that has
emerged out of the ﬁeld of cognitive science (see Smolensky, 1988; Boden, 1991) and has
intersected with the problem of animal orientation (Gallistel, in press). Taking up
Gallistel's challenge that symbolic computations provide the best explanation for the sun

compass learning problem, I have established several models of how subsymbolic

163

164
connectionist models could underlie the sun compass learning problem. In addition, I have

formalized a conceptual model of how the problem can be solved in the symbolic domain.

Both classes of models account for some of the aspects of the observed behavior that
were not present in the previous models. This suggests that the actual neural
computations that underlie the estimation of the sun's course may involve equivalent
mechanisms. The computational mechanisms hypothesized by these models may
contribute to the elucidation of the neural mechanisms that insects use to compute the
position of the sun and indeed how brains represent information about the environment.
The symbolic and connectionist ﬁameworks posit fundamentally diﬁ‘erent views of how
the brain processes information. The symbolic framework lacks a correspondence with
neuroscience: the fundamental elements of a symbol-processing mechanism have yet to be
identiﬁed (Gallistel, in press). On the other hand, the connectionist framework lacks a
correspondence with major areas of psychological theory: processes that appear to be
characterized by the logical manipulation of symbols (Boden, 1991; Gallistel, in press).

By examining how the sun-azimuth estimation process can be accomplished by these two
broad classes of models, I have shown what types of symbol-manipulating mechanisms are
necessary to solve the problem, and how connectionist models can solve this apparently
symbolic process. In Table 6.1, I present a synopsis of the features of the four main
models that have emerged from this endeavor. Included in the table are the main
distinguishing features of the models. I will consider each of these aspects in greater detail

in the text.

165

 

Table 6.1 Comparison of New Models

 

 

Model Representation Modification Symmetry Nocturnal
with Experience Assumptions Compensation

Ellipse Continuous Yes (Averaging) Yes Yes

Pattern Matching Binary No (Matching) No No

Network

Learning Advantage Binary Yes (Learning) Implicit No

Network

Interpolation Continuous Yes (Leaming) Some Yes

Network

 

6.1 Innate Structure

All of the models are consistent with the conclusion (W ehner & Muller, 1993; Dyer &
Dickinson, 1994) that there is innate structure to the mechanism employed by the bees and
ants to estimate the position of the sun at new times of day. This statement is based on
the observation that all of the models require global information about the task to come
close to simulating the behavior of the bees and ants in the experiments. That is, the local
information (the observed positions of the sun's azimuth at certain times of day) appears to
be insufﬁcient to determine the position of the sun at vastly different times of day.

Instead, either more information or some means of interpreting the available information is
needed. In the symbolic framework, the additional information is embodied in the
functional relationship between time and azimuth position. This even applies somewhat to
the linear models (New & New, 1962; Gould, 1980; Wehner & Lanﬁanconi, 1981; Dyer,

1985, 1987) of compensation, but it is more clearly evident in the nonlinear (ellipse-based)

166
model. The global information is that the azimuth changes at a varying rate that
approximates an elliptic function of time. In the connectionist models, the global
information resides in the connection weights of the preconﬁgured networks or in the

constraints imposed upon the learning process (interpolation network).

6.2 Templates

Dyer and Dickinson (1994) suggested that the implied innate structure might be analogous
to a sensory template after Marler (1976, 1984). Marler invoked the sensory template
concept as a heuristic model to describe the development of species-speciﬁc song in birds,
and he suggested that it may apply to other types of behavior in other organisms, though
he focused on vocal behavior. The template is a genetic constraint on learning that makes
the organism particularly sensitive to particular types of stimuli at particular times. Marler

(1976, p. 328) states:

Sensory templates provide a structural ﬁ'amework for the perceptual
analysis of arrays of stimuli that is both plastic and yet constrained. After
more or less extensive modiﬁcation by experience, with their number added
to or subjected to attrition, and changed in speciﬁcation so that their
properties may now be both species-speciﬁc and also population-, group-,
or even individual-speciﬁc, they then guide motor development by a
process of sensory feedback. Phases of this multistage process may
interdigitate in time or they may be temporally separated, proceeding most
readily at particular developmental stages or "sensitive periods."

