Modeling some properties of circadian rhythms

Previous Article
Designing neural networks for modeling biological data: A statistical perspective
MBE Home
This Issue
Next Article
Local versus nonlocal barycentric interactions in 1D agent dynamics

2014, 11(2): 317-330. doi: 10.3934/mbe.2014.11.317

Modeling some properties of circadian rhythms

Miguel Lara-Aparicio ^1, , Carolina Barriga-Montoya ^2, , Pablo Padilla-Longoria ^3, and Beatriz Fuentes-Pardo ^2,

1.	Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México,, Mexico
2.	Laboratorio de Cronobiología, Departamento de Fisiología, Facultad de Medicina, Universidad Nacional Autónoma de México, Mexico, Mexico
3.	Departamento de Matemáticas y Mecánica, Instituto de Investigaciones, en Matemáticas Aplicadas y en Sistemas. Universidad Nacional Autónoma de México, Mexico

Received September 2012 Revised January 2013 Published October 2013

Abstract
Related Papers
Under a Creative Commons license

Mathematical models have been very useful in biological research. From the interaction of biology and mathematics, new problems have emerged that have generated advances in the theory, suggested further experimental work and motivated plausible conjectures. From our perspective, it is absolutely necessary to incorporate modeling tools in the study of circadian rhythms and that without a solid mathematical framework a real understanding of them will not be possible. Our interest is to study the main process underlying the synchronization in the pacemaker of a circadian system: these mechanisms should be conserved in all living beings. Indeed, from an evolutionary perspective, it seems reasonable to assume that either they have a common origin or that they emerge from similar selection circumstances. We propose a general framework to understand the emergence of synchronization as a robust characteristic of some cooperative systems of non-linear coupled oscillators. In a first approximation to the problem we vary the topology of the network and the strength of the interactions among oscillators. In order to study the emergent dynamics, we carried out some numerical computations. The results are consistent with experiments reported in the literature. Finally, we proposed a theoretical framework to study the phenomenon of synchronization in the context of circadian rhythms: the dissipative synchronization of nonautonomous dynamical systems.

Keywords: circadian rhythms, network architecture., cooperative networks, Synchronization, ultradian rhythms.

Mathematics Subject Classification: Primary: 92B25, 34C15; Secondary: 68U2.

Citation: Miguel Lara-Aparicio, Carolina Barriga-Montoya, Pablo Padilla-Longoria, Beatriz Fuentes-Pardo. Modeling some properties of circadian rhythms. Mathematical Biosciences & Engineering, 2014, 11 (2) : 317-330. doi: 10.3934/mbe.2014.11.317

References:

