Data

We survey peer-reviewed publications covering the animal kingdom (Fig 1C) and find an abundance of communication tempos in the 0.5–4 Hz range, also known as the “delta wave” band in neuroscience. As a Discovery Report, this work is based on an initial nonexhaustive selection of data from the literature as well as random sampling from a database (described further below; Fig 2).

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Fig 2. Distribution of randomly sampled data from a wildlife acoustics database.

We randomly selected 50 unique examples of isochronous audio signals from the xeno-canto [1] database, 10 from each of the five animal groups compiled there: birds, bats, frogs, grasshoppers, and land mammals. The left panel shows a scatter plot of each measured tempo (color indicates animal group). Note that there is no sorting along the horizontal axis (body weights are not reported in the database and may overlap). The right panel shows a marginal histogram indicating the distribution of points (vertical axis) along with a log-normal fit. Data and code underlying this Figure can be found in https://doi.org/10.5281/zenodo.19069908.

https://doi.org/10.1371/journal.pbio.3003735.g002

We focus on isochronous communication in which signals are repeated multiple times with stable intervals between signal onsets (i.e., metronome-like signals: see Methods for details). Although trills (e.g., from stridulation in insects [2,3]) can also appear to be isochronous, we exclude them since they are the faster timescale of the signal, which could be thought of as analogous to pitch or color. We also stress that, although we limit the search to isochrony, this need not mean that a signal has less information content: information may be conveyed by modulating other aspects, such as the amplitude or the higher frequency components.

Our data reveal an apparently universal motif. Across 8 orders of magnitude in body weight, across modalities (vision versus audition), and across the tree of life (insects including fireflies [4,5] and crickets [6]; crustaceans [6,7]; amphibians [6,8]; birds [9–11]; fish [12,13]; and mammals including apes [14,15], humans [16], and sea lions [17]), communicating at a “carrier frequency” of ~0.5–4 Hz seems to be the norm. The instances we found are widespread—they come from all around the world and occur in creatures from the air, land, and sea.

We note that there is a risk of selection bias in our data, which could come about in two ways: we may have chosen an unrepresentative set of examples from published literature, and the literature itself could be unrepresentative of the full set of biological communication tempos. The latter issue may also be connected to inherent human bias due to our perceptual limitations (for instance, sound above 20 Hz becomes audible to us, but not to many other animals). Regardless, it is not possible to fully exclude either of these explanations for our observations, but we did make every effort to avoid inadvertently selecting examples of species consistent with our expectations. For example, we exclude bird song (e.g., thrush nightingale and zebra finch) that some consider isochronous but doesn’t fit our criteria, though it would also lie within the delta range [18]. We similarly exclude dog vocalizations around 2 Hz [19]. However, we do consider bushcrickets, which have been reported to communicate at 11–14 Hz [20], and Saccopteryx bilineata bat calls at 10–14 Hz [21], both of which lie outside the delta band (though bats might be expected to communicate differently because of their use of echolocation).

As a second method of combating potential bias in our choice of example species, we demonstrate the results of a random species selection process using the established wildlife sound database xeno-canto [1]. Analysis of that data again reveals a peak of the tempo distribution in the delta band (Fig 2), consistent with what is shown in Fig 1. It appears roughly unimodal with a peak around 3 Hz and a median of 3.45 Hz; a log-normal distribution appears plausible and cannot be rejected at the 2% level of significance ( with best fit parameters , , implying peak (mode) of 1.4 Hz and median of 3.0 Hz). See the Methods section for a detailed explanation of how we conducted this analysis and its limitations.

Possible explanation

The abundance of cases within the 0.5–4 Hz range suggests that there could be some adaptive value to this frequency band. Note that the signalers we’ve listed are most likely all physically capable of producing signals at higher tempos (and sometimes do, e.g., [4]) and obviously could always go slower. Thus, they appear to “opt” for this frequency band, further suggesting that a communication advantage is present in this range.

We speculate about the possible underlying reason for this widespread abundance. Music psychologists have suggested that humans favor musical tempos in the neighborhood of 2 Hz (or 120 beats per min—BPM) because of their gait [16]. Here we argue that this tempo is actually much more widespread, and thus perhaps it is utilized for a different reason—there might be a deep evolutionary root. We hypothesize, given the wide variety of signal production apparatuses, that it is likely not a biomechanical constraint on production; the reason could stem from the receiver’s end. Most likely, this isn’t something to do with sensory organs, as again, the channels of communication are diverse. On the other hand, all the receivers rely on some form of neural machinery for processing these signals—this is the common factor.