In the sun compass learning problem, there are clearly speciﬁc classes of stimuli that are
important. These correspond to azimuth position and time of day. In honey bees, the

learning occurs rapidly and during a speciﬁc period of the individual's life (Lindauer,

167
1959), though it is not clear that it cannot occur at other times. There are physiological
changes in the brain that correspond with this period in the bee's life, which are

hypothesized to be related to the many things she has to learn as she begins to forage

(Withers et al., 1993).

Perhaps the most critical correspondence between Marler‘s (1976) sensory template
concept and the sun compass learning problem is the feature of the template model that
suggests that the response to certain stimuli is both plastic and constrained. I would argue
that this is true for all four of the speciﬁc models I have considered. In general terms there
are speciﬁc stimuli (azimuth angle and time) that must be represented in a ﬂexible enough
framework to correspond to the range of azimuth ﬁrnctions encountered on the earth, but

they may also be constrained to fall into a particular range.

This is most evident in the pattern matching network. Encoded in the connection weights
of this network are a range of patterns of azimuth movement that correspond to the range
of natural patterns. If a spurious combination of azimuth position and time is entered into
the network, it has no effect on the response of the network. From the noisy information
is generated a complete pattern of azimuth-time mappings. This is seen in the response of
simulated network to one of the test patterns that included a spurious time-azimuth

coordinate not included in any of the training patterns.

168
With the learning advantage network, this may not be as much of an issue. Spurious
azimuth-time coordinates will be included in the learning process. If they are consistent
the network will releam the new pattern. It is conceivable that the network would learn
the new pattern faster than it would have without the preconﬁgured connection weights.
This would occur even if the spurious coordinates were beyond the range of the natural
patterns of azimuth-time coordinates, since by the Hamming distance rule, only the
amount of overlap matters and not the Pythagorean distance between the points.
However, the network would nonetheless be biased toward the step-ﬁrnction response in
the early stages of retraining. Since the step function is encoded in the initial connection
weights, the initial responses of the network would be biased toward a step function, but
as new inputs are encountered, the connection weights would shift to represent the new

pattern.

The interpolation network provides an excellent example of the need for global
information in the sun compass learning problem. A network trained on a subset of
time-azimuth coordinates can estimate the azimuth angle for new times of day. If the
training set is restricted to a portion of the day, the estimates in the other part of the day
would vary considerably from their actual values. In order to solve the problem a global
constraint is needed. One such constraint that reduces this problem is to include in the
learning set some universal relationships, such as that at 6:00 it is 180° ﬁom its position at

18:00 and vice versa.

169
In the symbolic model, the observable quantities of azimuth position and time are ﬂexible,
but the ﬁrnctional relationship is strictly constrained. The values must ﬁt an ellipse.
Spurious values may lead to ellipses that are considerably different in shape from the true
seasonal and latitudinal azimuth rate ﬁrnction. This may underlie the behavior of the bees
in the experiments presented in Chapter 3 (Dyer & Dickinson, 1994). In these
experiments the bees' representation of the sun's course corresponds to a curve
approximating a 180° step ﬁlnction. If something akin to the ellipse azimuth ﬁrnction
were the underlying mechanism, the ellipse being used to approximate the rate of change
of the azimuth would be a much more squashed ellipse than the ellipse corresponding to

the true azimuth rate function of the season and latitude of the experiments.

