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Synchronization and bursting transition of the coupled Hindmarsh-Rose systems with asymmetrical time-delays

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Abstract

The study of synchronization and bursting transition is very important and valuable in cognitive activities and action control of brain as well as enhancement for the reliability of the cortex synapses. However, we wonder how the synaptic strength and synaptic delay, especially the asymmetrical time-delays between different neurons can collectively influence their synchronous firing behaviors. In this paper, based on the Hindmarsh-Rose neuronal systems with asymmetrical time-delays, we investigate the collective effects of various delays and coupling strengths on the synchronization and bursting transition. It is shown that the interplay between delay and coupling strength can not only enhance or destroy the synchronizations but also can induce the regular transitions of bursting firing patterns. Specifically, as the coupling strength or time-delay increasing, the firing patterns of the time-delayed coupling neuronal systems consistently present a regular transition, that is, the periods of spikes during the bursting firings increase firstly and then decrease slowly. In particular, in contrast to the case of symmetrical time-delays, asymmetrical time-delays can lead to the paroxysmal synchronizations of coupling neuronal systems, as well as the concentration level of synchronization for the non-identically coupled system is superior to the one of identical coupling. These results more comprehensively reveal the rich nonlinear dynamical behaviors of neuronal systems and may be helpful for us to have a better understanding of the neural coding.

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References

  1. Xie Y, Kang Y M, Liu Y, et al. Firing properties and synchronization rate in fractional-order Hindmarsh-Rose model neurons. Sci China Tech Sci, 2014, 57: 914–922

    Article  Google Scholar 

  2. Ye W J, Liu S Q, Liu X L. Synchronization of two electrically coupled inspiratory pacemaker neurons. Sci China Tech Sci, 2014, 57: 929–935

    Article  Google Scholar 

  3. Gray C M, Konig P, Engel A K, et al. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature, 1989, 338: 334–337

    Article  Google Scholar 

  4. Steinmeta P N, Roy A, Fitzgerald P J, et al. Attention modulates synchronized neuronal firing in primate somatosensory cortex. Nature, 2000, 404: 187–190

    Article  Google Scholar 

  5. Han Q K, Sun X Y, Yang X G, et al. External synchronization of two dynamical systems with uncertain parameters. Sci China Tech Sci, 2010, 53: 731–740

    Article  MATH  Google Scholar 

  6. Fell J, Fernandez G, Klaver P, et al. Is synchronized neuronal gamma activity relevant for selective attentions. Brain Res Rev, 2003, 42: 265–272

    Article  Google Scholar 

  7. Ma J, Tang J. A review for dynamics of collective behaviors of network of neurons. Sci China Tech Sci, 2015, 58: 2038–2045

    Article  Google Scholar 

  8. Izhikevich E M. Neural excitability, spiking and bursting. Intl J Bifurcat Chaos, 2012, 10: 1171–1266

    Article  MathSciNet  MATH  Google Scholar 

  9. Wu C, Martel D T, Shore S E. Increased synchrony and bursting of dorsal cochlear nucleus fusiform cells correlate with tinnitus. J Neurosci: Official J Soc Neurosci, 2016, 36: 2068–2073

    Article  Google Scholar 

  10. Song X L, Wang C N, Ma J, et al. Transition of electric activity of neurons induced by chemical and electric autapses. Sci China Tech Sci, 2015, 58: 1–8

    Article  Google Scholar 

  11. Wang H X, Wang Q Y, Zheng Y H. Bifurcation analysis for Hindmarsh-Rose neuronal model with time-delayed feedback control and application to chaos control. Sci China Tech Sci, 2014, 57: 872–878

    Article  Google Scholar 

  12. Postnova S, Voigt K, Braun H A. Neural synchronization at tonicto-bursting transitions. J Biol Phys, 2007, 33: 129–143

    Article  Google Scholar 

  13. Wang H X, Lu Q S, Wang Q Y. Bursting and synchronization transition in the coupled modified ML neurons. Commun Nonlinear Sci Nume Simul, 2007, 13: 1668–1675

