Dynamical analysis reveals individuality of locomotion in goldfish
1 Department of Neuroscience, Albert Einstein College of Medicine of Yeshiva
University, 1410 Pelham Parkway South, Bronx, NY 10461, USA
2 Department of Physics, Ursinus College, Collegeville, PA 19426,
USA
3 Department of Pharmacology and Physiology, Drexel University College of
Medicine, Philadelphia, PA 19102, USA
4 Institute Pasteur, 28 Rue du Dr Roux, Paris, Cedex 15, France
* Author for correspondence (e-mail: hneumeis{at}aecom.yu.edu)
Accepted 24 November 2003
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Summary |
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Key words: fish, behaviour, individuality, locomotion, velocity, nonlinear dynamics, complexity, discriminant analysis
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Introduction |
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Swimming is actually composed of highly organized spatial and temporal
patterns even in a relatively homogeneous environment
(Kleerekoper et al., 1974;
Steele, 1983
). Some of these
patterns are complex and cannot be characterized with the tools of classical
kinematics, as they may exhibit nonlinear properties, such as persistence (the
tendency to repeat a given sequence), redundancy (the relationship between the
uncertainty of a signal and its length) and scale invariance (a tendency for a
signal to have the same structure when observed on different temporal or
spatial scales) (Faure et al.,
2003
). Indeed, nonlinear measures have been used to characterize
locomotion and the behavioural repertoires in various species, including
invertebrates (Dicke and Burrough,
1988
; Cole, 1995
),
fish (Coughlin et al., 1992
;
Alados and Weber, 1999
;
Brewer et al., 2001
), birds
(Viswanathan et al., 1996
;
Ferriere et al., 1999
) and
mammals (Paulus et al., 1990
;
Marghitu et al., 1996
;
Alados et al., 1996
;
Alados and Huffman, 2000
).
The present study was designed to (1) apply five nonlinear measures and one linear measure as descriptors of goldfish swimming trajectories in order to quantify this locomotor behaviour and (2) to develop a discriminant analysis that would allow us to ask if a given trajectory could be assigned to an individual within the experimental pool. It was found that, despite the apparent variability of trajectories, our protocol could reliably achieve such a classification.
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Materials And Methods |
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Swimming environment
A cylindrical Plexiglas tank (20 litre, 50 cm diameter) was used for the
experiments. The water column was comparatively shallow (10 cm deep) to
prevent fish from swimming out of the camera's focal plane and to minimise
errors due to changing swimming depth. To reduce mechanosensory and visual
cues, the tank was mounted on an anti-vibration table and its wall and lid
were translucent white. Its bottom was clear to allow video recording from
below.
Translucent white plastic sheets were mounted on the inside frame of the table with a small hole in the bottom sheet for the camera lens. Illumination was from above with a circular fluorescent bulb (approximately 350 lux at the water surface) and from below with four floodlights (approximately 250 lux at the bottom of the tank). New conditioned water was used for each recording session.
Data acquisition and experimental design
Approximately 30 min prior to all recording sessions, fish were transferred
to a translucent white container (20 cmx15 cmx10 cm) filled with
aerated, conditioned water, to be marked for automated motion tracking. Two
markers were applied with instant adhesive (Quick Tite; Locktite Corp., Avon,
OH, USA) along the ventral midline of the fish to specify its position on the
video image. They were made of double-sided black tape (1 cmx1 cm) with
a dot (approximately 4 mm diameter) of white nail polish painted in the
centre. For this purpose, the fish was removed from the water, the ventral
midline was exposed and the skin was gently dried. The markers were applied
between the paired pelvic and pectoral fins and onto the lower jaw in less
than 1.5 min, after which the fish recovered in fresh aerated water for at
least 10 min. This procedure had no obvious impact on behaviour and, in most
cases, the marker remained in place for several days.
To analyse locomotion (see also Faure
et al., 2003), recordings of the ventral view of the fish were
obtained from below at 30 Hz using a digital camcorder (Canon Optura; Canon
USA, Jamesburg, NJ, USA). Each recording session started 30 s after the fish
was introduced into the experimental tank and lasted 15 min. Video capturing
software (Adobe Premiere; Adobe Systems Inc., San José, CA, USA) was
used to subdivide a recording session into three 5-min trajectories. Five such
recording sessions, each obtained on a different day, were collected from five
fish and used to construct a library of 75 trajectories.