6.3 Symbolic versus Nonsymbolic

I have maintained throughout that the models differed in their underlying representational
structure. The major distinction I have used is between symbolic and nonsymbolic
processes. At the level of behavior, it is very hard to distinguish between these
alternatives. The indeterminacy of behavioral data certainly has contributed to much of
the debate among cognitive scientists regarding this distinction (Smolensky, 1988; Boden,
1991; Gallistel, in press). Table 6.1 suggests some of the distinctions between the models
that may be amenable to behavioral test. Two of these features make very speciﬁc
predictions about the positions of the sun estimated during the night. The fact that many
animals are able to estimate the position of the sun at night enforces a major constraint on

the types of processes that may underlie the behavior. In their current state, this fact

170
appears to eliminate the possibility of the two preconﬁgured networks with the binary
mapping. With the binary mapping there can be no interpolation for points not in the
training set. This seems to limit models of this sort. However, there may be several ways
around this problem. For the preconﬁgured pattern matching network, it is assumed that
responses of the ancestors of the bees and ants responded appropriately to observed
positions of the sun at different seasons and latitudes. Through natural selection an
appropriately conﬁgured network could conceivably emerge. But since, the ancestors
could not see a range of values for azimuth functions during the night, they would not
evolve a preconﬁgured network to deal with a nocturnal position of the sun. However,
there are seasons and latitudes in which the sun is visible at night. Moreover, there are
populations of arthropods that live in such areas and that orient by the sun throughout the
night (Papi & Syrjamaki, 1963), e. g, Finnish populations of wolf spiders (Arctosa
cinerea). It is true, however, that there is not a range of azimuth functions visible at night.
But one of the results from many studies of the nocturnal compensation is that a gradual
constant rate of compensation is observed rather than a varying rate like during the day

(Pardi & Ercolini, 1986).

There are two problems with the hypothesis the Arctic sun could determine the nocturnal
patterns in an preconﬁgured network, however. First, Italian populations of Arctosa
cinerea do not compensate during the night, suggesting that the Finnish populations are

learning the nocturnal course. Second, some arthropods, including, other spiders,

 

171
compensate for the nocturnal period by reversing the apparent movement of the azimuth,

and this is not how the Arctic sun moves (Tongiorgi, 1959; Papi & Syrjamaki, 1963).

For the learning advantage network, it is not clear that ancestors of an animal using this
mechanism would actually have to see the position of the sun at night for the network to
have it encoded in its connections. Since one of the advantages of this type of network is
decreased learning time, having some representation of the nocturnal sun's course may be
better than having none. The representation could be arbitrary, it could be a step function
extended through the night, or it could be an observed gradual function extended through

the night.

Both the nonsymbolic interpolation network and the symbolic ellipse model could achieve
nocturnal compensation, but they differ signiﬁcantly in their predictions, and both differ
somewhat from reality. In the ellipse model, the nocturnal movement of the sun is
assumed to be in the same direction as its movement during the day, and the pattern of
change in rate would be symmetrical with the daytime pattern. This is not necessarily true
of actual sun azimuth functions. In many cases the azimuth reverses directions during the
night. This pattern appears throughout the tropics. In addition, the pattern of movement
during the night (the azimuth's rate of change) is not necessarily symmetrical with the

pattern during the day. In fact, only during the equinox is it truly symmetrical.

172
There are also speciﬁc predictions for the pattern of nocturnal compensation if the
underlying mechanism resembles the connectionist interpolation network. The predictions
in this case depend on the constraints placed on the learning. Ifno constraints are placed
on the position of the sun at night, the network will extrapolate nonlinearly into the night.
At midnight, there will be an abrupt shift to a pattern that resembles backward

extrapolation (Dyer, 1985) although it too is nonlinear (see Figure 5.10 A).

The diverse predictions of the models with respect to nocturnal compensation suggest that
these models can be distinguished experimentally. The models also differ in predictions
and assumptions about the symmetry of the pattern of solar movement. In most of the
models there is some symmetry imposed that is not seen in the natural pattern of
movement of the sun's azimuth. In the ellipse model, this symmetry comes from the
pattern of movement assumed for the night, as discussed above. This is one of the most
substantial predictions of the ellipse model, and it is one that can be addressed
experimentally. There is evidence suggesting that this might be the case. The explanatory
power of the 180° step ﬁrnction (Dyer & Dickinson, 1994) rests on this symmetry. The
behavior of many systematic errors made in sun compensation also support this conclusion

(Wehner & Lanfranconi, 1981).