Neural circuits across distinct species might share similar characteristics if they had been selected to process similar information from the world. For example, vertebrate visual systems across species show an excess of dark spot detectors (OFF cells) as compared to bright spot detectors (ON cells) because of an asymmetry in the distribution of light in natural scenes [22]. Likewise, the rarity of retinal blue cones across species, and the large variance in red to green cone ratio in individual trichromats, can be understood as an adaptation to the structure of chromatic information in natural scenes combined with the constraints of lens-based eyes [23]. In these cases, neural circuits behave similarly because they are adapted to the structure of the world.

This reasoning is similar to the “sensory drive” hypothesis and other closely related ideas [24–27]. There it was suggested that males evolved signal designs that are efficient in that they fit the females’ sensory capacities (which are adapted to local environmental conditions). Hence, the direction of causality is suggested to be from the receiver to the producer; the producer evolved signals to accommodate the receiver’s evolved sensory constraints.

Another possibility is that neural circuits across species share characteristics because they are adapted to the basic biophysics of neurons. This may explain, for example, the fact that all mammalian brains show similar neural rhythms [28]. Indeed, a recent study with rats found that sound patterns in the range of 2 Hz produced the largest neural responses, based on data from the auditory cortex [16].

Thus, we suggest that the 0.5–4 Hz communication tempo we uncover across scales and taxa may be an adaptation to the (resonant) frequencies of typical small circuits in the brain. Below we show that a sensory “receiver” circuit (i.e., a neural circuit in the brain of the individual perceiving the signal) consisting of model neurons with characteristic biophysical integration times of a few hundred milliseconds [29–33] will respond best (“resonate,” i.e., lead to a strong response) to external stimuli in the tempo range of 0.5–4 Hz (we note that neural integration times on this order of magnitude appear to be conserved across a wide range of taxa and across different neuron types; future research might also attempt to test our suggestion that neural circuit properties have had a causal impact on signal tempo in the senders).

Computational experiments

To test our hypothesis that neural circuits in the receiver are best suited for the 0.5–4 Hz tempo range, we perform two computational experiments. First, we investigate whether circuits composed of simple model neurons would typically “inherit” their fundamental frequency (i.e., will individual neuron integration times set the tempo, or will there will be emergent frequencies, as can be observed in the brain [34]). Second, assuming that this is indeed the case, we attempt to derive “resonance curves” for these circuits, following the neural-resonance theory framework [35–38] (resonance, a concept borrowed from the physical sciences, refers to the extent to which an entity responds more strongly to external driving at a particular frequency as compared to others). This demonstrates to what extent these circuits are entrainable (i.e., responsive / could be influenced) to external stimuli around 2 Hz (which we use as a representative mid-range value for the delta band) [39–41], and how that depends on parameters. See Fig 3 for a visual explanation of the methodology.

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Fig 3. Schematic of the modeling methodology.

Explanation of the questions we address with our modeling approach: we ask about the “inheritance” of frequencies from the individual to the circuit level (and if, and how, it depends on the specifics of the connectivity/architecture), and whether we can construct “neural resonance curves” by forcing our system with an external stimulus (since we measure circuit-level response (R), this could also be interpreted as “circuit resonance").

https://doi.org/10.1371/journal.pbio.3003735.g003

In our experiments, we use an order parameter, R, to quantify how “in sync” each circuit is with the external signal. This may also be interpreted as a form of “circuit amplitude,” since more neurons fire together when the order is higher. See Methods for a detailed explanation of how the order parameter is calculated.

The global coupling (how much the neurons respond to input, whether from another neuron or the forcing) was set so the unforced system was slightly “subcritical,” i.e., just below the critical coupling strength for a phase transition. This means that the neurons themselves won’t sync without input—it is explicitly the influence of the forcing that might sync them

(It is well known in physics that coupled oscillator models like the Kuramoto model can undergo phase transitions from an incoherent, or disordered, state to a partially synchronized state when coupling exceeds a critical threshold. That a natural system would evolve to operate near a critical point would be consistent with much work on “self-organized criticality” see, e.g., [42–45].)

For the first computational experiment, we exhaustively test (for certain small circuits) all possible topologies to see if the emergent behavior is sensitive to how the circuit is wired. In other words, are some circuits much more or less entrainable? The assumption here is that neuronal wiring is most likely not fine-tuned. Fig 4A and B show that, for the vast majority of circuit topologies, response to a 2 Hz driving signal is not sensitive to topology—fine-tuning the graph structure appears to have little effect on the induced order R.

