2014, 11(1): 125-138. doi: 10.3934/mbe.2014.11.125

Generation of slow phase-locked oscillation and variability of the interspike intervals in globally coupled neuronal oscillators

1. 

Graduate School of Engineering, Kyoto University, Kyoto 615-8510, Japan, Japan

2. 

Faculty of Human Relation, Kyoto Koka Women's University, Kyoto 615-0882, Japan

Received  December 2012 Revised  July 2013 Published  September 2013

To elucidate how a biological rhythm is regulated, the extended (three-dimensional) Bonhoeffer-van der Pol or FitzHugh-Nagumo equations are employed to investigate the dynamics of a population of neuronal oscillators globally coupled through a common buffer (mean field). Interesting phenomena, such as extraordinarily slow phase-locked oscillations (compared to the natural period of each neuronal oscillator) and the death of all oscillations, are observed. We demonstrate that the slow synchronization is due mainly to the existence of ``fast" oscillators. Additionally, we examine the effect of noise on the synchronization and variability of the interspike intervals. Peculiar phenomena, such as noise-induced acceleration and deceleration, are observed. The results herein suggest that very small noise may significantly influence a biological rhythm.
Citation: Ryotaro Tsuneki, Shinji Doi, Junko Inoue. Generation of slow phase-locked oscillation and variability of the interspike intervals in globally coupled neuronal oscillators. Mathematical Biosciences & Engineering, 2014, 11 (1) : 125-138. doi: 10.3934/mbe.2014.11.125
References:
[1]

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L. Cheng and B. Ermentrout, Analytic approximations of statistical quantities and response of noisy oscillators,, Physica D, 240 (2011), 719.   Google Scholar

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E. J. Doedel and B. E. Oldeman, et al., AUTO-07P: Continuation and bifurcation software for ordinary differential equations,, Concordia University, (2009).   Google Scholar

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S. Doi and J. Inoue, Chaos and variability of inter-spike intervals in neuronal models with slow-fast dynamics,, AIP Conf. Proc., 1339 (2011), 210.   Google Scholar

[6]

S. Doi and S. Kumagai, Generation of very slow neuronal rhythms and chaos near the Hopf bifurcation in single neuron models,, J. Comp. Neurosci., 19 (2005), 325.  doi: 10.1007/s10827-005-2895-1.  Google Scholar

[7]

S. Doi and S. Sato, Regulation of differentiation in a population of cells interacting through a common pool,, J. Math. Biol., 26 (1988), 435.  doi: 10.1007/BF00276372.  Google Scholar

[8]

B. Ermentrout and M. Wechselberger, Canards, clusters, and synchronization in a weakly coupled interneuron model,, SIAM J. Appl. Dyn. Syst., 8 (2009), 253.  doi: 10.1137/080724010.  Google Scholar

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R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophy. J., 1 (1961), 445.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

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L. Glass, Synchronization and rhythmic processes in physiology,, Nature, 410 (2001), 277.  doi: 10.1038/35065745.  Google Scholar

[11]

B. Gutkin and B. Ermentrout, Dynamics of membrane excitability determine interspike interval variability: A link between spike generation mechanisms and cortical spike train statistics,, Neural Comput., 10 (1998), 1047.  doi: 10.1162/089976698300017331.  Google Scholar

[12]

B. Gutkin, J. Jost and H. Tuckwell, Inhibition of rhythmic neural spiking by noise: The occurrence of a minimum in activity with increasing noise,, Naturwiss., 96 (2009), 1091.  doi: 10.1007/s00114-009-0570-5.  Google Scholar

[13]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve,, J. Physiol., 117 (1952), 500.   Google Scholar

[14]

J. Honerkamp, G. Mutschler and R. Seitz, Coupling of a slow and a fast oscillator can generate bursting,, Bull. Math. Biol., 47 (1985), 1.  doi: 10.1016/S0092-8240(85)90002-3.  Google Scholar

[15]

G. Katriel, Synchronization of oscillators coupled through an environment,, Physica D, 237 (2008), 2933.  doi: 10.1016/j.physd.2008.04.015.  Google Scholar

[16]

H. Kori, Y. Kawamura and N. Masuda, Structure of cell networks critically determines oscillation regularity,, J. Theor. Biol., 297 (2012), 61.  doi: 10.1016/j.jtbi.2011.12.007.  Google Scholar

[17]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Springer Series in Synergetics, 19 (1984).  doi: 10.1007/978-3-642-69689-3.  Google Scholar

[18]

B. Lindner, A. Longtin and A. Bulsara, Analytic expressions for rate and CV of a type I neuron driven by white Gaussian noise,, Neural Comput., 15 (2003), 1761.  doi: 10.1162/08997660360675035.  Google Scholar