    Article  MathSciNet  MATH  Google Scholar 

  14. Yuan G Y, Zhang G C, Wang G R, et al. Synchronization and asynchronization in two coupled excitable systems. Commun Theor Phys, 2005, 43: 459–465

    Article  MathSciNet  Google Scholar 

  15. Nikola B, Ines G, Nebojsa V. Type I vs. type II excitable systems with delayed coupling. Chaos Soliton Fract, 2005, 23: 1221–1233

    Article  MathSciNet  MATH  Google Scholar 

  16. Dhamala M, Viktor K J, Ding M Z. Enhancement of neural synchrony by time delay. J Phys Rev Lett, 2004, 92: 74104

    Article  Google Scholar 

  17. Rossoni E, Chen Y H, Ding M Z, et al. Stability of synchronous oscillations in a system of Hodgkin-Huxley neurons with delayed diffusive and pulsed coupling. Phys Rev E, 2005, 71: 061904

    Article  MathSciNet  Google Scholar 

  18. Duan L X, Fan D G, Lu Q S. Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling. Cognitive Neurodyn, 2013, 7: 341–349

    Article  Google Scholar 

  19. Zhang J Q, Huang S, Pang S, et al. Optimizing calculations of coupling matrix in Hindmarsh-Rose neural network. Nonlinear Dyn, 2016, 84: 1303–1310

    Article  MathSciNet  Google Scholar 

  20. Juang J, Liang Y H. Cluster synchronization in networks of neurons with chemical synapses. Chaos An Interdiscip J Nonlinear Sci, 2014, 24: 013110

    Article  MathSciNet  Google Scholar 

  21. Qin H X, Ma J, Jin W Y, et al. Dynamics of electric activities in neuron and neurons of network induced by autapses. Sci China Tech Sci, 2014, 57: 936–946

    Article  Google Scholar 

  22. Sun Z K, Yang X L, Xu W. Taming complexity in nonlinear dynamical systems by recycled signal. Sci China Tech Sci, 2016, 59: 403–410

    Article  Google Scholar 

  23. Lu Q S, Shi X. Complete synchronization of coupled Hindmarsh Rose neurons with ring structure. Chin Phys Lett, 2004, 21: 1695–1698

    Article  Google Scholar 

  24. Belykh I, de Lange E, Hasler M. Synchronization of bursting neurons: What matters in the network topology. Phys Rev Lett, 2005, 94: 188101

    Article  Google Scholar 

  25. Fan D G, Wang Q Y. Improving desynchronization of parkinsonian neuronal network via triplet-structure coordinated reset stimulation. J Theor Bio, 2015, 370: 157–170

    Article  MathSciNet  MATH  Google Scholar 

  26. Fan D G, Wang Q Y, Perc M. Disinhibition-induced transitions between absence and tonic-clonic epileptic seizures. Scient Rep, 2015, 5: 12618

    Article  Google Scholar 

  27. Sun Z K, Yang X L. Generating and enhancing lag synchronization of chaotic systems by white noise. Chaos, 2011, 21: 033114

    Article  MathSciNet  MATH  Google Scholar 

  28. Fan D G, Wang Z H, Wang Q Y. Optimal control of directional deep brain stimulation in the parkinsonian neuronal network. Commun Nonlinear Sci Numer Simul, 2016, 36: 219–237

    Article  MathSciNet  Google Scholar 

  29. Kwon O, Moon H T. Coherence resonance in small-world networks of excitable cells. Phys Lett A, 2002, 298: 319–324

    Article  MATH  Google Scholar 

  30. Wang Q Y, Duan Z S, Perc M, et al. Synchronization transitions on small-world neuronal networks: Effects of information transmission delay and rewiring probability. Europhys Lett, 2008, 83: 50008

    Article  Google Scholar 

  31. Tanakaa G, Ibarz B, Sanjuan M A F, et al. Synchronization and propagation of bursts in networks of coupled map neurons. Chaos, 2006, 16: 013113

