Home Nonlinear analysis of pupillary dynamics
Article
Licensed
Unlicensed Requires Authentication

Nonlinear analysis of pupillary dynamics

  • Francesco Onorati EMAIL logo , Luca Tommaso Mainardi , Fabiola Sirca , Vincenzo Russo and Riccardo Barbieri
Published/Copyright: August 6, 2015

Abstract

Pupil size reflects autonomic response to different environmental and behavioral stimuli, and its dynamics have been linked to other autonomic correlates such as cardiac and respiratory rhythms. The aim of this study is to assess the nonlinear characteristics of pupil size of 25 normal subjects who participated in a psychophysiological experimental protocol with four experimental conditions, namely “baseline”, “anger”, “joy”, and “sadness”. Nonlinear measures, such as sample entropy, correlation dimension, and largest Lyapunov exponent, were computed on reconstructed signals of spontaneous fluctuations of pupil dilation. Nonparametric statistical tests were performed on surrogate data to verify that the nonlinear measures are an intrinsic characteristic of the signals. We then developed and applied a piecewise linear regression model to detrended fluctuation analysis (DFA). Two joinpoints and three scaling intervals were identified: slope α0, at slow time scales, represents a persistent nonstationary long-range correlation, whereas α1 and α2, at middle and fast time scales, respectively, represent long-range power-law correlations, similarly to DFA applied to heart rate variability signals. Of the computed complexity measures, α0 showed statistically significant differences among experimental conditions (p<0.001). Our results suggest that (a) pupil size at constant light condition is characterized by nonlinear dynamics, (b) three well-defined and distinct long-memory processes exist at different time scales, and (c) autonomic stimulation is partially reflected in nonlinear dynamics.


Corresponding author: Francesco Onorati, Department of Electronics, Information and Bioengineering (DEIB), Politecnico of Milan, Milan, Italy; and Behavior and Brain Lab, IULM University, Milan, Italy, E-mail:

References

[1] Andreassi JL. Pupillary response and behavior. Mahwah, NJ, USA: Lawrence Erlbaum 2000; 218–233.Search in Google Scholar

[2] Benarroch EE. Functional anatomy of the central autonomic nervous system. In: Appenzeller O, editor. Autonomic nervous system part 1, Handbook of clinical neurology series. Amsterdam, Netherlands: Elsevier 1999: 53–86.Search in Google Scholar

[3] Bitsios P, Szabadi E, Bradshaw CM. The fear-inhibited light reflex: importance of the anticipation of an aversive event. Int J Psychophysiol 2004; 52: 87–95.10.1016/j.ijpsycho.2003.12.006Search in Google Scholar

[4] Bond CVA. Personal relevance is an important dimension for visceral reactivity in emotional imagery. Cogn Emotion 1998; 12: 231–242.10.1080/026999398379736Search in Google Scholar

[5] Borgdorff P. Respiratory fluctuations in pupil size. Am J Physiol 1975; 228: 1094–1102.10.1152/ajplegacy.1975.228.4.1094Search in Google Scholar

[6] Bradley MM, Miccoli L, Escrig MA, Lang PJ. The pupil as a measure of emotional arousal and autonomic activation. Psychophysiology 2008; 45: 602–607.10.1111/j.1469-8986.2008.00654.xSearch in Google Scholar

[7] Bressloff PC, Wood CV. Spontaneous oscillations in a nonlinear delayed-feedback shunting model of the pupil light reflex. Phys Rev E 1998; 58: 3597–3606.10.1103/PhysRevE.58.3597Search in Google Scholar

[8] Bryce RM, Sprague KB. Revisiting detrended fluctuation analysis. Sci Rep 2012; 2.10.1038/srep00315Search in Google Scholar

[9] Calcagnini G, Censi F, Lino S, Cerutti S. Spontaneous fluctuations of human pupil reflect central autonomic rhythms. Method Inf Med 2000; 39: 142–145.10.1055/s-0038-1634277Search in Google Scholar

