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Circle maps with gaps: Understanding the dynamics of the two-process model for sleep–wake regulation

Published online by Cambridge University Press:  02 May 2018

M. P. BAILEY ,
G. DERKS  and
A. C. SKELDON
M. P. BAILEY
Affiliation:
Department of Mathematics, University of Surrey, Surrey GU2 7XH, UK emails: matthew.bailey@surrey.ac.uk, g.derks@surrey.ac.uk, a.skeldon@surrey.ac.uk
G. DERKS
Affiliation:
Department of Mathematics, University of Surrey, Surrey GU2 7XH, UK emails: matthew.bailey@surrey.ac.uk, g.derks@surrey.ac.uk, a.skeldon@surrey.ac.uk
A. C. SKELDON
Affiliation:
Department of Mathematics, University of Surrey, Surrey GU2 7XH, UK emails: matthew.bailey@surrey.ac.uk, g.derks@surrey.ac.uk, a.skeldon@surrey.ac.uk
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Abstract

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For more than 30 years the ‘two-process model’ has played a central role in the understanding of sleep/wake regulation. This ostensibly simple model is an interesting example of a non-smooth dynamical system, whose rich dynamical structure has been relatively unexplored. The two-process model can be framed as a one-dimensional map of the circle, which, for some parameter regimes, has gaps. We show how border collision bifurcations that arise naturally in maps with gaps extend and supplement the Arnold tongue saddle-node bifurcation set that is a feature of continuous circle maps. The novel picture that results shows how the periodic solutions that are created by saddle-node bifurcations in continuous maps transition to periodic solutions created by period-adding bifurcations as seen in maps with gaps.

Keywords

37E10 Maps of the circle92B25 Biological rhythms and synchronisation37G15 Bifurcations of limit cycles and periodic orbits37E05 Maps of the interval37N25 Dynamical systems in biology
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 5: Theory and applications of nonsmooth dynamical systems , October 2018 , pp. 845 - 868
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†This work was partially supported by the Engineering and Physical Sciences Research Council (Grant number EP/M506655/1).

