Journal of Computational Neuroscience

, Volume 31, Issue 1, pp 61–71

Firing responses of bursting neurons with delayed feedback



Thalamic neurons, which play important roles in the genesis of rhythmic activities of the brain, show various bursting behaviors, particularly modulated by complex thalamocortical feedback via cortical neurons. As a first step to explore this complex neural system and focus on the effects of the feedback on the bursting behavior, a simple loop structure delayed in time and scaled by a coupling strength is added to a recent mean-field model of bursting neurons. Depending on the coupling strength and delay time, the modeled neurons show two distinct response patterns: one entrained to the unperturbed bursting frequency of the neurons and one entrained to the resonant frequency of the loop structure. Transitions between these two patterns are explored in the model’s parameter space via extensive numerical simulations. It is found that at a fixed loop delay, there is a critical coupling strength at which the dominant response frequency switches from the unperturbed bursting frequency to the loop-induced one. Furthermore, alternating occurrence of these two response frequencies is observed when the delay varies at fixed coupling strength. The results demonstrate that bursting is coupled with feedback to yield new dynamics, which will provide insights into such effects in more complex neural systems.


Bursting neurons Delayed feedback Mean-field modeling Thalamic neurons Corticothalamic system 


  1. Attay, F. M., & Hutt, A. (2006). Neural fields with distributed transmission speeds and long-range feedback delays. SIAM Journal of Applied Dynamical Systems, 5, 670–698.CrossRefGoogle Scholar
  2. Belykh, I., de Lange, E., & Hasler, M. (2005). Synchronization of bursting neurons: What matters in the network topology. Physical Review Letters, 94, 188101:1–4.CrossRefGoogle Scholar
  3. Campbell, S. A. (2007). Time delays in neural systems. Neuronal dynamics and brain connectivity. Berlin: Springer.Google Scholar
  4. Contreras, D., Destexhe, A., Sejnowski, T. J., & Steriade, M. (1996). Control of spatiotemporal coherence of a thalamic oscillation by corticothalamic feedback. Science, 274, 771–774.PubMedCrossRefGoogle Scholar
  5. Contreras, D., & Steriade, M. (1996). Spindle oscillation in cats: The role of corticothalamic feedback in a thalamically generated rhythm. Journal of Physiology, 490, 159–179.PubMedGoogle Scholar
  6. Cui, J., Canavier, C. C., & Butera, R. J. (2009). Functional phase response curves: A method for understanding synchronization of adapting neurons. Journal of Neurophysiology, 102, 387–398.PubMedCrossRefGoogle Scholar
  7. Deco, G., Jirsa, V. K., Robinson, P. A., Breakspear, M., & Friston, K. (2008). The dynamics brain: From spiking neurons to neural masses and cortical fields. PLoS Computational Biology, 4, e1000092.PubMedCrossRefGoogle Scholar
  8. Dhamala, M., Jirsa, V. K., & Ding, M. (2004). Enhancement of neural synchrony by time delay. Physical Review Letters, 92, 074104:1–4.Google Scholar
  9. Doiron, B., Charcron, M. J., Maler, L., Longtin, A., & Bastian, J. (2003). Inhibitory feedback required for network oscillatory responses to communication but not prey stimuli. Nature, 421, 539–543.PubMedCrossRefGoogle Scholar
  10. Ermentrout, G. B., & Kopell, N. (1998). Fine structure of neural spiking and synchronization in the presence of conduction delays. Proceedings of the National Academy of Sciences of the United States of America, 95, 1259–1264.PubMedCrossRefGoogle Scholar
  11. Foss, J., & Milton, J. (2000). Multistability in recurrent neural loops arising from delay. Journal of Neurophysiology, 84, 975–985.PubMedGoogle Scholar
  12. Gutierrez, C., Cox, L. C., Rinzel, J., & Sherman, S. M. (2001). Dynamics of low-threshold spike activation in relay neurons of the cat lateral geniculate nucleus. Journal of Neuroscience, 21, 1022–1032.PubMedGoogle Scholar
  13. Izhikevich, E. M. (2007). Dynamical systems in neuroscience: The geometry of excitability and bursting. London: MIT Press.Google Scholar
  14. Lopes da Silva, F. H., Hoeks, A., Smits, H., & Zetterberg, L. H. (1974). Model of brain rhythmic activity, the alpha rhythm of the thalamus. Kybernetik, 15, 270–237.CrossRefGoogle Scholar
  15. McCormick, D. A., & Contreras, D. (2001). On the cellular and network bases of epileptic seizures. Annual Review of Physiology, 63, 815–846.PubMedCrossRefGoogle Scholar
  16. McCormick, D. A., & Feeser, H. R. (1990). Functional implications of burst firing and single spike activity in lateral geniculate relay neurons. Neuroscience, 39, 103–113.PubMedCrossRefGoogle Scholar
