# Nonlinear acoustic phenomena affect the perception of pain in human baby cries

Siloé Corvin, Mathilde Massenet, Angélique Hardy, Hugues Patural, Roland Peyron, Camille Fauchon, Nicolas Mathevon

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Siloé Corvin, Mathilde Massenet, Angélique Hardy, Hugues Patural, Roland Peyron, et al.. Nonlinear acoustic phenomena affect the perception of pain in human baby cries. Philosophical Transactions of the Royal Society B: Biological Sciences, 2025, 380 (1923), pp.20240023. 10.1098/rstb.2024.0023. hal-05033758

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Submitted on 14 Apr 2025

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<figure><img src="image_1.png" /></figure>
Nonlinear acoustic phenomena affect the perception of pain
in human baby cries

Siloé Corvin<fnref n="1" />,<fnref n="2" />*, Mathilde Massenet<fnref n="1" />, Angélique Hardy<fnref n="2" />,<fnref n="3" />, Hugues Patural<fnref n="4" />, Roland Peyron<fnref n="2" />,
Camille Fauchon<fnref n="2" />,<fnref n="3" />, Nicolas Mathevon<fnref n="1" />,<fnref n="5" />,<fnref n="6" />

ORCIDs: SC 0000-0002-9281-6087; MM 0000-0002-0085-1871; HP 0000-0002-5359-6544;
RP 0000-0002-2706-0594; CF 0000-0002-5848-8681; NM 0000-0003-0219-6601

**Keywords:** baby cry, nonlinear phenomena, playback experiments, pain information, acoustic analysis, parametric sound synthesis

<footnote n="1">¹ ENES Bioacoustics Research Lab, CRNL, CNRS, Inserm, University of Saint-Etienne, Saint-Etienne, France.</footnote>

<footnote n="2">² NEUROPAIN Team, CRNL, CNRS, Inserm, University of Saint-Etienne, Saint-Etienne, France.</footnote>

<footnote n="3">³ Neuro-Dol, Inserm, University of Clermont Auvergne, University Hospital of Clermont-Ferrand, Clermont-Ferrand, France</footnote>

<footnote n="4">⁴ Neonatal and Pediatric Intensive Care Unit, SAINBIOSE laboratory, Inserm, University Hospital of Saint-Etienne, University of Saint-Etienne, Saint-Etienne, France</footnote>

<footnote n="5">⁵ Institut universitaire de France</footnote>

<footnote n="6">⁶ Ecole Pratique des Hautes Etudes, CHArt lab, EPHE - PSL University, Paris, France</footnote>

<footnote>*Author for correspondence (siloe.corvin@yahoo.fr).</footnote>
2 Summary

3 What makes the pain cries of human babies so difficult to ignore? Vocal traits known as "nonlinear phenomena"
4 are prime candidates. These acoustic irregularities are common in babies' cries and typically associated with high
5 levels of distress or pain. Despite the vital importance of cries for a baby's survival, how these nonlinear
6 phenomena drive pain perception in adult listeners has not been systematically investigated. Here, by combining
7 acoustic analyses of cries recorded in different contexts with playback experiments using natural and synthetic
8 cries, we show that baby cries expressing acute pain are characterized by a pronounced presence of different
9 nonlinear phenomena, and that these nonlinear phenomena drive pain evaluation by adult listeners. While adult
10 listeners rated all cries presenting any of these nonlinear phenomena as expressing more pain, they were
11 particularly sensitive to the presence of chaos. Our results thus show that nonlinear phenomena, especially chaos,
12 encode pain information in baby cries and may be critically helpful for the development of vocal-based tools for
13 monitoring babies' needs in the context of paediatric care.
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14
15 Introduction

The cry of the human baby is a vital communication signal produced in contexts associated with hunger, discomfort or pain, and meant to attract the attention of caregivers [1]. Hence, decrypting the information encoded in babies' cries and understanding how this information is captured by caregivers have become key objectives for improving parent-baby relationships [2-9]. On one hand, babies' cries carry relatively stable information, linked to idiosyncratic traits of the emitter, such as their identity, age, weight, and height. This "static" information enables a given baby to be vocally identified by human caregivers, notably their parents [2-4]. On the other hand, cries also carry "dynamic" information related to the current physiological and emotional state of the baby, informing caregivers about the baby's immediate level of needs [5]. While cries do not seem to carry any evident information about their specific cause [4], they yet remain particularly informative regarding the baby's level of distress or pain.

Previous studies have shown that adult listeners, particularly those highly experienced with babies care, such as parents of young children, are able to assess the level of pain conveyed by a cry and accurately distinguish between discomfort and pain cries [5,6]. Indeed, in both human and non-human mammals, an individual's emotional state strongly affects the configuration of the vocal apparatus and the mechanisms of vocal production responsible for the acoustic characteristics of the emitted vocalisations. Increasing distress typically results in a stronger airflow through the larynx, leading to faster and possibly irregular and unstable vibrations of the vocal folds [10,11]. This configuration of the vocal apparatus is often associated with an increase in the fundamental frequency ($f_0$ corresponding to the rate at which the vocal folds vibrate and the perceived pitch), intensity, and duration of vocalisations [12,13], as well as the production of acoustic irregularities, known as nonlinear phenomena.

These nonlinear phenomena are commonly observed in the distress calls of infant mammals ([14-16]; Massenet et al., this issue [update once DOI is available]), including those of human babies ([5,17], and also see Figure 1 and supplementary materials for audio examples). Specifically, cries of human babies can contain several types of nonlinear phenomena: (1) *frequency jumps*, corresponding to abrupt changes in vocal fold vibrations resulting in a sudden jump in $f_0$, as in the breaking voices of teenage boys [18]; (2) *subharmonics*, often resulting from a period-doubling or tripling in the vocal fold vibration, visible as parallel spectral bands that are harmonically related to the $f_0$ (e.g., appearing at half $f_0$) and responsible for lowering the perceived pitch of vocalisations [10,11,19]; (3) *deterministic chaos* (hereafter chaos), corresponding to aperiodic, irregular vibration of the vocal folds, visible as 'noisy' sections in vocalisations, as in very rough roars [10]; (4) *vocal fry*, corresponding to relatively slow, irregular glottal pulses where the vocal folds typically vibrate below 70 Hz, as in low-pitched creaky voices, visible as pulses on spectrograms [20-22]; (5) *amplitude modulation*, corresponding to the modulation of the laryngeal frequency $f_0$ by a lower supralaryngeal frequency $j_0$, visible as spectral bands at $m*f_0 \pm n*j_0$ (where m and n are integers, Massenet et al., this issue [update once DOI is available]), creating vocal roughness as in Louis Armstrong's voice and (6) *vibrato*, corresponding to both frequency and amplitude modulations of the laryngeal $f_0$, as in the bleats of a goat [23,24]. Nonlinear phenomena, and especially chaos,
are responsible for the perceived roughness and instability of vocal signals [25,26]. Among these various types of nonlinear phenomena, chaos is the most commonly observed type in the distress calls of infant mammals, such piglets [27], infant giant pandas [15], infant elephants [14], kittens [16], puppies (Massenet et al., this issue [update once DOI is available]) or marmot pups [28].

