Physica A 443 (2016) 221–230

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Fractal scaling in bottlenose dolphin (Tursiops truncatus) echolocation: A case study Shaun T. Perisho a,∗ , Damian G. Kelty-Stephen b , Alen Hajnal a , Dorian Houser c , Stan A. Kuczaj II a a

Department of Psychology, The University of Southern Mississippi, 118 College Dr. #5025, Hattiesburg, MS 39406, United States

b

Grinnell College, 1115 8th Ave., Grinnell, IA, 50112, United States

c

National Marine Mammal Foundation, 2240 Shelter Island Drive, #200, San Diego, CA 92109, United States

highlights • • • •

We investigate fractal scaling behavior in two aspects of dolphin echolocation. We employ two widely used fractal analysis methods and compare their results. Results indicate persistent fractal scaling in both echolocation measures. Possible explanations for observed between-subject differences are discussed.

article

info

Article history: Received 11 November 2014 Received in revised form 14 May 2015 Available online 21 September 2015 Keywords: Echolocation Tursiops truncatus Fractal Analysis Cascade dynamics Detrended Fluctuation Analysis Fractal dimension

abstract Fractal scaling patterns, which entail a power-law relationship between magnitude of fluctuations in a variable and the scale at which the variable is measured, have been found in many aspects of human behavior. These findings have led to advances in behavioral models (e.g. providing empirical support for cascade-driven theories of cognition) and have had practical medical applications (e.g. providing new methods for early diagnosis of medical conditions). In the present paper, fractal analysis is used to investigate whether similar fractal scaling patterns exist in inter-click interval and peak–peak amplitude measurements of bottlenose dolphin click trains. Several echolocation recordings taken from two male bottlenose dolphins were analyzed using Detrended Fluctuation Analysis and Higuchi’s (1988) method for determination of fractal dimension. Both animals were found to exhibit fractal scaling patterns near what is consistent with persistent long range correlations. These findings suggest that recent advances in human cognition and medicine may have important parallel applications to echolocation as well. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Research on bottlenose dolphin echolocation has focused primarily on the physical properties of echolocation clicks such as frequency distribution, bandwidth, amplitude, beam width, directionality, and inter-click interval (ICI) [1–7]. While this approach has led to a greater understanding of many aspects of echolocation, collapsing click trains to average values overlooks the organization and structure of the click train as a whole. Because perception is a continuous process requiring

∗

Corresponding author. E-mail address: [email protected] (S.T. Perisho).

http://dx.doi.org/10.1016/j.physa.2015.09.012 0378-4371/© 2015 Elsevier B.V. All rights reserved.

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organisms to interpret and react to new information about their environment in real time, it cannot be assumed that an animal’s awareness of its surroundings and perceptual strategies remain constant throughout the periods in which we observe them. Continuous perceptual feedback, occurring over multiple time scales, likely results in changes in the click train as it is being produced, making echolocation a dynamic process [8]. The temporal structure of click trains, and the way in which that structure evolves, must be taken into account if we are to better understand the dynamics of echolocation. Although echolocation is a product of internal cognitive processes, it is simultaneously a reflection of the animal’s external environment. The physical characteristics of click trains are influenced by interactions on scales ranging from microscopic (e.g., neural firing) to macroscopic (e.g., the animal’s position in its habitat) [5,9,10]. In this sense, animal and environment form a single cohesive system from which the measurable properties of click trains emerge. We propose an approach that views echolocation as the emergent result of interactions unfolding within a dynamic, complex system. We first introduce an important class of methods known as fractal analysis and review previous research regarding fractality in biological systems. With this foundation in place, we apply fractal analytical methods to measures of dolphin echolocation and explore ways in which our findings might be applied to future cognitive, behavioral and veterinary research. 1.1. Fractal methods for describing fluctuations A fractal is a pattern, either spatial or temporal, that exhibits self-similarity on all scales [11]. In the case of a fractal time series, fluctuations in values over long periods (e.g., hours or days) resemble fluctuations over smaller periods (e.g., minutes or seconds). This self-similarity between different scales is referred to as fractal scaling. In mathematically ideal fractals, fractal scaling is exact and extends across an infinite range of scales. In the real world however, fractal analysis must deal with noise in the environment and a finite range of scales due to limiting factors such as sampling rate and length of observation. As a result, empirical fractals are said to exhibit statistical (as opposed to exact) self-similarity [12]. Because this statistical self-similarity is not always readily apparent through qualitative inspection, it is typically described via the following quantitative relationship: mn = pnα