[1]	D. J. Amit and N. Brunel, Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex, Cerebral Cortex, 7 (1997), 237-252. doi: 10.1093/cercor/7.3.237.
[2]	M. Barbi, S. Chillemi, A. Di Garbo and L. Reale, Stochastic resonance in a sinusoidally forced LIF model with noisy threshold, Biosystems, 71 (2003), 23-28. doi: 10.1016/S0303-2647(03)00106-0.
[3]	O. Bernander, C. Koch and M. Usher, The effect of synchronized inputs at the single neuron level, Neural Computation, 6 (1994), 622-641.
[4]	A. Buonocore, A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A Markov chain-based model for actomyosin dynamics, Sci. Math. Japon., 70 (2009), 159-174.
[5]	A. Buonocore, A. Di Crescenzo, B. Martinucci and L. M. Ricciardi, A stochastic model for the stepwise motion in actomyosin dynamics, Sci. Math. Japon., 58 (2003), 245-254.
[6]	M. J. Berry and M. Meister, Refractoriness and neural precision, J. Neurosci., 18 (1998), 2200-2211.
[7]	A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input, Biol. Cybern., 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6.
[8]	A. N. Burkitt, A review of the integrate-and-fire neuron model. II. Inhomogeneous synaptic input and network properties, Biol. Cybern., 95 (2006), 97-112. doi: 10.1007/s00422-006-0082-8.
[9]	H. P. Chan and W.-L. Loh, Some theoretical results on neural spike train probability models, Ann. Stat., 35 (2007), 2691-2722. doi: 10.1214/009053607000000280.
[10]	M. Crowder, Classical Competing Risks, Chapman & Hall/CRC, Boca Raton, 2001. doi: 10.1201/9781420035902.
[11]	M. Deger, S. Cardanobile, M. Helias and S. Rotter, The Poisson process with dead time captures important statistical features of neural activity, BMC Neuroscience, 10 (2009), P110. doi: 10.1186/1471-2202-10-S1-P110.
[12]	A. Di Crescenzo, E. Di Nardo, A. G. Nobile, E. Pirozzi and L. M. Ricciardi, On some computational results for single neurons' activity modeling, BioSystems, 58 (2000), 19-26.
[13]	A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, Stochastic population models with interacting species, J. Math. Biol., 42 (2001), 1-25. doi: 10.1007/PL00000070.
[14]	A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A note on birth-death processes with catastrophes, Stat. Prob. Lett., 78 (2008), 2248-2257. doi: 10.1016/j.spl.2008.01.093.
[15]	A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, On time non-homogeneous stochastic processes with catastrophes, in Cybernetics and Systems 2010 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2010, 169-174.
[16]	A. Di Crescenzo and M. Longobardi, On the NBU ageing notion within the competing risks model, J. Stat. Plann. Infer., 136 (2006), 1638-1654. doi: 10.1016/j.jspi.2004.08.022.
[17]	A. Di Crescenzo and M. Longobardi, Competing risks within shock models, Sci. Math. Japon., 67 (2008), 125-135.
[18]	A. Di Crescenzo, B. Martinucci, E. Pirozzi and L. M. Ricciardi, On the interaction between two Stein's neuronal units, in Cybernetics and Systems 2004 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2004, 205-210.
[19]	G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophy. J., 4 (1964), 41-68. doi: 10.1016/S0006-3495(64)86768-0.
[20]	D. Hampel and P. Lansky, On the estimation of refractory period, J. Neurosci. Meth., 171 (2008), 288-295. doi: 10.1016/j.jneumeth.2008.03.003.
[21]	D. H. Johnson, Point process models of single-neuron discharges, J. Comput. Neurosci., 3 (1996), 275-299. doi: 10.1007/BF00161089.
[22]	D. H. Johnson and A. Swami, The transmission of signals by auditory-nerve fiber discharge patterns, J. Acoust. Soc. Am., 74 (1983), 493-501. doi: 10.1121/1.389815.
[23]	R. E. Kass and V. Ventura, A spike-train probability model, Neural Comput., 13 (2001), 1713-1720. doi: 10.1162/08997660152469314.
[24]	A. Mazzoni, F. D. Broccard, E. Garcia-Perez, P. Bonifazi, M. E. Ruaro and V. Torre, On the dynamics of the spontaneous activity in neuronal networks, PLoS ONE, 2 (2007), e439. doi: 10.1371/journal.pone.0000439.
[25]	M. I. Miller, Algorithms for removing recovery-related distortion from auditory nerve discharge patterns, J. Acoust. Soc. Am., 77 (1985), 1452-1464. doi: 10.1121/1.392040.
[26]	U. Picchini, S. Ditlevsen, A. De Gaetano and P. Lansky, Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal, Neural Comput., 20 (2008), 2696-2714. doi: 10.1162/neco.2008.11-07-653.
[27]	L. M. Ricciardi, Diffusion Processes and Related Topics in Biology, Notes taken by Charles E. Smith, Lecture Notes in Biomathematics, Vol. 14, Springer-Verlag, Berlin-New York, 1977.
[28]	L. M. Ricciardi, Modeling single neuron activity in the presence of refractoriness: New contributions to an old problem, in Imagination and Rigor. Essays on Eduardo R. Caianiello's Scientific Heritage (ed. S. Termini), Springer-Verlag Italia, 2006, 133-145. doi: 10.1007/88-470-0472-1_11.
[29]	L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, On the instantaneous return process for neuronal diffusion models, in Structure: from Physics to General Systems - Festschrift Volume in Honour of E.R. Caianiello on his Seventieth Birthday (eds. M. Marinaro and G. Scarpetta), World Scientific, Singapore, 1992, 78-94.
[30]	L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Math. Japon., 50 (1999), 247-322.
[31]	W. R. Softky and C. Koch, The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs, The Journal of Neuroscience, 13 (1993), 334-350.
[32]	R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J., 5 (1965), 173-194. doi: 10.1016/S0006-3495(65)86709-1.
[33]	T. Tateno, S. Doi, S. Sato and L. M. Ricciardi, Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach, J. Stat. Phys., 78 (1995), 917-935. doi: 10.1007/BF02183694.
[34]	K. Yoshino, T. Nomura, K. Pakdaman and S. Sato, Synthetic analysis of periodically stimulated excitable and oscillatory membrane models, Phys. Rev. E (3), 59 (1999), 956-969. doi: 10.1103/PhysRevE.59.956.