While previous work showed that the acoustic 'roughness' of a baby cry signals the level of pain [5,29], to the best of our knowledge, there is no empirical study that has explored which of the different types of nonlinear phenomena encode pain information and experimentally tested how these affect pain perception in adult listeners. Here, we fill this gap combining acoustic analyses of cries recorded in painful and discomfort contexts, with playback experiments where parents and non-parents assessed the level of pain conveyed by natural cries (Experiment 1) and highly realistic synthetic cries (Experiment 2) for which we controlled the type of nonlinear phenomena using state-of-the-art sound synthesis tools ([25,30–32], Valente et al., this issue [update once DOI is available]).

At the emitter's end, we predict that pain cries would be characterized by a greater amount and diversity of nonlinear phenomena than discomfort cries. At the receiver end, we also predict that listeners' assessment of the pain carried by a cry would depend on the type of nonlinear phenomena. Specifically, given that chaos is the most common type of nonlinear phenomena observed in the vocalisations of highly distressed animals ([1,14,15,25,28]; Massenet et al., this issue [update once DOI is available], Valente et al., this issue [update once DOI is available]), we expect chaos to be the predominant nonlinear phenomena, both for pain coding by the baby and for assessment by adult receivers. Finally, we predict that parents, due to their experience of babies cries, are more inclined than nonparents to consider nonlinear phenomena in cries as a vocal indicator of pain [6].

## Methods

Our experimental protocol includes (1) the selection of 304 sequences of baby cries (durations of 6.3 ± 1.1 sec) recorded in contexts of discomfort and pain, (2) analysing nonlinear phenomena in these cries, (3) creating synthetic cries with and without nonlinear phenomena, (4) conducting two online psychoacoustic experiments each testing a total of 40 parents and 40 non-parents.

Participants were presented with natural sequences of discomfort and pain cries (Experiment 1), or synthetic sequences of cries containing or not containing nonlinear phenomena whose nature was controlled (Experiment 2). Participants were asked to rate the level of pain conveyed by each of the cries they heard.

### 1. Cries selection

We selected babies' cry sequences from a database at the ENES Bioacoustics Research Laboratory (see [5] for details on the methods used for the cry recordings). This database includes discomfort cries recorded during bathing, undressing, or dressing the baby, and pain cries recorded during a scheduled routine vaccination in which each baby received two vaccine injections (N = 24 babies, 11 boys and 13 girls; age = 60.3 ± 3.3 days on
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the vaccination day; delay between recording sessions of discomfort and pain cries for a given baby = 6.8 ± 3.3
days). Pain cries refer to two types of cry: acute pain cries emitted at the vaccine injection, and residual pain cries
emitted around 6 seconds after the injection (as described in [5]).

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Using Audacity open-source software (www.audacityteam.org), we carefully inspected these recordings and extracted 304 cry sequences (6.3 ± 1.1 sec) that contained a low level of background noise, i.e., no overlapping voices, dripping water or slamming doors. These 304 sequences were distributed as follows: 210 sequences of discomfort cries (2-19 sequences/baby), 46 sequences of residual pain cries (1-2 sequences/baby), and 48 sequences of acute pain cries (2 sequences/baby).

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## 2. Acoustic analyses

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We quantified the presence of nonlinear phenomena in each of the 304 selected cry sequences, and then
compared the nature and quantity of these nonlinear phenomena between the different conditions (discomfort,
acute pain, residual pain). We characterized the presence of nonlinear phenomena, namely frequency jumps,
chaos, vocal fry, subharmonics, amplitude modulation and vibrato, by visually inspecting and annotating
spectrograms provided by Praat software (frequency range 0.2 to 10 kHz and window length 0.05 to 0.08
seconds; [33]).

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Specifically, within each of the 304 cry sequences, we identified between 1 and 12 cry syllables, a syllable being
defined as a vocal utterance produced during a single exhalation and separated from the other syllables by
silences [34]. For each syllable of each cry sequence, we measured the occurrence and proportion relative to the
total syllable duration of each of the nonlinear phenomena. As frequency jumps correspond to sudden changes
in the fundamental frequency, there is no duration associated with these events, and we only measured their
occurrence.

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## 3. Natural and synthetic cry stimuli

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To test whether the presence of nonlinear phenomena in babies' cries modulates pain perception by adult
 listeners, we prepared sound stimuli for two psychoacoustic acoustic experiments.

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In *Experiment 1*, these stimuli corresponded to unmodified natural cry sequences, with amplitudes normalized
to 100% of the maximal amplitude ("normalize" function of tuneR package; cran.r-project.org/web/packages/tuneR/index.html). In *Experiment 2*, we aimed to test the perceptual effects of
nonlinear phenomena, independently of any other acoustic features. To achieve this, we used highly realistic
synthetic cry stimuli with or without nonlinear phenomena. Specifically, to create these synthetic stimuli, we first
selected 10 sequences of discomfort cries produced by 10 different babies (5 boys, 5 girls), with an average $f_0$
that fell within the observed natural distribution to limit strong inter-individual variation in cry $f_0$ and its possible
effect on pain ratings (Figure S1). We then used the selected natural cry sequences as templates to create their
fully synthetic copies without nonlinear phenomena (= synthetic cry prototypes; Figure 3A) with the R package
Soundgen [30]). Finally, we derived these synthetic cry prototypes into variants containing the different types of nonlinear phenomena. We did not consider adding vocal fry to cry prototypes because our acoustic analyses revealed that its presence does not considerably vary between discomfort and pain cries (Figure 2A), and playbacks of natural cry sequences showed that this phenomenon is not strongly correlated with listeners' rating of pain (Experiment 1, see Figure 2B). The timing of insertion and the proportion of nonlinear phenomena affecting the synthetic cries were informed by our acoustic analyses. Thus, we synthesized naturalistic variants, i.e., with acoustic characteristics in terms of nonlinear phenomena that remained within the range of variation of natural cries. In total, we created 60 stimuli for Experiment 2: 10 prototypes x 6 conditions (no nonlinear phenomenon, chaos, frequency jump, amplitude modulation, subharmonics and vibrato). All the sound stimuli used for experiments 1 and 2, as well as R codes used to synthesize the stimuli are available at https://osf.io/4yexz/.