(1)

where mn is some property measured at scale n, p is a factor of proportionality, and α is an exponent used to characterize the scaling properties of the data set [12]. Values of α near 0.5 indicate random uncorrelated noise, values closer to 1.0 suggest the presence of persistent fractal scaling, and values approaching 1.5 are indicative of uncorrelated non-stationary random-walk processes [13]. Spatial and temporal measures of biological systems have often been found to exhibit fractal scaling. Examples include stride interval in human gait [14], displacement of center-of-pressure during upright stance [15], human eyemovement [16–18], wielding behaviors underlying haptic perception [19,20], tree growth [21], vascular structure [22], albatross search patterns [23,24], marine predator foraging patterns [25], human heartbeat [13], human respiration [26], wolf search paths [27], mammalian social hierarchies [28], copepod movement patterns [29], and Tursiops aduncus dive durations [30]. The prevalence of fractal scaling in biological systems raises the question of what process or processes might be driving the formation of these complex patterns. 1.2. Fractal scaling as the product of underlying cascade processes In cascade-driven processes, energy is dispersed and transmitted by interactions that unfold across multiple temporal and spatial scales [31–33]. A common example involves a sand pile that is formed by an experimenter dropping a single grain of sand at a time on the floor [31]. As the pile grows and ultimately becomes unstable, it reaches a critical state in which the addition of a one more grain will trigger an avalanche. Depending on the conditions, this avalanche may die out quickly or it may increase in size multiplicatively, triggering larger and larger avalanches as it rolls down the side of the pile. Either way, the sand quickly settles into a more stable configuration and the experimenter resumes dropping grains one at a time (which continues to drive further cascading avalanches). Both the magnitude of these avalanches, and the time interval between consecutive avalanches exhibit strong fractal scaling over time [31]. The sand pile model is limited in that it only accounts for interactions between two distinct time scales (the slow local addition of individual grains versus the fast global cascades of avalanche activity) when processes in the real world unfold at a nearly infinite range of spatiotemporal scales. However, it clearly illustrates an important characteristic of cascade processes that we will revisit later—they are composed of a large number of strongly interconnected components. Through physical contact with their neighbors, the behavior of individual grains of sand can potentially affect the behavior of every other grain in the pile. Conversely, the behavior of every grain in the pile can potentially affect the behavior of each individual piece of sand. This strong interconnectivity between components is crucial to the emergence of fractal patterns in the behavior of the system as a whole. If interconnectivity were to break down (e.g., if partitions were inserted to separate distinct regions of the sand pile) the transfer of energy would be halted and the fractal characteristics of the avalanches would diminish or disappear [31,34]. Although they were initially developed for purely mathematical and physical applications [35,11], cascade models offer a useful theoretical framework for explaining how the complex, flexible behaviors that are characteristic of living organisms