[19]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proc. IRE, 50 (1962), 2061.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

[20]

A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences,", Cambridge Nonlinear Science Series, 12 (2001).  doi: 10.1017/CBO9780511755743.  Google Scholar

[21]

K. Sugimoto, Y. Nii, S. Doi and S. Kumagai, Frequency variability of neural rhythm in a small network of pacemaker neurons,, Proc. of AROB 7th '02, (2002), 54.   Google Scholar

show all references

References:
[1]

R. Borisyuk, D. Chik and Y. Kazanovich, Visual perception of ambiguous figures: Synchronization based neural models,, Biol. Cybern., 100 (2009), 491.  doi: 10.1007/s00422-009-0301-1.  Google Scholar

[2]

L. Cheng and B. Ermentrout, Analytic approximations of statistical quantities and response of noisy oscillators,, Physica D, 240 (2011), 719.   Google Scholar

[3]

H. Daido, Why circadian rhythms are circadian: Competitive population dynamics of biological oscillators,, Phys. Rev. Lett., 87 (2001).  doi: 10.1103/PhysRevLett.87.048101.  Google Scholar

[4]

E. J. Doedel and B. E. Oldeman, et al., AUTO-07P: Continuation and bifurcation software for ordinary differential equations,, Concordia University, (2009).   Google Scholar

[5]

S. Doi and J. Inoue, Chaos and variability of inter-spike intervals in neuronal models with slow-fast dynamics,, AIP Conf. Proc., 1339 (2011), 210.   Google Scholar

[6]

S. Doi and S. Kumagai, Generation of very slow neuronal rhythms and chaos near the Hopf bifurcation in single neuron models,, J. Comp. Neurosci., 19 (2005), 325.  doi: 10.1007/s10827-005-2895-1.  Google Scholar

[7]

S. Doi and S. Sato, Regulation of differentiation in a population of cells interacting through a common pool,, J. Math. Biol., 26 (1988), 435.  doi: 10.1007/BF00276372.  Google Scholar

[8]

B. Ermentrout and M. Wechselberger, Canards, clusters, and synchronization in a weakly coupled interneuron model,, SIAM J. Appl. Dyn. Syst., 8 (2009), 253.  doi: 10.1137/080724010.  Google Scholar

[9]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophy. J., 1 (1961), 445.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[10]

L. Glass, Synchronization and rhythmic processes in physiology,, Nature, 410 (2001), 277.  doi: 10.1038/35065745.  Google Scholar

[11]

B. Gutkin and B. Ermentrout, Dynamics of membrane excitability determine interspike interval variability: A link between spike generation mechanisms and cortical spike train statistics,, Neural Comput., 10 (1998), 1047.  doi: 10.1162/089976698300017331.  Google Scholar

[12]

B. Gutkin, J. Jost and H. Tuckwell, Inhibition of rhythmic neural spiking by noise: The occurrence of a minimum in activity with increasing noise,, Naturwiss., 96 (2009), 1091.  doi: 10.1007/s00114-009-0570-5.  Google Scholar

[13]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve,, J. Physiol., 117 (1952), 500.   Google Scholar

[14]

J. Honerkamp, G. Mutschler and R. Seitz, Coupling of a slow and a fast oscillator can generate bursting,, Bull. Math. Biol., 47 (1985), 1.  doi: 10.1016/S0092-8240(85)90002-3.  Google Scholar

[15]

G. Katriel, Synchronization of oscillators coupled through an environment,, Physica D, 237 (2008), 2933.  doi: 10.1016/j.physd.2008.04.015.  Google Scholar

[16]

H. Kori, Y. Kawamura and N. Masuda, Structure of cell networks critically determines oscillation regularity,, J. Theor. Biol., 297 (2012), 61.  doi: 10.1016/j.jtbi.2011.12.007.  Google Scholar

[17]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Springer Series in Synergetics, 19 (1984).  doi: 10.1007/978-3-642-69689-3.  Google Scholar

[18]

B. Lindner, A. Longtin and A. Bulsara, Analytic expressions for rate and CV of a type I neuron driven by white Gaussian noise,, Neural Comput., 15 (2003), 1761.  doi: 10.1162/08997660360675035.  Google Scholar

[19]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proc. IRE, 50 (1962), 2061.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

[20]

A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences,", Cambridge Nonlinear Science Series, 12 (2001).  doi: 10.1017/CBO9780511755743.  Google Scholar

[21]

K. Sugimoto, Y. Nii, S. Doi and S. Kumagai, Frequency variability of neural rhythm in a small network of pacemaker neurons,, Proc. of AROB 7th '02, (2002), 54.   Google Scholar

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