Data analysis
Commercial motion analysis software (WinAnalyze; Mikromak GmbH, Erlangen,
Germany) provided frame-to-frame data on the X and Y
position of the markers (Fig.
1A). Only the central marker data were used for the calculations
reported in this paper. The X and Y position data as
functions of time were used as primary data for the multivariate analysis
described below. The five nonlinear measures chosen for this study were
computed with software of our own construction; the mean velocity was then
taken as the 5-min average of the instantaneous velocity
[(dx/dt)2+(dy/dt)2]0.5.
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A pragmatic criterion for choosing each nonlinear measure was that it
should provide a quantitative value that could be assigned to a trajectory,
allowing for statistical comparisons between groups of data. A brief
description of these nonlinear measures is presented here; a more detailed
mathematical description, including additional references to the primary
literature, is given in Rapp et al.
(2002).
1. The characteristic fractal dimension (CFD)
The CFD measures the degree to which a trajectory departs from a
straight-line path (Katz and George,
1985; Katz, 1988
).
It is a measure of the total distance travelled from point to point (or frame
to frame) relative to the maximum separation of any two points in the series.
In other words, it is an approximation, equal to the distance travelled
divided by the diameter of the experimental tank. It is sensitive to the
duration of the observation period and to the speed of motion (see
Rapp et al., 2002
). It has a
minimum value of 1 but does not have an upper limit. Since, in the present
application, the fish is swimming in a cylindrical tank, a circular motion of
constant velocity would be equivalent to a straight line. As the trajectory
deviates from circular motion, the CFD increases. This measure has been used
to analyse a variety of complex geometrical patterns
(Rinaldo et al., 1993
;
Rodriguez-Iturbe and Rinaldo,
1997
).
2. The Richardson dimension (DR)
The DR is also an estimate of the degree to which a trajectory
departs from a straight line (Richardson,
1960; Mandelbrot,
1983
). In contrast with the CFD, DR also quantifies how
the estimate of a curve changes with the precision of the measurement. It is
an example of the generic class of dimension measures that have been applied
to the analysis of the classical problem of fractal geometry, namely `How long
is the coast line of Britain?'
(Mandelbrot, 1967
). Stated
operationally, for a fixed step length one counts the number of steps required
to walk around the coast (or, as in our application, along the fish's
trajectory). The length of the stride, i.e. the distance covered with each
step, is then reduced and the number of steps required using this new step
length is determined. The process is repeated and the log of the number of
steps required is plotted as a function of the log of the step length. Thus,
DR is a measure for scale invariance. The slope of this curve is
used to determine DR. As for the CFD, a value of 1 is obtained from
a straight line. The value of 2 is the maximum possible DR and it
represents a theoretical limit when a trajectory covers the entire
two-dimensional surface. Given the differences between the factors influencing
the CFD and DR, they can diverge.
Measures of fractal analysis comparable to CFD and DR have been
used to describe behavioural sequences, such as swimming and foraging in
clownfish (Coughlin et al.,
1992), trails in mites (Dicke
and Burrough, 1988
), reproductive behaviour in fathead minnows
(Alados and Weber, 1999
),
social behaviour in chimpanzees (Alados and
Huffmann, 2000
) and head lifting during feeding behaviour in ibex
(Alados et al., 1996
).
3. The LempelZiv complexity (LZC)
The LZC is a sequence-sensitive measure that characterizes the structure of
time-varying signals as a series of symbols
(Lempel and Ziv, 1976;
Ziv and Lempel, 1978
). The
spatial difference in the fish's position between two consecutive points in
time is compared, generating a time series of incremental distance travelled.
This distance function is simplified by partitioning it into a binary symbol
sequence about the median increment size. For example, a typical sequence
might be `aabaabbab' where `a' symbolises values less than the median and `b'
symbolises those greater. Then, the LZC is calculated for the resulting symbol
sequence. It reflects the number of sub-strings in the sequence (e.g. aab) and
the rate at which they occur. This measure will therefore give information
about the redundancy (or lack thereof) of a trajectory, for example about the
irregularity of its velocity. Kurths et al.
