American Institute of Mathematical Sciences

Advanced
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

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]

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[2]

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A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input,, Biol. Cybern., 95 (2006), 1.  doi: 10.1007/s00422-006-0068-6.  Google Scholar

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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), (2004), 205.   Google Scholar

[19]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophy. J., 4 (1964), 41.  doi: 10.1016/S0006-3495(64)86768-0.  Google Scholar

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D. Hampel and P. Lansky, On the estimation of refractory period,, J. Neurosci. Meth., 171 (2008), 288.  doi: 10.1016/j.jneumeth.2008.03.003.  Google Scholar

[21]

D. H. Johnson, Point process models of single-neuron discharges,, J. Comput. Neurosci., 3 (1996), 275.  doi: 10.1007/BF00161089.  Google Scholar

[22]

D. H. Johnson and A. Swami, The transmission of signals by auditory-nerve fiber discharge patterns,, J. Acoust. Soc. Am., 74 (1983), 493.  doi: 10.1121/1.389815.  Google Scholar

[23]

R. E. Kass and V. Ventura, A spike-train probability model,, Neural Comput., 13 (2001), 1713.  doi: 10.1162/08997660152469314.  Google Scholar

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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).  doi: 10.1371/journal.pone.0000439.  Google Scholar

[25]

M. I. Miller, Algorithms for removing recovery-related distortion from auditory nerve discharge patterns,, J. Acoust. Soc. Am., 77 (1985), 1452.  doi: 10.1121/1.392040.  Google Scholar

[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.  doi: 10.1162/neco.2008.11-07-653.  Google Scholar

[27]

L. M. Ricciardi, Diffusion Processes and Related Topics in Biology,, Notes taken by Charles E. Smith, (1977).   Google Scholar

[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), (2006), 133.  doi: 10.1007/88-470-0472-1_11.  Google Scholar

[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), (1992), 78.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[32]

R. B. Stein, A theoretical analysis of neuronal variability,, Biophys. J., 5 (1965), 173.  doi: 10.1016/S0006-3495(65)86709-1.  Google Scholar

[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.  doi: 10.1007/BF02183694.  Google Scholar

[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.  doi: 10.1103/PhysRevE.59.956.  Google Scholar

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.  doi: 10.1093/cercor/7.3.237.  Google Scholar

[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.  doi: 10.1016/S0303-2647(03)00106-0.  Google Scholar

[3]

O. Bernander, C. Koch and M. Usher, The effect of synchronized inputs at the single neuron level,, Neural Computation, 6 (1994), 622.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[6]

M. J. Berry and M. Meister, Refractoriness and neural precision,, J. Neurosci., 18 (1998), 2200.   Google Scholar

[7]

A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input,, Biol. Cybern., 95 (2006), 1.  doi: 10.1007/s00422-006-0068-6.  Google Scholar

[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.  doi: 10.1007/s00422-006-0082-8.  Google Scholar

[9]

H. P. Chan and W.-L. Loh, Some theoretical results on neural spike train probability models,, Ann. Stat., 35 (2007), 2691.  doi: 10.1214/009053607000000280.  Google Scholar

[10]

M. Crowder, Classical Competing Risks,, Chapman & Hall/CRC, (2001).  doi: 10.1201/9781420035902.  Google Scholar

[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).  doi: 10.1186/1471-2202-10-S1-P110.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/PL00000070.  Google Scholar

[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.  doi: 10.1016/j.spl.2008.01.093.  Google Scholar

[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), (2010), 169.   Google Scholar

[16]

A. Di Crescenzo and M. Longobardi, On the NBU ageing notion within the competing risks model,, J. Stat. Plann. Infer., 136 (2006), 1638.  doi: 10.1016/j.jspi.2004.08.022.  Google Scholar

[17]

A. Di Crescenzo and M. Longobardi, Competing risks within shock models,, Sci. Math. Japon., 67 (2008), 125.   Google Scholar

[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), (2004), 205.   Google Scholar

[19]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophy. J., 4 (1964), 41.  doi: 10.1016/S0006-3495(64)86768-0.  Google Scholar

[20]

D. Hampel and P. Lansky, On the estimation of refractory period,, J. Neurosci. Meth., 171 (2008), 288.  doi: 10.1016/j.jneumeth.2008.03.003.  Google Scholar

[21]

D. H. Johnson, Point process models of single-neuron discharges,, J. Comput. Neurosci., 3 (1996), 275.  doi: 10.1007/BF00161089.  Google Scholar

[22]

D. H. Johnson and A. Swami, The transmission of signals by auditory-nerve fiber discharge patterns,, J. Acoust. Soc. Am., 74 (1983), 493.  doi: 10.1121/1.389815.  Google Scholar

[23]

R. E. Kass and V. Ventura, A spike-train probability model,, Neural Comput., 13 (2001), 1713.  doi: 10.1162/08997660152469314.  Google Scholar

[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).  doi: 10.1371/journal.pone.0000439.  Google Scholar

[25]

M. I. Miller, Algorithms for removing recovery-related distortion from auditory nerve discharge patterns,, J. Acoust. Soc. Am., 77 (1985), 1452.  doi: 10.1121/1.392040.  Google Scholar

[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.  doi: 10.1162/neco.2008.11-07-653.  Google Scholar

[27]

L. M. Ricciardi, Diffusion Processes and Related Topics in Biology,, Notes taken by Charles E. Smith, (1977).   Google Scholar

[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), (2006), 133.  doi: 10.1007/88-470-0472-1_11.  Google Scholar

[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), (1992), 78.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[32]

R. B. Stein, A theoretical analysis of neuronal variability,, Biophys. J., 5 (1965), 173.  doi: 10.1016/S0006-3495(65)86709-1.  Google Scholar

[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.  doi: 10.1007/BF02183694.  Google Scholar

[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.  doi: 10.1103/PhysRevE.59.956.  Google Scholar

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