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Fig 4. Modeling circuits of neurons.

Circuits of Kuramoto oscillators with N = 5 oscillators driven by an external stimulus of 2 Hz. (A) The mean order parameter R (quantifying synchrony) of each circuit (with 10 edges, we ran this for all 1,665 possible directed graphs, excluding isomorphic graphs). The data is sorted from the lowest to the highest R. The curve is a mean of 100 curves obtained from 100 realizations. The red shaded area is to draw attention to the lower R regime. (B) Example topologies: we show the topologies of the best and worst in terms of entrainability (highest and lowest R). (C) Neural resonance curves exploring the parameter space of K_ext and (in this case these are all-to-all topologies). We vary the forcing (i.e., external) tempo f_ext from 0 to 4 Hz. These curves are averages obtained from 10 different realizations (10 different initial seeds). The dashed line in the leftmost plot is at 2 Hz—the mean of the natural tempos of the circuits. (D) A heatmap showing the response of the system as a function of the input coupling strength and the forcing tempo. This reveals an “Arnold tongue" at 2 Hz that widens as the input strength increases. We see no additional tongues at the harmonics (highlighted with white dashed lines) or subharmonics. Result obtained from 10 realizations (initial seeds). Data and code underlying this Figure can be found in https://doi.org/10.5281/zenodo.19069908.

https://doi.org/10.1371/journal.pbio.3003735.g004

For the second computational experiment, we examine how different forcing tempos affect the receiving circuit. Fig 4C shows that a form of resonance is present: the strongest response is to external signals that are around the intrinsic tempo of the circuit (in this case 2 Hz). Greater neural diversity allows for stronger induced order even at tempos different from the mean intrinsic tempo of the neuron population. This widening of the resonance curve may suggest an evolutionary role for heterogeneity in neurons of the receiver circuit: larger heterogeneity allows for a strong response even to signals that are slightly detuned from the expected external forcing tempo. But the peak of the response is still at the intrinsic underlying tempo of the circuit, meaning that external tempos much different from the delta band will be much less influential, if at all. Another way to view this effect is by visualizing the “Arnold tongues" of the system—V-shaped regions in parameter space in which the system responds strongly (Fig 4D). The absence of such tongues in the harmonics or subharmonics (as would be expected, e.g., for mechanically resonant systems) implies that the circuit cannot be driven at the harmonics of the fundamental frequency. So, at least for this model, entrainment to much higher frequencies is not possible.

Field data

Cricket and firefly data presented in Fig 1 come from analysis of original video and audio recordings collected with a Sony Alpha 7 SII camera in Amphawa, Thailand in summer 2022. Firefly flash timings were obtained via a computer vision algorithm we developed as described in [54]. To obtain a global picture of synchronous firefly tempos, we summed all flashes per frame (we want the overall group dynamics) and then smoothed this signal (Savitzky-Golay filter of polynomial order 2 and frame length 9), subtracted the mean, and plotted the spectrogram (time window 100 samples = 3.33 s for 29.97 FPS video). All computational scripts described here and throughout were written in Matlab v2020a [55].

For the cricket (audio) data, we first applied a highpass filter (cutoff 5 kHz), then computed the peak envelope of the absolute value of the filtered data using a Hilbert filter of window size 3,000 samples (68 ms). The mean was then subtracted and spectrogram plotted.

Data from previously published work and established databases

We began our search by focusing on widely accepted isochronous animal signals. However, because the definition of isochrony varies in the published literature, challenges immediately arose. For example, for inclusion in our dataset, it was necessary to establish a minimum number of signal repetitions (5) and a minimum level of IOI (inter-onset-interval) consistency (std. dev. 25%). Neither of these are typically made explicit in relevant publications, so there was often a need for manual estimation from figures or close reading of text and tables. Furthermore, some isochronous (or quasi-isochronous) signals are not flagged as such in relevant publications. Therefore, our dataset is nonexhaustive.

We wished to test our hypothesis of the universality of the 0.5–4 Hz communication band by seeing if and when it broke down: are there limitations related to body sizes, taxa, or communication modalities? This motivated collection of data across the broadest possible range of species.

In Fig 1 we show approximate values for the weights and tempos of the given species. Exact weights and tempos were rarely reported in the referenced works. For weights, we estimated based on searches for typical values for each species, ignoring distinctions between males and females. We note that, even if our estimated weights were off by a factor of two, our figure would remain largely unchanged due to the many orders of magnitude covered. For tempos, we approximated them from what was presented in figures and/or data shared in each study (e.g., a time series plot).

