Skip to main content
SpringerLink
Log in

A review for dynamics of collective behaviors of network of neurons

  • Review
  • Published:
Science China Technological Sciences Aims and scope Submit manuscript

Abstract

The nervous system is composed of a large number of neurons, and the electrical activities of neurons can present multiple modes during the signal transmission between neurons by changing intrinsic bifurcation parameters or under appropriate external forcing. In this review, the dynamics for neuron, neuronal network is introduced, for example, the mode transition in electrical activity, functional role of autapse connection, bifurcation verification in biological experiments, interaction between neuron and astrocyte, noise effect, coherence resonance, pattern formation and selection in network of neurons. Finally, some open problems in this field such as electromagnetic radiation on electrical activities of neuron, energy consumption in neurons are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hodgkin A L, Huxley A F. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol, 1952, 117: 500–544

    Article  Google Scholar 

  2. Fromherz P, Müller C O. Cable properties of a straight neurite of a leech neuron probed by a voltage-sensitive dye. Proc. Natl. Acad. Sci. USA, 1994, 91: 4604–4608.

    Article  Google Scholar 

  3. Hsagawa H. Responses of a Hodgkin-Huxley neuron to various types of spike-train inputs. Phys Rev E, 2000, 61: 718–726

    Article  Google Scholar 

  4. Hindmarsh J L, Rose R M. A model of neuronal bursting using three coupled first-order differential equations. Proc R Soc Lond B, 1984, 221: 87–102

    Article  Google Scholar 

  5. Izhikevich E M. Which model to use for cortical spiking neurons? IEEE T Neural Netw, 2004, 15: 1063–1070

    Article  Google Scholar 

  6. Fitzhugh R. Mathematical models of threshold phenomena in the nerve membrane. Bull Math Biophys, 1955, 17: 257–278

    Article  Google Scholar 

  7. Izhikevich E M. Dynamical Systems in Neuroscience: The geometry of excitability and bursting. The MIT Press, Cambridge, MA, 2007

    Google Scholar 

  8. Liu F, Wang J F, Wang W. Frequency sensitivity in weak signal detection. Phys Rev E, 1999, 59: 3453–3460

    Article  Google Scholar 

  9. Mark D M, Derek A. What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology. PLoS Comput Biol, 2009, 5: e1000348

    Article  Google Scholar 

  10. Wang W, Chen G, Wang Z D. 40-Hz coherent oscillations in neuronal systems. Phys Rev E, 1997, 56: 3728–3731

    Article  Google Scholar 

  11. Hongeycutt R L. Stochastic Runge-Kutta algorithms. I. White noise. Phys Rev A, 1992, 45: 600–603

    Google Scholar 

  12. Sagués F, Sancho J M, García- Ohalvo J. Spatiotemporal order out of noise. Rev Mod Phys, 2007, 79: 829–882

    Article  Google Scholar 

  13. Yang J, Zhou W N, Shi P, et al. Adaptive synchronization of delayed Markovian switching neural networks with Levy noise. Neurocomput, 2015, 156: 231–238

    Article  Google Scholar 

  14. Xu Y, Wang X Y, Zhang H Q. Stochastic stability for nonlinear systems driven by Levy noise. Nonlinear Dyn, 2012, 68: 7–15

    Article  MathSciNet  MATH  Google Scholar 

  15. Bazsó F, Zalányi, Csárdi G. Channel noise in Hodgkin–Huxley model neurons. Phys Lett A, 2003, 311: 13–20

    Article  MathSciNet  MATH  Google Scholar 

  16. Parpura V, Basarsky T A, Liu F, et al. Glutamate-mediated astrocyte-neuron signalling. Nature, 1994, 369: 744–747

    Article  Google Scholar 

  17. Pitta M D, Volman V, Berry H, et al. A tale of two stories: Astrocyte regulation of synaptic depression and facilitation. PLoS Comput Biol, 2011, 7: e1002293

    Article  Google Scholar 

  18. Sinha S, Saramaki J, Kaski K. Emergence of self-sustained patterns in small-world excitable media. Phys Rev E, 2007, 76: 015101

    Article  Google Scholar 

  19. Erichsen R, Brunnet L G. Multistability in networks of Hindmarsh-Rose neurons. Phys Rev E, 2008, 78: 061917

    Article  Google Scholar 

  20. Bekkers J M. Synaptic transmission: Functional autapses in the cortex. Curr Biol, 2003, 13: 433–435

    Article  Google Scholar 

  21. Volman V, Bazhenov M, Sejnowski T J. Computational models o neuron-astrocyte interaction in epilepsy. Front Comput Neurosci, 2012, 6: 58