(1995
) used this method for
analysing heart rate variability in an investigation of predictors of sudden
cardiac death, while Gu et al.
(1994
) and Xu et al.
(1998
) found differences in
the electroencephalograms (EEGs) of healthy controls and psychotics with
symbolic dynamics. Subsequently, it was shown that the complexity of
multichannel EEGs of healthy controls is sensitive to changes in behaviour
(Watanabe et al., 2002
; this
reference includes a review of the associated literature). The value of LZC
increases approximately linearly with the number of measurements in the time
series and attains a maximum with random numbers
(Rapp et al., 2001a
). For data
sets of the length used in this study (9000), a maximum of approximately 700
would be expected.
4. The Hurst exponent (HE)
The HE measures persistence the tendency of large displacements to
be followed by large displacements (e.g. an increase is followed by an
increase) and small displacements to be followed by small displacements
and anti-persistence, which is the tendency of large displacements to
be followed by small displacements (e.g. an increase is followed by a
decrease) and vice versa (Hurst,
1951; Hurst et al.,
1965
; Feder, 1988
;
Bassingthwaighte et al., 1994
).
In other words, it describes how deterministic a trajectory is, i.e. the
extent to which a future component of the trajectory is specified by
components of its past. Theoretically, its range of possible values is 0 to 1,
with 0.5 as the crossover point between anti-persistence and persistence
(since it is estimated from the loglog plot of variability
versus epoch length, uncertainty in curve fitting can expand this
range slightly). An HE of 0.5 would be obtained if the trajectory was
indistinguishable from a random walk. Biological applications of the HE have
included investigations of heart interbeat interval sequences
(DePetrillo et al., 1999
;
Sherman et al., 2000
) and
pulmonary dynamics (Zhang and Bruce,
2000
).
5. Relative dispersion (R. Disp.)
R. Disp. measures the dependence of signal variance on the duration of the
dataset. It ranges from 1.0 to 1.5
(Boffetta et al., 1999) and
quantifies the change in the uncertainty in a time series' mean value as the
observation period increases. Practically, the R. Disp. is the slope of the
linear region of a loglog plot of the coefficient of variation of a
signal versus the length of the data set. Its primary applications
have been in the analysis of the physics of turbulent flow
(Pedersen et al., 1996
;
Willis et al., 1997
) but it
has also been used in the quantitative characterization of pulmonary perfusion
(Klocke et al., 1995
;
Capderou et al., 2000
).
All of the algorithms used to calculate these measures are sensitive to
noise in the data, non-stationarities in the underlying dynamics and the
temporal duration of the examined epoch. For example, filtered noise can mimic
low-dimensional chaotic attractors (Rapp
et al., 1993) and, if inappropriately applied, the method of
surrogate data (which is used to validate dynamical calculations) can give
false-positive indications of non-random structure (Rapp et al.,
1994
,
2001b
). These are central
concerns if one is trying to establish the absolute value of one of these
measures, such as the true value of the DR. However, this is a less
crucial consideration in the present investigation because we do not presume
to calculate the value of any measure in an absolute sense. Rather, we are
computing approximations of these empirical measures, which nonetheless may be
of value in the classification of these signals. The efficacy of these
computed values in the classification was assessed quantitatively in the
course of the discriminant analysis, as described below.
Discriminant analysis
A multivariate discrimination was constructed to ask specific questions
about the behavioural data. For example, can locomotor performance be
distinguished between individual fish? For this purpose, each swimming
trajectory was represented by its set of values calculated for the five
nonlinear measures described above plus its mean velocity. Since it was
possible that no measure alone would provide consistent results for such
discrimination, all the measures were incorporated into the discriminant
analysis and then their relative contributions to the classification process
were assessed, as described in the Results section. The discriminant analysis
is thus based on these six measures, and calculations were made between the
sets of values defining individual trajectories in a matrix consisting of a
six-dimensional space. All statistical procedures used are explained in
mathematical detail by Rapp et al.
(2002).
