Application of the Kelvin-Voigt viscoelastic model to hippus reveals major insights into the autonomic nervous system activity

Abstract

This study characterized the autonomic nervous system impulses underlying the iris spontaneous cyclic activity, which is characterized by hippus occurrence, by taking into account also its viscoelastic properties using the Kelvin-Voigt model. Hippus recordings were carried out in young active individuals, in the supine and standing position to test the model sensitivity, simultaneously with cardiac activity. This was followed by the recording of a photomotor reflex sequence. A clustering analysis allowed the selection of relatively low-noise hippus, where the model showed a high degree of fit (mean error = 3.5 ± 1.1%). Despite the modest reproducibility of autonomic nervous system impulses, ranging from 0.35 ± 0.33 to 0.88 ± 30 in each participant, hippus showed strong similarities, suggesting the existence of an autonomic activity signature. Parasympathetic impulses were significantly (p = 3.4 × 10^− 3) higher in the supine than in the standing position. The amount of expended energy was 7.5 times lower (p = 1.7 × 10^− 7) during hippus than the photomotor reflex, indicating a less costly activity. The application of the model to hippus and the photomotor reflex provided information on the autonomic nervous system basal state and adaptive reserve, respectively. These analyses allow better understanding of the autonomic nervous system activity in both clinical and athletic contexts.

Introduction

Hippus is characterized by a decrease followed by an increase of the pupil diameter. This phenomenon is rhythmic, synchronous, bilateral, spontaneous, and irregular¹, occurring at an average frequency of 0.2 Hz with large variability in both frequency and amplitude^2,3,4. Hippus onset is mediated centrally via parasympathetic pathways and can be modulated by respiratory-related brainstem activity⁵. Studies on pupillary signal analysis have primarily considered the duration and amplitude of pupil diameter oscillations^6,7,8 to provide insights into autonomic nervous system activity. These studies do not consider the viscoelastic forces involved in the stretching and shortening spontaneous activities of the iridal muscles. In addition to being controlled by the autonomic nervous system, hippus is influenced by viscoelastic forces that partially originate from the striated type I and III collagen fibers in the iridal stroma⁹, which is anatomically connected to the iridal muscles¹⁰. These fibers play a crucial role in the iris repetitive and natural oscillatory activity^11,12. The physics of oscillatory motion makes it possible to quantify the respective contributions of intrinsic contractile forces, restoring forces, and viscous resistive forces. This type of modeling allows the characterization of the consequences of chronic diseases, such as alterations in iris tissue compliance in diabetes, or impairments in contractile activity regulated by the autonomic nervous system¹³. Viscoelastic models are crucial for the analysis of natural oscillatory movements within the organism, such as those of the iris, because they faithfully capture the dynamic behavior of these organs. Regarding the reflex activity of the iridal muscles induced by a light stimulus, it has, for its part, been analyzed using the Kelvin-Voigt viscoelastic model. This approach enables the characterization of parasympathetic and sympathetic nervous system impulses by taking into account the viscoelastic properties of the iridal muscles^13,14. This type of analysis could be applied to hippus, given its repetitive, complex pattern that does not appear random and can be studied, modelled, and understood (i.e. individual signature)¹⁵. Compared with the amplitude of pupil diameter variations during the photomotor reflex, hippus presents lower variation amplitudes, potentially leading to a lower signal-to-noise ratio¹⁶. This could pose a challenge for the analysis unless the hippus signals are sorted in function of their quality. To better characterize the autonomic nervous system activity, Bufo et al.¹⁷ highlighted the importance of analyzing cardiac and pupillary activity simultaneously. In a sports context, such as training¹⁸, or in a clinical context, such as burnout¹⁹ or chronic disease²⁰, assessing the autonomic nervous system activity allows the adaptation and guidance of the type and quantity of work. In this study, we hypothesized that (i) the hippus signal quality may influence its reproducibility and the constants obtained from the Kelvin-Voigt model, suggesting that noise in lower-quality hippus signals is not entirely randomly distributed and may introduce a significant bias in the results; (ii) the parasympathetic and sympathetic activities underlying hippus are sensitive to positional changes, providing valuable information on the autonomic nervous system basal state; and (iii) the autonomic activity that generates the hippus signal is lower and therefore less energy-consuming than that responsible for the photomotor reflex, which is by nature more constrained as it is a reflex.

Results

Selection of hippus kinetics

The Kelvin-Voigt model, which characterizes the parasympathetic and sympathetic impulses and also the viscoelastic forces, was applied to a total of 479 hippus kinetics in the supine position and 309 in the standing position. The least noisy kinetics were selected based on their fit quality with the Kelvin-Voigt model, evaluated by computing the mean error (ME) (Fig. 1a). The ME for the 378 selected kinetics in the supine position ranged from 1% to 18% (R² from 0.01 to 0.99), and the ME for the 208 selected kinetics in the standing position ranged from 1% to 12% (R² from 0.15 to 0.99). For these selected kinetics, three fit quality levels, based on the mean error values were identified: good, moderate, and poor (Fig. 1b). In the supine position, 200 kinetics were of good quality (R² = 0.91 ± 0.12, ME = 3.80 ± 1.19%), 98 of moderate quality (R² = 0.86 ± 0.17, ME = 7.66 ± 1.36%), and 80 of poor quality (R² = 0.86 ± 0.15, ME = 13.36 ± 1.93%). In the standing position, 98 kinetics were of good quality (R² = 0.89 ± 0.09, ME = 3.00 ± 0.79%), 72 of moderate quality (R² = 0.80 ± 0.18, ME = 6.03 ± 0.95%), and 38 of poor quality (R² = 0.70 ± 0.20, ME = 9.92 ± 1.15%). Our analysis detected a systematic bias for moderate- and poor-quality hippus compared with good-quality hippus kinetics. The Kelvin-Voigt model and data on the timing of the parasympathetic and sympathetic nervous system activation (see Methods) allowed calculating the parasympathetic impulse ($\:Imp1),\:$the sympathetic impulse ($\:Imp3$), and the parasympathetic-sympathetic co-activation ($\:Imp2$). The $\:Imp2$ of good-quality kinetics (− 0.17 ± 0.45 mN.s/g) was significantly different (p = 7.38 × 10^− 6 and p = 5.38 × 10^− 5) compared with the $\:Imp2$ of moderate-quality kinetics (0.01 ± 0.59 mN.s/g) and with the $\:Imp2$ of poor-quality kinetics (0.02 ± 0.66 mN.s/g). The $\:Imp3$ of good-quality kinetics (0.28 ± 0.27 mN.s/g) also was significantly different (p = 3.19 × 10^− 3 and p = 7.29 × 10^− 5) compared with the $\:Imp3$ of moderate-quality kinetics (0.39 ± 0.33 mN.s/g) and with the $\:Imp3$ of poor-quality kinetics (0.43 ± 0.34 mN.s/g). We did not detect any significant difference between the $\:Imp2$ values of moderate- and poor-quality kinetics (p = 0.99) and between the $\:Imp3\:$values of moderate- and poor-quality kinetics (p = 0.25). Regarding the first impulse ($\:Imp1$), we did not find any significant difference (p = 0.30) between the good-quality (0.41 ± 0.42 mN.s/g), moderate-quality (0.43 ± 0.55 mN.s/g), and poor-quality (0.48 ± 0.59 mN.s/g) groups. Consequently, we decided to consider only the kinetics from the good-quality category where noise would not introduce bias in the estimation of the autonomic nervous system impulses, despite the significant loss of hippus kinetics and therefore, statistical power.

