[en] The general purpose of this paper is to build up on our understanding of the basic mathematical principles that underlie the emergence of synchronous biological rhythms, in particular, the circadian clock. To do so, we study the role that the coupling strength, coupling type, and noise play in the synchronization of a system of coupled, non-linear oscillators. First, we study a deterministic model based on Van der Pol coupled oscillators, modeling a population of diffusively coupled cells, to find regions in the parameter space for which synchronous oscillations emerge and to provide conditions under which diffusive coupling kills the synchronous oscillation. Second, we study how noise and coupling interact and lead to synchronous oscillations in linearly coupled oscillators, modeling the interaction between various pacemaker populations, each having an endogenous circadian clock. To do so, we use the Fokker-Planck equation associated to the system. We show how coupling can tune the frequency of the emergent synchronous oscillation, which provides a general mechanism to make fast (ultradian) pacemakers slow (circadian) and synchronous via coupling. The basic mechanisms behind the generation of oscillations and the emergence of synchrony that we describe here can be used to guide further studies of coupled oscillations in biophysical non-linear models.
Disciplines :
Mathematics
Author, co-author :
Franci, Alessio ; Université de Liège - ULiège > Département d'électricité, électronique et informatique (Institut Montefiore) > Brain-Inspired Computing ; Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico
Herrera-Valdez, Marco Arieli; Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico
Lara-Aparicio, Miguel; Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico
Padilla-Longoria, Pablo; Departamento de Matemáticas y Mecánica, Instituto de Investigación en Matemáticas Aplicadas y Sistemas, Universidad Nacional Autónoma de Mexico, Mexico City, Mexico
Language :
English
Title :
Synchronization, Oscillator Death, and Frequency Modulation in a Class of Biologically Inspired Coupled Oscillators
Bünsow RC. The circadian rhythm of photoperiodic responsiveness in Kalanchoe. In: Stillman B, Stewart D, Grodzicker T, editors. Cold Spring Harbor Symposia on Quantitative Biology. Vol. 25. Cold Spring Harbor Laboratory Press (1960). p. 257–60.
Pol B. Biological rhythms considered as relaxation oscillations. J Intern Med. (1940) 103:76–88.
Sholl D. Regularities in growth curves, including rhythms and allometry. Dyn Growth Process. (1954) 224–241.
Pavlidis T, Zimmerman W, Osborn J. A mathematical model for the temperature effects on circadian rhythms. J Theor Biol. (1968) 18:210–21. 5647132
Klotter K. Theoretical analysis of some biological models. In: Stillman B, Stewart D, Grodzicker T, editors. Cold Spring Harbor Symposia on Quantitative Biology. Vol. 25. Cold Spring Harbor Laboratory Press (1960). p. 189–96.
Klotter K. General properties of oscillating systems. In: Stillman B, Stewart D, Grodzicker T, editors. Cold Spring Harbor Symposia on Quantitative Biology. Vol. 25. Cold Spring Harbor Laboratory Press (1960). p. 185–87.
Sweeney BM. The photosynthetic rhythm in single cells of Gonyaulax polyedra. In: Stillman B, Stewart D, Grodzicker T, editors. Cold Spring Harbor Symposia on Quantitative Biology. Vol. 25. Cold Spring Harbor Laboratory Press (1960). p. 145–48.
Winfree AT. The Geometry of Biological Time. Vol. 12. New York, NY: Springer Science & Business Media (2001).
