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

Figures

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  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…

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  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.

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  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 (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:Imp1$$\end{document}), orange rectangle = parasympathetic-sympathetic co-activation impulse (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}…

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  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.

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  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…

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  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, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:Imp1$$\end{document} = impulse 1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}…

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  Fig. 7. Schematic representation of the time delays (τ) and forces (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:f$$\end{document}) 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. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:Imp1$$\end{document} = Impulse 1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:Imp2$$\end{document} = Impulse 2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}…