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Journal of Computational Neuroscience

, Volume 43, Issue 3, pp 203–225 | Cite as

Stimulus-induced transitions between spike-wave discharges and spindles with the modulation of thalamic reticular nucleus

  • Denggui Fan
  • Qingyun WangEmail author
  • Jianzhong Su
  • Hongguang Xi
Article

Abstract

It is believed that thalamic reticular nucleus (TRN) controls spindles and spike-wave discharges (SWD) in seizure or sleeping processes. The dynamical mechanisms of spatiotemporal evolutions between these two types of activity, however, are not well understood. In light of this, we first use a single-compartment thalamocortical neural field model to investigate the effects of TRN on occurrence of SWD and its transition. Results show that the increasing inhibition from TRN to specific relay nuclei (SRN) can lead to the transition of system from SWD to slow-wave oscillation. Specially, it is shown that stimulations applied in the cortical neuronal populations can also initiate the SWD and slow-wave oscillation from the resting states under the typical inhibitory intensity from TRN to SRN. Then, we expand into a 3-compartment coupled thalamocortical model network in linear and circular structures, respectively, to explore the spatiotemporal evolutions of wave states in different compartments. The main results are: (i) for the open-ended model network, SWD induced by stimulus in the first compartment can be transformed into sleep-like slow UP-DOWN and spindle states as it propagates into the downstream compartments; (ii) for the close-ended model network, weak stimulations performed in the first compartment can result in the consistent experimentally observed spindle oscillations in all three compartments; in contrast, stronger periodic single-pulse stimulations applied in the first compartment can induce periodic transitions between SWD and spindle oscillations. Detailed investigations reveal that multi-attractor coexistence mechanism composed of SWD, spindles and background state underlies these state evolutions. What’s more, in order to demonstrate the state evolution stability with respect to the topological structures of neural network, we further expand the 3-compartment coupled network into 10-compartment coupled one, with linear and circular structures, and nearest-neighbor (NN) coupled network as well as its realization of small-world (SW) topology via random rewiring, respectively. Interestingly, for the cases of linear and circular connetivities, qualitatively similar results were obtained in addition to the more irregularity of firings. However, SWD can be eventually transformed into the consistent low-amplitude oscillations for both NN and SW networks. In particular, SWD evolves into the slow spindling oscillations and background tonic oscillations within the NN and SW network, respectively. Our modeling and simulation studies highlight the effect of network topology in the evolutions of SWD and spindling oscillations, which provides new insights into the mechanisms of cortical seizures development.

Keywords

Spike-wave discharges (SWD) UP-DOWN state Spindles Thalamic reticular nucleus (TRN) Cortex Stimulation 

Notes

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant Nos. 11325208, 11572015 and 11172017), the Project funded by China Postdoctoral Science Foundation (Grant No. 2016M600037) and the Fundamental Research Funds for the Central Universities (FRF-TP-16-068A1).

Author Contributions

DF, QW, JS and HX conceived, designed and performed the research as well as wrote the paper.

Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.

Competing financial interests

The authors declare no competing financial interests.