In addition to the above, we also collected a large random sample of species present in the established wildlife sound database xeno-canto [1]. This database includes many taxa with wide geographic dispersal, though it is limited to sound signals. Specifically, we sampled uniformly at random from each of the five animal groups that exist in the database (birds, grasshoppers, bats, frogs, and land mammals) until we found 10 in each group that were isochronous based on our criteria (≥5 intervals with ≤25% standard deviation). When a recording came from a species or genus already covered, we discarded it. The database includes a grading system for the quality of the recordings (from highest A to lowest E); we restricted our selection by choosing only samples that were graded A or B.

Once the samples were collected, for each case we used the associated plots provided by xeno-canto. In other words, we relied on their handling of the raw data and how they produced these time-series plots. We manually annotated the spectrograms to locate each call and then computed the mean and standard deviation of the implied IOIs (see Fig C in S1 Text for an example).

Overall, we considered 124 recordings (20 bird, 19 grasshopper, 27 bat, 16 frog, and 42 land mammal) to arrive at 10 unique isochronous cases from each group. For all groups except land mammals, at least half of the recordings were deemed isochronous.

Computational experiments

To test our hypotheses, we simulate the Kuramoto model [56–58]. It is well known that neurons respond to synaptic currents by spiking intermittently and often rhythmically; this spiking can be regarded as a form of oscillation and modeled at different levels of accuracy via systems of differential equations. The Kuramoto model offers simplicity and yet it was used previously to model neural circuits—see, e.g., [59,60]. It can be seen as a limit of the Stuart–Landau model—recently utilized by Zalta and colleagues [38]—where amplitudes are nearly constant.

We modeled circuits of Kuramoto oscillators according to

(1)

where adjacency matrix A=[A_ij] defines the coupling circuit, and represent the internal phase and natural frequency of the ith oscillator, respectively, N is the number of oscillators, K is the inter-neuron coupling strength, and K_ext is the strength of coupling to the external forcing (expressed in units of K so that K_ext = 1 corresponds to external forcing being equal in magnitude to internal coupling).

The critical coupling K_c was determined using the large-N formula (exact as ), where is the probability distribution function (PDF) for the oscillator natural frequencies. When the natural frequencies are normally distributed with standard deviation , this becomes . In practice, we fixed K just below this theoretical critical point at with . Note that the fact that K_c depends on means that the critical coupling changes as the oscillator diversity changes.

Numerical integration of system (1) was performed with an explicit fourth order Runge–Kutta method (ode45 in Matlab). We ran multiple realizations where, in each realization, both oscillator initial phases and natural frequencies were selected randomly, with phases chosen uniformly at random and frequencies chosen from a Gaussian distribution of mean and standard deviation . When we varied the graphs A_ij, we kept the mean frequency fixed at Hz.

We use the order parameter R to quantify (on a zero to one scale) the synchrony or “orderliness” of the group at each point in time [58]:

(2)

This can be understood as the distance to the center of mass of all oscillators if one imagines each one represented by a point-like mass located on the unit circle at an angle corresponding to its phase . Then the “no sync” state (where oscillators have random or uniformly distributed phases around the circle) will be located at the origin of the coordinate system, while the center of mass for the “perfect sync” state (where all oscillators have the same phase) will lie on the unit circle.

Note that we calculate R including the phase of the external forcing (i.e., treating it as a sixth node); by doing so, we incorporate information about entrainment to the forcing and not just synchronization of the neurons (higher R means the forcing and the circuit are in sync).

In this study, we choose to focus on circuits of five vertices and exactly 10 directed edges (a total of 1,665 unique nonisomorphic circuits). This is because the number of graphs grows super-exponentially, and thus full computational exploration of all graphs of size N quickly becomes prohibitively expensive. We exclude self-coupling (loops of length 1) and choose a fixed value of 10 edges to maximize the number of distinct circuits covered and focus on the coupling structure rather than the dependence of entrainment/resonance on the number of edges.

When deriving “neural resonance curves,” we fix the coupling topology to be global (all-to-all, i.e., fully connected) and evaluate the effects of varying both the heterogeneity of the oscillators, quantified by standard deviation , and the magnitude (relative weighting) of the forcing K_ext. As Fig 4C shows, we found that the response peaked at the mean oscillator frequency and that increasing K_ext and both widened and amplified the peak.