References

Archer, S. N., Laing, E. E., Möller-Levet, C. S., van der Veen, D. R, Bucca, G., Lazar, A. S., Santhi, N., Slak, A., Kabiljo, R., von Schantz, M., Smith, C. P. & Dijk, D.-J. (2014) Mistimed sleep disrupts circadian regulation of the human transcriptome. Proc. Natl. Acad. Sci. USA 111, E682–E691.Google Scholar
Arnold, V. I. (1991) Cardiac arrhythmias and circle mappings. Chaos 1, 20–21, reprinted from Arnold's thesis (Moscow 1959).Google Scholar
Aschoff, J. (1965) Circadian rhythms in man. Science 148, 1427–1432.Google Scholar
Avrutin, V., Schantz, M. & Banerjee, S. (2006) Multi-parametric bifurcations in a piecewise-linear discontinuous map. Nonlinearity 19, 1875–1906.Google Scholar
Barone, T. L. (2000) Is the siesta an adaptation to disease? Hum. Nat. 11, 233–258.Google Scholar
Booth, V. & Diniz Behn, C. G. (2014) Physiologically-based modeling of sleep-wake regulatory networks. Math. Biosci. 250, 54–68.Google Scholar
Booth, V., Xique, I. & Diniz Behn, C. G. (2017) One-dimensional map for the circadian modulation of sleep in a sleep-wake regulatory network model for human sleep. SIAM J. Appl. Dyn. Syst. 16, 1089–1112.Google Scholar
Borbély, A. A. (1982) A two-process model of sleep regulation. Hum. Neurobiol. 1, 195–204.Google Scholar
Daan, S., Beersma, D. G. M. & Borbély, A. A. (1984) Timing of human sleep: Recovery process gated by a circadian pacemaker. Am. J. Physiol. 246, R161–R178.Google Scholar
Derks, G., Glendinning, P. A. & Skeldon, A. C. (2018) Transitions to gaps and non-monotonicity in circle maps. In preparation.Google Scholar
Diniz Behn, C. G. & Booth, V. (2010) Simulating microinjection experiments in a novel model of the rat sleep-wake regulatory network. J. Neurophysiol. 103, 1937–1953.Google Scholar
Diniz Behn, C. G., Brown, E. N., Scammell, T. E. & Kopell, N. J. (2007) Mathematical model of network dynamics governing mouse sleep-wake behaviour. J. Neurophysiol. 97, 3828–3840.Google Scholar
Ekirch, A. R. (2005) At Day's Close: A History of Nighttime. W.W. Norton & Company, New York.Google Scholar
Galland, B. C., Taylor, B. J., Elder, D. E. & Herbison, P. (2012) Normal sleep patterns in infants and children: A systematic review of observational studies. Sleep Med. Rev. 16, 213–222.Google Scholar
Granados, A., Alsedà, L. & Krupa, M. (2017) The period adding and incrementing bifurcations: From rotation theory to applications. SIAM Rev. 59, 225–292.Google Scholar
Keener, J. P. (1980) Chaotic behaviours in piecewise continuous difference equations. Trans. Am. Math. Soc. 261, 589–604.Google Scholar
Knutson, K. L. (2010) Sleep duration and cardiometabolic risk: A review of the epidemiologic evidence. Best Pract. Res.: Clin. Endocrinol. Metab. 24, 731–743.Google Scholar
Kumar, R., Bose, A. & Mallick, B. N. (2012) A mathematical model towards understanding the mechanism of neuronal regulation of wake-NREMS-REMS states. PLoS One 7, e42059.Google Scholar
Luyster, F. S., Strollo, P. J. Jr., Zee, P. C. & Walsh, J. K. (2012) Sleep: A health imperative. Sleep 35, 727–734.Google Scholar
Mackay, R. & Tresser, C. (1986) Transition to topological chaos for circle maps. Physica D 19, 206–237.Google Scholar
Möller-Levet, C. S., Archer, S. N., Bucca, G., Laing, E., Slak, A., Kabiljo, R., Lo, J., Santhi, N., von Schantz, M., Smitth, C. P. & Dijk, D.-J. (2013) Effects of insufficient sleep on circadian rhythmicity and expression amplitude of the human blood transcriptome. Proc. Natl. Acad. Sci. USA 110, E1132–E1141.Google Scholar
Nakao, M., Sakai, H. & Yamamoto, M. (1997) An interpretation of the internal desynchronisations based on dynamics of the two-process model. Methods Inform. Med. 36 (4–5), 282–285.Google Scholar
Nakao, M. & Yamamoto, M. (1998) Bifurcation properties of the two-process model. Psychiatry Clin. Neurosci. 52, 131–133.Google Scholar
Nielsen, L. S., Danielsen, K. & Sorensen, T. (2011) Short sleep duration as a possible cause of obesity: Critical analysis of the epidemiological evidence. Obes. Rev. 12, 78–92.Google Scholar
Phillips, A. J. K., Czeisler, C. A. & Klerman, E. B. (2011) Revisiting spontaneous internal desynchrony using a quantitative model of sleep physiology. J. Biol. Rhythms 26, 441–453.Google Scholar
Phillips, A. J. K. & Robinson, P. A. (2007) A quantitative model of sleep-wake dynamics based on the physiology of the brainstem ascending arousal system. J. Biol. Rhythms 22, 167–179.Google Scholar
Phillips, A. J. K., Robinson, P. A., Kedziora, D. J. & Abeysuriya, R. G. (2010) Mammalian sleep dynamics: How diverse features arise from a common physiological framework. PLoS Comput. Biol. 6, e1000826.Google Scholar
Postnova, S., Voigt, K. & Braun, H. (2009) A mathematical model of homeostatic regulation of sleep-wake cycles by hypocretin/orexin. J. Biol. Rhythms 24 (6), 523–535.Google Scholar
Pring, S. R. & Budd, C. J. (2011) The dynamics of a simplified pin-ball machine. IMA J. Appl. Math. 76, 67–84.Google Scholar
Rempe, M. J., Best, J. & Terman, D. (2010) A mathematical model of the sleep/wake cycle. J. Math. Biol. 60, 615–644.Google Scholar
Rhodes, F. & Thompson, C. L. (1986) Rotation numbers for monotone functions on the circle. J. Lond. Math. Soc. 34, 360–368.Google Scholar
Saper, C. B., Scammell, T. E. & Lu, J. (2005) Hypothalamic regulation of sleep and circadian rhythms. Nature 437, 1257–1263.Google Scholar
Skeldon, A. C. & Derks, G. (2017a) Nonsmooth Maps and the Fast-Slow Regulation of Sleep-Wake Regulation: Part I, Vol. 8, Birkhauser, Cham, pp. 167–170.Google Scholar
Skeldon, A. C., Derks, G. & Booth, V. (2017b) Nonsmooth Maps and the Fast-Slow Regulation of Sleep-Wake Regulation: Part II, Vol. 8, Birkhauser, Cham, pp. 171–175.Google Scholar
Skeldon, A. C., Dijk, D.-J. & Derks, G. (2014) Mathematical models for sleep-wake dynamics: Comparison of the two-process model and a mutual inhibition neuronal model. PLoS One 10, e103877.Google Scholar
Tamakawa, Y., Karashima, A., Koyama, Y., Katayama, N. & Nakao, M. (2006) A quartet neural system model orchestrating sleep and wakefulness mechanisms. J. Neurophysiol. 95 (4), 2055–2069.Google Scholar
Wever, R. (1979) The Circadian System of Man: Results of Experiments Under Temporal Isolation. Springer, New York.Google Scholar