PSAME(Fish A, Fish B) is defined as the probability that
the six-dimensional measurement distributions corresponding to Fish A and Fish
B were drawn from the same parent distribution. The estimate of failure in a
pairwise discrimination is PERROR(Fish A, Fish B). This is
the theoretically estimated probability that a trajectory from Fish A will be
incorrectly classified as a Fish B trajectory and vice versa. Note
that PERROR is not the same as PSAME
and can be much larger. For example, a previous report
(Rapp et al., 2002
) included
an example in which PSAME=3.2x1013
while PERROR=0.32, which is relatively large, given that
the maximum possible error in a pairwise discrimination (the error rate
corresponding to random assignment) is PERROR=0.5. A
disparity between PERROR and PSAME
occurs because they address different questions. PSAME
determines if the means of two multivariate distributions are significantly
different. For cases where only one measure is used, PSAME
is identical to the probability calculated in a t-test.
PSAME can be very small even when the two distributions
overlap. However, if the distributions do overlap, which is the case here,
there can be considerable error in a between-group classification.
Two classification criteria were used for PSAME and
PERROR. The first classification is based on the minimum
Mahalanobis distance (Lachenbruch,
1975). In the context of the six-dimensional measure space, the
Mahalanobis distance is a generalized mathematical distance between the vector
from the single trajectory that is to be classified and the collection of
measure vectors calculated from all of the trajectories obtained from one of
the fish. The test trajectory is deemed to be a member of the group
corresponding to the smallest Mahalanobis distance. The second procedure for
classifying a trajectory is based on the Bayesian likelihood
(McLachlan, 1992
). The
trajectory's vector is classified into the group corresponding to the maximum
Bayesian membership probability. Both classification schemes incorporate a
correction for correlations between the measures, ensuring that dynamically
similar measures do not bias the classification results. In practice, the two
procedures usually give identical results. Cases where results differ
correspond to classification with low confidence levels. Finally, as the
descriptive analysis did not reveal consistent time-dependent differences
between three successive 5-min trajectories for most measures, this variable
was not incorporated into the discriminant analysis.
A distinction should be made between the out-of-sample classifications used
in this study and within-sample classification. When an out-of-sample
classification is performed, the trajectory to be classified is removed from
the library before the classification was calculated. For this reason, the
error rates of classifications are always greater than, or at best equal to,
the error rates obtained using within-sample classifications, where the
trajectory to be classified remains in the library during the calculation. If
the number of elements in each group is small (here, there are 15 trajectories
for each fish), the disparity between within-sample and out-of sample
classifications can be large. A comparison showing how within-sample
classifications can give unrealistically optimistic results is given in
Watanabe et al. (2002).
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Results |
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Swimming trajectories of different fish are dissimilar in appearance (Fig. 2). The distribution of path components in the centre versus the periphery of the tank seems to be most variable. For example, the three consecutive 5-min trajectories of Fish 2 in Fig. 2A show more time spent in the centre of the tank than do the three consecutive trajectories of Fish 5 in Fig. 2B, which indicate relatively little time spent in the centre or traversing it. Instead, in Fig. 2B there is rather more accumulation of path components near the wall, sometimes forming dense patches, which are not seen in the trajectories of Fig. 2A. In addition, there is session-to-session variation in an individual fish's trajectories, as seen by comparing the first (Fig. 2B) and fifth (Fig. 2C) recording sessions of Fish 5. In the fifth session, there is a greater tendency to explore the centre than in the first session. This difference is reflected in a significant difference between the mean velocities of the two 15-min sessions (62.6 mm s1 vs 58.8 mm s1; P<0.002). Also, there is greater variability between the three successive trajectories of the fifth session than between those in the first session; indeed, the third 5-min trajectory of Fig. 2C more closely represents those of the first session (2B) than the two trajectories preceding it, as it is denser at the periphery and exhibits a smaller number of excursions to the centre of the tank.