[10] Damasio AR. Emotion in the perspective of an integrated nervous system. Brain Res Rev 1998; 26: 83–86.10.1016/S0165-0173(97)00064-7Search in Google Scholar

[11] Dietz J, Bradley MM, Okun MS, Bowers D. Emotion and ocular responses in Parkinson’s disease. Neuropsychologia 2011; 49: 3247–3253.10.1016/j.neuropsychologia.2011.07.029Search in Google Scholar

[12] Diks C, Van Houwelingen JC, Takens F, DeGoede J. Reversibility as a criterion for discriminating time series. Phys Lett A 1995; 201: 221–228.10.1016/0375-9601(95)00239-YSearch in Google Scholar

[13] Draper NR, Smith H. Applied regression analysis, 2nd ed. New York: John Wiley and Sons, 1981.Search in Google Scholar

[14] Einhäuser W, Stout J, Koch C, Carter O. Pupil dilation reflects perceptual selection and predicts subsequent stability in perceptual rivalry. Proc Natl Acad Sci USA 2008; 105: 1704.10.1073/pnas.0707727105Search in Google Scholar

[15] Gamlin PDR. The pretectum: connections and oculomotor-related roles. Progr Brain Res 2006; 151: 379–405.10.1016/S0079-6123(05)51012-4Search in Google Scholar

[16] Glass L. Introduction to controversial topics in nonlinear science: is the normal heart rate chaotic? Chaos 2009; 19: 028501.10.1063/1.3156832Search in Google Scholar

[17] Granholm E, Steinhauer SR. Pupillometric measures of cognitive and emotional processes. Int J Psychophysiol 2004; 52: 1–6.10.1016/j.ijpsycho.2003.12.001Search in Google Scholar

[18] Grassberger P, Procaccia I. Characterization of strange attractors. Phys Rev Lett 1983; 50: 346–349.10.1103/PhysRevLett.50.346Search in Google Scholar

[19] Grassberger P, Procaccia I. Measuring the strangeness of strange attractors. Physica D 1983; 9: 189–208.10.1016/0167-2789(83)90298-1Search in Google Scholar

[20] Jones RH, Dey I. Determining one or more change points. Chem Phys Lipids 1995; 76: 1–6.10.1016/0009-3084(94)02422-2Search in Google Scholar

[21] Koss MC. Pupillary dilation as an index of central nervous system α2-adrenoceptor activation. J Pharmacol Method 1986; 15: 1–19.10.1016/0160-5402(86)90002-1Search in Google Scholar

[22] Koss MC, Gherezghiher T, Nomura A. A CNS adrenergic inhibition of parasympathetic oculomotor tone. J Autonom Nerv Syst 1984; 10: 55–68.10.1016/0165-1838(84)90067-5Search in Google Scholar

[23] Kreibig SD. Autonomic nervous system activity in emotion: a review. Biol Psychol 2010; 84: 394–421.10.1016/j.biopsycho.2010.03.010Search in Google Scholar PubMed

[24] Lake DE, Richman JS, Griffin MP, Moorman JR. Sample entropy analysis of neonatal heart rate variability. Am J Physiol-Reg I 2002; 283: R789–R797.10.1152/ajpregu.00069.2002Search in Google Scholar

[25] Lerman PM. Fitting segmented regression models by grid search. Appl Stat 1980; 29: 77–84.10.2307/2346413Search in Google Scholar

[26] Loewenfeld IE, Lowenstein O. The pupil: anatomy, physiology, and clinical applications. Ames, IA: Iowa State University Press, 1993.Search in Google Scholar

[27] Loewy AD, Araujo JC, Kerr FWL. Pupillodilator pathways in the brain stem of the cat: anatomical and electrophysiological identification of a central autonomic pathway. Brain Res 1973; 60: 65–91.10.1016/0006-8993(73)90851-2Search in Google Scholar

[28] Longtin A. Nonlinear oscillations, noise and chaos in neural delayed feedback. PhD thesis 1989.Search in Google Scholar