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## 4. Psychoacoustic experiments

The two online psychoacoustic experiments involved a total of 160 participants (n = 80 listeners aged 32.3 ± 5.7 years old (range [23–43])) in Experiment 1; n = 80 listeners aged 31.0 ± 6.9 years old (range [19–49])) in Experiment 2). Via the online platform Prolific (https://www.prolific.com/), we recruited 40 parents and 40 non-parents for each experiment, distributed equally between the sexes (40 men, 40 women). When parents participated to the experiments, they had at least one child under the age of 2 (mean child age = 11.6 ± 5.9 months in Experiment 1, 9.0 ± 5.4 months in Experiment 2). The recruited non-parent participants had no experience of caring for babies. We excluded paediatric care professionals such as nannies, midwives and paediatricians, as well as people who occasionally babysat or were in contact with a baby in their close circle. All participants reported having normal hearing.

Once recruited on the Prolific platform, participants were directed to the Labvanced online platform ([35], https://www.labvanced.com/) where the playback experiments were conducted. Before starting an experiment, participants were asked to fill a short questionnaire about their age, sex, and experience with babies. Participants were asked to take part in the experiment in a quiet room, with headphones, and with a sound volume set at a high but still comfortable level.

In each of the two playback experiments, participants were presented with a total of 60 sequences of natural or synthetic cries. Specifically, in *Experiment 1*, we exposed participants to natural cry sequences randomly selected from 3 conditions (discomfort, acute pain, residual pain). For each participant, 20 cry sequences from 20 different babies were selected for each condition. In *Experiment 2*, participants were exposed to sequences of synthetic cry, chosen at random from 6 conditions: no nonlinear phenomena, chaos, frequency jump, amplitude modulation, subharmonics, vibrato (each condition balanced across synthetic cry prototypes).

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After each cry exposure, participants were asked to rate their perceived pain in cries by answering the question "How does this baby feel?" with a continuous scale ranging from 0 (no pain and no distress) to 100 (worst pain
157 imaginable). A "pain threshold" at 40/100, as used in studies on pain neurophysiology [36,37], was also indicated
158 visually by adding a mark to the scale. Participants were neither informed about the type of cry (discomfort,
159 residual pain, acute pain cries in *Experiment 1*; condition type in *Experiment 2*), nor the cry quality (natural or
160 synthetic). After completion of the experiment, participants were paid at the recommended rate of 8.0 GBP per
161 hour.
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163 5. Statistical analysis

164 We performed statistical analyses using Bayesian mixed models fitted with the brms R package ([38], R (v4.2.2)).
165 The advantages of using the Bayesian approach are manifold, including its high flexibility, quantification of
166 uncertainty in estimates, use of priors for regularization, intuitive interpretation of confidence intervals, and
167 ability to handle unbalanced datasets and complex models [38–40].

168 Our first aim was to test whether the context of crying and the presence of nonlinear phenomena are correlated.
169 We first ran a statistical model including the presence of any nonlinear phenomena (i.e., independently from the
170 nonlinear phenomena type, absence/presence coded as 0/1), as a response variable modelled by a logistic
171 function (Bernoulli family), and the crying context (discomfort, residual pain, acute pain) as a fixed factor. We
172 then ran another similar model, adding the type of nonlinear phenomena (i.e., amplitude modulation, chaos,
173 frequency jumps, subharmonics, vibrato and vocal fry) as fixed factor. We then built additional models to
174 investigate how the proportion of non-discrete nonlinear phenomena (chaos, vocal fry, amplitude modulation,
175 subharmonics and vibrato) and the number of discrete frequency jump events vary in cries with zero-one-inflated
176 beta and zero-inflated Poisson distributions, respectively [41]. Finally, in a last model, we used lognormal
177 distribution to investigate whether the duration of pain cries differ from discomfort cries. All models examining
178 acoustic data also included the baby's identity as random factor.

179 Our second aim was to test whether the presence of nonlinear phenomena drives how adult listeners rate the
180 pain expressed by babies' cries. We built two independent statistical models, the first using pain ratings of natural
181 cry sequences (*Experiment 1*), and the second using pain ratings of synthetic cry sequences (*Experiment 2*). In
182 each of the two models, response variables were the ratings (expressed as percentages), recoded from 0 to 1 to
183 model our data as a zero-one-inflated beta distribution. These models included parental status (parent or non-
184 parent) and the condition (no nonlinear phenomena and all nonlinear phenomena types except vocal fry for the
185 second model) as fixed factors. The identity of the participant and the identity of the baby (i.e., the origin of the
186 stimulus) were included as random factors.

187 All models were fitted with mildly informative conservative priors. Four thousand iterations were run over four
188 MCMC chains (Markov Chain Monte Carlo), the first 500 of each chain being used to adjust the algorithm.
189 Convergence was checked [38] and achieved for every model. Model predictions expressed from 0 to 1 were
190 reconverted into percentages for an easier interpretation of our results. We summarised our results with the
191 medians of the posterior distributions and their 95% credible intervals (CIs). Credible intervals contain the
192 parameter values of highest probability and span the 95% most probable value. Credible intervals for estimates
that do not include zero (or any threshold value of interest) indicate a credible effect given the observed data and model structure. In the same way, distribution of posteriors value over a given threshold can be quantified and indicate a credible difference when more than 97.5% of the posteriors are over the threshold value. When contrasting two conditions, Cls of this difference excluding the null value (i.e. only positive values or only negative values within the CI) indicate a credible difference between the conditions [40]. We present either the median and Cls of parameter values (Figure 2A) or contrasts between two conditions (Figures 2B and 3C, differences between cries with and without nonlinear phenomena).

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## 201 Results

### 202 1. Pain cries are longer and characterised by nonlinear phenomena

From the 304 cry sequences, we identified a total of 1084 cry syllables distributed as follows: 831 of discomfort, 148 of residual pain and 106 of acute pain. The duration of these syllables increases with the level of pain expressed by the baby. Specifically, cry syllables of acute pain cries lasted 2.1 sec [1.7, 2.7] (median and 95% credible interval CI), and were longer by 0.62 sec [0.29, 1.01] and 0.86 sec [0.55, 1.24] than residual pain cry syllables (1.5 sec [1.2, 1.9]) and discomfort cry syllables (1.3 sec [1.0, 1.5]), respectively. Residual pain cry syllables were also 0.24 sec [0.06, 0.45] longer than discomfort cry syllables.

We observed nonlinear phenomena in the majority of syllables of all cry types: 71.0% [62.1, 78.7] of discomfort cries, 82.5% [73.3, 89.1] of residual pain cries, 83.7% [73.7, 90.8] of acute pain cries. However, nonlinear phenomena were more present in residual pain cries and acute pain cries compared to discomfort cries (+11.3% [4.1, 18.6] and +12.5% [4.3, 20.5] respectively; these Cls excluding the null value correspond to a credible difference between the conditions; see Table 1 for details). Specifically, when comparing to the presence of nonlinear phenomena in discomfort cries, chaos and subharmonics were more present in residual and acute pain cries. Frequency jumps, amplitude modulation and vibrato were more present in acute pain cries but not in residual pain cries, compared to discomfort cries. Acute pain cries were also characterized by an increased presence of amplitude modulation compared to residual pain cries (Figure 2A; see Table 1 for details of model data and Table S1 for raw data). Finally, we found only marginal or no variation in the duration of episodes of nonlinear phenomena between the three types of cries (see Table S2).