Fig. 1

Representation of the two-step clustering procedure for hippus selection and the average kinetics of the three hippus types as a function of their duration. (a) First clustering step performed on the full dataset (n total = 788 hippus). Each point represents a single hippus. The rectangle highlights the hippus selected during this first clustering step using the k-means algorithm based on the quality of their fit with the Kelvin-Voigt model. (b) Second clustering step applied to the hippus retained after the first selection (n total = 586). Each point represents one hippus. Green triangles () indicate the subset of good-quality hippus selected for subsequent analyses (n total = 298), blue crosses () represent moderate-quality hippus (blue crosses ), and red diamonds () represent poor-quality hippus. (c) Representative short-duration hippus kinetics in the supine position (n = 99, filled orange circles ) and in the standing position (n = 36, empty green circles ). (d) Representative intermediate-duration hippus kinetics in the supine position (n = 73, filled orange circles ) and in the standing position (n = 40, empty green circles ). (e) Representative long-duration hippus kinetics in the supine position (n = 28, filled orange circles ) and in the standing position (n = 22, empty green circles ). The amplitude variations in pupil radius were normalized using a min-max function. Note that the constriction phase duration was shorter in the supine position than in the standing position, regardless of the hippus duration. To avoid overloading the graph, panels (a) and (b) only present the fit selection for the supine position because the procedure was the same for the standing position.

Identification of three hippus duration categories

The clustering analysis of good-quality hippus kinetics identified three hippus types based on their cycle duration (short, intermediate, or long) (Fig. 1c-e). In the supine position, 99 hippus kinetics had a short duration (1.21 ± 0.28 s), 73 had an intermediate duration (1.80 ± 0.19 s), and 28 had a long duration (2.74 ± 0.40 s). In the standing position, 36 hippus kinetics had a short duration (1.37 ± 0.20 s), 40 an intermediate duration (1.80 ± 0.19 s), and 22 a long duration (2.55 ± 0.33 s). All these kinetic types are individually represented in Fig. 2.

Fig. 2

Superposition of individual kinetics. The top panels show hippus kinetics in the supine position with (a) short-duration (1.21 ± 0.28 s), (b) intermediate-duration (1.80 ± 0.19 s), and (c) long-duration (2.74 ± 0.40 s). The bottom panels show hippus kinetics in the standing position with (d) short-duration (1.37 ± 0.20 s), (e) intermediate-duration (1.80 ± 0.19 s), and (f) long-duration (2.55 ± 0.33 s). All hippus kinetics were aligned relative to the start of the constriction phase and were normalized in terms of amplitude using the min-max function. Note the hippus good reproducibility, highlighting an individual signature. To avoid overloading the figure, participants with fewer than 3 usable hippus were not represented.

Model fit quality across hippus duration categories and positions

In the supine position, the ME values of short-, intermediate-, and long-duration hippus kinetics were 3.6 ± 1.3% (R² = 0.87 ± 0.15), 3.8 ± 1.1% (R² = 0.95 ± 0.06), and 4.5 ± 0.89% (R² = 0.95 ± 0.04), respectively. In the standing position, the ME values of short-, intermediate-, and long-duration hippus kinetics were 2.7 ± 0.79% (R² = 0.9 ± 0.07), 3.0 ± 0.73% (R² = 0.89 ± 0.09), and 3.6 ± 0.53% (R² = 0.89 ± 0.11), respectively (Fig. 3).

Fig. 3

Typical representation of a hippus kinetic in each duration category from the same participant. The raw hippus kinetics in the supine position with (a) short-, (b) intermediate-, and (c) long-durations are shown in orange, and the modeled kinetics are superimposed in brick red dashed lines. (d-f) The same analysis in the standing position, with raw kinetics shown in green. Note the good fit quality, with r² values ranging from 0.88 (c) to 0.99 (b) for this participant, despite the measurement noise inherent to the small amplitudes of spontaneous pupil variations present in more than half of the hippus. Moreover, the duration of the constriction phase was systematically shorter in the supine than in the standing position.  Blue rectangle = parasympathetic impulse ($\:Imp1$),  orange rectangle = parasympathetic-sympathetic co-activation impulse ($\:Imp2$),  green rectangle = sympathetic impulse ($\:Imp3$).

Individual signatures of autonomic nervous system activity revealed by hippus and their intra-individual variability

In each participant, the hippus kinetics displayed a relatively reproducible characteristic shape, forming, to some extent, an individual signature (Fig. 2). This signature nevertheless showed variability in the sympathetic and parasympathetic impulse and stiffness constant values. Specifically, in the short-duration hippus kinetics, the coefficients of variation (CVs) of $\:Imp1$, $\:Imp2\:$and $\:Imp3$ in the supine and standing positions were 0.60 ± 0.28 and 0.54 ± 0.21, − 7.13 ± 17.66 and − 10.09 ± 31.37, and 0.88 ± 0.30 and 0.44 ± 0.28, respectively. The CVs of $\:{k}_{d1}$ and $\:{k}_{d2}$ in the supine and standing position were 0.27 ± 0.11 and 0.28 ± 0.13, and 0.10 ± 0.05 and 0.07 ± 0.05, respectively. In the intermediate-duration hippus kinetics, the CVs of $\:Imp1,$ $\:Imp2\:$and $\:Imp3$ in the supine and standing position were 0.54 ± 0.38 and 0.69 ± 0.37, − 3.11 ± 9.64 and − 1.00 ± 1.23, and 0.75 ± 0.29 and 0.79 ± 0.39, respectively. The CVs of $\:{k}_{d1}$ and $\:{k}_{d2}$ in the supine and standing position were 0.23 ± 0.11 and 0.18 ± 0.11, and 0.10 ± 0.05 and 0.07 ± 0.06 respectively. In the long-duration hippus kinetics, the CVs of $\:Imp1$, $\:Imp2\:$and $\:Imp3$ in the supine and standing positions were 0.68 ± 0.43 and 0.61 ± 0.48, 7.10 ± 19.87 and 0.13 ± 0.66, and 0.60 ± 0.33 and 0.35 ± 0.33, respectively. The CVs of $\:{k}_{d1}$ and $\:{k}_{d2}$ in the supine and standing position were 0.38 ± 0.25 and 0.17 ± 0.12, and 0.17 ± 0.07 and 0.07 ± 0.05, respectively.