show all references

References:

[1]	D. J. Amit and N. Brunel, Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex, Cerebral Cortex, 7 (1997), 237-252. doi: 10.1093/cercor/7.3.237.
[2]	M. Barbi, S. Chillemi, A. Di Garbo and L. Reale, Stochastic resonance in a sinusoidally forced LIF model with noisy threshold, Biosystems, 71 (2003), 23-28. doi: 10.1016/S0303-2647(03)00106-0.
[3]	O. Bernander, C. Koch and M. Usher, The effect of synchronized inputs at the single neuron level, Neural Computation, 6 (1994), 622-641.
[4]	A. Buonocore, A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A Markov chain-based model for actomyosin dynamics, Sci. Math. Japon., 70 (2009), 159-174.
[5]	A. Buonocore, A. Di Crescenzo, B. Martinucci and L. M. Ricciardi, A stochastic model for the stepwise motion in actomyosin dynamics, Sci. Math. Japon., 58 (2003), 245-254.
[6]	M. J. Berry and M. Meister, Refractoriness and neural precision, J. Neurosci., 18 (1998), 2200-2211.
[7]	A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input, Biol. Cybern., 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6.
[8]	A. N. Burkitt, A review of the integrate-and-fire neuron model. II. Inhomogeneous synaptic input and network properties, Biol. Cybern., 95 (2006), 97-112. doi: 10.1007/s00422-006-0082-8.
[9]	H. P. Chan and W.-L. Loh, Some theoretical results on neural spike train probability models, Ann. Stat., 35 (2007), 2691-2722. doi: 10.1214/009053607000000280.
[10]	M. Crowder, Classical Competing Risks, Chapman & Hall/CRC, Boca Raton, 2001. doi: 10.1201/9781420035902.
[11]	M. Deger, S. Cardanobile, M. Helias and S. Rotter, The Poisson process with dead time captures important statistical features of neural activity, BMC Neuroscience, 10 (2009), P110. doi: 10.1186/1471-2202-10-S1-P110.
[12]	A. Di Crescenzo, E. Di Nardo, A. G. Nobile, E. Pirozzi and L. M. Ricciardi, On some computational results for single neurons' activity modeling, BioSystems, 58 (2000), 19-26.
[13]	A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, Stochastic population models with interacting species, J. Math. Biol., 42 (2001), 1-25. doi: 10.1007/PL00000070.
[14]	A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A note on birth-death processes with catastrophes, Stat. Prob. Lett., 78 (2008), 2248-2257. doi: 10.1016/j.spl.2008.01.093.
[15]	A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, On time non-homogeneous stochastic processes with catastrophes, in Cybernetics and Systems 2010 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2010, 169-174.
[16]	A. Di Crescenzo and M. Longobardi, On the NBU ageing notion within the competing risks model, J. Stat. Plann. Infer., 136 (2006), 1638-1654. doi: 10.1016/j.jspi.2004.08.022.
[17]	A. Di Crescenzo and M. Longobardi, Competing risks within shock models, Sci. Math. Japon., 67 (2008), 125-135.