  17. McCormick, D. A., & Huguenard, J. R. (1992). A model of the electrophysiological properties of thalamocortical relay neurons. Journal of Neurophysiology, 68, 1384–1400.PubMedGoogle Scholar
  18. Meeren, K. M., Pijn, J. P. M., van Luijtelaar, E. L. J. M., Coenen, A. M. L., & Lopes da Silva, F. H. (2002). Cortical focus drives widespread corticothalamic networks during spontaneous absence seizures in rats. Journal of Neuroscience, 22, 1480–1495.PubMedGoogle Scholar
  19. Nunez, P. L. (1974). Wave-like properties of the alpha rhythm. IEEE Transactions on Biomedical Engineering, 21, 473–482.CrossRefGoogle Scholar
  20. Plant, R. E. (1981). A Fritzhugh differential-difference equation modeling recurrent neural feedback. SIAM Journal of Applied Mathematics, 40, 150–162.CrossRefGoogle Scholar
  21. Robinson, P. A. (2005). Propagator theory of brain dynamics. Physical Review E, 72, 011904:1–13.CrossRefGoogle Scholar
  22. Robinson, P. A., Loxley, P. N., O’Connor, S. C., & Rennie, C. J. (2001). Modal analysis of corticothalamic dynamics, electroencephalographic spectra, and evoked potentials. Physical Review E, 63, 041909:1–13.Google Scholar
  23. Robinson, P. A., Rennie, C. J., & Rowe, D. L. (2002). Dynamics of large-scale brain activity in normal arousal states and epileptic seizures. Physical Review E, 65, 041924:1–9.CrossRefGoogle Scholar
  24. Robinson, P. A., Rennie, C. J., & Wright, J. J. (1997). Propagation and stability of waves of electrical activity in the cerebral cortex. Physical Review E, 56, 826–840.CrossRefGoogle Scholar
  25. Robinson, P. A., Rennie, C. J., Rowe, D. L., & O’Connor, S. C. (2004). Estimation of multiscale neurophysiologic parameters by electroencephalographic means. Human Brain Mapping, 23, 53–72.PubMedCrossRefGoogle Scholar
  26. Robinson, P. A., Rennie, C. J., Rowe, D. L., O’Connor, S. C., Wright, J. J., Gorden, E., et al. (2003a). Neurophysical modeling of brain dynamics. Neuropsychopharmacology, 28, S74–S79.PubMedCrossRefGoogle Scholar
  27. Robinson, P. A., Whitehouse, R. W., & Rennie, C. J. (2003b). Nonuniform corticothalamic continuum model of electroencephalographic spectra with application to split-alpha peaks. Physical Review E, 68, 021922:1–10.CrossRefGoogle Scholar
  28. Robinson, P. A., Wu, H., & Kim, J. W. (2007). Neural rate equations for bursting dynamics derived from conductance-based equations. Journal of Theoretical Biology, 250, 663–672.PubMedCrossRefGoogle Scholar
  29. Shepherd, G. M. (2004). The synaptic organization of the brain. New York: Oxford University Press.CrossRefGoogle Scholar
  30. Sillito, A. M., & Jones, H. E. (2002). Corticothalamic interactions in the transfer of visual information. Philosophical Transactions of the Royal Society B, 357, 1739–1752.CrossRefGoogle Scholar
  31. Steriade, M., Jones, E. G., & Llinàs, R. R. (1990). Thalamic oscillations and signaling. New York: Wiley.Google Scholar
  32. Tóth, T., & Crunelli, V. (1992). Computer simulation of the pacemaker oscillations of thalamocortical cells. Neuroreport, 3, 65–68.PubMedCrossRefGoogle Scholar
  33. Wallenstein, G. V. (1994). A model of the electrophysiological properties of nucleus reticularis thalamic neurons. Biophysical Journal, 66, 978–988.PubMedCrossRefGoogle Scholar
  34. Wang, X. J. (1994). Multiple dynamical modes of thalamic relay neurons: Rhythmic bursting and intermittent phase-locking. Neuroscience, 59, 21–31.PubMedCrossRefGoogle Scholar
  35. Wilson, H. R. (1999a). Simplified dynamics of human and mammalian neocortical neurons. Journal of Theoretical Biology, 200, 375–388.PubMedCrossRefGoogle Scholar
  36. Wilson, H. R. (1999b). Spikes, decisions, and actions. New York: Oxford University Press.Google Scholar
  37. Wright, J. J., & Liley, D. T. J. (1996). Dynamics of the brain at global and microscopic scales: Neural networks and the EEG. Behavioral and Brain Sciences, 19, 285–309.CrossRefGoogle Scholar
  38. Wu, H., Kim, J. W., Robinson, P. A., & Drysdale, P. M. (2009). Firing pattern of bursting neurons under sinusoidal drive in mean-field modeling. Journal of Theoretical Biology, 259, 101–108.PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Hui-Ying Wu
    • 1
    • 2
  • Peter A. Robinson
    • 1
    • 2
    • 3
  • Jong Won Kim
    • 1
    • 2
    • 3
  1. 1.School of PhysicsThe University of SydneySydneyAustralia
  2. 2.Brain Dynamics Center, Sydney Medical School–WesternThe University of SydneyWestmeadAustralia
  3. 3.Center for Integrated Research and Understanding of SleepWoolcock Institute of Medical ResearchGlebeAustralia

Personalised recommendations