Hirota T, Lee JW, Lewis WG, Zhang EE, Breton G, Liu X, et al. High-throughput chemical screen identifies a novel potent modulator of cellular circadian rhythms and reveals CKIα as a clock regulatory kinase. PLoS Biol. (2010) 8:e1000559. 10.1371/journal.pbio.100055921179498
Partch CL, Green CB, Takahashi JS. Molecular architecture of the mammalian circadian clock. Trends Cell Biol. (2014) 24:90–9. 10.1016/j.tcb.2013.07.002.23916625
Welsh DK, Yoo SH, Liu AC, Takahashi JS, Kay SA. Bioluminescence imaging of individual fibroblasts reveals persistent, independently phased circadian rhythms of clock gene expression. Curr Biol. (2004) 14:2289–95. 10.1016/j.cub.2004.11.05715620658
Hastings MH, Maywood ES, O'Neill JS. Cellular circadian pacemaking and the role of cytosolic rhythms. Curr Biol. (2008) 18:R805–15. 10.1016/j.cub.2008.07.02118786386
Nader N, Chrousos GP, Kino T. Interactions of the circadian CLOCK system and the HPA axis. Trends Endocrinol Metab. (2010) 21:277–86. 10.1016/j.tem.2009.12.01120106676
Abruzzi KC, Rodriguez J, Menet JS, Desrochers J, Zadina A, Luo W, et al. Drosophila CLOCK target gene characterization: implications for circadian tissue-specific gene expression. Genes Dev. (2011) 25:2374–86. 10.1101/gad.174110.11122085964
Manoogian EN, Panda S. Circadian rhythms, time-restricted feeding, and healthy aging. Ageing Res Rev. (2017) 39:59–67. 10.1016/j.arr.2016.12.006.28017879
Tahara Y, Aoyama S, Shibata S. The mammalian circadian clock and its entrainment by stress and exercise. J Physiol Sci. (2017) 67:1–10. 10.1007/s12576-016-0450-727084533
Rusak B, Zucker I. Neural regulation of circadian rhythms. Physiol Rev. (1979) 59:449–526. 379886
Guo H, Brewer JM, Lehman MN, Bittman EL. Suprachiasmatic regulation of circadian rhythms of gene expression in hamster peripheral organs: effects of transplanting the pacemaker. J Neurosci. (2006) 26:6406–12. 10.1523/JNEUROSCI.4676-05.200616775127
Tahara Y, Shibata S. Circadian rhythms of liver physiology and disease: experimental and clinical evidence. Nat Rev Gastroenterol Hepatol. (2016) 13:217. 10.1038/nrgastro.2016.826907879
Oster H, Damerow S, Kiessling S, Jakubcakova V, Abraham D, Tian J, et al. The circadian rhythm of glucocorticoids is regulated by a gating mechanism residing in the adrenal cortical clock. Cell Metab. (2006) 4:163–73. 10.1016/j.cmet.2006.07.00216890544
Ishida A, Mutoh T, Ueyama T, Bando H, Masubuchi S, Nakahara D, et al. Light activates the adrenal gland: timing of gene expression and glucocorticoid release. Cell Metab. (2005) 2:297–307. 10.1016/j.cmet.2005.09.00916271530
Kiessling S, Eichele G, Oster H. Adrenal glucocorticoids have a key role in circadian resynchronization in a mouse model of jet lag. J clin Invest. (2010) 120:2600–9. 10.1172/JCI4119220577050
Damiola F, Le Minh N, Preitner N, Kornmann B, Fleury-Olela F, Schibler U. Restricted feeding uncouples circadian oscillators in peripheral tissues from the central pacemaker in the suprachiasmatic nucleus. Genes Dev. (2000) 14:2950–61. 10.1101/gad.18350011114885
Kronauer RE, Czeisler CA, Pilato SF, Moore-Ede MC, Weitzman ED. Mathematical model of the human circadian system with two interacting oscillators. Am J Physiol Regul Integr Comp Physiol. (1982) 242:R3–17. 7058927
olde Scheper T, Klinkenberg D, Pennartz C, Van Pelt J. A mathematical model for the intracellular circadian rhythm generator. J Neurosci. (1999) 19:40–47.
Glass L. Synchronization and rhythmic processes in physiology. Nature (2001) 410:277. 10.1038/3506574511258383
Ueda HR, Hagiwara M, Kitano H. Robust oscillations within the interlocked feedback model of Drosophila circadian rhythm. J Theor Biol. (2001) 210:401–6. 10.1006/jtbi.2000.222611403560
Jiang J, Liu Q, Niu L. Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM. Math Biosci Eng. (20170 14:1247–59. 10.3934/mbe.201706429161859
Lara-Aparicio M, de Medrano SL, Fuentes-Pardo B, Moreno-Sáenz E. A qualitative mathematical model of the ontogeny of a circadian rhythm in crayfish. Bull Math Biol. (1993) 55:97–110.
Fanjul-Moles ML, Moreno-Sáenz E, Villalobos-Hiriart N, Fuentes-Pardo B. ERG circadian rhythm in the course of ontogeny in crayfish. Comp Biochem Physiol A Physiol. (1987) 88:213–9.