Together, our results suggest that, the mere presence of nonlinear phenomena, rather than their duration, is positively correlated with the pain expressed by the baby. Residual pain cries are predominantly characterized by chaos, subharmonics, and amplitude modulation, whereas acute pain cries exhibit various types of nonlinear phenomena (except vocal fry).

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### 225 2. Nonlinear phenomena drive the perception of babies' pain in adult listeners

In *Experiment 1*, we played natural cries with and without nonlinear phenomena to parents and non-parents. Our results show that natural cries without nonlinear phenomena led to a pain rating of 33.7% (95% CI [30.3, 37.4]) on a perceptual scale ranging from 0 to 100%. This score was statistically below the pain threshold of 40%
specified to the listeners (99.9% of the posterior distribution below the 40% threshold), suggesting that they perceived cries without nonlinear phenomena as expressing relatively low pain levels. In contrast, listeners rated natural cries with any combination of nonlinear phenomena as expressing pain above the 40% threshold (average rating across all nonlinear phenomena 44.8% [41.2, 48.4], 99.5% of the posterior distribution was above the 40% threshold), which was statistically higher than for cries without nonlinear phenomena (11.1% higher, 95% CI [9.4, 12.8], 100% of the posterior distribution of the difference was positive). Specifically, both parents and non-parents rated natural cries with chaos, frequency jumps, subharmonics or amplitude modulation as expressing more pain. They indeed attributed higher scores to cries with nonlinear phenomena compared to cries without nonlinear phenomena (cries with chaos: 9.6% [8.1, 11.0], frequency jumps: 3.7% [2.3, 5.2], subharmonics: 3.0% [1.6, 4.3], amplitude modulation: 4.8% [3.3, 6.4]; coefficients averaged between parents and non-parents; Figure 2B, Table 2 for model data; Figure S2 and Table S3 for raw data). Furthermore, our results suggest that parents tend to show greater sensitivity to cries with nonlinear phenomena than do non-parents, especially for cries with vibrato (Table 2).

*Experiment 2*, carried out with synthetic cries, clarifies the effect of nonlinear phenomena on the perception of pain expressed by babies' cries. In line with the results of *Experiment 1*, synthetic cries without nonlinear phenomena were rated as expressing little pain (median pain rating = 30.8% [23.1, 39.1]). In contrast, synthetic cries with nonlinear phenomena were rated as expressing greater pain level (median pain rating = 37.0% [28.5, 45.8]), confirming that nonlinear phenomena are indeed responsible for the perception of pain expressed by cries. While this perceptual effect is observed in all listeners, parents are more sensitive to nonlinear phenomena than non-parents when assessing pain in synthetic cries (Table 3). In parents, the experimental addition of any type of nonlinear phenomenon increased pain perception compared to cries without nonlinear phenomena in parents (Figure 3C and Table 3 for model data details on the perceptual effects of each type of nonlinear phenomena; Figure S3 and Table S4 for raw data) – the addition of chaos causing the strongest perceptual effect (perceived pain increased by +12.9% [10.1, 15.5]). In non-parents, similar perceptual effects were observed in non-parents, but to lesser extent: for them, the presence of frequency jumps in cries did not affect perception of pain (0.5% [-1.6, 2.7]; Table 3).

## Discussion

By combining acoustic analysis of natural signals and playback experiments with natural and artificial stimuli, the present study clarifies how nonlinear phenomena encode pain in babies' cries. Our analyses show that various types of nonlinear phenomena are frequent in babies' cries, and that their presence is more pronounced when cries are recorded in a painful context. Our playback experiments show that adult listeners, both parents and non-parents, largely rely on the presence and nature of nonlinear phenomena in cries to assess the level of pain expressed by the baby.

Pain cries, whether acute (i.e., at the time of a vaccine injection) or residual (i.e., a few seconds after the vaccine injection), more often contain episodes of chaos and subharmonics than cries recorded in a situation of
265 discomfort (e.g., at the time of the baby's bath). Cries indicating acute pain are also more likely to contain 266 episodes of amplitude modulation, vibrato and frequency jumps than other cries (residual pain cries and 267 discomfort cries). In contrast, vocal fry, which occurs less in cries compared to the other types of nonlinear 268 phenomena, does not considerably vary with the baby's level of discomfort or pain. Taken together, the diversity 269 of these nonlinear phenomena therefore contributes to make the baby's cry a graded signal, expressing the 270 baby's distress according to a continuous variation. Such gradation of pain encoded by nonlinear phenomena is 271 also found in vocalisations emitted during the different phases of epidural-free childbirth, where woman 272 produced more chaos and amplitude modulation during the final stages of labour (Valente et al., this issue 273 [update once DOI is available]). The variability in the type of nonlinear phenomena in pain cries can probably be 274 explained by the physiological modulations a baby undergoes during a pain episode, with activation of the 275 autonomic nervous system and a decrease of the vagal tone [42]. This results in increased subglottal pressure, 276 laryngeal constriction, and increased tension in the vocal folds, a vocal configuration often associated with 277 irregular or asynchronous vibration of the vocal folds from which nonlinear phenomena derive [11,43].

278 Importantly, echoing the increased presence of nonlinear phenomena in pain cries, both parents and nonparents 279 rated natural cries with nonlinear phenomena, as expressing more pain compared to natural cries without 280 nonlinear phenomena. Particularly, chaos, which is key acoustic feature of pain cries, is also the nonlinear 281 phenomenon associated with the greatest perceived pain in natural cries. In contrast, the occurrence of vocal 282 fry, which does not considerably vary across all types of cries, is not associated with an increase in listeners' 283 ratings of pain. In the same way, vibrato episodes show the smallest difference in presence between pain and 284 non-pain cries, and is linked to an increased pain perception as observed in parents only. While the results of 285 our acoustic analyses and first playback experiment suggest that nonlinear phenomena are strong candidates 286 to signal pain via babies' cries, this is further confirmed by our second playback experiment, in which we used 287 synthetic cries. Indeed, the use of synthetic cries eliminates any confounding effect that might be caused by any 288 acoustic feature other than a nonlinear phenomenon. By independently adding each type of nonlinear 289 phenomenon to synthetic copies of natural cries, and comparing pain ratings between these stimuli, we 290 demonstrate that the presence of nonlinear phenomena increases pain ratings by adult listeners. As with natural 291 cries, the results of our playback experiments with synthetic cries show that chaos is responsible for the strongest 292 perceptual effect of pain. 293

