Bottlenose Dolphins’ Clicks Comply with Three Laws of Efficient Communication

Abstract

Bottlenose dolphins’ broadband click vocalisations are well-studied in the literature concerning their echolocation function. Their potential use for communication among conspecifics has long been speculated but has yet to be conclusively established. In this study, we first categorised dolphins’ click production based on their amplitude contour and then analysed the distribution of individual clicks and click sequences against their duration and length. The results show that the repertoire and composition of clicks and click sequences adhere to the three essential linguistic laws of efficient communication: Zipf’s rank–frequency law, the law of brevity, and the Menzerath–Altmann law. Conforming to the rank–frequency law suggests that clicks may form a linguistic code subject to selective pressures for unification, on the one hand, and diversification, on the other. Conforming to the other two laws also implies that dolphins use clicks according to the compression criterion or minimisation of code length without losing information. Such conformity of dolphin clicks might indicate that these linguistic laws are more general, which produces an exciting research perspective on animal communication.

1. Introduction

The communicative aspect of various animal vocalisations is commonly evaluated through the degree of conforming to the laws of efficient communication, a series of observed regularities in the communicative use of linguistic code that go back to Zipf’s influential work [1] and have been further elaborated in the context of information theory [2,3]. In the following, we consider

• Zipf–Mandelbrot’s rank–frequency law—also known as Zipf’s law [1], formulated as

$$f\left(r,\alpha ;\beta \right)=c\times {\left(r+\beta \right)}^{-\alpha }$$

where r is the frequency rank of a word, f is its frequency in a natural corpus, α is a positive parameter, and β is a rank-shifting constant introduced by Mandelbrot [3]. Under Mandelbrot’s condition, Zipf’s distribution is necessary, but not sufficient, for optimal information transmission [4]. Zipf’s law formalises a negative correlation between f and r for α > 0, which converges into a linear dependency with a negative slope on a logarithmic scale. Zipf’s law has been observed at various levels of structural organisation in natural language, and in the broader context of the biological organisation of living systems, as well as at macro levels, such as larger ecological systems [5].

• Zipf’s law of abbreviation—broadly states that the length of a linguistic unit reversely correlates with its frequency of use; viz., more frequent sequences tend to be shorter [1]. Language studies on the relation between word length in terms of letters in English and Swedish and their frequencies [6] and the length of sentences in terms of the number of words for American English [7] found that it follows the gamma distribution for probability density:

$$f\left(L,\alpha ;\beta \right)=K\times L^{\alpha -1}\times e^{{\displaystyle \frac{-L}{\beta }}}$$

where L is the sequence length, α is the shape parameter determining the function’s peak existence and location, and β is the scale parameter determining the spread of the distribution.

• Menzerath–Altmann’s law—indicates a reverse correlation between a sequence’s size and its constituent parts. In other words, the longer a sequence, the shorter its constituents, and vice versa. In natural language, the law applies at different levels of linguistic description, e.g., sentences (size measured in terms of number of clauses), words (measured in terms of number of syllables), and other linguistic units. Altmann [8] proposed the following formulation of this law:

$$f\left(L,a,b,c\right)=a\times L^b\times e^{-cx},$$

(1)

where f is the size of a constituent, L is the size of the entire sequence, and a, b, and c are parameters.

From the information-theoretic perspective, Zipf’s and Menzerath–Altmann’s laws are commonly seen as realisations of the tendency to compress or minimise the linguistic code without losing information [9]. Compression was even suggested as a universal principle guiding animal behaviour in general [9], as well as communication, although, concerning the latter, the results were somewhat dependent on the species and selected unit of analysis [10,11,12,13,14,15,16,17,18,19,20,21]. These and other studies have reported that the laws of communication efficiency and structure apply to certain primates [8,12,16], mammals [14], cetaceans [15], and birds [17,18], but not to others [13,15]. This can be attributed to evolutionary, social, cognitive, environmental, phylogenetic, and other factors, which collectively contribute to the diversity in animal sound communication systems and may condition the need to align with linguistic laws for some species, but not for others. These factors are briefly outlined below:

(i)

Evolutionary Pressures: Species that rely heavily on vocal communication for survival, such as primates, cetaceans, and certain birds, have evolved complex vocal systems optimised for efficiency and structure to enhance communication effectiveness.