Sensitivity of the Kelvin-Voigt model parameters to position changes

The application of the Kelvin-Voigt model allowed characterizing the parasympathetic and sympathetic impulses and the viscoelastic forces. The sum of all hippus impulses provided an energy of 0.52 ± 0.46 mN.s/g. This value was 0.41 ± 0.42 mN.s/g for $\:Imp1$, − 0.17 ± 0.45 mN.s/g for $\:Imp2$, and 0.28 ± 0.27 mN.s/g for $\:Imp3$. The subject-level mean values for $\:Imp1$, $\:Imp2$, and $\:Imp3$, derived from the Kelvin-Voigt model, were significantly higher ($\:Imp1$: p = 3.4 × 10^− 3, 95% CI [− 0.44, − 0.11], Cohen’s dz = − 0.96; $\:Imp2$: p = 1.2 × 10^− 2, 95% CI [0.05, 0.34], Cohen’s dz = 0.78; $\:Imp3$: p = 3.6 × 10^− 3, 95% CI [− 0.21, − 0.05], Cohen’s dz = − 0.95) in the supine than in the standing position ($\:Imp1$ = 0.51 ± 0.44 mN.s/g vs. 0.21 ± 0.26 mN.s/g; $\:Imp2$ = −0.25 ± 0.49 mN.s/g vs. −0.02 ± 0.27 mN.s/g; $\:Imp3$ = 0.33 ± 0.30 mN.s/g vs. 0.19 ± 0.16 mN.s/g). The stiffness constant $\:{k}_{d1}$ and $\:{k}_{d2}$ did not vary ($\:{k}_{d1}$: p = 0.077, 95% CI [− 0.01, 0.16], Cohen’s dz = 0.51; $\:{k}_{d2}$: p = 0.083, 95% CI [−0.01, 0.12], Cohen’s dz = 0.50) between supine and standing position ($\:{k}_{d1}$: 1.15 ± 0.33 mN/g.mm vs. 1.20 ± 0.09 mN/g.mm; $\:{k}_{d2}$ 1.66 ± 0.09 mN/g.mm² vs. 1.72 ± 0.07 mN/g.mm²).

Sensitivity of the Kelvin-Voigt model applied to the photomotor reflex

The sum of the impulses from all photomotor reflexes provided an energy of 3.89 ± 1.37 mN.s/g, which was significantly (p = 1.7 × 10^− 7, 95% CI [2.84, 4.39], Cohen’s dz = 2.69) higher than the energy expended during hippus. The subject-level mean values for $\:Imp1$ (3.18 ± 1.05 mN.s/g) and $\:Imp3\:$(0.83 ± 0.33 mN.s/g) were significantly higher ($\:Imp1$: p = 6.0 × 10^− 8, 95% CI [2.48, 3.70], Cohen’s dz = 2.93; $\:Imp3$: p = 8.04 × 10^− 38, 95% CI [0.23, 0.62], Cohen’s dz = 1.24) than what observed for hippus. The subject-level mean values of $\:Imp1$, $\:Imp2$ and $\:Imp3$, $\:{k}_{d1}$, $\:{k}_{d2}$ for the photomotor reflexes, obtained with the Kelvin-Voigt model, did not differ significantly between the supine and standing positions ($\:Imp1$: 3.19 ± 1.72 mN.s/g vs. 3.45 ± 1.17 mN.s/g, $\:Imp2$: −0.22 ± 0.39 mN.s/g vs. −0.04 ± 0.22 mN.s/g, $\:Imp3$: 0.82 ± 0.32 mN.s/g vs. 0.88 ± 0.24 mN.s/h, $\:{k}_{d1}$: 1.19 ± 0.34 mN/g.mm vs. 1.00 ± 0.33 mN/g.mm, $\:{k}_{d2}$: 1.93 ± 0.25 mN/g.mm² vs. 1.81 ± 0.28 mN/g.mm²). Their confidence intervals encompassed zero, and effect sizes were small to moderate. Figure 4 illustrates the comparisons between the hippus and photomotor reflex kinetics.

Fig. 4

Comparison between the hippus and the photomotor reflex. (a) Superposition of all normalized photomotor reflex kinetics, using the min-max function, in the supine position (the mean kinetic is in bold). The light impulse induces a very homogeneous and highly reproducible response, particularly during the constriction phase, while the redilation phase, partially under sympathetic control, shows greater variability. (b) Representation of the normalized and averaged kinetics of photomotor reflexes in the supine position (empty pink squares ) and standing position (filled purple squares ) and of all hippus types in the supine position (filled orange circles ) and standing position (empty green circles ). Note that the hippus kinetics showed smaller amplitude variations than expected, between 0 and 1, because this represents the average of normalized hippus. The hippus kinetics displayed greater variability because they are not constrained by an external stimulus, unlike the photomotor reflex. To avoid overloading the graph, panel (a) presents only photomotor reflexes in the supine position.

Distinct correlation patterns in hippus and photomotor reflex parameters

The subject-level mean values of the hippus parameters showed significant correlations with each other, which were not necessarily found between the photomotor reflex parameters, and vice versa (Fig. 5). The initial pupil diameter was not correlated (r = 0.38, p = 0.15) with $\:Imp1$ of hippus, and strongly correlated (r = 0.95, p = 3.69 × 10^− 8) with $\:Imp1$ of the photomotor reflex. $\:Imp1$ was significantly (r = − 0.70, p = 2.72 × 10^− 3) and strongly negatively correlated with $\:Imp2$ in hippus, but not in the photomotor reflex (r = 0.06, p = 0.83). No significant correlation was observed between $\:Imp1$ and $\:Imp3$ in hippus (r = 0.46, p = 0.07) or in the photomotor reflex (r = 0.45, p = 0.08). The autonomic nervous system impulses from hippus showed no correlation with the autonomic nervous system impulses from the photomotor reflex.

