{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:46:57Z","timestamp":1760237217061,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,11]],"date-time":"2020-03-11T00:00:00Z","timestamp":1583884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia","award":["UID\/MAT\/04674\/2013  e","UID\/Multi\/04466\/2019"],"award-info":[{"award-number":["UID\/MAT\/04674\/2013  e","UID\/Multi\/04466\/2019"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>This article aims to examine the dynamical characteristics of the pupillary light reflex and to provide a contribution towards their explanation based on the nonlinear theory of dynamical systems. To introduce the necessary concepts, terminology, and relevant features of the pupillary light reflex and its associated delay, we start with an overview of the human eye anatomy and physiology with emphasis on the iris, pupil, and retina. We also present the most highly regarded models for pupil dynamics found in the current scientific literature. Then we consider the model developed by Longtin and Milton, which models the human pupillary light reflex, defined by a nonlinear differential equation with delay, and present our study carried out on the qualitative and quantitative dynamic behavior of that neurophysiological control system.<\/jats:p>","DOI":"10.3390\/math8030394","type":"journal-article","created":{"date-parts":[[2020,3,12]],"date-time":"2020-03-12T04:13:57Z","timestamp":1583986437000},"page":"394","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Searching for Complexity in the Human Pupillary Light Reflex"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2669-2581","authenticated-orcid":false,"given":"Ros\u00e1rio D.","family":"Laureano","sequence":"first","affiliation":[{"name":"Department of Mathematics, ISTAR-IUL Information Sciences, Technologies and Architecture Research Center, ISCTE-IUL Lisbon University Institute, Avenida das For\u00e7as Armadas, 1649-026 Lisboa, Portugal"}]},{"given":"Diana","family":"Mendes","sequence":"additional","affiliation":[{"name":"Department of Quantitative Methods for Management and Economics, BRU-IUL Business Research Unit, ISCTE-IUL Lisbon University Institute, Avenida das For\u00e7as Armadas, 1649-026 Lisboa, Portugal"}]},{"given":"Clara","family":"Gr\u00e1cio","sequence":"additional","affiliation":[{"name":"Department of Mathematics, CIMA-Research Centre for Mathematics and Applications, Universidade de \u00c9vora, Rua Rom\u00e3o Ramalho, 59,7000-585 \u00c9vora, Portugal"}]},{"given":"F\u00e1tima","family":"Laureano","sequence":"additional","affiliation":[{"name":"Instituto de Microcirurgia Ocular, Torres de Lisboa, Rua Tom\u00e1s da Fonseca, 1600-209 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,11]]},"reference":[{"key":"ref_1","unstructured":"Baker, C.T.H., Paul, C.A.H., and Will, D.R. (1995). A Bibliography on the Numerical Solution of Delay Differential Equations, Mathematics Department, University of Manchester. Numerical Analysis Report 269."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Erneux, T. (2009). Applied Delay Differential Equations, Springer.","DOI":"10.1007\/978-0-387-74372-1_8"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1016\/S0022-247X(02)00135-X","article-title":"Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays","volume":"271","author":"Huanga","year":"2002","journal-title":"J. Math. Anal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.jde.2006.05.006","article-title":"Nonmonotone travelling waves in a single species reaction-diffusion equation with delay","volume":"228","author":"Faria","year":"2006","journal-title":"J. Differ. Equ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1702","DOI":"10.1063\/1.458052","article-title":"Differential delay equations in chemical kinetics: Some simple linear model systems","volume":"92","author":"Epstein","year":"1990","journal-title":"J. Chem. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"114309","DOI":"10.1063\/1.5006923","article-title":"Climate models with delay differential equations","volume":"27","author":"Keane","year":"2017","journal-title":"Chaos"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1142\/S0218339007002313","article-title":"Three types of simple dde\u2019s describing tumor growth","volume":"15","author":"Bodnar","year":"2007","journal-title":"J. Biol. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Atay, F.M. (2010). Delay-Induced Stability: From Oscillators to Networks. Complex Time-Delay Systems, Theory and Applications, Springer.","DOI":"10.1007\/978-3-642-02329-3"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1007\/BF02459969","article-title":"Modelling autonomous oscillations in the human pupil light reflex using non-linear delay-differential equations","volume":"51","author":"Longtin","year":"1989","journal-title":"Bull. Math. Bio."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1364\/JOSA.34.000319","article-title":"On the stiles-crawford effect","volume":"34","author":"Moon","year":"1979","journal-title":"J. Opt. Soc. Am."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1364\/JOSA.42.000492","article-title":"Pupil size as determined by adapting luminancet","volume":"42","author":"Gebhard","year":"1952","journal-title":"J. Opt. Soc. Am."},{"key":"ref_12","first-page":"491","article-title":"The verriest lecture. How much light reaches the retina?","volume":"59","author":"Pokorny","year":"1997","journal-title":"Colour Vis. Defic. XIII. Doc. Ophth. Proc. Ser."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1109\/10.1365","article-title":"Latency of the pupillary response","volume":"35","author":"Link","year":"1988","journal-title":"IEEE Trans. Bio. Eng"},{"key":"ref_14","first-page":"521","article-title":"Pupillary control pathways","volume":"Volume 1","author":"Masland","year":"2008","journal-title":"The Senses: A Comprehensive Reference"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1515\/phys-2019-0047","article-title":"Modeling Human Pupil Dilation to Decouple the Pupillary Light Reflex","volume":"17","author":"Napieralski","year":"2019","journal-title":"Open Phys."},{"key":"ref_16","unstructured":"Tilmant, C., Gindre, G., Sarry, L., and Boire, J.-Y. (2003, January 17\u201321). Monitoring and modeling of pupillary dynamics. Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE Cat. No.03CH37439), Cancun, Mexico."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Pamplona, V., Oliveira, M., and Baranoski, V. (2009). Photorealistic Models for Pupil Light Reflex and Iridal Pattern Deformation. ACM Trans. Graph., 4.","DOI":"10.1145\/1559755.1559763"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1167\/tvst.8.5.29","article-title":"The flicker Pupil Light Response (fPLR)","volume":"8","author":"Adhikari","year":"2019","journal-title":"Transl. Vis. Sci. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Roussel, M. (2019). Nonlinear Dynamics Hands-On Introductory Survey, Morgan & Claypool Publishers.","DOI":"10.1088\/2053-2571\/ab0281"},{"key":"ref_20","first-page":"3593","article-title":"Measuring and controlling the chaotic motion of profits","volume":"11","author":"Mendes","year":"2009","journal-title":"Int. J. Bifurc. Chaos Appl. Sci. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1103\/PhysRevE.75.046215","article-title":"Approximating chaotic saddles for delay differential equations","volume":"75","author":"Taylor","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1299","DOI":"10.1007\/s10208-017-9369-5","article-title":"Algorithm for Rigorous Integration of Delay Differential Equations and the Computer-Assisted Proof of Periodic Orbits in the Mackey-Glass Equation","volume":"18","author":"Szczelina","year":"2018","journal-title":"Found Comput. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1016\/0167-2789(82)90042-2","article-title":"Chaotic attractors of an infinite-dimensional dynamical system","volume":"4","author":"Farmer","year":"1982","journal-title":"Phys. D"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1925","DOI":"10.1109\/JRPROC.1959.287206","article-title":"Oscillation and Noise in the Human Pupil Servomechanism","volume":"47","author":"Stark","year":"1959","journal-title":"Proc. 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