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might arise. In cascade-driven systems, scale-invariant interactions between components lead the system toward critical states (i.e., fluid, flexible arrangements of components serving as an intermediate configuration hovering between many distinct, crystallized, latent states). From these critical states, the system can quickly adapt to outside perturbations as the environment imposes different constraints on it. A real-world analogue of this process can be found in the ways foraging animals adapt their behavioral states as they receive new information about their surroundings. Dolphins, for example, have been known to adjust various parameters of their click trains (such as interclick interval, amplitude, and frequency content) in response to perceptual feedback from their environment [36,37]. From this perspective, the ‘‘avalanches’’ are the physical characteristics of the produced click train, while the ‘‘sand pile’’ is made up of the animal, its environment, and all the physical and biological components therein. When applying cascade models to biological systems, the scale-invariant interactions responsible for driving the emergence of critical states should manifest as fractal fluctuations in measurements of the organism’s behavior [31,38,39,34,40]. 1.3. Fractal perceptual strategies allow for efficient scanning of the environment Fractal fluctuations are not unique to biological systems—natural scenery has been shown to exhibit fractal scaling [41–43] and a review by Billock et al. [16] found an average scaling exponent of α = 1.04 for 1176 images with published statistics. The fact that physical features of the environment exhibit fractal scaling suggests that the distribution of salient information in the natural world is largely scale-invariant. Basic activities like foraging and navigation require the quick and efficient assimilation of perceptual details spanning a wide range of scales. Marine predators must sift through information unfolding at relatively short time scales (e.g., abrupt changes in a fleeing animal’s trajectory) as well as those occurring over much longer time scales (e.g., migratory patterns of prey species). Spatial examples of this scale-invariance range from the shapes, textures and coloration patterns that identify prey, to the larger geological features that identify their habitats. Geographic distributions of phytoplankton and krill, for example, have been shown to exhibit fractal scaling characteristics [44,45,25]. Due to the fact that planktivorous fish movements are heavily influenced by plankton densities [46,47], it should come as no surprise that the foraging behaviors of these fish (and the animals that feed on them) exhibit fractal scaling as well [44,25,48]. Due to the space-filling properties of fractal patterns, the adoption of fractal perceptual strategies allows predators to efficiently scan a wide range of scales for important information [49]. In fact, humans have already been shown to employ fractal exploratory strategies across a range of sensory modes [50,16,51,19,20,52]. Because echolocation is the product of a continuous interaction between animal and environment, its physical properties should reflect the fractal scaling characteristics of the very system in which it is embedded. For this reason, we hypothesize that quantitative measurements of bottlenose dolphin echolocation, specifically peak–peak amplitude and interclick interval (ICI), will exhibit quantitative evidence of fractal scaling over time. 2. Methods 2.1. Materials and experimental design Data were collected from two male bottlenose dolphins performing free-swimming target detection tasks. The dolphins were maintained by the US Navy Marine Mammal Program located at the Space and Naval Warfare Systems Center Pacific in San Diego, CA. Echolocation recordings were made using the Biosonar Measurement Tool (BMT), the design and analysis of which have been previously published [53,54]. Here we briefly describe the data collection process previously reported. The reader is referred to the previously referenced articles for detailed information on the BMT design and dolphin echolocation strategies. Following this overview, details of the fractal domain analysis are presented. The BMT was designed to be held by a dolphin via a neoprene-covered bite plate and was capable of recording acoustic emissions and locations of free-swimming dolphins in target detection tasks [54]. The device consisted of one Reson TC4013 hydrophone for recording outgoing clicks and two receivers for recording incoming click returns. The TC4013 hydrophone sat 1 m in front of the animal, in-line with the maximum response axis of its outgoing clicks, and recorded outgoing clicks with a peak–peak sound pressure limit of 215 dB re 1 µPa [54]. All recordings were sampled at ∼314 kHz with 16-bit resolution [53,54]. Experimental trials took place between July 2001 and January 2003 at two separate open-water sites within San Diego Bay, San Diego, CA. The test field for the first dolphin, LUT (24 years old), was an approximately 132 × 99 m rectangular area with a depth ranging from 0 m on the shoreline boundary to 23.8 m near the bayward boundary [53]. The test field for the second dolphin, FLP (25 years old), was an approximately 231 × 112 m rectangle with depths ranging from 0 m on the shoreline boundary to 14 m on the bayward boundary [53]. During each trial, the animal was asked to determine whether a spherical aluminum target (10 in. diameter, 0.19 in. wall thickness) was present within a given search area marked by a floating buoy. If the target was detected the animal was trained to whistle, return to the boat, and report to a response paddle representing the target-present condition. If the target was not found, the animal completed a path around the buoy, returned to the boat, and reported to a response paddle corresponding to the ‘‘target absent’’ condition [53]. The distance between the boat and buoy varied between 25 and 60 m

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ξ 101

2.5

FLP Click Train

2

Pressure (dB re: 1 uPa)

1.5 1 0.5 0 -0.5 -1 -1.5 -2

0

0.5

1

1.5

2

2.5 Sample

3

3.5

4

4.5

5 ξ 101

Fig. 1. Example of a representative click train from one of FLP’s trials.

