Nonlinear Dynamics

, Volume 84, Issue 3, pp 1541–1551 | Cite as

A neural network model of spontaneous up and down transitions

  • Xuying Xu
  • Li Ni
  • Rubin Wang
Original Paper


Spontaneous periodic up and down transitions are considered to be a significant phenomenon that is characteristic of slow-wave sleep. We studied a neural network model of spontaneous up and down transitions based on our former study of a single-neuron model. We expanded the model by using two types of neurons—excitatory and inhibitory neurons—and redefining the connecting function between two neurons instead of assuming a constant connection, so that the model is closer to the in vivo network. Using this model, we studied the relationship between the transitions and network parameters such as size, structure and the ratio of excitatory to inhibitory neurons. We found that the network parameters have little impact on these spontaneous periodic up and down transitions. However, the intrinsic currents were found to play a leading role in the process. Then, we studied the transitions in the presence of stimulation and found that the addition of stimulation did have an effect on the network transitions. Through the observation and analysis of the findings, we have tried to explain the dynamics of up and down transitions and to lay the foundation for future work on the role of these transitions in cortex activity.


Spontaneity Up and down transition Ionic channel Chemical synapse Subthreshold activity 



This work is supported by the National Natural Science Foundation of China (No. 11232005) and The Ministry of Education Doctoral Foundation (No. 20120074110020).