293 In both playback experiments, parents and non-parents were highly sensitive to the presence of nonlinear 294 phenomena when rating the level of pain conveyed by cries. Specifically, both parents and non-parents judged 295 cries with chaos, and to a lesser extent, those with amplitude modulation, subharmonics and vibrato, as 296 expressing more pain than cries without nonlinear phenomena. While this result indicates that parenthood is 297 not a prerequisite for being able to use nonlinear phenomena to assess pain level, in a previous study, we also 298 showed that the ability to identify a pain or discomfort cry from a familiar or unfamiliar baby is highly correlated 299 with the listener's level of prior experience with babies [6]. In the experiments we report here, non-parents 300 showed some ability at identifying pain in cries with nonlinear phenomena, but parents tended to rate the same
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cries as expressing more pain, thus suggesting that baby caring experience may strengthen the effect of nonlinear phenomena on perceived pain.

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While our experiments demonstrate that the presence of nonlinear phenomena in baby cries drives pain ratings in adult listeners, we cannot rule out the possibility that, in real-life conditions with natural cries, other acoustic features, such as $f_0$, duration of cry syllable or sound intensity, may also modulate this rating. Interactions between these acoustic features and nonlinear phenomena might also change perception. In addition, non-parents may be particularly sensitive to these other acoustic properties, such as the $f_0$ of cries or their duration. Indeed, as these parameters are controlled in synthetic cries and do not vary among variants (cries with vs. without nonlinear phenomena), this may explain why synthetic cries are less likely to evoke pain for non-parents compared to natural cries. Future work could systematically test this hypothesis by replicating our experiment using synthetic cries in which the cry $f_0$ or duration vary, as these two acoustic features have also been demonstrated to affect perceived distress [32,44].

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Furthermore, the cries of acute pain considered here were recorded during a simple vaccination and therefore probably do not correspond to the maximum pain a baby can feel. They therefore represent only a subspace of the "acoustic space of pain" that can be expressed by babies cries [5,45]. We can hypothesize that the expression of extreme pain by a baby's cry may combine several acoustic features, such as nonlinear phenomena, strong modulations of $f_0$, and the duration of the cry syllables.

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In humans, nonlinear phenomena are also present in nonverbal vocalisations of adults. More specifically, their presence is common in sounds emitted in negative contexts with high arousal, like screams [31,46]. Recent studies have also demonstrated that nonlinear phenomena affect pain perception in volitional nonverbal vocalisations expressing different levels of pain produced by actors [47] or in vocalisations emitted during the different phases of epidural-free childbirth (Valente et al., this issue [update once DOI is available]). While nonlinear phenomena are key features for the communication of distress and pain in human babies and adults, these features are also common in the distress calls of non-human infant mammals ([14–16,48]. Yet, whether the function of nonlinear phenomena in communicating distress is potentially shared across mammals remains to be clarified as only few studies have so far systematically tested the effect of nonlinear phenomena on distress perception in the vocalisations of non-human animals. For example, Massenet and colleagues showed that nonlinear phenomena, and particularly chaos, characterize the whines of domestic dog puppies (Massenet et al. this issue [update once DOI is available]), and that experimentally adding chaos to synthetic whines increases perceived distress as judged by both experienced and non-experienced human listeners in puppy caregiving (such as breeders or veterinarians) [32]. Note that here too, experience counts: only experienced listeners were sensitive to whines in which authors added amplitude modulation and subharmonics [32]. Finally, the interspecific function of nonlinear phenomena for distress vocal communication may extend even to taxa that are phylogenetically distant from mammals. A recent study has shown experimentally that adult Nile crocodiles are attracted by the cries of baby hominids (bonobo, chimpanzee and human), and that the intensity of the crocodiles' response depends mainly on nonlinear phenomena, in particular chaos [49].
By demonstrating the importance of various nonlinear phenomena as markers of pain in the cries of human babies, the present study makes an important contribution to our understanding of this communication signal. Various avenues of investigation now lie ahead. A first objective will be to gain a more global understanding of the acoustic space of pain encoded by infant cries, by considering more critical situations than a simple vaccination session, such as when babies have to receive care known to be painful at the hospital. Indeed, accurate assessment of pain in newborns, both premature and full-term, remains a challenge even today [50]. A second objective will be to continue exploring the variability of pain expression between different babies. In recent work, we have shown that the idiosyncratic characteristics of cries (the "vocal signature" specific to each baby) are less prominent in pain cries than in discomfort cries [9]. This result indicates that intense pain causes the cries of different babies to converge in a constricted acoustic space. This convergence of cries due to pain could open up possibilities for the implementation of automated pain detection systems in hospital settings. A third objective will be to characterize the ontogeny of pain coding in the first weeks of life. How do changes in the vocal apparatus accompanying infant growth translate into the production of nonlinear acoustic phenomena? We know that vocal signatures drift predictively with age [4], but are nonlinear phenomena reliable indicators of pain in babies of all ages? While this may be the case, longitudinal investigations on this topic are still lacking. A final objective relates more specifically to listeners. Although we have considered various nonlinear phenomena in the present study, we have only tested a small number of experimental signals. Among other possibilities, it would be interesting, for example, to test how adult listeners react to different combinations of nonlinear phenomena. It is indeed possible that some of these phenomena have a potentiating effect on others (e.g. the presence of vibrato, which is not very effective on its own, perhaps reinforces the perception of pain in a cry with chaos). In this respect, modern methods for analysing and synthesizing sound signals, such as those used in the present study, will be indispensable tools for exploring the coding of pain information in babies' cries, as well as for exploring the decoding of this information by caregivers.

## Conclusions

Here, we tested whether nonlinear phenomena encode pain in babies cries, and found that baby cries expressing acute pain are characterized by a pronounced presence of nonlinear phenomena, and that these nonlinear phenomena drive pain evaluation by adult listeners. While adult listeners rated all cries presenting any of nonlinear phenomena as expressing more pain, they were particularly sensitive to the presence of chaos. Our results thus show that nonlinear phenomena, especially chaos, encode pain information in baby cries and may be critically helpful for the development of vocal-based tools for monitoring babies' needs in the context of paediatric care.

## Acknowledgments

We thank Prof. David Reby for his valuable advice and for reading early drafts of the manuscript.
374

375 Fundings

376 Funding was provided by the ANR (BABYCRY project; SCREAM project), the AXA Foundation (NM), the Labex
377 CeLyA (France), the University of Saint-Etienne (France), the Centre National de la Recherche Scientifique (CNRS,
378 France), the Institut National de la Santé et de la Recherche Médicale (Inserm, France), and the Institut
379 universitaire de France (NM). SC was a recipient of a PhD grant from the Fondation pour la Recherche Médicale
380 (FRM; code ECO202006011648). CF was a recipient of a research grant from the FRM (ARF202110014013).