(ii)

Social Structure: Animals with complex social structures, like dolphins and primates, benefit from efficient communication to maintain social bonds, coordinate group activities, and navigate social hierarchies.

(iii)

Cognitive Abilities: Species with higher cognitive abilities are more likely to develop sophisticated communication systems that follow linguistic laws. This includes the ability to learn, remember, and produce a wide range of sound vocalisations.

(iv)

Environmental Factors: The habitat and environmental conditions can influence the development of communication systems. For example, animals in dense forests may develop shorter, more frequent calls to avoid sound attenuation, while those in open environments may use longer calls.

(v)

Phylogenetic Constraints: Some species may have inherent biological limitations that prevent the development of complex vocal systems. These constraints can be anatomical, neurological, or related to the species’ evolutionary history.

(vi)

Functional Needs: The specific needs of a species, such as hunting, mating, or territory defence, can shape their communication systems. Species that rely on vocalisations for these critical functions are more likely to develop efficient and structured communication.

In this regard, Youngblood [15] studied sixteen whale and dolphin species and found that only for eleven of these does the structure of communication align well with Menzerath’s law and Zipf’s law of abbreviation, similarly to human speech.

Dolphins produce three primary types of acoustic signals: frequency-modulated whistles for social communication, echolocation clicks for navigation and hunting and burst-pulsed sounds for social interactions [20]. Among these, whistles and clicks are the most extensively studied in bottlenose dolphins (Tursiops truncatus) [20]. Whistles occur within the human audible range and are closely associated with intraspecies communication [22]. A controlled study has shown that their distribution conforms to Zipf’s frequency–rank law [23]. Notably, bottlenose dolphins produce individually distinctive signature whistles that function analogously to names, facilitating individual recognition and maintaining group cohesion [24,25]. These signature whistles are learned early in life and remain stable over time, serving as a key component of their social communication system.

Clicks, in contrast, are brief broadband pulses generated by phonic lips located beneath the blowhole. The sound is focused through the lipid-rich melon into a narrow, directional beam projected into the water [26,27,28]. These echolocation clicks typically last 40 to 150 microseconds and span a frequency range of approximately 20 to 150 kHz—well beyond human auditory capabilities [29,30]. They can be emitted as isolated pulses or in sequences known as click trains. Studies on dolphin clicks highlighted the complexity and adaptability of dolphin echolocation, providing insights into their sonar system and communication abilities [19,20,21]. The role of clicks in echolocation is well documented [31,32,33,34]. Additionally, emerging research suggests that click sequences may also serve communicative functions among conspecifics at varying distances [11,12], although this remains to be confirmed by controlled studies.

In the realm of dolphin cognition research, various studies [35,36,37,38,39,40,41] used clustering techniques to categorise whistles produced by adult and infant bottlenose dolphins into discrete types. Subsequently, they analysed their distributions concerning Zipf’s rank–frequency law. According to these studies, this distribution manifests a linear dependency on a log–log scale with a slope varying from about −0.82 for newborns to up to −1.07 within 2–8 months and converging on the adult value of about −0.95 afterwards. This indicates that dolphins’ use of whistles is broadly in line with the predictions of Zipf’s law and the functional tension hypothesis [1]. It is, therefore, reasonable to verify whether similar considerations apply regarding dolphin click signals.

In this work, we explore the distribution of click-to-click sequences produced by adult bottlenose dolphins living in captivity and demonstrate that it properly conforms to three linguistic laws: (i) Zipf’s frequency–rank law, (ii) the law of abbreviation, and (iii) Menzerath–Altmann’s law, as outlined above. We test whether dolphins’ click emissions comply with these laws of communication. Specifically, we are interested in testing this hypothesis because if a fit to Zipf’s distribution is not observed, this immediately implies that the observed click sequences do not originate from within a communication system. Conversely, while a good fit to Zipf’s distribution does not conclusively entail that the dolphin clicks are part of a language-like communication system, such a result might encourage further studies on the clicks’ function in various scenarios.