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An initial overall impression of the nonlinear dynamical analysis can be obtained by determining the range and variability of each measure determined across all five fish. These results are stated in Table 1. The coefficient of variation, CV=(S.D./mean)x100%, provides a quantitative characterization of the degree of spread in the observed dynamical measures. A high degree of variation is observed for some measures. The mean velocity has the highest CV (30.2%) and a nearly 10-fold range in values, and the CV of the LZC is also high (25.7%). By contrast, the CVs of the CFD, the DR and the R. Disp. are less than 7%. With the exception of DR, the mean values of the nonlinear measures are all consistent with properties of a complex dynamical behaviour. The data summarized in Table 1 are displayed separately for each fish in Table 2. The latter results were obtained by averaging the values from all recording sessions (five per fish) and all trajectories (three for each recording session). Appreciably different values were obtained for each fish. Nevertheless, given the large S.D.s, the between-fish distributions overlap. Mean velocity values are similar for Fish 2 and Fish 4 and for Fish 3 and Fish 5. This pattern was repeated for two of the nonlinear measures, CFD and DR, but not for the other three. In general, there did not seem to be a consistent relationship between the mean values of different parameters and individual fish, suggesting that the measures, which, with the exception of mean velocity, are empirical, reflect different properties of the swimming trajectories.
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Three of the six measures have time-dependent changes during the 15-min recording periods. Mean velocity decreased by 77% from the first to the last 5-min recordings (from 58.41 mm s1 to 45.01 mm s1), and the mean CFD decreased by 5% from 1.62 to 1.54. By contrast, the average LZC increased by 15% from 214 to 248, while the other measures did not change appreciably. Since the data were pooled for multiple exposures of the five fish, a repeated-measures analysis of variance (ANOVA) was used to ask if there were significant changes in a given measure between the three subsequent 5-min epochs of a 15-min recording session. The results, shown in Table 3, indicate significant differences (P<0.015) between subsequent 5-min trajectories for mean velocity and CFD. Also, in the case of LZC, the first 5-min trajectory was significantly different from both the second and third ones. These time-dependent changes in the six measures relative to each other during a 15-min recording are illustrated in Fig. 3. Values for each measure are normalized with respect to the corresponding values obtained in the first 5-min trajectory. The repeated-measures ANOVA was also used to ask if there were differences between the five subsequent sessions in which data were collected from each fish, and the results were negative. Since the changes that occurred within a 15-min recording session were minimal, the discriminant analysis did not treat successive 5-min trajectories separately.
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Discriminant analysis classifies individual fish
Three questions were addressed in the discriminant analysis: (1) based on
the application of these six dynamical measures, would it be possible to
conclude that the five fish are different; (2) given a trajectory and its
dynamical characterization, would it be possible to correctly determine which
fish produced the trajectory and (3) of the six measures used, which ones were
the most effective in discriminating between different fish? These questions
were addressed by performing a discriminant analysis based on the six
measures, with each fish providing a total of 15 trajectories. For this
analysis, no distinction was made between first, second and third 5-min
trajectories. Using these measures, we calculated
PSAME(Fish A, Fish B), which is the probability that the
six-dimensional measurement distributions corresponding to Fish A and Fish B
were drawn from the same parent distribution (see Materials and methods). The
results from the 10 possible pairwise discriminations are shown in
Table 4. As an example from
that table, it is seen that PSAME(1,
2)=0.19x105; that is, the probability that Fish 1 and
Fish 2 trajectories were produced by the same fish is
0.19x105. We conclude that Fish 1 and Fish 2 have very
different dynamical profiles. The largest value of PSAME
is PSAME(3, 4)=0.9x102. While Fish
3 and Fish 4 are the most similar, even in this case the probability that
these trajectories were obtained from the same fish is less than 1%. Given the
very low value of PSAME, it might be supposed that a
classification of a single trajectory amongst the five fish would be highly
accurate. However, this is not necessarily the case.
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PERROR is a theoretical prediction of the pairwise classification error, using the between-group Mahalanobis distance. In the present study, using six measures, the theoretical PERROR for the 10 pairwise calculations was less than 0.07 in eight cases and ranged from 0.003 (Fish 2 vs Fish 4) to a maximum of 0.1118 (Fish 3 vs fish 4).