[29] Longtin A, Milton JG. Complex oscillations in the human pupil light reflex with “mixed” and delayed feedback. Math Biosci 1988; 90: 183–199.10.1016/0025-5564(88)90064-8Search in Google Scholar

[30] Longtin A, Milton JG, Bos JE, Mackey MC. Noise and critical behavior of the pupil light reflex at oscillation onset. Phys Rev A 1990; 41: 6992.10.1103/PhysRevA.41.6992Search in Google Scholar PubMed

[31] Mesin L, Monaco A, Cattaneo R. Investigation of nonlinear pupil dynamics by recurrence quantification analysis. Biomed Res Int 2013; 2013: 1–11.10.1155/2013/420509Search in Google Scholar PubMed PubMed Central

[32] Milton J, Bayer W, an der Heiden U. Modeling the pupil light reflex with differential delay equations. Zeitschrift für Angewandte Mathematik und Mechanik 1998; S625–S628.Search in Google Scholar

[33] Moloney KP, Jacko JA, Vidakovic B, Sainfort F, Leonard VK, Shi B. Leveraging data complexity: pupillary behavior of older adults with visual impairment during hci. ACM T Comput Hum Interaction 2006; 13: 376–402.10.1145/1183456.1183460Search in Google Scholar

[34] Murphy PR, O’Connell RG, O’Sullivan M, Robertson IH, Balsters JH. Pupil diameter covaries with bold activity in human locus coeruleus. Hum Brain Mapp 2014; 35: 4140–4154.10.1002/hbm.22466Search in Google Scholar PubMed PubMed Central

[35] Onorati F, Barbieri R, Mauri M, Russo V, Mainardi LT. Characterization of affective states by pupillary dynamics and autonomic correlates. Front Neuroengineering 2013; 6.10.3389/fneng.2013.00009Search in Google Scholar PubMed PubMed Central

[36] Onorati F, Mauri M, Russo V, Mainardi LT. Reconstruction of pupil dilation signal during eye blinking events. In: Cerutti S, Dickhaus H, Mainardi LT, Yana K, editors. Proceeding of the VII international workshop on biosignal interpretation, BSI2012. Como, Italy, 2012: 117–120.Search in Google Scholar

[37] Park G, Thayer JF. From the heart to the mind: cardiac vagal tone modulates top-down and bottom-up visual perception and attention to emotional stimuli. Front Psychology 2014; 5.10.3389/fpsyg.2014.00278Search in Google Scholar PubMed PubMed Central

[38] Partala T, Surakka V. Pupil size variation as an indication of affective processing. Int J Hum-Comput St 2003; 59: 185–198.10.1016/S1071-5819(03)00017-XSearch in Google Scholar

[39] Peng C-K, Havlin S, Stanley HE, Goldberger AL. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos: An Interdisciplinary Journal of Nonlinear Science 1995; 5: 82–87.10.1063/1.166141Search in Google Scholar

[40] Penzel T, Kantelhardt JW, Grote L, Peter J-H, Bunde A. Comparison of detrended fluctuation analysis and spectral analysis for heart rate variability in sleep and sleep apnea. IEEE T Biomed Eng 2003; 50: 1143–1151.10.1109/TBME.2003.817636Search in Google Scholar

[41] Pincus SM, Gladstone IM, Ehrenkranz RA. A regularity statistic for medical data analysis. J Clin Monitor Comput 1991; 7: 335–345.10.1007/BF01619355Search in Google Scholar

[42] Pittasch D, Lobmann R, Behrens-Baumann W, Lehnert H. Pupil signs of sympathetic autonomic neuropathy in patients with type 1 diabetes. Diabetes Care 2002; 25: 1545–1550.10.2337/diacare.25.9.1545Search in Google Scholar

[43] Plutchik R. Theories of emotion. Washington, DC, USA: American Psychological Association 2013.Search in Google Scholar

[44] Provenzale A, Smith LA, Vio R, Murante G. Distinguishing between low-dimensional dynamics and randomness in measured time series. Physica D: Nonlinear Phenomena 1992; 58: 31–49.10.1016/0167-2789(92)90100-2Search in Google Scholar