Fig. 5

Correlation analysis between parameters obtained from hippus and photomotor reflex. Each data point represents a single hippus or photomotor reflex event; therefore, multiple points may originate from the same participant. The correlation coefficients and p-values shown in the panels are computed across events. For statistical inference, parameter values were averaged for each participant, and complementary correlation analyses were performed using one observation per participant (n = 16), ensuring statistical independence. The corresponding correlation coefficients and p-values are reported in the figure legend. (a, d) Correlation analysis between the initial pupil diameter and Impulse 1 in (a) hippus (r = 0.38, p = 1.47e − 01) and (d) photomotor reflex (r = 0.95, p = 3.69e − 08). (b, e) Correlation analysis between Impulse 2 and Impulse 1 in (b) hippus (r = − 0.70, p = 2.72e − 03) and (e) photomotor reflex (r = 0.06, p = 8.27e − 01). (c, f) Correlation analysis between Impulse 3 and Impulse 1 in (c) hippus (r = 0.46, p = 7e − 02) and (f) photomotor reflex (r = 0.45, p = 8.08e − 02). These analyses included good-quality hippus and photomotor reflex in both the supine and standing positions. Note the strong correlation between the parasympathetic impulse and the initial diameter in the photomotor reflex (d) consistent with retinal protection and visual adaptation, but not in hippus (a) because retinal structures are not under any specific stress. Also note the close correlation between the parasympathetic impulse (impulse 1) and the co-activation impulse (impulse 2), indicating that the activation of the co-activation impulse is proportional to the parasympathetic impulse in hippus (b). Conversely, the photomotor reflex is characterized by an extremely powerful parasympathetic impulse to protect the retina from light stress, and the co-activation impulse seems insufficient to determine the end of constriction (e). The solid line in each panel represents the best-fit line. The Spearman correlation coefficient (ρ) and p-value (p) are provided to show the significance of each correlation. Impulse 1 = parasympathetic impulse, Impulse 2 = co-activation impulse, Impulse 3 = sympathetic impulse.

Biological significance of the hippus, photomotor reflex, and heart rate variability parameters

The principal component analysis of the hippus parameters ($\:Imp1$, $\:Imp2$ and $\:Imp3)\:$(Fig. 6a) revealed that the first two components collectively explained 68.1% of the total variance. The loadings of the first principal component showed a strong positive association with $\:Imp2$ in the supine position (0.67) and in the standing position (0.78) and with $\:Imp3$ in the supine position (0.76), and in the standing position (0.64). These results indicated that the sympathetic activity during co-activation ($\:Imp2$) and to some extent, the pure sympathetic activity ($\:Imp3$) were associated with the first principal component. The second principal component was associated mainly with the parasympathetic activity, as indicated by the high negative association with $\:Imp1$ in the supine position (− 0.68) and strong positive association with $\:Imp1$ in the standing position (0.73). The principal component analysis of the photomotor reflex parameters ($\:Imp1$, $\:Imp2$ and $\:Imp3)\:$(Fig. 6b) revealed that the first two principal components together explained 70.8% of the total variance. The eigenvectors for each impulse were similar in the supine and standing position. The loadings of impulses in the supine and standing positions on the first principal component were 0.57 and 0.52 for $\:Imp1$, 0.47 and 0.65 for $\:Imp2$, and 0.84 and 0.88 for $\:Imp3$. The loadings in the supine and standing positions on the second principal component were − 0.54 and − 0.55 for $\:Imp1$, 0.77 and 0.56 for $\:Imp2$, and − 0.15 and − 6 × 10^− 4 for $\:Imp3$. These principal component analysis results for hippus and the photomotor reflex showed that hippus impulses in the supine position were inversely associated with impulses in the standing position on the second principal component, a pattern not observed for the photomotor reflex, indicating a greater sensitivity of hippus to positional changes. The principal component analysis results of the heart rate variability parameters (LF, HF, LF + HF, MeanRR) (Fig. 6c) revealed that the first two principal components together explained 75.8% of the total variance. The eigenvectors of all variables in the supine position occupied a very distinct and narrow sector compared with those in the standing position. In the first principal component, the loadings between the supine and standing positions were very similar for all variables: LF (0.78 and 0.79, respectively), HF (0.83 and 0.79, respectively), LF + HF (0.83 and 0.80, respectively), and MeanRR (0.42 and 0.46, respectively). The parameters in the supine position showed a similar but opposite association compared with those in the standing position on the second principal component: LF (0.53 and − 0.58, respectively), HF (0.50 and − 0.50, respectively), LF + HF (0.53 and − 0.58, respectively), and MeanRR (0.25 and − 0.14, respectively). The autonomic nervous system impulses from hippus showed no correlation with the heart rate variability parameters. In the supine position, the analyses did not show any significant correlation between the hippus autonomic nervous system impulses and the heart rate variability parameters. $\:Imp1$ was not significantly correlated with MeanRR (ρ = −0.09, p = 0.738), LF (ρ = −0.38, p = 0.143), HF (ρ = −0.42, p = 0.109), and LF + HF (ρ = −0.45, p = 0.083). $\:Imp2$ was not significantly correlated with MeanRR (ρ = −0.14, p = 0.594), LF (ρ = 0.24, p = 0.373), HF (ρ = 0.29, p = 0.278), and LF + HF (ρ = 0.28, p = 0.299). $\:Imp3$ did not show significant correlations with MeanRR (ρ = −0.12, p = 0.664), LF (ρ = −0.02, p = 0.945), HF (ρ = −0.07, p = 0.805), and LF + HF (ρ = −0.03, p = 0.900). In the standing position, no significant correlation was observed between the hippus autonomic nervous system impulses and the heart rate variability parameters. $\:Imp1$ was not significantly correlated with MeanRR (ρ = −0.06, p = 0.822), LF (ρ = 0.09, p = 0.730), HF (ρ = 0.05, p = 0.865), and LF + HF (ρ = 0.11, p = 0.697). $\:Imp2$ was not significantly correlated with MeanRR (ρ = −0.24, p = 0.367), LF (ρ = −0.20, p = 0.456), HF (ρ = −0.27, p = 0.315), and LF + HF (ρ = −0.21, p = 0.443). $\:Imp3$ did not show significant correlations with MeanRR (ρ = 0.16, p = 0.541), LF (ρ = 0.16, p = 0.541), HF (ρ = 0.13, p = 0.641), and LF + HF (ρ = 0.19, p = 0.484).

Fig. 6

Results of the principal component analysis of autonomic nervous system parameters obtained from hippus kinetics, photomotor reflexes, and heart rate variability. (a) The eigenvectors of the autonomic nervous system impulses from hippus kinetics were clearly distinct from each other in a given position and also distinct between the supine and standing positions. (b) Conversely, the eigenvectors of the autonomic nervous system impulses of photomotor reflexes were not sensitive to positional changes. (c) For a given position, the eigenvectors of the descriptive parameters of the autonomic activity from the heart rate variability were similar to each other, but clearly distinct between the supine and standing positions. st = standing position, su = supine position, PR = photomotor reflex, Hip = hippus, $\:Imp1$ = impulse 1, $\:Imp2$ = impulse 2, $\:Imp3$ = impulse 3, LF = low frequencies, HF = high frequencies, LFplusHF = low frequencies + high frequencies, Dim = dimension.

Discussion

The main observations of our study include (i) the existence of an identifiable individual signature within hippus kinetics that exhibit a good fit quality with the Kelvin-Voigt model; (ii) this signature is related to the autonomic nervous system state because it is sensitive to positional changes; and (iii) the characterization of autonomic nervous system impulses during hippus and during the photomotor reflex reveals that they provide different information.