381

382

383 References

384 1. Lingle S, Wyman MT, Kotrba R, Teichroeb LJ, Romanow CA. 2012 What makes a cry a cry? A review of infant distress vocalizations. *Curr Zool* **58**, 698–726. (doi:10.1093/czoolo/58.5.698)

386 2. Gustafsson E, Levréro F, Reby D, Mathevon N. 2013 Fathers are just as good as mothers at recognizing the cries of their baby. *Nat Commun* **4**, 1698. (doi:10.1038/ncomms2713)

388 3. Bouchet H, Plat A, Levréro F, Reby D, Patural H, Mathevon N. 2020 Baby cry recognition is independent of motherhood but improved by experience and exposure. *Proc R Soc Lond B Biol Sci* **287**, 20192499. (doi:10.1098/rspb.2019.2499)

391 4. Lockhart-Bouron M *et al*. 2023 Infant cries convey both stable and dynamic information about age and identity. *Commun Psychol* **1**, 26. (doi:10.1038/s44271-023-00022-z)

393 5. Koutseff A, Reby D, Martin O, Levrero F, Patural H, Mathevon N. 2017 The acoustic space of pain: cries as indicators of distress recovering dynamics in pre-verbal infants. *Bioacoustics* **27**, 313–325. (doi:10.1080/09524622.2017.1344931)

396 6. Corvin S, Fauchon C, Peyron R, Reby D, Mathevon N. 2022 Adults learn to identify pain in babies’ cries. *Curr Biol* **32**, R824–R825. (doi:10.1016/j.cub.2022.06.076)

398 7. Levrero F, Mathevon N, Pisanski K, Gustafsson E, Reby D. 2018 The pitch of babies’ cries predicts their voice pitch at age 5. *Biol Lett* **14**, 20180065. (doi:10.1098/rsbl.2018.0065)

400 8. Wermke K, Robb MP. 2010 Fundamental Frequency of Neonatal Crying: Does Body Size Matter? *J Voice* **24**, 388–394. (doi:10.1016/j.jvoice.2008.11.002)

402 9. Corvin S, Fauchon C, Patural H, Peyron R, Reby D, Theunissen FE, Mathevon N. 2024 Pain cues override identity cues in baby cries. *iScience* (doi:10.1016/j.isci.2024.110375)

404 10. Wilden I, Herzel H, Peters G, Tembrock G. 1998 Subharmonics, biphonation, and deterministic chaos in mammal vocalization. *Bioacoustics* **9**, 171–196. (doi:10.1080/09524622.1998.9753394)

406 11. Fitch WT, Neubauer J, Herzel H. 2002 Calls out of chaos: the adaptive significance of nonlinear phenomena in mammalian vocal production. *Anim Behav* **63**, 407–418. (doi:10.1006/anbe.2001.1912)

408 12. Briefer EF. 2012 Vocal expression of emotions in mammals: mechanisms of production and evidence. *J Zool* **288**, 1–20. (doi:10.1111/j.1469-7998.2012.00920.x)

410 13. Briefer EF. 2020 Coding for ‘dynamic’ information: vocal expression of emotional arousal and valence in non-human animals. In *Coding strategies in vertebrate acoustic communication*, pp. 137–162. Springer International Publishing.

413 14. Stoeger AS, Charlton BD, Kratochvil H, Fitch WT. 2011 Vocal cues indicate level of arousal in infant African elephant roars. *J Acoust Soc Am* **130**, 1700–1710. (doi:10.1121/1.3605538)

415 15. Stoeger AS, Baotic A, Li D, Charlton BD. 2012 Acoustic Features Indicate Arousal in Infant Giant Panda Vocalisations. *Ethology* **118**, 896–905. (doi:10.1111/j.1439-0310.2012.02080.x)
417
418
419

16. Scheumann M, Roser A-E, Konerding W, Bleich E, Hedrich H-J, Zimmermann E. 2012 Vocal correlates of sender-identity and arousal in the isolation calls of domestic kitten (Felis silvestris catus). *Front Zool* 9, 1–14. (doi:10.1186/1742-9994-9-36)

420
421

17. Mende W, Herzel H, Wermke K. 1990 Bifurcations and chaos in newborn infant cries. *Phys Lett A* 145, 418–424. (doi:10.1016/0375-9601(90)90305-8)

422
423

18. Riede T, Wilden I, Tembrock G. 1997 Subharmonics, biphonations, and frequency jumps—common components of mammal vocalization or indicators for disorders? *Mamm Biol* 62, 198–203.

424

19. Steinecke I, Herzel H. 1995 Bifurcations in an asymmetric vocal-fold model. *J Acoust Soc Am* 97, 1874–1884.

425
426

20. Herbst CT. 2020 Registers—the snake pit of voice pedagogy Part 1: Proprioception, perception, and laryngeal mechanisms. *J Sing* 77, 175–190.

427
428

21. Anikin A, Canessa-Pollard V, Pisanski, Katarzyna, Massenet M, Reby D. 2023 Beyond speech: Exploring diversity in the human voice. *iScience* 26. (doi:10.1016/j.isci.2023.108204)

429
430

22. Berry DA, Herzel H, Titze IR, Story BH. 1996 Bifurcations in excised larynx experiments. *Journal of Voice* 10, 129–138. (doi:10.1016/S0892-1997(96)80039-7)

431
432

23. Horii Y. 1989 Frequency Modulation Characteristics of Sustained /a/ Sung in Vocal Vibrato. *J Speech Lang Hear Res* 32, 829–836. (doi:10.1044/jshr.3204.829)

433
434

24. Horii Y. 1989 Acoustic analysis of vocal vibrato: a theoretical interpretation of data. *J Voice* 3, 36–43. (doi:10.1016/S0892-1997(89)80120-1)

435
436

25. Anikin A, Pisanski K, Massenet M, Reby D. 2021 Harsh is large: nonlinear vocal phenomena lower voice pitch and exaggerate body size. *Proc R Soc Lond B Biol Sci* 288, 20210872. (doi:10.1098/rspb.2021.0872)

437
438

26. Bergan CC, Titze IR. 2001 Perception of Pitch and Roughness in Vocal Signals with Subharmonics. *J Voice* 15, 165–175. (doi:10.1016/S0892-1997(01)00018-2)

439
440

27. Linhart P, Ratcliffe VF, Reby D, Špinka M. 2015 Expression of Emotional Arousal in Two Different Piglet Call Types. *PLoS One* 10, e0135414. (doi:10.1371/journal.pone.0135414)

441
442

28. Blumstein DT, Winternitz J, Daniel JC. 2008 The structure, meaning and function of yellow-bellied marmot pup screams. *Anim Behav* 76, 1055–1064. (doi:10.1016/j.anbehav.2008.06.002)

443
444

29. Fauchon C, Corvin S, Faillenot I, Patural H, Reby D, Peyron R, Mathevon N. submitted Parenting and personality drive brain response to pain cues in baby cries.