for each trial and the order in which target-present versus target-absent stations were presented was determined using a Gellerman series. The series was balanced to ensure that, although each condition had an equal probability of being chosen, no more than three trials of the same type would be assigned in succession [53]. Data collection was interrupted with FLP after 26 trials due to animal health issues, while LUT was able to complete a total of 431 trials [53]. Fig. 1 shows a typical click train produced by FLP. Trial start time was defined as the point at which the animal emitted its first echolocation click. Trial end times for targetpresent trials were defined as the time at which the animal whistled, signaling detection of the target. Trial end times for target-absent trials were defined as the point at which the last click emitted by the animal was detected. Start and end times of false-positive trials (in which the animal reported finding the target when it was not there) were defined the same way as target-present trials, and misses (in which the animal failed to detect a target that actually was present) were treated like target-absent trials [53]. Houser et al. [53] selected trials for analysis based on the following criteria: (1) unambiguous animal response; (2) no failure of system hardware during data collection or data upload; and (3) absence of system or environmental noise prohibiting the detection of clicks. Individual clicks were detected using a 1000-point sliding window instantaneous peakpressure detector with a 175-dB re: 1-µPa threshold. Each detected click was then recorded as a 256-point waveform with a 32-point buffer prior to the pressure peak. Because the instantaneous peak-pressure detector was occasionally triggered by extraneous environmental or system noise, each waveform was inspected by a human observer and irrelevant detections were thrown out [53]. Our analysis is centered on two distinct variables: inter-click interval (ICI) and peak–peak amplitude. The ICI was determined by measuring the distance between the maximum instantaneous peak pressures of successive clicks. Intervals greater than 500 ms were thrown out and defined as the termination point of a click train [53]. Peak–peak amplitude was calculated as the difference between the maximum and minimum sound pressures of each click measured in dB re: 1 µPa. Amplitudes below the 175 dB re: 1 µPa threshold mentioned above were not included in the analysis. The reader is referred to Houser et al. [53] for a more detailed treatment of the initial data processing steps described above. 2.2. Analysis Because trials in which the animals reported the target present were over as soon as the animal correctly or incorrectly identified the target, they were consistently shorter than trials in which the animal continued searching for the target (e.g. LUT produced around thirty clicks per trial when he reported the target present). Due to the fact that long range correlations are not detectable at such short resolutions, we chose to focus our analysis on instances in which the animals reported the target absent (as these were the trials that produced the longest time series). This resulted in 14 trials for FLP (8 correct and 6 incorrect) with an average of 568 clicks per trial, and 27 trials for LUT (18 correct and 9 incorrect) with an average of 190 clicks per trial. FLP’s trials contained a minimum of 431 and a maximum of 718 clicks per trial, while LUT’s trials contained a minimum of 135 to a maximum of 380 clicks per trial.

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We created randomized ‘‘surrogate’’ signals by shuffling the data points of each original recording. The random shuffling process destroyed any long-term correlations that may have existed in the original unshuffled series and, as a result, should have produced scaling exponents near α = 0.5 (the value which, as stated earlier, corresponds to random uncorrelated noise). If the original unshuffled series had a scaling exponent near α = 1.0 while its counterpart surrogate produced a scaling exponent near α = 0.5, we could assume that the difference was due solely to the correlations destroyed by the shuffling process and not by some other artifact of the recording. Because we expected scaling exponents of unshuffled series to be higher (near α = 1.0) than their shuffled surrogates (near α = 0.5), a two-tailed paired samples t-test was used to check for a significant difference between the two. ICI outliers greater than one standard deviation from the mean were removed prior to further analysis. Only 3.3% of the data points exceeded this threshold due to the fact that the few outliers within the data set were many orders of magnitude greater than the rest of the data points. Inclusion of these values produced highly inflated standard deviations that exceeded the vast majority (96.7%) of ICI values. Because the underlying theme of fractality concerns correlations on many different scales, it has been traditionally assumed to require very long data sets—on the order of 212 points [12,55]. Unfortunately, this requirement excludes many real world data sets for which lengths of that magnitude are unrealistic or very difficult to collect. In light of this problem, Delignières et al. [55] conducted an extensive investigation of the reliability of a wide range of fractal analyses when applied to short time series. One of the methods tested, Detrended Fluctuation Analysis (DFA) was found to be particularly robust, providing accurate scaling estimates with no apparent bias for time series as short as 64 (26 ) points. Because click trains in our data set tended to be on the order of hundreds (as opposed to thousands) of clicks long, DFA was the most appropriate choice for our analysis. The standard DFA method, as outlined by Peng et al. [56], was applied to both ICI and peak amplitude measurements of each click train. Data were first detrended by removing the mean and the time series was divided into a series of bins of length n. A least-squares fit was calculated for the data points in each bin and the root-mean-square fluctuation of the entire time series (averaged across all bins) with a bin size of n was calculated using the equation:

  N 1  [y(k) − yn (k)]2 F (n) = 

(2)

N k−1

where F (n) represents the average standard deviation unaccounted for by the least squares fit in each bin. This process was repeated for all possible bin sizes n and a double log plot comparing each bin size with its corresponding root-mean-square fluctuation was constructed. The slope of the linear fit of log(F (n)) to log(n) was then used to determine the scaling exponent α , described in (1) above [13]. Because questions have been raised concerning DFA’s vulnerability to spurious curvature at smaller time windows [57], we conducted a parallel analysis using Higuchi’s fractal dimension estimation method [58]. We begin with a finite series of observations taken at regular intervals: X (1) , X (2) , X (3) , . . . , X (N ) .

(3)

From this original series, we construct a new series Xkm , defined as follows: Xkm ; X (m) , X (m + k) , X (m + 2k) , . . . , X



 m+

N −m k

  k

(m = 1, 2, 3, . . . , k)

(4)

where [] denotes Gauss’ notation. Both k and m are integers and indicate initial time and interval time, respectively. For a given value of k, we obtain k sets of new times series. The length of the curve associated with each of these k series is defined as: Lm (k) =

1 k

 N −m   k  N −1    N −m  (X (m + ik) − X (m + (i − 1) k)) i=1

k

k

(5)

where the term N −1  N −m  k

k

(6)

represents a normalization factor. Higuchi [58] defines the length of the curve for the time interval k, ⟨L(k)⟩, as the average value over k sets of Lm (k). If this average value follows a power law:

⟨L(k)⟩ ∝ k−D

(7)

then the curve is fractal with dimension D. The scaling exponent α is related to fractal dimension by the following relationship:

α = 2 − D.

(8)

Results of both methods were checked for significance using a linear mixed effects model [59] with between-animal comparisons to check for any individual differences that might exist. Response type (correct vs. incorrect response) and trial number were entered as random variables.

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DFA Output: FLP vs LUT 1.5

log F(n)

1

0.5 LUT FLP

0

1

1.1

1.2

1.3

1.4

1.5 log n

1.6

1.7

1.8

1.9

2

Fig. 2. A log–log plot comparing DFA output of FLP vs. LUT peak–peak dB time series. Although both series display a linear correlation between log F (n) and log(n) scaling across scales, the plot belonging to LUT exhibits a more gradual slope (and therefore a smaller scaling exponent) than that of FLP.

3. Results 3.1. ICI results Average scaling exponents for ICIs of each animal are given in Tables 1 and 2. Values were in the fractal (H ≈ 1.0) range, indicating the presence of persistent fractal scaling in both conditions (see Fig. 2 for an example). It should be noted that our use of the term ‘‘persistent’’ in regards to our results involves the assumption that the time series at hand can be categorized as fractional Gaussian noise (fGn). If one were to assume fractional Brownian motion (fBm), our time series would be better classified as exhibiting anti-persistent scaling. Because the boundary between fGn and fBm is traditionally considered to be α = 1.0, time series with scaling coefficients falling directly at or near that boundary (such as the ones found in this study) can be difficult to classify definitively. We assume fGn in the discussion that follows but regardless of classification, we hope to demonstrate that the fluctuations observed are not simply uncorrelated white noise. As expected, surrogate signals produced estimates near α = 0.5, corresponding to random uncorrelated noise. Comparisons between the original time series and their surrogates showed significant differences regardless of animal or accuracy of the response (p < 0.01). Although FLP (α = 1.028) produced higher α estimates than LUT (α = 0.929), the linear mixed-effects model indicated that this difference was not significant. However, our linear-mixed effects model did indicate a significant (p < 0.01) difference between-subjects when considering fractal dimension estimates. FLP produced significantly lower fractal dimension estimates (D = 1.043) than LUT (D = 1.104), corresponding to α values of 0.957 for FLP and 0.896 for LUT. Differences between conditions (correct vs. incorrect response) were not significant, regardless of analysis type. 3.2. Peak–peak amplitude results Average scaling exponents for peak–peak amplitude levels are given in Table 2. The results showed values in the 1.0 < α < 1.5 range, indicating the presence of fractal scaling in both conditions. As with the ICI analysis, the generated surrogate signals produced α estimates in the range expected for random uncorrelated processes. Comparisons between the original time series and their surrogates showed significant differences across both conditions (p < 0.01). The linear mixedeffects model found a significant difference between animals (p < 0.05), but not condition, for both analysis types. Again, FLP produced slightly higher average DFA scaling estimates (α = 1.238) than LUT (α = 1.098), along with significantly lower fractal dimension estimates (D = 1.030) than LUT (D = 1.120), corresponding to α values of for 0.970 FLP and for 0.880 LUT (see Fig. 3). 4. Conclusions Our results provide statistical evidence that persistent fractal scaling is present in two separate characteristics of bottlenose dolphin echolocation. In the same way that the pervasiveness of fractal scaling in human behavioral measures