[18]	A. Di Crescenzo, B. Martinucci, E. Pirozzi and L. M. Ricciardi, On the interaction between two Stein's neuronal units, in Cybernetics and Systems 2004 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2004, 205-210.
[19]	G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophy. J., 4 (1964), 41-68. doi: 10.1016/S0006-3495(64)86768-0.
[20]	D. Hampel and P. Lansky, On the estimation of refractory period, J. Neurosci. Meth., 171 (2008), 288-295. doi: 10.1016/j.jneumeth.2008.03.003.
[21]	D. H. Johnson, Point process models of single-neuron discharges, J. Comput. Neurosci., 3 (1996), 275-299. doi: 10.1007/BF00161089.
[22]	D. H. Johnson and A. Swami, The transmission of signals by auditory-nerve fiber discharge patterns, J. Acoust. Soc. Am., 74 (1983), 493-501. doi: 10.1121/1.389815.
[23]	R. E. Kass and V. Ventura, A spike-train probability model, Neural Comput., 13 (2001), 1713-1720. doi: 10.1162/08997660152469314.
[24]	A. Mazzoni, F. D. Broccard, E. Garcia-Perez, P. Bonifazi, M. E. Ruaro and V. Torre, On the dynamics of the spontaneous activity in neuronal networks, PLoS ONE, 2 (2007), e439. doi: 10.1371/journal.pone.0000439.
[25]	M. I. Miller, Algorithms for removing recovery-related distortion from auditory nerve discharge patterns, J. Acoust. Soc. Am., 77 (1985), 1452-1464. doi: 10.1121/1.392040.
[26]	U. Picchini, S. Ditlevsen, A. De Gaetano and P. Lansky, Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal, Neural Comput., 20 (2008), 2696-2714. doi: 10.1162/neco.2008.11-07-653.
[27]	L. M. Ricciardi, Diffusion Processes and Related Topics in Biology, Notes taken by Charles E. Smith, Lecture Notes in Biomathematics, Vol. 14, Springer-Verlag, Berlin-New York, 1977.
[28]	L. M. Ricciardi, Modeling single neuron activity in the presence of refractoriness: New contributions to an old problem, in Imagination and Rigor. Essays on Eduardo R. Caianiello's Scientific Heritage (ed. S. Termini), Springer-Verlag Italia, 2006, 133-145. doi: 10.1007/88-470-0472-1_11.
[29]	L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, On the instantaneous return process for neuronal diffusion models, in Structure: from Physics to General Systems - Festschrift Volume in Honour of E.R. Caianiello on his Seventieth Birthday (eds. M. Marinaro and G. Scarpetta), World Scientific, Singapore, 1992, 78-94.
[30]	L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Math. Japon., 50 (1999), 247-322.
[31]	W. R. Softky and C. Koch, The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs, The Journal of Neuroscience, 13 (1993), 334-350.
[32]	R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J., 5 (1965), 173-194. doi: 10.1016/S0006-3495(65)86709-1.
[33]	T. Tateno, S. Doi, S. Sato and L. M. Ricciardi, Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach, J. Stat. Phys., 78 (1995), 917-935. doi: 10.1007/BF02183694.
[34]	K. Yoshino, T. Nomura, K. Pakdaman and S. Sato, Synthetic analysis of periodically stimulated excitable and oscillatory membrane models, Phys. Rev. E (3), 59 (1999), 956-969. doi: 10.1103/PhysRevE.59.956.