Fanjul-Moles ML, Miranda-Anaya M, Fuentes-Pardo B. Effect of monochromatic light upon the erg crcadian rhythm during ontogeny in crayfish (Procambarus clarkii). Comp Biochem Physiol A Physiol. (1992) 102:99–106.
Fuentes-Pardo B, Fanjul-Moles ML, Moreno-Sáenz E. Synchronization by light of the ERG circadian rhythm during ontogeny in the crayfish. Biol Rhythm Res. (1992) 23:81–91.
Van der Pol B. On “relaxation-oscillations.” Lond Edinb Dublin Philos Mag J Sci. (1926) 2:978–92.
Fuentes-Pardo B, Lara-Aparicio M, de Medrano SL. Perturbation of a circadian rhythm by single and periodic signals and its mathematical simulation. Bull Math Biol. (1995) 57:175–89.
Dowse HB, Hall JC, Ringo JM. Circadian and ultradian rhythms in period mutants of Drosophila melanogaster. Behav Genet. (1987) 17:19–35. 3109369
Goldbeter A. From Ultradian Biochemical Oscillations to Circadian Rhythms. In: Vanden Driessche T, Guisset JL, Petiau-de Vries GM, editors. Membranes and Circadian Rythms. Springer (1996). p. 67–93.
Lloyd D, Rossi EL. Ultradian Rhythms in Life Processes: An Inquiry Into Fundamental Principles of Chronobiology and Psychobiology.London: Springer Science & Business Media (2012).
Barriga-Montoya C, Padilla-Longoria P, Lara-Aparicio M, Fuentes-Pardo B. Ultradian rhythms underlying the dynamics of the circadian pacemaker. In: Vonend O, editior. Aspects of Pacemakers-Functions and Interactions in Cardiac and Non-Cardiac Indications. InTech (2011).
Agaev R, Chebotarev P. On the spectra of nonsymmetric Laplacian matrices. Linear Algebra Appl. (2005) 399:157–68. 10.1016/j.laa.2004.09.003
La Salle JP. An invariance principle in the theory of stability, differential equations and dynamical systems. In: Proceedings of the International Symposium, Puerto Rico (1967). p. 277–86.
Guckenheimer J, Holmes P. Local bifurcations. In: Guckenheimer J, Holmes P, editors. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York, NY: Springer (1983), p. 117–165. 10.1007/978-1-4612-1140-2_3
Hassard BD, Kazarinoff ND, Wan YH. Theory and Applications of Hopf Bifurcation. Vol. 41. Cambridge: CUP Archive (1981).
Golubitsky M, Stewart I, Schaeffer DG. Singularities and Groups in Bifurcation Theory. Vol. 1. Verlag: Springer (1985).
Jiao X, Wang R. Synchronous firing patterns of neuronal population with excitatory and inhibitory connections. Int J Non Linear Mech. (2010) 45:647–51. 10.1016/j.ijnonlinmec.2008.11.020
Le Minh N, Damiola F, Tronche F, Schütz G, Schibler U. Glucocorticoid hormones inhibit food-induced phase-shifting of peripheral circadian oscillators. EMBO J. (2001) 20:7128–36. 10.1093/emboj/20.24.7128
Hammond C, Bergman H, Brown P. Pathological synchronization in Parkinson's disease: networks, models and treatments. Trends Neurosci. (2007) 30:357–64. 10.1016/j.tins.2007.05.00417532060
Schnitzler A, Gross J. Normal and pathological oscillatory communication in the brain. Nat Revi Neurosci. (2005) 6:285–96. 10.1038/nrn165015803160
Engel Jr J, Bragin A, Staba R, Mody I. High-frequency oscillations: what is normal and what is not? Epilepsia (2009) 50:598–604. 10.1111/j.1528-1167.2008.01917.x19055491
Velazquez JLP, Carlen PL. Gap junctions, synchrony and seizures. Trends Neurosci. (2000) 23:68–74. 10.1016/S0166-2236(99)01497-6
Herrera-Valdez MA, McKiernan EC, Berger SD, Ryglewski S, Duch C, Crook S. Relating ion channel expression, bifurcation structure, and diverse firing patterns in a model of an identified motor neuron. J comput Neurosci. (2013) 34:211–29. 10.1007/s10827-012-0416-622878689
McKiernan EC. A genetic manipulation of motor neuron excitability does not alter locomotor output in Drosophila larvae. PeerJ PrePrints (2015) 3:e469v3. 10.7287/peerj.preprints.469v3