References

  1. Achermann, P., & Borbely, A.A. (1997). Low-frequency (< 1Hz) oscillations in the human sleep electroencephalogram. Neuroscience, 81(1), 213–222.CrossRefPubMedGoogle Scholar
  2. Amzica, F., & Steriade, M. (1997). The K-complex: its slow (< 1-Hz) rhythmicity and relation to delta waves. Neurology, 49(4), 952–959.CrossRefPubMedGoogle Scholar
  3. Baier, G., Goodfellow, M., Taylor, P.N., Wang, Y., & Garry, D.J. (2012). The importance of modeling epileptic seizure dynamics as spatio-temporal patterns. Frontiers in physiology, 3, 281.CrossRefPubMedPubMedCentralGoogle Scholar
  4. Baier, G., Taylor, P.N., & Wang, Y. (2017). Understanding epileptiform after-discharges as rhythmic oscillatory transients. Frontiers in Computational Neuroscience, 11, 25.CrossRefPubMedPubMedCentralGoogle Scholar
  5. Barnes, G.N., & Paolicchi, J.M. (2008). Neuropsychiatric comorbidities in childhood absence epilepsy. Nature Clinical Practice Neurology, 4(12), 650–651.CrossRefPubMedGoogle Scholar
  6. Blik, V. (2015). Electric stimulation of the tuberomamillary nucleus affects epileptic activity and sleepCwake cycle in a genetic absence epilepsy model. Epilepsy Research, 109, 119–125.CrossRefPubMedGoogle Scholar
  7. Boly, M., Jones, B., Findlay, G., Plumley, E., Mensen, A., Hermann, B., & Maganti, R. (2017). Altered sleep homeostasis correlates with cognitive impairment in patients with focal epilepsy. Brain: A Journal of Neurology, 140(4), 1026–1040.CrossRefGoogle Scholar
  8. Breakspear, M., Roberts, J.A., Terry, J.R., Rodrigues, S., Mahant, N., & Robinson, P.A. (2006). A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. Cerebral Cortex, 16(9), 1296–1313.CrossRefPubMedGoogle Scholar
  9. Caplan, R., Siddarth, P., Stahl, L., Lanphier, E., Vona, P., Gurbani, S., et al. (2008). Childhood absence epilepsy: behavioral, cognitive, and linguistic comorbidities. Epilepsia, 49(11), 1838–1846.CrossRefPubMedGoogle Scholar
  10. Chatburn, A., Coussens, S., Lushington, K., Kennedy, D., Baumert, M., & Kohler, M. (2013). Sleep spindle activity and cognitive performance in healthy children. Sleep, 36(2), 237–243.CrossRefPubMedPubMedCentralGoogle Scholar
  11. Chen, B., Detyniecki, K., Choi, H., Hirsch, L., Katz, A., Legge, A., & Farooque, P. (2017). Psychiatric and behavioral side effects of anti-epileptic drugs in adolescents and children with epilepsy. European Journal of Paediatric Neurology, 21(3), 441–449.CrossRefPubMedGoogle Scholar
  12. Chen, M., Guo, D., Li, M., Ma, T., Wu, S., Ma, J., & Yao, D. (2015). Critical roles of the direct GABAergic pallido-cortical pathway in controlling absence seizures. PLoS Computational Biology, 11(10), e1004539.CrossRefPubMedPubMedCentralGoogle Scholar
  13. Chen, M., Guo, D., Wang, T., Jing, W., Xia, Y., Xu, P., & et al. (2014). Bidirectional control of absence seizures by the basal ganglia: a computational evidence. PLoS Computational Biology, 10(3), e1003495.CrossRefPubMedPubMedCentralGoogle Scholar
  14. Christensen, J.A.E., Nikolic, M., Warby, S.C., Koch, H., Zoetmulder, M., Frandsen, R., et al. (2014). Sleep spindle alterations in patients with parkinson’s disease. Frontiers in Human Neuroscience, 9(233), S47–S47.Google Scholar
  15. Contreras, D., Destexhe, A., & Steriade, M. (1997). Spindle oscillations during cortical spreading depression in naturally sleeping cats. Neuroscience, 77(4), 933–936.CrossRefPubMedGoogle Scholar
  16. Crunelli, V., David, F., Lörincz, M.L., & Hughes, S.W. (2015). The thalamocortical network as a single slow wave-generating unit. Current Opinion in Neurobiology, 31, 72–80.CrossRefPubMedGoogle Scholar
  17. Da Silva, F.L., Blanes, W., Kalitzin, S.N., Parra, J., Suffczynski, P., & Velis, D.N. (2003). Epilepsies as dynamical diseases of brain systems: basic models of the transition between normal and epileptic activity. Epilepsia, 44(s12), 72–83.CrossRefGoogle Scholar