    Article  Google Scholar 

  22. Sjöström P J, Rancz E A, Roth A. Häusser M. Dendritic excitability and synaptic plasticity. Physiol Rev, 2008, 88: 769–840

    Google Scholar 

  23. Gerstner W, Kistler W M. Spiking Neuron Models Single Neurons, Populations, Plasticity. Cambridge University Press, 2002

    Google Scholar 

  24. Holtmaat A, Svoboda K. Experience-dependent structural synaptic plasticity in the mammalian brain. Nature Rev Neurosci, 2009, 10: 647–658

    Article  Google Scholar 

  25. Huang X H, Hu G. Plasticity-induced characteristic changes of pattern dynamics and the related phase transitions in small-world neuronal networks. Chin Phys B, 2014, 23: 0108703

    Article  MathSciNet  Google Scholar 

  26. 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 

  27. Yang Z Q, Hao L J. Dynamics of different compound bursting in two phantom bursting mechanism models. Sci China Tech Sci, 2014, 57: 885–892

    Article  Google Scholar 

  28. Duan L X, Wang Q Y, Lu Q S. Two-parameter bifurcation analysis of firing activities in the Chay neuronal model. Neurocomput, 2008, 72: 341–351

    Article  Google Scholar 

  29. 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 

  30. Shi M, Wang Z H. Abundant bursting patterns of a fractional-order Morris–Lecar neuron model. Commun Nonlinear Sci Numer Simul, 2014, 19: 1956–1969

    Article  MathSciNet  Google Scholar 

  31. Storace M, Linaro D, de Lange E. The Hindmarsh–Rose neuron model: Bifurcation analysis and piecewiselinear approximations. Chaos, 2008, 18: 033128

    Article  MathSciNet  Google Scholar 

  32. Rich P C. Dynamics in the parameter space of a neuron model. Chin Phys Lett, 2012, 29: 060506

    Article  Google Scholar 

  33. Gu H G, Pan B B, Xu J. Experimental observation of spike, burst and chaos synchronization of calcium concentration oscillations. EPL, 2014, 106: 50003

    Article  Google Scholar 

  34. Gu H G, Chen S G. Potassium-induced bifurcations and chaos of firing patterns observed from biological experiment on a neural pacemaker. Sci China Tech Sci, 2014, 57: 864–871

    Article  Google Scholar 

  35. Zheng H W, Wang R B, Qiao L K, et al. The molecular dynamics of neural metabolism during the action potential. Sci China Tech Sci, 2014, 57: 857–863

    Article  Google Scholar 

  36. Bélanger M, Allaman I, Magistretti P J. Brain energy metabolism: focus on astrocyte-neuron metabolic cooperation. Cell Metab, 2011, 14: 724–738

    Article  Google Scholar 

  37. Sarasola C, Torrealdea FJ, D’Anjou A, et al. Energy balance in feedback synchronization of chaotic systems. Phys Rev E, 2004, 69: 011606

    Article  Google Scholar 

  38. Kobe D H. Helmholtz’s theorem revisited. Am J Phys, 1986, 54: 552–554

    Article  Google Scholar 

  39. Schmid G, Goychuk I, Hänggi P. Controlling the spiking activity in excitable membranes via poisoning. Physica A 2004, 344: 665–670

    Article  Google Scholar 

  40. Crotty P, Schult D, Segall K. Josephson junction simulation of neurons. Phys Rev E, 2010, 82: 011914

    Article  Google Scholar 

  41. Nägler K, Mauch D H, Pfrieger F W. Glia-derived signals induce synapse formation in neurons of the rat central nervous system. J Physiol, 2001, 533: 665–679

    Article  Google Scholar 

  42. Herrmann C S, Klaus A. Autapse turns neuron into oscillator. Int J Bifurcat Chaos, 2004, 14: 623–633

    Article  MathSciNet  MATH  Google Scholar 

  43. Yun Y L, Schmid G, Hänggi P, et al. Spontaneous spiking in an autaptic Hodgkin-Huxley setup. Phys Rev E, 2010, 82: 061907

    Article  MathSciNet  Google Scholar 

  44. 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: 1007–1014

    Article  Google Scholar 

  45. Wang H X, Wang Q Y, Lu Q S, et al. Equilibrium analysis and phase synchronization of two coupled HR neurons with gap junction. Cogn Neurodyn, 2013, 7: 121–131