The symmetry assumptions in the other models may not be as substantial. Symmetry
assumptions are encoded in the preconﬁgured networks, although symmetry may not be

necessary. Some kind of symmetry assumption is required to impose the necessary

173
constraints on the interpolation network. In the current model, an assumption of 180°
symmetry is necessary for the network to accurately estimate the position of the sun in the

morning with only aﬁemoon experience with azimuth-time pairings.

6.4 A False Dichotomy?

Throughout this dissertation I have maintained a dichotomy between symbolic and
nonsymbolic solutions to the problem of estimating the position of the sun during
unexperienced times of day. This dichotomy has a tradition in the literature (Smolensky,
1988; Gallistel, in press), but there are also those who suggest that the approaches are not
as different as they have been portrayed (Boden, 1991; Oden, 1994). The distinction may
be more an issue of what is computed versus how it is computed (Boden, 1991).
Symbolic models provide an hypothesis about what is computed while connectionist
models provide an hypothesis about how distributed networks of simple elements
compute. Attempts to integrate these levels of explanation are emerging (Smolensky et
al., 1994). I favor Marr's (1982) three levels of explanation (computation, algorithm, and
implementation). Resolving the symbolic-nonsymbolic dichotomy into Marr's trichotomy

would be a worthwhile challenge.

For the sun compass learning problem, traditional symbolic models have provided a
framework for investigating the problem (New & New, 1962; Gould, 1980; Wehner &
Lanfranconi, 1981; Dyer, 1985, 1987), after all, they made speciﬁc predictions that could

be falsiﬁed by behavioral experiments (Dyer & Dickinson, 1994). It could be argued that

174
to a certain extent, they are phenomenological models that are meant to describe the
behavior rather than the neural implementation of the behavior. Yet they have been
interpreted as the latter (Gallistel, 1990, in press). Even New and New (1962) suggested
this in the following statement: "in fact they appear to have an innate mechanism that can

divide angles by time" (p. 287).

At a fundamental level there may be no difference between these approaches. The
interpolation network looks very much like a symbolic model. The equations that describe
it are indeed quite simple because of the limited number of neural elements. Equations 5.]
and 5.2 can be combined in a relatively simple form to describe the output of this network.
In this sense, the connection weights (w), become the parameters of the ﬁrnctional
relationship between inputs and outputs, and in this case there are only four of them,
which is in the range of the parameters required for the symbolic model based on the
equation of the ellipse. Since connectionist networks are universal approximators
(Hornick et al., 1989), a network not ﬁrndamentally different from the interpolation
network could approximate any ﬁmctional relationship the variables in this case (azimuth
position and time). Thus over a given range, the nonsymbolic model would have exactly
the same characteristics as a symbolic model. The network would then be ﬁrnctionally
isomorphic with the symbolic model in a very real sense. Clever experimental
manipulations may be able to distinguish whether the animal's representation is truly
isomorphic with the true parameters of the problem or whether it is ﬁrnctionally

isomorphic over the range in which the problem is normally solved.

175

6.5 Caveat

The modeling effort cannot stand alone. It must be thoroughly based in data, and it must
be amenable to experimental falsiﬁcation. The models presented here conform to these
requirements. It is clear that there may be multiple ways to model the sun compass
learning problem, but without models of the possible mechanisms it is unclear exactly what

questions to ask with experiments.

 

6.6 Representing the Regularities of the World

I began with a very broad statement about the regularities of the world. Throughout this
dissertation, I have focused on a very narrow aspect of one environmental regularity, the
sun's course, and how it could be represented in the brain of an insect. In this case it is
clear that the regularities of solar movement are encoded in the brain of the insect. This
regularity is not completely genetically hard wired in the insect's brain, but aspects of the
regularity are encoded in the insect's brain. This allows an insect with very little
experience with the sun's course to have a relatively accurate representation of the sun's
course. But it is ﬂexible enough to allow a more accurate representation develop with
more experience. I have attempted to show in very broad terms how this could be
accomplished with models of two general classes. The nature-nurture debate has been
raging for centuries, but far too infrequently are problems at the cusp addressed

systematically.

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