The Kelvin-Voigt viscoelastic model (Eq. (1)) accurately depicted hippus in a population of young active individuals, as evidenced by the high coefficient of determination (> 0.90) and the low ME (< 4%) (Fig. 3). In our study, parasympathetic and sympathetic impulses were extracted from hippus (Eq. (12)), taking into account the viscoelastic properties of the iris. The application of the model to hippus revealed that only low-noise hippus should be selected. Hippus categories associated with a high signal-to-noise ratio exhibit a systematic bias in the estimation of autonomic nervous system impulses, as evidenced by the significant differences observed between the three signal-quality levels. Our recommendation for future research in this field is to consider only good-quality signals with a mean error < 6% between the value predicted by the viscoelastic model and the actual kinetics of the pupil radius during hippus. Our analysis focused on good-quality signals, representing 38% of all observed hippus (i.e. 298 kinetics). This approach made it possible to distinguish three categories of hippus duration that displayed invariant parameters and good reproducibility for each young active individual (Fig. 2). For each participant, we could identify a relatively constant profile among their hippus kinetics. This profile represents an individual signature corresponding to the viscoelastic properties of the iris and to the contractile impulses exerted by the iris muscles under autonomic nervous system control. This is in agreement with a previous study showing that in six healthy participants, the spontaneous pupillary activity over a 3-minute period exhibited complex repetitive patterns, and that these patterns were not merely random but followed deterministic chaotic rules¹⁵.

The initial constriction phase of hippus is primarily controlled by the parasympathetic activity, characterized by $\:Imp1$ (Eq. (10)). Quantitatively, the parasympathetic impulse values are 61% higher in the supine position than in the standing position. This is consistent with the literature, where parasympathetic activity, evaluated by the high frequencies band of spontaneous iridal muscle oscillations, is greater in the supine than in the standing position²¹. Qualitatively, the parasympathetic impulses of hippus collected in the supine position provide different information from those in the standing position (Fig. 6a), as do the heart rate variability parameters (Fig. 6c). However, for a given position (i.e. supine or standing), the eigenvectors of the heart rate variability parameters occupy a narrow sector, consistent with the study by Schmitt et al.²². The latter reported, on the one hand, strong correlations among the frequency-domain parameters of heart rate variability, suggesting redundancy between them, and, on the other hand, that the changes in these parameters induced by a change in position are related to mean heart rate (i.e. the inverse of the mean R-R interval duration). The availability of a sensitive method applicable to hippus for detecting state changes in the balance of the autonomic nervous system presents interesting prospects, both for healthy individuals such as athletes and for patients. This method could assist professionals in guiding their decisions regarding workload management. Furthermore, it has been shown that the analysis of pupil diameter fluctuations is a relevant tool for assessing pain in patients²³. Applying the Kelvin-Voigt model under these conditions could provide clinicians with a method to adjust their patients’ analgesic treatments. In contrast with hippus, the primarily functional role of photomotor reflex impulses, imposed by the strong constraint of the light stimulus, explains the narrow eigenvector domain associated with each impulse, indicating the absence of position-dependent adaptation (Fig. 6b).

Although hippus is noisier than the photomotor reflex due to the smaller amplitude of pupil diameter variations, the distinctiveness of hippus kinetics between supine and standing position contrasts with the similarity of constriction kinetics during the photomotor reflex in these two conditions (Figs. 2 and 4b). This is explained by the fact that the parasympathetic impulse of hippus is only weakly dependent on the initial pupil diameter (Fig. 5a), unlike in the photomotor reflex (Fig. 5d). This correlation between the parasympathetic impulse of the photomotor reflex and the initial diameter, regardless of the position, is related to the protective role of retinal structures²⁴ and the adjustment of visual quality ensured by post-stimulus constriction²⁵. As the autonomic activity extracted from hippus reflects spontaneous activity, it is consistent that there is no correlation with the impulses extracted from the photomotor reflex that primarily provide information on the adaptive reserve of the autonomic nervous system. Due to the functional role of the constrictor’s contractile impulse during the photomotor reflex (i.e. $\:Imp1$, parasympathetic impulse), it provides an amount of energy 7.8 times greater than that of hippus (3.18 mN.s/g versus 0.41 mN.s/g). Subsequently, the co-contraction impulse of both iris muscles (i.e. $\:Imp2$), which represents sympathetic activity superimposed on parasympathetic activity, allows slowing down the constriction phase to initiate the redilation phase. During this co-activation phase in the photomotor reflex, the contractile impulse of the dilator is not sufficient to slow down constriction and restore the initial diameter, as it is 26.5 times smaller than the contractile impulse generated by the constrictor (Fig. 5e). In contrast, during hippus, the contractile impulse of the dilator during co-activity is only 2.4 times smaller than the constrictor impulse (Fig. 5b). To achieve redilation, the contractile impulse of the dilator (i.e. $\:Imp3$, pure sympathetic activity) in the photomotor reflex plays a major role, as evidenced by a value 6.9 times greater than the co-activation impulse, whereas in hippus it is only 1.7 times greater. This major role of the dilator’s contractile impulse during the photomotor reflex is consistent with its association with the functional impulse of the constrictor across individual photomotor reflex events (Fig. 5f), an association that is not observed for hippus (Fig. 5c). Following the initial parasympathetic activity that reduces pupil diameter, the sympathetic activity during the photomotor reflex responds accordingly to redilate the pupil. This functional coupling between pupillary constriction and dilation during changes in luminance promotes optimal visual adaptation²⁶. Conversely, for hippus, the iridal muscles seem to function with minimal energy expenditure during the redilation phase. This economy in spontaneous activity of the iris is explained (i) by the well-proportioned co-activation impulse relative to the initial parasympathetic impulse (Fig. 5b) that allows a relatively smaller pure sympathetic impulse compared with the photomotor reflex, and (ii) by the elastic energy restitution made possible in part by collagen fibers^11,12.