2. Materials and Methods

2.1. Recordings

In a series of six underwater recording sessions that took place between November 2019 and February 2021, we recorded vocalisations occurring within a family of five Caribbean bottlenose dolphins (2 males, 2 females, and 1 male infant) living in a controlled aquatic environment at the Varna Dolphinarium (Varna, Bulgaria). The characteristics of the recording equipment and recording parameters used in the sessions are described in Appendix A. Each recording session lasted 5 min due to the limited battery capacity of the equipment (AC power is not allowed due to safety concerns). In the recording setup, we used a sampling frequency of 250 kHz (cf. Appendix A), corresponding to a sampling interval of 4 µs, to capture the frequency range of echolocation clicks. Throughout all six recording sessions (total duration 30 min), the dolphins consistently swam around both sections of their pool, interacting naturally with each other without any signs of distraction or engagement in human-initiated activities. There were no humans near the pool during the recording sessions. Our recording setup did not allow the attribution of specific acoustic emissions to individuals as we used a signal hydrophone.

2.2. Ethics

No invasive procedures or substances were applied to the animals during data collection. The recordings were performed in full compliance with the existing regulations for animal care in Bulgaria and the EU, including Directive 2010/63/EU, following recommendations of the Ethical Committee of the University of Nova Gorica. Bulgarian law does not require specific permissions regarding passively collecting animal vocalisation data non-invasively.

2.3. Signal Analysis

The raw 5 min long recordings containing vocalisation streams were denoised via a digital 4th-order IIR Butterworth high-pass filter with a cut-off frequency of 100 Hz. The sequences of clicks were extracted from the records using the short-time (frames of 100 msec.) median frequency: any signal frame with a median frequency above 5 kHz and a maximum level above 0.05 V was considered active, as extracted sequences were amplitude-normalised. Individual clicks were identified via envelope detection using the following routine: (i) filtration was conducted with a 2nd-order IIR Butterworth high-pass filter with a cut-off frequency of 5 kHz; (ii) sequence segmentation was carried out with a frame duration of 2.5 msec; (iii) coarse envelope detection was performed via short-time kurtosis computation—every frame with kurtosis above 30 was marked as one containing an individual click; (iv) finally, all individual peaks were processed further to reduce reverberation from the aquarium walls (cf. Figure 1 and Figure 2). Refer to Appendix A for details. All steps of the signal analysis were carried out in the MATLAB R2017b environment (the software code is publicly available at https://www.mathworks.com/matlabcentral/fileexchange/68476-data-acquisition-and-logging-with-matlab).

We focused on the amplitude of the clicks as the principal acoustic feature because we hypothesised that it could encode communicative information. The rationale was the following: In spoken language, listeners rely on several low-level acoustic dimensions, namely, fundamental frequency (pitch), spectral shape (formants), temporal extent, and amplitude (intensity), to recognise phonemic contrasts and to recover higher-level prosody. Amplitude modulations, in particular, signal lexical stress (e.g., the louder syllable in ‘PERmit’ vs. ‘perMIT’), cue rhythmic grouping for speech segmentation, and combine with pitch and duration to convey sentence focus and discourse prominence.

Because intensity can carry meaning while remaining largely independent of spectral content, we reasoned that dolphins might also exploit the energy envelopes of their broadband clicks as a communicative variable. Moreover, amplitude is robust to orientation-dependent frequency shifts under water and straightforward to measure at the sub-millisecond scale at which clicks occur, making it both biologically accessible to dolphins and analytically tractable. For these theoretical and practical reasons, amplitude was selected as a key parameter in our study of dolphin click communication.

We opted for a 148 µs window (37 × 4 µs) for principled, data- and literature-driven reasons [19,38,39]. Using the same window and temporal resolution prevents the introduction of artefacts due to resampling, while being large enough to encompass most of the envelopes in our dataset (Appendix A), yet short enough to exclude low-amplitude reverberation tails. Fixed-length amplitude vectors of similar dimensionality are also typical in automated click-classification work [19], because they place clicks of slightly different duration in a common acoustic space without time-warping. Correspondingly, each click from the dataset is represented as a 1 × 37 vector of amplitude values, which can have either positive or negative values.

We also calculated the number of clicks constituting each sequence (sequence size), overall sequence durations, and the inter-click intervals between two click onsets within a single click sequence.