[45] Rainville P, Bechara A, Naqvi N, Damasio AR. Basic emotions are associated with distinct patterns of cardiorespiratory activity. Int J Psychophysiol 2006; 61: 5–18.10.1016/j.ijpsycho.2005.10.024Search in Google Scholar

[46] Rao RKA, Yeragani VK. Decreased chaos and increased nonlinearity of heart rate time series in patients with panic disorder. Auton Neurosci 2001; 88: 99–108.10.1016/S1566-0702(01)00219-3Search in Google Scholar

[47] Richman JS, Moorman JR. Physiological time-series analysis using approximate entropy and sample entropy. Am J Physiol-Heart C 2000; 278: H2039–H2049.10.1152/ajpheart.2000.278.6.H2039Search in Google Scholar PubMed

[48] Rosenberg ML, Kroll MH. Pupillary hippus: an unrecognized example of biologic chaos. J Biol Syst 1999; 7: 85–94.10.1142/S0218339099000085Search in Google Scholar

[49] Rosenstein MT, Collins JJ, De Luca CJ. A practical method for calculating largest Lyapunov exponents from small data sets. Physica D: Nonlinear Phenomena 1993; 65: 117–134.10.1016/0167-2789(93)90009-PSearch in Google Scholar

[50] Samuels ER, Szabadi E. Functional neuroanatomy of the noradrenergic locus coeruleus: its roles in the regulation of arousal and autonomic function. Part II: physiological and pharmacological manipulations and pathological alterations of locus coeruleus activity in humans. Current Neuropharmacology 2008; 6: 254.10.2174/157015908785777193Search in Google Scholar

[51] Schreiber T, Schmitz A. Discrimination power of measures for nonlinearity in a time series. Phys Rev E 1997; 55: 5443.10.1103/PhysRevE.55.5443Search in Google Scholar

[52] Schreiber T, Schmitz A. Surrogate time series. Physica D: Nonlinear Phenomena 2000; 142: 346–382.10.1016/S0167-2789(00)00043-9Search in Google Scholar

[53] Soares VCG, Souza JKS, Ginani GE, Pompéia S, Tierra-Criollo CJ, Melges DB. Identification of drowsiness and alertness conditions by means of spectral f-test applied to pupillometric signals. In Journal of Physics: Conference Series. IOP Publishing 2013; 477: 12–27.Search in Google Scholar

[54] Stanten SF, Stark L. A statistical analysis of pupil noise. IEEE T Biomed Eng 1966; 13: 140–152.Search in Google Scholar

[55] Stark L. Stability, oscillations, and noise in the human pupil servomechanism. Proceedings of the IRE 1959; 47: 1925–1939.10.1109/JRPROC.1959.287206Search in Google Scholar

[56] Stark L, Campbell FW, Atwood J. Pupil unrest: an example of noise in a biological servomechanism. Nature 1958; 182: 857–858.10.1038/182857a0Search in Google Scholar

[57] Steinhauer SR, Siegle GJ, Condray R, Pless M. Sympathetic and parasympathetic innervation of pupillary dilation during sustained processing. Int J Psychophysiol 2004; 52: 77–86.10.1016/j.ijpsycho.2003.12.005Search in Google Scholar

[58] Sterpenich V, D’Argembeau A, Desseilles M, et al. The locus ceruleus is involved in the successful retrieval of emotional memories in humans. J Neurosci 2006; 26: 7416–7423.10.1523/JNEUROSCI.1001-06.2006Search in Google Scholar

[59] Szabadi E, Bradshaw CM. Autonomic pharmacology of α2-adrenoceptors. J Psychopharm 1996; 10: 6–18.Search in Google Scholar

[60] Thayer JF, Hansen AL, Saus-Rose E, Johnsen BH. Heart rate variability, prefrontal neural function, and cognitive performance: the neurovisceral integration perspective on self-regulation, adaptation, and health. Ann Behav Med 2009; 37: 141–153.10.1007/s12160-009-9101-zSearch in Google Scholar