The error rate also can be determined empirically by performing a classification. The results of an out-of-sample classification are shown in Table 5 for both minimum Mahalanobis distance and maximum Bayesian likelihood criteria, respectively. For example, the entry 13/12 in the Fish 1Fish 1 box means that 13 out of 15 Fish 1 trajectories were classified as Fish 1 using the minimum distance criterion and 12 were correctly classified as Fish 1 using the maximum likelihood criterion. The entry 2/3 in the Fish 1Fish 5 box means that two Fish 1 trajectories were classified as Fish 5 using minimum distance and three Fish 1 trajectories were classified as Fish 5 using maximum likelihood as the criterion. Thus, more than 75% of the trajectories from Fish 1, 2 and 5 were correctly classified with both criteria. Also, a comparison based on mean velocity alone suggests similarities between Fish 1 and 4 and between Fish 3 and 5; the discriminant analysis, which uses six measures, does not often confuse these fish.
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The expectation error rate is the error rate that would be observed if the classifications were performed randomly. There are five fish. If trajectories were assigned randomly, four out of five trajectories would be misclassified. This gives an expectation error rate of 80%. For these data, the overall error rate using minimum Mahalanobis distance as the classification criterion is 36%. The overall error rate using the maximum Bayesian likelihood is 28%.
The third question to be addressed with discrimination analysis asked, `of
the measures used, which were the most effective in discriminating between
different fish?'. This question is not easily answered when there are five
groups (five fish) as opposed to only two. In the case of a pairwise,
two-group comparison, a measure's coefficient of determination establishes the
amount of total between-group variance that can be accounted for by the
measure (Flury and Riedwyl,
1988). Then, the larger a measure's coefficient of determination,
the more effective it is in discriminating between groups. A large coefficient
of determination corresponds to a large between-group Mahalanobis distance
(specifically, the partial derivative of the coefficient of determination with
respect to the Mahalanobis distance is positive). The effectiveness of the six
measures in the 10 pairwise between-group discriminations has been assessed
empirically. Table 6 gives the
rank ordering of the coefficients of determination for each measure for each
pairwise discrimination (ordered from the largest to the smallest). For
example, when Fish 1 and Fish 2 are compared, the HE is most effective in
discriminating between the two groups while the DR is the least
effective. When the rank ordering of the 10 pairwise discriminations is
compared, none of the measures stands out as being exceptionally effective.
However, if the rank order is treated as a score for each pair, the data
indicate that the DR and the HE have the lowest cumulative scores,
suggesting they are the most effective. Interestingly, the mean values of
these two measures (Table 1) are consistent with trajectories that are relatively stable or determined
(i.e. mean of HE=0.82 indicates a high degree of persistence and mean
DR=1.06 indicates high similarity to a straight line trajectory).
The lack of a consistent pattern in the results presented in
Table 6 is not surprising,
since our results established that the fish trajectories are highly
individualistic (Table 4) using
a statistic, PSAME, that combines all six measures.
Another approach for obtaining an estimate of the comparative effectiveness of
each dynamical measure is to calculate each measure's average coefficient of
determination, taking the average over the 10 pairwise discriminations. These
average values are shown in Table
7 and again suggest that DR and the HE are the most
effective measures when used alone.
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Discussion |
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Interpretation of fish locomotion with nonlinear measures
Although they are empirical, the tools of nonlinear dynamical analysis are
increasingly being used in the analysis of biological phenomena
(Faure and Korn, 2001;
Giesinger, 2001
), including
continuously recorded behavioural sequences. One rationale is that, since
these measures are sensitive to the spatio-temporal structure of a sequence,
they might reveal hidden structures in those continuous signals. Indeed,
studies have shown that the examination of behavioural data that appeared to
be random can reveal highly non-random components when analysed with
sequence-sensitive nonlinear measures. For example, a number of behaviours
have been described as fractal, from spontaneous locomotion
(Dicke and Burrough, 1988
;
Coughlin et al., 1992
;
Motohashi et al., 1993
;
Cole, 1995
) and foraging
(Alados and Weber, 1999
) in
diverse species to social behaviour in chimpanzees
(Alados and Huffman, 2000
) and
feeding-related activities in goats (Alados
et al., 1996
). Related tools have also been used to successfully
analyse the pattern of transitions between periods of active swimming and
inactivity (Faure et al.,
2003
). As discussed below, this type of analysis might be
effectively employed to reveal subtle changes in locomotion not revealed with
classical means.