[1]	Jifa Jiang, Qiang Liu, Lei Niu. Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1247-1259. doi: 10.3934/mbe.2017064
[2]	Kunwen Wen, Lifang Huang, Qiuying Li, Qi Wang, Jianshe Yu. The mean and noise of FPT modulated by promoter architecture in gene networks. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2177-2194. doi: 10.3934/dcdss.2019140
[3]	Daniel M. N. Maia, Elbert E. N. Macau, Tiago Pereira, Serhiy Yanchuk. Synchronization in networks with strongly delayed couplings. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3461-3482. doi: 10.3934/dcdsb.2018234
[4]	Sujit Nair, Naomi Ehrich Leonard. Stable synchronization of rigid body networks. Networks and Heterogeneous Media, 2007, 2 (4) : 597-626. doi: 10.3934/nhm.2007.2.597
[5]	Inmaculada Leyva, Irene Sendiña-Nadal, Stefano Boccaletti. Explosive synchronization in mono and multilayer networks. Discrete and Continuous Dynamical Systems - B, 2018, 23 (5) : 1931-1944. doi: 10.3934/dcdsb.2018189
[6]	Eleonora Catsigeras. Dynamics of large cooperative pulsed-coupled networks. Journal of Dynamics and Games, 2014, 1 (2) : 255-281. doi: 10.3934/jdg.2014.1.255
[7]	Stefano Fasani, Sergio Rinaldi. Local stabilization and network synchronization: The case of stationary regimes. Mathematical Biosciences & Engineering, 2010, 7 (3) : 623-639. doi: 10.3934/mbe.2010.7.623
[8]	David Hales, Stefano Arteconi. Motifs in evolving cooperative networks look like protein structure networks. Networks and Heterogeneous Media, 2008, 3 (2) : 239-249. doi: 10.3934/nhm.2008.3.239
[9]	Shirin Panahi, Sajad Jafari. New synchronization index of non-identical networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1359-1373. doi: 10.3934/dcdss.2020371
[10]	Xiwei Liu, Tianping Chen, Wenlian Lu. Cluster synchronization for linearly coupled complex networks. Journal of Industrial and Management Optimization, 2011, 7 (1) : 87-101. doi: 10.3934/jimo.2011.7.87
[11]	Leslaw Skrzypek, Yuncheng You. Feedback synchronization of FHN cellular neural networks. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6047-6056. doi: 10.3934/dcdsb.2021001
[12]	Joseph D. Skufca, Erik M. Bollt. Communication and Synchronization in Disconnected Networks with Dynamic Topology: Moving Neighborhood Networks. Mathematical Biosciences & Engineering, 2004, 1 (2) : 347-359. doi: 10.3934/mbe.2004.1.347
[13]	Roberta Sirovich, Laura Sacerdote, Alessandro E. P. Villa. Cooperative behavior in a jump diffusion model for a simple network of spiking neurons. Mathematical Biosciences & Engineering, 2014, 11 (2) : 385-401. doi: 10.3934/mbe.2014.11.385
[14]	Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1927-1941. doi: 10.3934/jimo.2019036
[15]	Hiroaki Uchida, Yuya Oishi, Toshimichi Saito. A simple digital spiking neural network: Synchronization and spike-train approximation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1479-1494. doi: 10.3934/dcdss.2020374
[16]	Tingting Zhu. Emergence of synchronization in Kuramoto model with frustration under general network topology. Networks and Heterogeneous Media, 2022, 17 (2) : 255-291. doi: 10.3934/nhm.2022005
[17]	Jin-Liang Wang, Zhi-Chun Yang, Tingwen Huang, Mingqing Xiao. Local and global exponential synchronization of complex delayed dynamical networks with general topology. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 393-408. doi: 10.3934/dcdsb.2011.16.393
[18]	Tingting Su, Xinsong Yang. Finite-time synchronization of competitive neural networks with mixed delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3655-3667. doi: 10.3934/dcdsb.2016115
[19]	Serap Ergün, Sirma Zeynep Alparslan Gök, Tuncay Aydoǧan, Gerhard Wilhelm Weber. Performance analysis of a cooperative flow game algorithm in ad hoc networks and a comparison to Dijkstra's algorithm. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1085-1100. doi: 10.3934/jimo.2018086
[20]	Jinsen Zhuang, Yan Zhou, Yonghui Xia. Synchronization analysis of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1607-1629. doi: 10.3934/dcdss.2020279

2018 Impact Factor: 1.313

Tools

Metrics

Other articles
by authors

on AIMS
Miguel Lara-Aparicio Carolina Barriga-Montoya Pablo Padilla-Longoria Beatriz Fuentes-Pardo
on Google Scholar
Miguel Lara-Aparicio Carolina Barriga-Montoya Pablo Padilla-Longoria Beatriz Fuentes-Pardo

[Back to Top]