2.4. Statistical Analysis

We grouped the collected dataset of clicks using the affinity propagation (AP) method [42]. Unlike the traditional k-means cluster analysis, AP does not require the number of clusters to be predetermined. Instead, by viewing each data point as a node in a network, the AP algorithm takes measures of similarity between pairs of data points as input. It recursively sends real-valued messages along the network’s edges until an optimal set of clusters and final exemplars are identified. The algorithm allows for the adjustment of the sensitivity or granularity of the clustering process by setting the input preference parameter or selecting a specific quantile of the input similarities. Using the apcluster package in R [43], we first computed a similarity matrix for the amplitude data in the final dataset. Negative squared errors (Euclidean distance) were used as the similarity measure. On the conservative side, we set the initial input preference parameter value at 0.8, which calls for a higher degree of similarity and potentially a higher number of clusters. We consider the cluster size a good approximation for the occurrence frequency of the corresponding click types.

To investigate whether the length of a sequence influences its frequency of use along the lines of Zipf’s law of brevity, we fitted the observed sequence-length distribution against theoretical gamma, Weibull, and log-normal distributions using the fitdistrplus package in R. The fits were evaluated via the Kolmogorov–Smirnov (K-S) criterion, whereby a larger p-level value indicates a better fit, and by a visual inspection of the respective cumulative distribution function plots.

To analyse the possible relation between the length of a click sequence and the duration of its components along the lines predicted by the Menzerath–Altmann law, we fitted linear mixed-effect models using the lme4 package [44] in R v.4.0.2 [45]. Click duration was included as a response variable, and sequence length was the only fixed factor. In addition, sequence code and recording session were included as random effects to account for repeated measurements. The original distribution of click durations showed a slight positive skew. Consequently, the dependent variable was square-root-transformed for the model; the model residuals were visually inspected for homoscedasticity and normal distribution via qqplot and their distribution against the fitted values. We used the likelihood-ratio test to evaluate the significance of the entire model, which included the fixed and random factors, against the null model, which only had the random factors. The 95% confidence intervals (CIs) and p-values for the obtained estimates were computed using the Wald approximation.

3. Results

Overall, we identified 3618 clicks organised in 98 sequences. The clicks were produced in the broadband frequency range of 25 ÷ 125 kHz, which is consistent with previous reports [26,27,28]. The length of sequences varied between 6 and 257 clicks (median = 23, IQR = 14 ÷ 48), and their duration varied within the range of 178 ÷ 9914 msec (median = 992, IQR = 615 ÷ 1656). The duration of individual clicks varied in the range of 0.37 ÷ 490 msec (median = 36, IQR = 27 ÷ 47).

The clustering algorithm applied to the array of amplitude vector representations of individual clicks converged after 543 iterations, yielding 848 clusters or click categories. The size of the resulting clusters varied between 1 and 120 clicks. A linear regression mapping the log-rank of the signal against the log frequency of occurrence (=log size of the cluster) showed a slope of −0.94, indicating that the distribution of clicks in our tested dolphins’ repertoire is non-random in accord with Zipf’s rank–frequency law. For comparison, we also generated a matrix of pseudo-random values within the numerical range of the actual amplitudes of the clicks in our database. We ran the same categorising algorithm, and the results indicated that most categories had similar sizes (frequencies of occurrence), as expected. The results of both cases are shown in Figure 3.

Figure 4 shows a probability density plot of sequence lengths regarding the number of pulses across the entire dataset, the fit curves, and the corresponding cumulative distribution (CDF) functions. We found that the distribution of sequence lengths indeed receives a good fit by the family of the generalised gamma distributions, particularly gamma (K-S goodness-of-fit test: D = 0.118, p = 0.1321), Weibull (K-S: D = 0.10248, p = 0.2548), and log-normal (K-S: D = 0.072666, p = 0.6788) distributions. The K-S criterion and the CDF visual inspection suggest that our click data best fit the log-normal distribution.