445
446

30. Anikin A. 2019 Soundgen: An open-source tool for synthesizing nonverbal vocalizations. *Behav Res Methods* 51, 778–792. (doi:10.3758/s13428-018-1095-7)

447
448

31. Anikin A. 2020 The perceptual effects of manipulating nonlinear phenomena in synthetic nonverbal vocalizations. *Bioacoustics* 29, 226–247. (doi:10.1080/09524622.2019.1581839)

449
450
451

32. Massenet M, Anikin A, Pisanski K, Reynaud K, Mathevon N, Reby D. 2022 Nonlinear vocal phenomena affect human perceptions of distress, size and dominance in puppy whines. *Proc R Soc B* 289, 20220429. (doi:10.1098/rspb.2022.0429)

452

33. Boersma P, Weenink D. 2022 *Praat, doing phonetics by computer* [Computer program]. See https://www.praat.org.

453
454

34. LaGasse LL, Neal AR, Lester BM. 2005 Assessment of infant cry: Acoustic cry analysis and parental perception. *Ment Retard Dev Disabil Res Rev* 11, 83–93. (doi:10.1002/mrdd.20050)

455
456

35. Finger H, Goeke C, Diekamp D, Standvoss K, König P. 2017 LabVanced: a unified javascript framework for online studies. pp. 1–3. Cologne, Germany.

457
458

36. Gélébart J, Garcia-Larrea L, Frot M. 2023 Amygdala and anterior insula control the passage from nociception to pain. *Cereb Cortex* 33, 3538–3547. (doi:10.1093/cercor/bhac290)
459
460

37. Perchet C, Hagiwara K, Salameh C, Garcia-Larrea L. 2023 Cold-evoked potentials in clinical practice: A head-to-head contrast with laser-evoked responses. *Eur J Pain* **27**, 1006–1022. (doi:10.1002/ejp.2142)

461
462

38. Bürkner P-C. 2017 **brms**: An R Package for Bayesian Multilevel Models Using Stan. *J Stat Softw* **80**, 1–28. (doi:10.18637/jss.v080.i01)

463

39. Kruschke JK. 2014 *Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan*. Academic Press.

464
465

40. Kruschke JK, Liddell TM. 2018 The Bayesian New Statistics: Hypothesis testing, estimation, meta-analysis, and power analysis from a Bayesian perspective. *Psychon Bull Rev* **25**, 178–206. (doi:10.3758/s13423-016-1221-4)

466
467

41. Ospina R, Ferrari SL. 2012 A general class of zero-or-one inflated beta regression models. *Comput Stat Data Anal* **56**, 1609–1623. (doi:10.1016/j.csda.2011.10.005)

468
469

42. Porter FL, Porges SW, Marshall RE. 1988 Newborn Pain Cries and Vagal Tone: Parallel Changes in Response to Circumcision. *Child Dev* **59**, 495–505. (doi:10.2307/1130327)

470
471

43. Tokuda IT. 2018 Non-linear dynamics in mammalian voice production. *Anthropol Sci* **126**, 35–41. (doi:10.1537/ase.171130)

472
473

44. Gustafson GE, Green JA. 1989 On the Importance of Fundamental Frequency and Other Acoustic Features in Cry Perception and Infant Development. *Child Dev* **60**, 772. (doi:10.2307/1131017)

474
475

45. Ipp M, Parkin PC, Lear N, Goldbach M, Taddio A. 2009 Order of vaccine injection and infant pain response. *Arch Pediatr Adolesc Med* **163**, 469–472. (doi:10.1001/archpediatrics.2009.35)

476
477

46. Anikin A, Pisanski K, Reby D. 2020 Do nonlinear vocal phenomena signal negative valence or high emotion intensity? *R Soc Open Sci* **7**, 201306. (doi:10.1098/rsos.201306)

478
479

47. Raine J, Pisanski K, Simner J, Reby D. 2019 Vocal communication of simulated pain. *Bioacoustics* **28**, 404–426. (doi:10.1080/09524622.2018.1463295)

480
481

48. Blumstein DT, Récapet C. 2009 The sound of arousal: The addition of novel non-linearities increases responsiveness in marmot alarm calls. *Ethology* **115**, 1074–1081. (doi:10.1111/j.1439-0310.2009.01691.x)

482
483

49. Thévenet J, Papet L, Coureaud G, Boyer N, Levréro F, Grimault N, Mathevon N. 2023 Crocodile perception of distress in hominid baby cries. *Proc R Soc Lond B Biol Sci* **290**, 20230201. (doi:10.1098/rspb.2023.0201)

484
485

50. Campbell-Yeo M, Eriksson M, Benoit B. 2022 Assessment and Management of Pain in Preterm Infants: A Practice Update. *Children* **9**, 244. (doi:10.3390/children9020244)

486

487

488

Supplementary material

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The natural and synthetic cries, the R codes to create the synthetic cries, the acoustic and playback datasets and the R code for analysing data can be downloaded from https://osf.io/4yexz/.

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<table><thead><tr><th>Nonlinear phenomena presence</th><th>residual pain vs discomfort</th><th>acute pain vs discomfort</th><th>acute pain vs residual pain</th></tr></thead><tbody><tr><td>any NLP</td><td>11.3 [4.1, 18.6]</td><td>12.5 [4.3, 20.5]</td><td>1.1 [-8.2, 10.4]</td></tr><tr><td>amplitude modulation</td><td>3.3 [-5.5, 12.3]</td><td>18.9 [8.5, 28.8]</td><td>15.6 [2.8, 28.1]</td></tr><tr><td>chaos</td><td>17.4 [10.2, 25.3]</td><td>22.2 [13.8, 31.4]</td><td>4.8 [-5.8, 15.7]</td></tr><tr><td>frequency jump</td><td>4.8 [-0.8, 12.0]</td><td>13.1 [5.2, 22.4]</td><td>8.2 [-1.5, 18.6]</td></tr><tr><td>subharmonics</td><td>10.5 [2.2, 19.5]</td><td>13.1 [3.4, 23.3]</td><td>2.5 [-9.8, 14.7]</td></tr><tr><td>vibrato</td><td>6.0 [-2.1, 14.7]</td><td>11.1 [1.5, 21.2]</td><td>5.1 [-7.0, 17.3]</td></tr><tr><td>vocal fry</td><td>-1.9 [-5.3, 2.6]</td><td>3.3 [-1.7, 10.4]</td><td>5.1 [-1.1, 12.7]</td></tr></tbody></table>

Table 1. Pairwise comparisons of nonlinear acoustic phenomena presence across the three types of cries (i.e., discomfort, residual pain and acute pain cries). Results are summarised by the median of the posterior distributions and their 95% credible interval. NLP = nonlinear phenomena. Values in bold correspond to comparisons for which more than 97.5% of the posteriors are positive and indicate a credible difference in the presence of nonlinear phenomena between two types of cries.