The absence of strong correlations between autonomic nervous system impulses extracted from iris movements and heart rate variability parameters was expected^17,27. This result is likely explained by the fact that autonomic nervous system activity assessed using the Kelvin-Voigt model applied to iris movements accounts for the viscoelastic properties of the tissue, whereas time-, frequency-, and non-linear analyses of heart rate variability do not allow such consideration. In other words, heart rate variability measurements are typically interpreted as being solely influenced by autonomic nervous system activity, whereas viscoelastic forces are necessarily also involved, as in any real oscillatory system. One potential solution lies in applying the viscoelastic model to beat-to-beat variations in left ventricular volume in order to discriminate between active and passive forces governing systolic and diastolic phases. A second reason for the weak correlations observed between parameters derived from the Kelvin-Voigt model applied to the iris and those derived from heart rate variability is that the heart has its own nodal tissue and therefore, its own intrinsic rhythm that is modulated by the activity of both branches of the autonomic nervous system²⁸. In contrast, the iridal muscles are innervated by distinct branches of the autonomic nervous system, with parasympathetic control of the constrictor muscle and sympathetic control of the dilator muscle²⁹. Furthermore, the origin of hippus is not located within the eye, but is the result of parasympathetic impulses³⁰. Exploring the autonomic nervous system from multiple perspectives, through hippus, the photomotor reflex, and heart rate variability, is in line with the findings by Bufo et al.¹⁷. These authors reported distinct autonomic nervous system activity profiles as a function of age, based on pupillary, electrodermal, and cardiac tone. Because the movements of the iris and the cardiovascular system are not influenced in exactly the same way by baroreceptors (i.e. the carotid sinus and the aortic arch) and by respiratory centers, it appears important to simultaneously measure the oscillatory movements of these two biological systems in order to more accurately capture the state of autonomic nervous system activity. In athletes, this is crucial during critical phases of training, particularly those involving overload and tapering.

The present findings are based on a homogeneous sample of adolescent male athletes and should therefore not be directly generalized to other populations, including females, older individuals, or patients. Regarding the photomotor reflex, sex- and disease-related effects on autonomic function have previously been reported. Specifically, a reduced parasympathetic impulse in females compared with males³¹, as well as an increased sympathetic impulse in individuals with diabetes compared with healthy subjects¹³, have been described. The limitation of using multiple approaches lies in the time required for the analysis; however, with task automation, this becomes a fully feasible procedure. Another limitation is that hippus presents a signal-to-noise ratio that is not always favorable. For this reason, we proposed a method to select good-quality hippus. Although an individual signature of autonomic nervous system activity is present in young and active subjects, there is a notable cycle-to-cycle variability. Therefore, a large number of hippus recordings must be collected, and a minimum recording duration of 5 min is required to obtain sufficiently robust results from the model application. Lastly, hippus analysis provides information on the autonomic nervous system basal state at a specific moment in time, in our case, at rest in the morning. The quality of this information is dependent on the environment because the autonomic nervous system is sensitive to many external stimuli. In our study, we strived to control these factors using a windowless room with controlled artificial lighting in terms of intensity and direction, and away from any noise sources.

In conclusion, the influence of the pupillary signal quality on the autonomic nervous system impulses and restoring forces highlights the need to select only hippus with a favorable signal-to-noise ratio, and we propose a method for this. Although good-quality hippus exhibit natural variability, an individual signature that provides information on the activity of both branches of the autonomic nervous system in a basal state can be identified. Conversely, the more reproducible photomotor reflex reflects autonomic activity that responds reflexively to a constraint imposed on the eye, with the aim of protecting retinal structures. The Kelvin-Voigt model applied to hippus has the advantage of characterizing both autonomic nervous system impulses and viscoelastic forces in a basal state, while its application to the photomotor reflex allows the characterization of the autonomic nervous system adaptive capacities in response to stress. This partly explains why we did not observe any correlation between the autonomic activity obtained from hippus and from the photomotor reflex. The analysis of hippus, the photomotor reflex and heart rate variability demonstrates a complementarity in the information provided on the autonomic nervous system activity.

Methods

Ethical approval

Participants and their parents received detailed information about the study objectives, benefits, risks, and procedures. They read and provided their signed informed consent to participate in the study. All procedures were conducted in accordance with the Helsinki Declaration, and ethical approval for the study was granted by the French Ethics Committee for Research into the Sciences and Techniques of Physical and Sports Activities (IRB00012476-2024-02-04–305).

Protocol execution

Participants entered the room, and setting up the equipment took 5 min. Then, they lay down for one minute. After this initial minute of adaptation, simultaneous recordings of the spontaneous pupillary signal and heart rate variability were performed for 5 min. At the end of this period, the participants stood up and remained in the standing position for one minute. Afterwards, simultaneous recordings of the spontaneous pupillary signal and heart rate variability resumed for 5 min. Recording, first in the supine position and then in the standing position, was repeated a second time for the photomotor reflexes.

Participants

Sixteen male handball and basketball players (n = 9 and n = 7) participated in the study. Their characteristics were as follows (mean ± SD): age = 15.9 ± 0.9 years; weight = 72.8 ± 9.1 kg; height = 1.82 ± 0.07 m. These young active individuals were in their post-season phase during which they had a training regimen of approximately 10 ± 2 h per week in the past year. All participants had normal vision or vision corrected to normal by lenses. The tests (pupil diameter variations and heart rate variability) were conducted between 08:00 and 13:00 at sea level. Participants were advised to avoid high altitudes in the seven days before testing³², abstain from caffeine³³, and avoid intense physical and mental exercises³⁴ 24 h before testing.

Procedures

Hippus recordings to assess autonomic nervous system activity and iris viscoelastic properties

Each participant entered the experimental room to be prepared and adapt to the ambient light for 5 min before lying down. Participants wore glasses equipped with two cameras (Pupil core by Pupil Labs – Berlin, Germany) to record the pupil diameter at a sampling frequency of 120 Hz. This device was validated against a reference system³⁵ They were placed in supine position on the back and then in standing position, with the head turned towards the television screen. The room was soundproof, windowless, and maintained at a constant temperature of ~ 24 °C. The walls and floor were covered in black. The basal illumination perpendicular to the center of the turned-on black screen (brightness set to 0), at a distance of 55 cm from the participant’s forehead³⁶, was adjusted to a value of 10 lx³⁷, measured with a luxmeter (Lightmeter turbotech TT1308 – China). Lighting was provided by three halogen lamps equipped with a dimmer with a color temperature of about 3000 K. The participant’s gaze was fixed on the central area of the television screen (Samsung QB50R LED 50 inches - Vietnam). After a 1-minute adaptation phase in the supine position, the analysis started and continued for 5 min. Then, the participant stood up, and after another 1-minute adaptation, recordings were taken in standing position for 5 min. The total test duration was 12 min, and the black screen remained continuously active throughout the entire test. Participants were required to minimize eye movements to avoid variations in accommodation and angle of vergence.

Hippus clustering to select good-quality signals and identify individual autonomic signatures

The pupil diameter signal was recorded over time. From this raw signal, all characteristic hippus were selected. The time required for one hippus cycle, as well as the accuracy of the viscoelastic model predictions compared with the actual observations, indicated by the mean error, were calculated. The K-means clustering algorithm, an unsupervised automatic grouping method, was used with three centers to identify groups based on the mean error values of raw hippus compared with the model. This first clustering step was performed on the full dataset (n = 788 hippus). An example of this procedure for the supine condition is shown in Fig. 1a, the same procedure was applied in the standing position (data not shown for clarity). In a second clustering step, the same procedure was applied to the hippus retained after the first selection (n = 586 hippus), resulting in a final subset of 298 good-quality hippus (Fig. 1b). This two-step clustering approach allowed the determination of the kinetics that were relatively less noisy, compared with the model, and associated with a favorable signal-to-noise ratio.