The five nonlinear measures applied in the present study are empirical measures of complexity of swimming behaviour, and each reduces a trajectory into a single value. With the exception of the Richardson dimension, the values of these nonlinear measures are consistent with the notion that goldfish swimming in even a relatively sparse environment is a mixture of random and nonlinear deterministic activities. Their empirical nature may explain the finding that two of the measures, the characteristic fractal dimension and Richardson dimension, which are expected to reflect similar properties, often diverged.
The degree of complexity exhibited in locomotor behaviour and other
behavioural patterns can depend on the environment
(Coughlin et al., 1992;
Motohashi et al., 1993
;
Anderson et al., 1997
). Spatial
and temporal complexity of foraging trajectories, for example, can be
correlated to the pattern of occurrence of food sources
(Cole, 1995
;
Viswanathan et al., 1996
).
Similarly, some bird species exhibit nonlinearities in vigilance behaviour
(Ruxton and Roberts, 1999
),
and correlations have been drawn between fractal complexity and the ability to
cope with the environment, such as in the presence of toxins or stress
(Alados et al., 1996
;
Alados and Weber, 1999
;
Alados and Huffman, 2000
). One
can thus speculate that fish exposed to an environment more heterogeneous than
that used in the present study would generate swimming trajectories with
higher values of CFD and DR, indicative of a more fractal nature.
Such an experimental design would give more insight into what extent the
environment might influence the nonlinear properties and their underlying
components.
The nonlinear measures and discriminant analysis employed here may then be
applied to detect subtle changes in behavioural sequences altered by changes
in the environment. Fish behaviour is increasingly important in toxicology,
and it has already been shown that fractal dimension could serve as a
sensitive measure for quantifying differences in locomotor activity during
sublethal exposure to toxic contaminants
(Motohashi et al., 1993;
Alados and Weber, 1999
;
Brewer et al., 2001
). The
application of multiple measures, including a linear one, may well enhance
such discriminations. Indeed, preliminary data, obtained using this
methodology to distinguish swimming trajectories of goldfish exposed to low
dosages of Malathion, a pesticide and neurotoxin, confirm this expectation
(Neumeister et al., 2001
).
Exposure to a novel environment for a continuous period or for several
discrete periods will, in general, result in a gradual decrease of locomotor
activity over the course of several days or weeks
(Russell, 1973;
Warren and Callaghan, 1976
;
Clark and Ehlinger, 1987
).
Novelty represents a potentially stressful situation
(Russell, 1973
;
Csányi and Tóth,
1985
; Gervai and
Csányi, 1986
). For example, male guppies initially show
high velocity swimming at the periphery of an open field, and it has been
suggested that this activity is related to some degree of fear
(Warren and Callaghan, 1976
).
In the present study, a relatively small but significant decrease during the
15-min period was not only detected in mean velocity but also in CFD and
LempelZiv complexity. The results in the CFD are consistent with
reports that fractal dimension decreases in conditions characterized as
stressful (Alados et al., 1996
;
Alados and Weber, 1999
;
Alados and Huffman, 2000
).
Nevertheless, this modification with time can be subtle, and it remains to be
seen if further development of the discriminant analysis would benefit by
treating successive 5-min trajectories separately.
Classifying trajectories
Multivariate discriminant analysis, which allowed us to classify swimming
trajectories to the fish that generated them, has a long and successful
history in the physical and biological sciences
(Lachenbruch, 1975;
McLachlan, 1992
). The
combination of discriminant analysis with nonlinear measures is, however,
comparatively recent (Rapp et al.,
2002
; Watanabe et al.,
2002
). In the present study, a discriminant analysis based on six
measures was used to characterize between-group differences and to classify
individuals amongst the groups, with each fish defining its own group. Five
fish were used and five recordings consisting of three consecutive 5-min
trajectories were obtained from each fish. Thus, in the language of
discriminant analysis, there are five groups, 15 elements in each group and
six-dimensional measure space.
As outlined above, we addressed a sequence of three questions. First, we asked if we are able to conclude that the fish are different, computing PSAME for each pair of fish. Although direct visual observation of the fish did not suggest that their swimming behaviour was dramatically different, the calculations of PSAME indicate that trajectories are highly individual, and each fish has a very different swimming profile.