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Table 1 ICI results for each animal and response type, calculated over two orders of magnitude. Standard deviations for each average value are given in parentheses. Fractal dimension estimates consistently exhibited lower variance than α estimates and trends were consistent between analysis type. FLP produced higher α estimates, and lower D estimates, than LUT across conditions. Analysis type

Animal

Correct response avg.

Incorrect response avg.

Total avg.

DFA DFA Fractal dimension Fractal dimension

FLP LUT FLP LUT

1.031 (0.27) 0.884 (0.28) 1.036 (0.03) 1.112 (0.09)

1.025 (0.23) 1.018 (0.29) 1.053 (0.01) 1.081 (0.05)

1.028 (0.22) 0.929 (0.29) 1.043 (0.02) 1.104 (0.08)

Table 2 Peak–peak amplitude results for each animal and response type, calculated over two orders of magnitude. Standard deviations for each average value are given in parenthesis. As was the case with the ICI series, fractal dimension estimates consistently exhibited lower variance than α estimates. Betweensubjects trends were similar to those seen in our ICI series, with FLP producing higher α estimates and lower D estimates than LUT. Analysis type

Animal

Correct response avg.

Incorrect response avg.

Total avg.

DFA DFA Fractal dimension Fractal dimension

FLP LUT FLP LUT

1.221 (0.07) 1.091 (0.19) 1.030 (0.01) 1.130 (0.07)

1.200 (0.11) 1.117 (0.22) 1.030 (0.01) 1.10 (0.03)

1.238 (0.11) 1.098 (0.20) 1.030 (0.01) 1.120 (0.06)

Fig. 3. Comparison of DFA and Higuchi’s method results for each animal. Error bars indicate ±1 standard deviation. Fractal dimension estimates produced by Higuchi’s method exhibited substantially smaller standard deviations than α estimates produced by DFA. Keeping in mind the relationship α = 2 − D, it is apparent that the between-subjects trends are preserved in both analyses—FLP produces time series with higher α estimates than LUT across both measures.