Photomotor reflex recordings to assess autonomic adaptive reserve

After the 10-minute recording of spontaneous pupillary muscle activity, participants lay down again, and after a 1-minute adaptation phase in the supine position, the television screen generated ten 200 ms flashes, each spaced 30 s apart, for a total test duration of 5 min. Then, the same procedure was performed in the standing position. The screen was continuously active during the adaptation phase and between flashes. Participants were required to minimize eye movements to avoid variations in accommodation and angle of vergence and to refrain from blinking during and for few seconds after each light flash. The flashes, with an intensity of 30 lx³⁷, were red. The color temperature and gamma number BT 1886 were set to 7000 K and − 5, respectively. The flash conditions and their timing were regulated by a Python-written program.

Kelvin-Voigt model to extract autonomic nervous system impulses from hippus and photomotor reflex

The Kelvin-Voigt viscoelastic model, adapted to the photomotor reflex by Yan et al.¹³, was used to characterize the viscoelastic forces and those related to sympathetic and parasympathetic activities present in the hippus and the photomotor reflex. The balance of forces normalized relative to the mass of the iridal muscle-stroma complex during pupil size variations was defined by the Kelvin-Voigt model:

$$\:\frac{{{d^2}r}}{{d{t^2}}} = {k_{d2}}{\left( {{l_0} - r} \right)^2} + \:{k_{d1}}\left( {{l_0} - r} \right) - D\frac{{dr}}{{dt}} - {F_n}\left( t \right)$$

(1)

where $\:{l}_{0}$ is the pupil radius equal to the real radius at the signal start and $\:r$ the pupil radius in function of time. $\:{k}_{d1}$ is a stiffness constant (mN/g.mm) applied over a length $\:\left({l}_{0}-r\right)$ in mm. $\:{k}_{d2}$ corresponds to a measure of pressure, a force applied over a surface in mN/g.mm². The set $\:{k}_{d2}{\left({l}_{0}-r\right)}^{2}+\:{k}_{d1}\left({l}_{0}-r\right)$ expresse the restoring force in mN/g. $\:D\frac{dr}{dt}$ represents the dynamic viscosity where $\:D$ is the viscous constant in g/s, defined at 4.3 mN.s/g.mm as proposed by Yan et al.¹³, and $\:\frac{dr}{dt}$ is the speed in mm/s. The force developed by the autonomic nervous system ($\:{F}_{n}$) in mN/g represents the sum of the forces from the branches of the parasympathetic nervous system ($\:{F}_{p}$) and the sympathetic nervous system ($\:{F}_{s}$), where the force due to the sympathetic nervous system is negative. This is defined by the following system of equations:

$$\:{\text{F}}_{\text{n}}\left(\text{t}\right)={\text{F}}_{\text{p}}\left(\text{t}\right)-\:{\text{F}}_{\text{s}}\left(\text{t}\right)$$

(2)

$$\:{\text{F}}_{\text{p}}\left(\text{t}\right)={\text{f}}_{\text{p}0}\text{*}[\text{u}\left(\text{t}-{{\uptau\:}}_{\text{p}1}\right)-\text{u}\left(\text{t}-{\text{t}}_{\text{d}}-{{\uptau\:}}_{\text{p}2}\right)]$$

(3)

$$\:{\text{F}}_{\text{s}}\left(\text{t}\right)={\text{f}}_{\text{s}0}\text{*}\left[\text{u}\left(\text{t}-{{\uptau\:}}_{\text{s}1}\right)-\text{u}\left(\text{t}-{\text{t}}_{\text{d}}-{{\uptau\:}}_{\text{p}2}\right)\right]+{\text{f}}_{\text{s}1}\text{*}[\text{u}\left(\text{t}-{\text{t}}_{\text{d}}-{{\uptau\:}}_{\text{p}2}\right)-\text{u}\left(\text{t}-{\text{t}}_{\text{d}}-{{\uptau\:}}_{\text{s}2}\right)]$$

(4)

$\:\text{u}$ corresponds to the unit step function, which represents the application of forces at specific moments in time. $\:{f}_{p0}$ is the intensity of the force originating from the parasympathetic nervous system. $\:{f}_{s0}$ and $\:{f}_{s1}$ are used to describe the force $\:{F}_{s}$. The first sympathetic phase ($\:{f}_{s0}$) acts in co-activation with the force $\:{f}_{p0}$ originating from the parasympathetic nervous system. It is assumed that the first sympathetic phase ($\:{f}_{s0}$) always acts with a greater force intensity than the second sympathetic phase ($\:{f}_{s1}$). $\:{\tau\:}_{p1}$ is the delay between the beginning of the light stimulus and the activation of the force $\:{F}_{p}$. $\:{\tau\:}_{p2}$ is the delay between the end of the light stimulus and the end of the force $\:{F}_{p}$. $\:{\tau\:}_{s1}$ is the delay between the beginning of the light stimulus and the activation of the force $\:{F}_{s}$. $\:{\tau\:}_{s2}$ is the delay between the end of the light stimulus and the end of the force $\:{F}_{s}$. $\:{t}_{d}$ is the duration of the light stimulus in the context of photomotor reflex (Fig. 7a). The timing of the parasympathetic and sympathetic activations in the photomotor reflex (Fig. 7a) is defined by the following system of equations and allows identifying the parasympathetic impulse named Impulse 1 ($\:Imp1$), the sympathetic impulse named Impulse 3 ($\:Imp3$), and the combination of the two, named Impulse 2 ($\:Imp2$):

Fig. 7

Schematic representation of the time delays (τ) and forces ($\:f$) obtained from the modelling of the photomotor reflex (a) and hippus (b). The mean real autonomic forces and mean real time delays of the impulses in the photomotor reflex and hippus are represented. $\:Imp1$ = Impulse 1, $\:Imp2$ = Impulse 2, $\:Imp3$ = Impulse 3, $\:{f}_{p0}$ = parasympathetic force, $\:{f}_{s0}$ = initial sympathetic force, $\:{f}_{s1}$ = secondary sympathetic force.

$$\:Imp1={\int\:}_{{\tau\:}_{p1}}^{{\tau\:}_{s1}}{F}_{n}dt$$

(5)

$$\:Imp2={\int\:}_{{\tau\:}_{s1}}^{{t}_{d}+{\tau\:}_{p2}\:}{F}_{n}dt$$

(6)

$$\:Imp3={\int\:}_{{t}_{d}+{\tau\:}_{p2}\:}^{{t}_{d}+{\tau\:}_{s2}}{F}_{n}dt$$

(7)

