# American Institute of Mathematical Sciences

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:
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##### 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. [2] L. Cheng and B. Ermentrout, Analytic approximations of statistical quantities and response of noisy oscillators,, Physica D, 240 (2011), 719. [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. [4] E. J. Doedel and B. E. Oldeman, et al., AUTO-07P: Continuation and bifurcation software for ordinary differential equations,, Concordia University, (2009). [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. [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. [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. [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. [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. [10] L. Glass, Synchronization and rhythmic processes in physiology,, Nature, 410 (2001), 277. doi: 10.1038/35065745. [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. [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. [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. [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. [15] G. Katriel, Synchronization of oscillators coupled through an environment,, Physica D, 237 (2008), 2933. doi: 10.1016/j.physd.2008.04.015. [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. [17] Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Springer Series in Synergetics, 19 (1984). doi: 10.1007/978-3-642-69689-3. [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. [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. [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. [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.
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