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has been interpreted as empirical support for cascade models of human behavior, the presence of scaling patterns in our data suggests that cascade models may be applicable to dolphin echolocation as well. An integral feature of cascade models lies in the fact that they view behavior not as a one-way process that begins in the brain and ends in the body, but as the emergent result of a larger system in which brain, body and environment are inextricably intertwined. As a concrete example, consider the two measurements used in this study. Although inter-click interval and peak–peak amplitude appear to reflect very different aspects of echolocation (one temporal and the other physical), they are strongly interconnected through the animal’s interaction with its environment. Due to increases in attenuation and travel time, the animal will produce clicks at longer inter-click intervals and higher peak–peak amplitudes when echolocating on a distant target than it would when echolocating on a nearby target. As a result, changes in inter-click interval will tend to correlate with changes in peak–peak amplitude, and both will tend to be positively correlated with target distance. Target distance is just one component (or ‘‘grain of sand’’) that, along with other variables like water temperature, salinity, target size, and target composition, contributes to the larger behavior (or ‘‘cascade’’) of echolocation. In this way, each individual value in our time series is more than a simple measurement of physical acoustic properties—it reflects countless interactions between a complex web of physical and temporal components. Cascade models provide us with the theoretical framework to tackle this complexity and fractal analysis provides us with the quantitative tools to investigate how changes in these interactions might affect behavior. The structural backbone of cascade-driven systems lies in the interactions between components. If the components of the system lose interconnectivity, cascading ‘‘avalanche’’ behaviors cannot propagate and the system will no longer exhibit fractal organization. Goldberger et al. [60] applied this idea in a study investigating changes in human heart function with age. The authors hypothesized that, as physiological structures in the human body break down due to aging ([61,62]), behavioral measures will show a subsequent decrease in fractal organization associated with declines in overall health. The authors found that, while young healthy subjects produced heartbeat intervals with strong fractal scaling, both elderly subjects and those suffering from congestive heart failure produced heartbeat intervals exhibiting significantly weaker fractal scaling over time. Hausdorff [63] used similar techniques to show that human subjects who were elderly or suffered from a degenerative disease produced walking stride-intervals with significantly weaker fractal organization than the stride-intervals of healthy subjects. In both studies, a decline in the physical structures of the body (whether due to aging or disease) correlated with an overall decrease in fractal organization of observable behavior. Recall that while strong connections between components lead to persistent long-range fractal fluctuations, a breakdown in that connectivity results in a corresponding breakdown of fractal scaling. As a result, we would expect any disruption in the flow of information (i.e., interconnectivity) between animal and environment to result in a decrease in fractal scaling of behavioral time series. The fact that both animals produced ICI series with scaling in the H ≈ 1.0 range suggests that the input/output relationships governing click spacing (the reception of an echo and the subsequent production of another click) are functioning with the efficiency one would expect from echolocating predators well adapted to a marine environment. Our results showed no significant difference in scaling exponent between conditions, suggesting the animals employed similar search strategies across trials. We did, however, observe a significant difference between scaling of successive peak–peak dB measurements produced by each animal. The scaling exponent of FLP’s peak–peak dB series (α = 1.238) was significantly higher than that of LUT’s (α = 1.098), suggesting a decline in persistent fractal scaling ([13]). While the exact cause of the difference is not clear, it is worth noting that hearing tests performed at a later date suggested FLP suffered from a significant hearing deficit at the time our data was collected [64,53,65]. Although hearing loss was observed across all frequencies tested, it was least severe at 30 kHz and FLP seemed to accommodate for this by reallocating click energy to the 30 kHz range [65]. While it is tempting to make inferences about fractal scaling and hearing loss based on these observations, the fact that our sample size is limited to two individuals prevents us from drawing any concrete conclusions. Although the animals involved exhibited substantial differences in hearing ability, we cannot rule out the possibility that the observed difference in fractal scaling might be wholly or partially due to other variables not controlled for in this study. Further research involving larger samples is needed to expand upon these preliminary findings. Our findings, though limited by sample size, provide the first evidence of persistent fractal scaling in bottlenose dolphin click trains—opening the door for future research into the viability of fractal analysis as a means of describing the dynamic characteristics of dolphin echolocation. However, the possible applications of fractal analysis extend far beyond the realm of acoustics. As a measure of the functional efficiency and interconnectivity of complex systems, fractal analysis has the potential to serve as a quick and non-invasive method of diagnosing the overall health of theoretically any biological function [60,63,66]. Because cascade models suggest that every aspect of a complex organism’s functioning should exhibit fractal scaling, we predict that further research will reveal pervasive fractal scaling in other measures of dolphin behavior. If this is the case, fractal analysis offers an accessible and relatively inexpensive method not only of monitoring an animal’s health, but investigating the complexity of behavioral observations in general. References [1] M. Andre, C. Kamminga, Rhythmic dimension in the echolocation click trains of sperm whales: A possible function of identification and communication, J. Mar. Biol. Assoc. UK 80 (1) (2000) 163–169. [2] W.L. Au, R.W. Floyd, J.E. Haun, Propagation of Atlantic bottlenose dolphin echolocation signals, J. Acoust. Soc. Am. 64 (2) (1978) 411–422. [3] W.L. Au, R.A. Kastelein, T. Rippe, N.M. Schoonerman, Transmission beam pattern and echolocation signals of a harbor porpoise (Phocoena phocoena), J. Acoust. Soc. Am. 106 (6) (1999) 3699–3705.

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