From a mechanical perspective, an impulse is defined as the time integral of force (i.e. the area under the force-time curve). In the present study, this mechanical quantity is interpreted in a physiological context and used as a proxy to quantify the contractile action exerted by the iris muscles under autonomic nervous system control. In the context of hippus, the autonomic forces have the same meanings as those during the photomotor reflex. As hippus occurs in the absence of an external stimulus, the model was adapted, and the timing of the parasympathetic and sympathetic activations is defined by the following system of equations (Fig. 7b):

$$\:{F}_{p}\left(t\right)={f}_{p0}*\left[u\left(t-{\tau\:}_{p1}\right)-u\left(t-{\tau\:}_{p1}-{\tau\:}_{p2}\right)\right]$$

(8)

$$\:{F}_{s}\left(t\right)={f}_{s0}*\left[u\left(t-{\tau\:}_{p1}-{\tau\:}_{s1}\right)-u\left(t-{\tau\:}_{p1}-{\tau\:}_{p2}\right)\right]+{f}_{s1}*[u\left(t-{\tau\:}_{p1}-{\tau\:}_{p2}\right)-u\left(t-{\tau\:}_{p1}-{\tau\:}_{s2}\right)]$$

(9)

For hippus, $\:{\tau\:}_{p1}$ identifies the beginning of the signal and does not have any biological significance. $\:{\tau\:}_{p2}$ is the delay between the start and the end of the parasympathetic activation. $\:{\tau\:}_{s1}$ is the delay between the start of the parasympathetic activation and the start of the activation of the force $\:{F}_{s}$. $\:{\tau\:}_{s2}$ is the delay between the start of the parasympathetic activation and the end of the activation of the force $\:{F}_{s}$ (Fig. 7b). The difference between the model applied to hippus and the one applied to the photomotor reflex is that the time constants ($\:\tau\:$) of hippus do not depend on the stimulus duration, unlike in the photomotor reflex.

$$\operatorname{Im} p1=\mathop \smallint \limits_{{{\tau _{p1}}}}^{{{\tau _{p1}}+{\tau _{s1}}}} {F_n}dt$$

(10)

$$\operatorname{Im} p2=\mathop \smallint \limits_{{{\tau _{p1}}+{\tau _{s1}}}}^{{{\tau _{p1}}+{\tau _{p2}}~}} {F_n}dt$$

(11)

$$\operatorname{Im} p3=\mathop \smallint \limits_{{{\tau _{p1}}+{\tau _{p2}}~}}^{{{\tau _{p1}}+{\tau _{s2}}}} {F_n}dt$$

(12)

Heart rate variability recording to obtain conventional autonomic nervous system indices

The recording and data processing of heart rate variability were carried out following the recommendations^38,39. Throughout the protocol, participants wore a chest heart rate monitor (Polar^® H9 by Polar - Finland) to measure the durations of the intervals between each R wave of cardiac electrical activity. Signals were recorded using the EliteHRV 5.5.6 application on an iPhone 11 and exported to Kubios^® HRV standard 3.5.0 for analysis. This is considered a valid method of heart rate variability analysis in athletes in supine position⁴⁰. After spending 5 min in the room for equipment set-up, participants lay down, and the 5-minute heart rate variability recording began after a 1-minute period in the supine position. Then, they stood up and after another 1-minute adaptation the recording continued for another 5 min in standing position. Within each of these 5-minute analysis windows, data analyses were based on 4 min of RR intervals, from the first to the fifth minute; the first minute allows a regular respiratory rhythm so that respiratory sinus arrhythmia does not drift during the heart rate variability test and does not lead to a misinterpretation of the frequency domain parameters⁴¹. Ectopic beats in the 4-minute RR interval recordings were corrected using very low threshold-based automatic inspections proposed by Kubios. From the RR intervals, temporal, frequency, and non-linear heart rate variability parameters were extracted and included mean heart rate in bpm, root mean square of successive differences (RMSSD) in ms, standard deviation of normal-to-normal intervals (SDNN) in ms, low frequencies (LF) in ms² (0.05 to 0.15 Hz) high frequencies (HF) in ms² (0.15 to 0.40 Hz), low frequencies (LF) + HF in ms², LF/HF ratio (unitless)³⁸, standard deviation 1 (SD1), the ratio of standard deviation 2 over standard deviation 1 (SD2/SD1)⁴², and the detrended fluctuation analysis alpha 1 (DFAα1)⁴³.

Statistical analysis

Evolutionary algorithms, complemented by the classical iterative nonlinear method, were employed to minimize the sum of squares between the Kelvin-Voigt model and each hippus (n = 788) and between the Kelvin-Voigt model and each photomotor reflex (n = 133). This approach was used to determine the model parameters that enable a thorough characterization of the parasympathetic and sympathetic activities in pupil variations specific to young active individuals, taking into consideration the viscoelastic parameters. The coefficient of determination and the mean error were used to assess the degree of fit of the Kelvin-Voigt model to the pupil variations. The K-means clustering algorithm, based on mean error, selected a total of 586 kinetics divided into three groups of fit quality (good-quality hippus = 298, moderate-quality hippus = 170, poor-quality hippus = 118). The normality of the autonomic nervous system impulses from the hippus in the three fit quality groups was tested using the Shapiro-Wilk test (p < 0.05), and the homogeneity of variances was checked with the Levene’s test (p < 0.05). When the conditions of normality or homogeneity of variances were not met, the Kruskal-Wallis test was used to compare the parameter distributions across the three fit quality groups. When a significant main effect was observed, post-hoc pairwise comparisons were performed using independent two-sample Wilcoxon rank-sum tests. Normality of the paired differences between positions (supine vs. standing) and between pupillary processes (hippus vs. photomotor reflex) was assessed using the Shapiro-Wilk test. When the assumption of normality was not met, the Wilcoxon signed-rank test was used to compare positions; otherwise, paired t-tests were applied. Position- and pupillary-process related analyses were conducted on subject-level mean values, with one value per participant and per position or pupillary process derived from the Kelvin-Voigt model. Effect sizes were quantified, 95% confidence intervals were reported, and corrections for multiple comparisons were consistently applied using the Holm procedure. The Shapiro-Wilk test was also used to verify the distribution normality (p < 0.05) of subject-level mean parameters for good-quality hippus (n = 16) and photomotor reflex parameters (n = 16). When normality conditions were not met (p < 0.05), the Spearman’s correlation coefficient was used to assess significant correlations (p < 0.05) between parameters. The coefficient of variation was calculated for each parameter to assess the relative variability. Three two-dimensional principal component analyses were used to highlight linear correlation patterns and explore the underlying structure of the data by reducing the dimensionality of the variable space studied between the averaged hippus parameters for each subject (n = 16), the averaged photomotor reflex parameters for each subject (n = 16), and the heart rate variability parameters (n = 16). A significance threshold of p< 0.05 was retained for all tests, and exact p-values for each test are provided in the Results section. All computations were performed with RStudio 2023.12.1³⁴.