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

Modeling some properties of circadian rhythms

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

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.
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.

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