Finally, we found that sequence length had a statistically significant (model full vs. null: χ²(1) = 65.79, p < 0.001) and negative effect on the click duration (beta= −1.03 × 10⁻⁴, 95% CI [−1.28 × 10⁻⁴, −7.85 × 10⁻⁵], t(3448) = −8.18, p < 0.001). Regarding the untransformed dependent variable, increasing the sequence length by one click leads to a decrease in individual click duration by about 0.4 msec.

4. Discussion

Our results indicate that the observed repertoire of broadband click sounds emitted by Tursiops truncatus conforms to three essential linguistic laws. The number of occurrences of a particular type of click in the dataset, as identified by the clustering algorithm, is inversely correlated with its frequency rank. This is consistent with the idea that, similarly to human languages, this repertoire is formed due to two opposing selectional pressures, one aiming for greater unification and a reduced number of repetitions, and the other aiming for greater token diversity. The observed slope value of −0.94 is on par with that previously reported in the domain of dolphins’ whistles [15,35,36,37,38,39,40,41]. This might indicate a similar strategy of use of both signal types in communication, which is conceivable given their common anatomical source, namely, nasal or phonic lips [26,27,28,46,47]. The distribution of sequence lengths followed a log-normal distribution typical for natural language by the law of brevity.

Let us suppose that clicks are used for communicative purposes in ways that Zipf’s law series predicts. In that case, one can expect structural sophistication in how click sequences are formed to express meanings in the dolphins’ putative language. We did not control the contextual variables in the reported study, so it remains impossible to correlate specific clicks or click sequences with conceivable meanings they could convey (cf. [48]). Nevertheless, the structural complexity of utterances can be fruitfully studied independently of associated meaning, as has been abundantly demonstrated for natural language by modern formal syntactic theory [49].

One pertinent question is what levels of linguistic building blocks are represented by individual clicks and click sequences. Human language is characterised by the duality of patterning, whereby meaningless units such as phonemes are combined to produce minimal meaning-bearing conglomerates, e.g., morphemes, and those further up the structural hierarchy, reaching ultimately the level of a complete message or utterance [50]. In animal communication, the duality of patterning has yet to be attested, but some aspects of it were affirmed in certain species [51]. The sheer variety of click types raises the valid question of whether individual clicks or groups of clicks may possibly serve as units conventionalised for use with a fixed lexical or functional meaning. Under this hypothesis, of particular interest are potential syntactic rules used to concatenate clicks into sequences [51,52]. Investigation of these syntactic rules requires a fine-grained analysis of the conditional probabilities of occurrence of various click combinations, possibly along the entropy computation suggested for dolphins’ whistles [37]. We deem that further studies on the structure of dolphin language are required before we can make conclusions on the multifunctionality of click series. The reader may refer to [53], where a fascinating hypothesis was formulated, and a thorough review of the dolphin repertoire is available in [54,55].

Furthermore, the size of a click sequence was shown to correlate inversely with the duration of clicks, in accord with Menzerath–Altmann’s law. Previous studies in corpus linguistics demonstrate that Menzerath–Altmann’s law holds for word length in terms of the number of letters as well as sentence lengths in terms of the number of words and can be modelled by a version of the generalised gamma distribution, in particular, gamma [6,7], Weibull [3], or log-normal [56] distributions. Similar shape and scale parameters characterise all three observed distributions, and choosing among them is challenging [57]. However, they all capture the exponential-like character of dependency between frequency and length.

5. Conclusions

We studied the distribution of click sequences produced by adult bottlenose dolphins to determine whether it conforms to Zipf’s frequency–rank law, the law of abbreviation, and the Menzerath–Altmann law. The limited amount of data (3618 clicks extracted from 30 min of data recorded in six sessions between 2019 and 2021) does not permit a fine-grained investigation of the proper distribution model, whether it is gamma, Weibull, or log-normal, and this was not the goal of our research. Here, we do not take a stand concerning the selection of a proper distribution, and we refer to previous results as an empirical guideline regarding the type of distribution we observed for the dolphin clicks, if they indeed show signs of a communicative system.

The results reported in this study indicate a good fit to the abovementioned three laws, which, however, does not conclusively entail that the dolphin clicks are part of a language-like communication system. Our results will hopefully encourage further research on the suspected multifunctionality of dolphin clicks and perhaps on the benefits of their use in various circumstances.