499

<table><thead><tr><th>Comparisons with natural cries without nonlinear phenomena</th><th>Rating by all participants</th><th>Rating by parents</th><th>Ratings by non-parents</th><th>Difference between parents and non-parents</th></tr></thead><tbody><tr><td>Cries with any combination of NLP types</td><td>11.1 [9.4, 12.8]</td><td>12.9 [10.7, 15.2]</td><td>9.2 [6.9, 11.5]</td><td>3.7 [0.6, 6.7]</td></tr><tr><td>Cries with amplitude modulation</td><td>4.8 [3.3, 6.4]</td><td>6.0 [4.0, 8.1]</td><td>3.6 [1.6, 5.6]</td><td>2.4 [-0.2, 5.1]</td></tr><tr><td>Cries with chaos</td><td>9.6 [8.1, 11.0]</td><td>10.6 [8.6, 12.5]</td><td>8.6 [6.7, 10.5]</td><td>2.0 [-0.6, 4.5]</td></tr><tr><td>Cries with frequency jumps</td><td>3.7 [2.3, 5.3]</td><td>2.8 [0.9, 4.8]</td><td>4.6 [2.7, 6.6]</td><td>-1.8 [-4.3, 0.7]</td></tr><tr><td>Cries with subharmonics</td><td>3.0 [1.6, 4.3]</td><td>3.6 [1.8, 5.4]</td><td>2.3 [0.6, 4.1]</td><td>1.3 [-1.1, 3.7]</td></tr><tr><td>Cries with vibrato</td><td>0.6 [-0.8, 2.0]</td><td>2.0 [0.1, 3.9]</td><td>-0.7 [-2.5, 1.1]</td><td>2.7 [0.3, 5.1]</td></tr><tr><td>Cries with vocal fry</td><td>-0.7 [-2.3, 1.0]</td><td>-0.5 [-2.6, 1.7]</td><td>-0.8 [-2.9, 1.2]</td><td>0.4 [-2.3, 3.1]</td></tr></tbody></table>

Table 2. Comparison between pain levels estimated by adult listeners listening to natural cries without nonlinear phenomena and natural cries with nonlinear phenomena. Results are summarised by the median of the posterior distributions and their 95% credible interval. NLP = nonlinear phenomena. Values in italic correspond to comparisons for which 95% to 97.5% of the posteriors are positive and indicate a tendency of higher pain estimation in parents compared to non-parents. Values in bold correspond to comparisons for which more than 97.5% of the posteriors are positive and indicate a credible difference.

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507

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<table><caption>Table 3. Comparison between pain levels estimated by adult listeners listening to synthetic cries without nonlinear phenomena and synthetic cries with nonlinear phenomena. Results are summarised by the median of the posterior distributions and their 95% credible interval. Values in bold correspond to comparisons for which more than 97.5% of the posteriors are positive and indicate a credible difference.</caption><thead><tr><th>Comparison with synthetic cries without nonlinear phenomena</th><th>Rating by all participants</th><th>Rating by parents</th><th>Rating by non-parents</th><th>Difference between parents and non-parents</th></tr></thead><tbody><tr><td>Synthetic cries with amplitude modulation</td><td>5.8 [4.0, 7.5]</td><td>7.7 [5.3, 10.3]</td><td>3.7 [1.0, 6.5]</td><td>4.0 [0.7, 7.2]</td></tr><tr><td>Synthetic cries with chaos</td><td>10.9 [8.9, 12.9]</td><td>12.9 [10.1, 15.5]</td><td>9.0 [6.5, 11.5]</td><td>3.8 [0.5, 7.2]</td></tr><tr><td>Synthetic cries with frequency jump</td><td>2.6 [1.0, 4.3]</td><td>4.6 [2.4, 7.1]</td><td>0.5 [-1.6, 2.7]</td><td>4.1 [0.9, 7.4]</td></tr><tr><td>Synthetic cries with subharmonics</td><td>6.0 [3.9, 8.2]</td><td>8.3 [5.3, 11.4]</td><td>3.8 [1.6, 6.1]</td><td>4.6 [0.6, 8.6]</td></tr><tr><td>Synthetic cries with vibrato</td><td>5.6 [3.6, 7.7]</td><td>8.3 [5.4, 11.5]</td><td>2.9 [0.1, 5.7]</td><td>5.4 [1.4, 9.6]</td></tr></tbody></table>

Figure

<figure><img src="image_2.png" /><figcaption>Figure 1. Nonlinear phenomena in babies' cries. The nonlinear phenomena present in natural baby cries are of 6 different types: amplitude modulation (A), chaos (B), frequency jump (C), subharmonics (D), vibrato (E) and vocal fry (F). The moment of the cry characterized by a nonlinear phenomenon is marked in red on each spectrogram.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Nonlinear phenomena as markers of pain expressed by babies' cries. A. Proportion of cries presenting nonlinear phenomena as a function of the recording context (blue square: discomfort; orange circle: residual pain; red triangle: acute pain). All NLP: all nonlinear phenomena combined. B. Pain perception by adult listeners as a function of the type of nonlinear phenomenon present in natural cries. The graph illustrates the differences in pain perception between cries containing different types of nonlinear phenomena and cries without nonlinear phenomena. Listeners were either non-parents (blue circle) or parents (orange triangle). Solid markers represent medians of posteriors distributions, with bars indicating the 95% CIs. NLP = Nonlinear phenomena.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Nonlinear phenomena drive pain assessment by adult listeners. A. Top: Example of a natural discomfort cry without nonlinear phenomena. Bottom: Synthetic copy of this natural cry. The synthetic copies of natural cries are used in Experiment 2 as control stimuli (synthetic cries without nonlinear acoustic phenomena) and as prototypes for the synthesis of stimuli with nonlinear phenomena. B. Variants of a prototype synthetic cry to which different types of nonlinear phenomena have been added (from top to bottom: amplitude modulation, chaos, frequency jumps, subharmonics and vibrato). C. Results of Experiment 2 (listening to synthetic cries).</figcaption></figure>
539 Nonlinear phenomena modulate the perception of pain by non-parents (blue circle) and parents (orange

540 triangle). Overall, parents are more sensitive to nonlinear phenomena than non-parents. Solid markers represent

541 the medians of the posterior distributions, with bars indicating the 95% CIs. NLP = Nonlinear phenomena.

542