We then addressed the problem of classification of individual 5-min
trajectories among the five possible groups, by calculating
PERROR for each pairwise classification. As expected (see
Results), PERROR is larger than PSAME,
with an average value of 5.7%. However, PERROR is a
theoretical estimate of the error in a pairwise classification based on the
between-group Mahalanobis distance
(Lachenbruch, 1975). An
empirical test of this classification was produced by computing an
out-of-sample classification that used the minimum individual-to-group
Mahalanobis distance as the classification criterion. It gave an error rate of
36%, in contrast to the expected error rate obtained with random assignment of
80%. The error rate using maximum Bayesian likelihood as the assignment
criterion was even less, 28%.
It might seem surprising that, while the average PERROR is 5.7%, the empirically determined classification error rate is greater. Yet, PERROR is the predicted error rate in a single pairwise classification. The empirically determined error rate is more appropriately compared against a classification procedure based on a sequence of pairwise classifications in which several individual pairwise errors accumulate to produce the overall result. When the distinction between pairwise and global error is taken into account, it is seen that the error rates are similar.
The third question concerned the identification of the measure or measures that were most successful in discriminating between fish. This was investigated by calculating the coefficient of determination in each pairwise classification for each measure. The results indicated that no single measure emerged as the most effective. However, it was possible to conclude that the nonlinear measures were more effective than the mean velocity, with the most effective being the HE and DR, values which are consistent with the general conclusion that fish swimming in a sparse environment have a relatively low degree of complexity.
It should be recognized that the ability to classify any given trajectory is limited. To introduce an analogy, we can prove that fingerprints are highly individual but we can't usually base a positive identification on a single fingerprint. We should point out that these conclusions are dependent on the measures used in this study. The application of additional measures to these data might result in an improvement in the classification calculations. Thus, the results presented here are, in a sense, a worst-case calculation.
Individuality
We have found that the discriminant analysis using swimming trajectories
and nonlinear dynamical measures established in a convincing manner that fish
locomotion is highly individualistic. Recent ethological and psychological
studies have revealed individual differences in many species
(Clark and Ehlinger, 1987;
Bell, 1991
;
Mather and Anderson, 1993
;
Boissy and Bouissou, 1995
). As
already mentioned, most of these studies concerned higher order behaviours. To
our knowledge, idiosyncratic variability in fish swimming has not been the
subject of previous investigations, although it has been noted
(Kleerekoper et al., 1974
).
Locomotion serves a range of behaviours in fish, including exploration,
foraging and social interactions. Individuality in these behaviours can be
expected to benefit survival of individuals and, therefore, of the population.
For example, it may increase access to food sources by enhancing the search
efficiency of shoaling fish (Gotceitas and
Colgan, 1988
; Colgan et al.,
1991
). Additionally, it can provide a competitive advantage to
some individuals, such as the dominant ones within a hierarchy based upon
boldness (Budaev, 1997
;
Wilson et al., 1993
). Again,
this would contribute to the fitness of the population by guaranteeing
survival of individuals in the case of limited resources
(Magurran, 1986a
;
Gotceitas and Colgan, 1988
).
Thus, the variations observed here may have functional relevance.
Three categories of mechanisms have been proposed to underlie behaviours
that are unique to one individual as opposed to another, namely a variable
environment, social effects and phenotypic variability (reviewed in
Magurran, 1986a). In that
context, the present study was designed to quantitatively characterise
swimming of one fish alone in a sparse and constant environment, minimising
any affective contribution to the resulting pattern. The results demonstrate
that, with the appropriate analytical tools, it is possible to conclude that
this elementary behaviour exhibits individuality. Thus, we suggest that this
property reflects phenotypic differences of either genetic or experiential
origin. Such differences are not simply related to environmental conditions,
body size or sex, as these factors were controlled in this study. Rather, they
may be embedded in underlying intrinsic processes. It has been suggested that
a population benefits from varying phenotypes, or differences in individuals,
by being better adapted to environmental conditions
(Clark and Ehlinger, 1987
). In
this context, it would be interesting to know how the individuality observed
in the present study would change in other conditions, such as a more
heterogeneous environment or one requiring social interactions.
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Acknowledgments |
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