    Article  Google Scholar 

  46. Wang L, Zeng Y J. Control of bursting behavior in neurons by autaptic modulation. Neurological Sci, 2013, 34: 1977194

    Google Scholar 

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

    Article  Google Scholar 

  48. Qin H X, Ma J, Wang C N, et al. Autapse-induced target wave, spiral wave in regular network of neurons. Sci China Phys Mech Astron, 2014, 57: 1918–1926

    Article  Google Scholar 

  49. Huang X Y, Xu W F, Liang J M, et al. Spiral wave dynamics in neocortex, Neuron, 2010, 60: 978–990

  50. Wang Q Y, Zheng Y H, Ma J. Cooperative dynamics in neuronal networks. Chaos Solitons Fractals, 2013, 56: 19–27

    Article  Google Scholar 

  51. Song Z G, Xu J. Stability switches and Bogdanov-Takens bifurcation in an inertial two-neuron coupling system with multiple delays. Sci China Tech Sci, 2014, 57: 893–904

    Article  Google Scholar 

  52. Sun X J, Shi X. Effects of channel blocks on the spiking regularity in clustered neuronal networks. Sci China Tech Sci, 2014, 57: 879–884

    Article  Google Scholar 

  53. Jiao X F, Zhu D F. Phase-response synchronization in neuronal population. Sci China Tech Sci, 2014, 57: 923–928

    Article  Google Scholar 

  54. 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 

  55. Zhou J, Wu Y J, Liu Z R. Distributed coordinated adaptive tracking in networked redundant robotic systems with a dynamic leader. Sci China Tech Sci, 2014, 57: 905–913

    Article  Google Scholar 

  56. Zhou J, Wu Q J, Xiang L. Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization. Nonlinear Dyn, 2012, 68: 1393–1403

    Article  MathSciNet  Google Scholar 

  57. Cao J D, Wang J. Global asymptotic stability of a general class of recurrent neural networks with time-varying delays. IEEE T Circ Sys I Fund Theo Appl, 2003, 50: 34–44

    Article  Google Scholar 

  58. Cao J D, Liang J L. Boundedness and stability for Cohen-Grossberg neural network with time-varying delays. J Math Anal Appl, 2004, 296: 665–685

    Article  MathSciNet  MATH  Google Scholar 

  59. Cao J D, Lu J Q. Adaptive synchronization of neural networks with or without time-varying delay. Chaos, 2006, 16: 013133

    Article  MathSciNet  Google Scholar 

  60. Marinov C A, Hopfield J J. Stable computational dynamics for a class of circuits with O(N) Interconnections capable of KWTA and rank extractions. IEEE T Circ Syst I, 2005, 52: 949–95

    Article  MathSciNet  Google Scholar 

  61. Armstrong R A. A spatial pattern analysis of beta-amyloid (A beta) deposition in the temporal lobe in Alzheimer’s disease. Folia Neuropathol, 2010, 48: 67–74

    Google Scholar 

  62. Itoh M, Chua L O. Memristor oscillators. Int J Bifurcation Chaos, 2008, 18: 3183–3206

    Article  MathSciNet  MATH  Google Scholar 

  63. Volos C K, Kyprianidis L M, Stouboulos I N, et al. Memristor: A new concept in synchronization of coupled neuromorphic circuits. J Engin Sci Tech Rev, 2015, 8: 157–173

    Google Scholar 

  64. Song X L, Jin W Y, Ma J. Energy dependence on the electric activities of neuron. Chin Phys B, 2015, 24: 128710

    Article  Google Scholar 

  65. Chen W, Rolls E T, Gu H G, et al. Autism: reduced connectivity between cortical areas involved in face expression, theory of mind, and the sense of self. Brain, 2015, 138: 1328–1393

    Google Scholar 

  66. Gu H G, Pan B B, Chen G R, et al. Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models. Nonlinear Dyn, 2014, 78: 391–407

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Lanzhou University of Technology, Lanzhou, 730050, China

    Jun Ma

  2. College of Science, China University of Mining and Technology, Xuzhou, 221116, China

    Jun Tang

Authors
  1. Jun Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Ma.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, J., Tang, J. A review for dynamics of collective behaviors of network of neurons. Sci. China Technol. Sci. 58, 2038–2045 (2015). https://doi.org/10.1007/s11431-015-5961-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11431-015-5961-6

Keywords

Search

Navigation