  1. 1.
    Parga, N., Abbott, L.F.: Network model of spontaneous activity exhibiting synchronous transitions between up and down states. Front. Neurosci. 1(1), 57 (2007)CrossRefGoogle Scholar
  2. 2.
    Anderson, J., Lampl, I., Reichova, I., Carandini, M., Ferster, D.: Stimulus dependence of two-state fluctuations of membrane potential in cat visual cortex. Nat. Neurosci. 3(6), 617–621 (2000)CrossRefGoogle Scholar
  3. 3.
    Lampl, I., Reichova, I., Ferster, D.: Synchronous membrane potential fluctuations in neurons of the cat visual cortex. Neuron 22(2), 361–374 (1999)CrossRefGoogle Scholar
  4. 4.
    Steriade, M., Nuñez, A., Amzica, F.: Intracellular analysis of relations between the slow (1 Hz) neocortical oscillation and other sleep rhythms of the electroencephalogram. J. Neurosci. 13(8), 3266–3283 (1993)Google Scholar
  5. 5.
    Petersen, C.C., Hahn, T.T., Mehta, M., Grinvald, A., Sakmann, B.: Interaction of sensory responses with spontaneous depolarization in layer 2/3 barrel cortex. Proc. Natl. Acad. Sci. 100(23), 13638–13643 (2003)CrossRefGoogle Scholar
  6. 6.
    Araki, O.: Computer simulations of synchrony and oscillations evoked by two coherent inputs. Cogn. Neurodyn. 7(2), 133–141 (2013)CrossRefGoogle Scholar
  7. 7.
    Ma, J., Hu, B., Wang, C., Jin, W.: Simulating the formation of spiral wave in the neuronal system. Nonlinear Dyn. 73(1), 73–83 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Wang, G., Jin, W., Wang, A.: Synchronous firing patterns and transitions in small-world neuronal network. Nonlinear Dyn. 81(3), 1453–1458 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhou, J., Wu, Q., Xiang, L.: Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization. Nonlinear Dyn. 69(3), 1393–1403 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Liu, Y., Wang, R., Zhang, Z., Jiao, X.: Analysis of stability of neural network with inhibitory neurons. Cogn. Neurodyn. 4(1), 61–68 (2010)CrossRefGoogle Scholar
  11. 11.
    Gu, H., Pan, B., Li, Y.: The dependence of synchronization transition processes of coupled neurons with coexisting spiking and bursting on the control parameter, initial value, and attraction domain. Nonlinear Dyn. 82(3), 1191–1210 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Njap, F., Claussen, J.C., Moser, A., Hofmann, U.G.: Modeling effect of GABAergic current in a basal ganglia computational model. Cogn. Neurodyn. 6(4), 333–341 (2012)CrossRefGoogle Scholar
  13. 13.
    Zheng, C., Zhang, T.: Alteration of phase-phase coupling between theta and gamma rhythms in a depression-model of rats. Cogn. Neurodyn. 7(2), 167–172 (2013)CrossRefGoogle Scholar
  14. 14.
    Wang, Q., Zheng, Y., Ma, J.: Cooperative dynamics in neuronal networks. Chaos Solions Fractals 56(7), 19–27 (2013)Google Scholar
  15. 15.
    Perc, M.: Spatial coherence resonance in excitable media. Phys. Rev. E 72(1), 016207 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gosak, M., Marhl, M., Perc, M.: Spatial coherence resonance in excitable biochemical media induced by internal noise. Biophys. Chem. 128(2), 210–214 (2007)CrossRefMATHGoogle Scholar
  17. 17.
    Perc, M., Gosak, M., Marhl, M.: Periodic calcium waves in coupled cells induced by internal noise. Chem. Phys. Lett. 437(1), 143–147 (2007)CrossRefGoogle Scholar
  18. 18.
    DiFrancesco, D.: Pacemaker mechanisms in cardiac tissue. Annu. Rev. Physiol. 55(1), 455–472 (1993)CrossRefGoogle Scholar
  19. 19.
    Lüthi, A., McCormick, D.A.: H-current: properties of a neuronal and network pacemaker. Neuron 21(1), 9–12 (1998)CrossRefGoogle Scholar
  20. 20.
    Pape, H.C.: Queer current and pacemaker: the hyperpolarization-activated cation current in neurons. Annu. Rev. Physiol. 58(1), 299–327 (1996)CrossRefGoogle Scholar
  21. 21.
    Dickson, C.T., Magistretti, J., Shalinsky, M.H., Fransén, E., Hasselmo, M.E., Alonso, A.: Properties and role of Ih in the pacing of subthreshold oscillations in entorhinal cortex layer II neurons. J. Neurophysiol. 83(5), 2562–2579 (2000)Google Scholar
  22. 22.
    Wilson, A., Cowan, J.D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12(1), 1–24 (1972)CrossRefGoogle Scholar
  23. 23.
    Compte, A., Sanchez-Vives, M.V., McCormick, D.A., Wang, X.J.: Cellular and network mechanisms of slow oscillatory activity (1 Hz) and wave propagations in a cortical network model. J. Neurophysiol. 89, 2707–2725 (2003)CrossRefGoogle Scholar
  24. 24.
    Holcman, D., Tsodyks, M.: The emergence of up and down states in cortical networks. PLOS Comput. Biol. 2, 174–181 (2006)CrossRefGoogle Scholar
  25. 25.
    Yang, Z., Wang, Q., Danca, M.F., Zhang, J.: Complex dynamics of compound bursting with burst episode composed of different bursts. Nonlinear Dyn. 70(3), 2003–2013 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ma, J., Tang, J.: A review for dynamics of collective behaviors of network of neurons. Sci. China Technol. Sci. 58, 2038–2045 (2015)CrossRefGoogle Scholar
  27. 27.
    Xu, X., Wang, R.: Neurodynamics of up and down transitions in a single neuron. Cogn. Neurodyn. 8(6), 509–515 (2014)CrossRefGoogle Scholar
  28. 28.
    Xu, X., Wang, R.: Neurodynamics of up and down transitions in network model. Abstr. Appl. Anal. 2013, 9 (2013). doi: 10.1155/2013/486178
  29. 29.
    Wang, R., Zhang, Z., Chen, G.: Energy coding and energy functions for local activities of the brain. Neurocomputing 73(1), 139–150 (2009)CrossRefGoogle Scholar
  30. 30.
    Wang, R., Zhang, Z., Chen, G.: Energy function and energy evolution on neuronal populations. Neural Netw. IEEE Trans. 19(3), 535–538 (2008)CrossRefGoogle Scholar
  31. 31.
    Wang, R., Zhang, Z.: Energy coding in biological neural networks. Cogn. Neurodyn. 1(3), 203–212 (2007)CrossRefGoogle Scholar
  32. 32.
    Wang, Q.Y., Murks, A., Perc, M., Lu, Q.S.: Taming desynchronized bursting with delays in the Macaque cortical network. Chin. Phys. B 20(4), 040504 (2011)CrossRefGoogle Scholar
  33. 33.
    Loewenstein, Y., Mahon, S., Chadderton, P., Kitamura, K., Sompolinsky, H., Yarom, Y., Häusser, M.: Bistability of cerebellar Purkinje cells modulated by sensory stimulation. Nat. Neurosci. 8(2), 202–211 (2005)CrossRefGoogle Scholar
  34. 34.
    Koch, C., Segev, I.: Methods in neuronal modeling: from ions to networks. MIT Press, Cambridge (1998)Google Scholar
  35. 35.
    Destexhe, A., Mainen, Z.F., Sejnowski, T.J.: Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism. J. Comput. Neurosci. 1(3), 195–230 (1994)CrossRefGoogle Scholar
  36. 36.
    Bazhenov, M., Timofeev, I., Steriade, M., Sejnowski, T.J.: Potassium model for slow (2–3 Hz) in vivo neocortical paroxysmal oscillations. J. Neurophysiol. 92(2), 1116 (2004)CrossRefGoogle Scholar
  37. 37.
    Ermentrout, T.J., Terman, D.H.: Mathematical foundations of neuroscience. Springer, Berlin (2010)Google Scholar
  38. 38.
    Li, C.Y., Poo, M.M., Dan, Y.: Burst spiking of a single cortical neuron modifies global brain state. Science 324(5927), 643–646 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institute for Cognitive NeurodynamicsEast China University of Science and TechnologyShanghaiPeople’s Republic of China

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