[61] Thayer JF, Lane RD. A model of neurovisceral integration in emotion regulation and dysregulation. J Affect Disorders 2000; 61: 201–216.10.1016/S0165-0327(00)00338-4Search in Google Scholar

[62] Theil H. Economic forecasts and policy. Number 15 in contributions to economic analysis. Amsterdam: North-Holland Pub. Co. 1961.Search in Google Scholar

[63] Theiler J. Estimating fractal dimension. J Opt Soc Am A 1990; 7: 1055–1073.10.1364/JOSAA.7.001055Search in Google Scholar

[64] Theiler J, Eubank S, Longtin A, Galdrikian B, Farmer JD. Testing for nonlinearity in time series: the method of surrogate data. Physica D: Nonlinear Phenomena 1992; 58: 77–94.10.1016/0167-2789(92)90102-SSearch in Google Scholar

[65] Usui S, Stark L. A model for nonlinear stochastic behavior of the pupil. Biol Cybern 1982; 45: 13–21.10.1007/BF00387209Search in Google Scholar

[66] Valenza G, Citi L, Lanatá A, Scilingo EP, Barbieri R. Revealing real-time emotional responses: a personalized assessment based on heartbeat dynamics. Scientific Reports 2014; 4.10.1038/srep04998Search in Google Scholar

[67] Vautard R, Yiou P, Ghil M. Singular-spectrum analysis: a toolkit for short, noisy chaotic signals. Physica D: Nonlinear Phenomena 1992; 58: 95–126.10.1016/0167-2789(92)90103-TSearch in Google Scholar

[68] Voss A, Schulz S, Schroeder R, Baumert M, Caminal P. Methods derived from nonlinear dynamics for analysing heart rate variability. Philos Tr R Soc A 2009; 367: 277–296.10.1098/rsta.2008.0232Search in Google Scholar

[69] Vuksanović V, Gal V. Nonlinear and chaos characteristics of heart period time series: healthy aging and postural change. Auton Neurosci 2005; 121: 94–100.10.1016/j.autneu.2005.06.004Search in Google Scholar

[70] Warga M, Lüdtke H, Wilhelm H, Wilhelm B. How do spontaneous pupillary oscillations in light relate to light intensity? Vision Res 2009; 49: 295–300.10.1016/j.visres.2008.09.019Search in Google Scholar

[71] Wierda SM, van Rijn H, Taatgen NA, Martens S. Pupil dilation deconvolution reveals the dynamics of attention at high temporal resolution. Proc Natl Acad Sci 2012; 109: 8456–8460.10.1073/pnas.1201858109Search in Google Scholar

[72] Wolf A, Swift JB, Swinney HL, Vastano JA. Determining lyapunov exponents from a time series. Physica D: Nonlinear Phenomena 1985; 16: 285–317.10.1016/0167-2789(85)90011-9Search in Google Scholar

[73] Yeragani VK, Radhakrishna RK, Tancer M, Uhde T. Nonlinear measures of respiration: respiratory irregularity and increased chaos of respiration in patients with panic disorder. Neuropsychobiology 2002; 46: 111–120.10.1159/000066388Search in Google Scholar PubMed

[74] Yoshida H, Mizuta H, Gouhara T, Suzuki Y, Yana K, Okuyama F. Statistical properties of simultaneously recorded fluctuations in pupil diameter and heart rate. In: Engineering in Medicine and Biology Society, 1995. IEEE 17th Annual Conference, IEEE 1995; 165–166.Search in Google Scholar

[75] Yoshida H, Yana K, Okuyama F, Tokoro T. Time-varying properties of respiratory fluctuations in pupil diameter of human eyes. Method Inform Med 1994; 33: 46–46.10.1055/s-0038-1634990Search in Google Scholar

Received: 2015-2-6
Accepted: 2015-6-10
Published Online: 2015-8-6
Published in Print: 2016-2-1

©2016 by De Gruyter

Downloaded on 8.8.2025 from https://www.degruyterbrill.com/document/doi/10.1515/bmt-2015-0027/html
Scroll to top button