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhou C S, Zemanov L, Zamora G, et al. Hierarchical organization unveiled by functional connectivity in complex brain networks. Phys Rev Lett, 2006, 97: 238103

    Article  Google Scholar 

  33. Leyva I, Sendina-Nadal I, Almendral J A, et al. Sparse repulsive coupling enhances synchronization in complex networks. Phys Rev E, 2006, 74: 056112

    Article  Google Scholar 

  34. Wu S, Wang D, Okubo S. Mechanical engineering control system for the linear time delay with disturbances. In: Proceedings of IEEE 10th International Conference on Computer-Aided Industrial Design & Conceptual Design. IEEE, 2009. 2267–2270

    Google Scholar 

  35. Karimi H R, Moshiri B, Lucas C. Robust fuzzy linear control of a class of stochastic nonlinear time-delay systems. Nonlinear Dyn Syst Theory, 2004, 3: 317–332

    MathSciNet  MATH  Google Scholar 

  36. Konen C S, Kleiser R, Bremmer F, et al. Global synchronization of general complex dynamical networks with time-varying delay. RO-MAN IEEE, 2007, 182: 833–838

    Google Scholar 

  37. Wu J S, Jiao L. Synchronization in dynamic networks with nonsymmetrical time-delay coupling based on linear feedback controllers. Phys A Stat Mech Its Appl, 2008, 387: 2111–2119

    Article  Google Scholar 

  38. Yang X L, Senthilkumar D V, Sun Z K, et al. Key role of time delay and connection topology in shaping the dynamics of noisy genetic regulatory networks. Chaos, 2011, 21: 047522

    Article  Google Scholar 

  39. Sun Z K, Fu J, Xiao Y Z, et al. Delay-induced stochastic bifurcations in a bistable system under white noise. Chaos, 2015, 25: 083102

    Article  MathSciNet  Google Scholar 

  40. Sun Z K, Yang X L, Xiao Y Z, et al. Modulating resonance behaviors by noise recycling in bistable systems with time delay. Chaos, 2014, 24: 023126

    Article  MathSciNet  MATH  Google Scholar 

  41. Ott W, Rivas M A, West J. Observing Lyapunov exponents of infinite-dimensional dynamical systems. J Stat Phys, 2015, 161: 1098–1111

    Article  MathSciNet  MATH  Google Scholar 

  42. Rao R P N, Sejnowski T J. Spike-time-dependent Hebbian plasticity as temporal difference learning. Neural Comp, 2001, 13: 2221–2237

    Article  MATH  Google Scholar 

  43. Masquelier T, Guyonneau R, Thorpe S J. Competitive STDP based spike pattern learning. Neural Comp, 2008, 21: 1–18

    MATH  Google Scholar 

  44. Shampine L F, Thompson S. Solving DDEs in Matlab. Appl Numer Math, 2001, 37: 441–458

    Article  MathSciNet  MATH  Google Scholar 

  45. Hu H Y, Wang Z H. The research progress of nonlinear dynamical systems with time-delay. Adv Mech, 1999, 29: 501–512

    Google Scholar 

  46. Wang Q Y, Shi X, Lu Q S. Synchronous Dynamics of the Coupled Neural System. Beijing: Science Press, 2008

    Google Scholar 

  47. Rosin B, Slovik M, Mitelman R, et al. Closed-loop deep brain stimulation is superior in ameliorating parkinsonism. Neuron, 2011, 72: 370–384

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Dynamics and Control, Beihang University, Beijing, 100191, China

    DengGui Fan & QingYun Wang

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  1. DengGui Fan
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  2. QingYun Wang
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Correspondence to QingYun Wang.

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Fan, D., Wang, Q. Synchronization and bursting transition of the coupled Hindmarsh-Rose systems with asymmetrical time-delays. Sci. China Technol. Sci. 60, 1019–1031 (2017). https://doi.org/10.1007/s11431-016-0169-8

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  • DOI: https://doi.org/10.1007/s11431-016-0169-8

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