  18. Destexhe, A. (1999). Can GABAA conductances explain the fast oscillation frequency of absence seizures in rodents?. European Journal of Neuroscience, 11(6), 2175–2181.CrossRefPubMedGoogle Scholar
  19. Destexhe, A., Neubig, M., Ulrich, D., & Huguenard, J. (1998). Dendritic low-threshold calcium currents in thalamic relay cells. Journal of Neuroscience, 18(10), 3574–3588.PubMedGoogle Scholar
  20. Drover, J.D., Schiff, N.D., & Victor, J.D. (2010). Dynamics of coupled thalamocortical modules. Journal of computational neuroscience, 28(3), 605–616.CrossRefPubMedGoogle Scholar
  21. Eschenko, O., Magri, C., Panzeri, S., & Sara, S.J. (2012). Noradrenergic neurons of the locus coeruleus are phase locked to cortical up-down states during sleep. Cerebral Cortex, 22(2), 426–435.CrossRefPubMedGoogle Scholar
  22. Evangelista, E., Bnar, C., Bonini, F., Carron, R., Colombet, B., Rgis, J., et al. (2015). Does the thalamo-cortical synchrony play a role in seizure termination?. Frontiers in Neurology, 6, 192.CrossRefPubMedPubMedCentralGoogle Scholar
  23. Fan, D., Wang, Q., & Perc, M. (2015). Disinhibition-induced transitions between absence and tonic-clonic epileptic seizures. Scientific Reports, 5, 12618.CrossRefPubMedPubMedCentralGoogle Scholar
  24. Ferrarelli, F. (2015). Sleep in patients with schizophrenia. Current Sleep Medicine Reports, 1(2), 150–156.CrossRefPubMedPubMedCentralGoogle Scholar
  25. Ferrarelli, F., & Tononi, G. (2017). Reduced sleep spindle activity point to a TRN-MD thalamus-PFC circuit dysfunction in schizophrenia. Schizophrenia Research, 180, 36–43.CrossRefPubMedGoogle Scholar
  26. Fong, G.C.Y., Shah, P.U., Gee, M.N., Serratosa, J.M., Castroviejo, I.P., Khan, S., et al. (1998). Childhood absence epilepsy with tonic-clonic seizures and electroencephalogram 3C4-Hz spike and multispikeCslow wave complexes: linkage to chromosome 8q24. The American Journal of Human Genetics, 63(4), 1117–1129.CrossRefPubMedGoogle Scholar
  27. Golomb, D., Wang, X.J., & Rinzel, J. (1996). Propagation of spindle waves in a thalamic slice model. Journal of Neurophysiology, 75(2), 750–769.PubMedGoogle Scholar
  28. Goodfellow, M., Schindler, K., & Baier, G. (2011). Intermittent spikeCwave dynamics in a heterogeneous, spatially extended neural mass model. NeuroImage, 55(3), 920–932.CrossRefPubMedGoogle Scholar
  29. Guo, D., Chen, M., Perc, M., Wu, S., Xia, C., Zhang, Y., & Yao, D. (2016a). Firing regulation of fast-spiking interneurons by autaptic inhibition. EPL (Europhysics Letters), 114(3), 30001.Google Scholar
  30. Guo, D., Wu, S., Chen, M., Perc, M., Zhang, Y., Ma, J., & Yao, D. (2016b). Regulation of irregular neuronal firing by autaptic transmission. Scientific Reports, 6, 26096.Google Scholar
  31. Jansen, B.H., & Rit, V.G. (1995). Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. Biological cybernetics, 73(4), 357–366.CrossRefPubMedGoogle Scholar
  32. Kandel, A., & Buzski, G. (1997). Cellular-synaptic generation of sleep spindles, spike-and-wave discharges, and evoked thalamocortical responses in the neocortex of the rat. Journal of Neuroscience, 17(17), 6783–97.PubMedGoogle Scholar
  33. Kostopoulos, G., Gloor, P., Pellegrini, A., & Gotman, J. (1981). A study of the transition from spindles to spike and wave discharge in feline generalized penicillin epilepsy: microphysiological features. Experimental Neurology, 73(1), 43–54.CrossRefPubMedGoogle Scholar
  34. Kostopoulos, G.K. (2000). Spike-and-wave discharges of absence seizures as a transformation of sleep spindles: the continuing development of a hypothesis. Clinical Neurophysiology, 111(s2), S27–38.CrossRefPubMedGoogle Scholar
  35. Krosigk, M.V., Bal, T., & Mccormick, D.A. (1993). Cellular mechanisms of a synchronized oscillation in the thalamus. Science, 261(5119), 361.CrossRefGoogle Scholar
  36. Latreille, V. (2015). Sleep spindles in parkinson’s disease may predict the development of dementia. Neurobiology of Aging, 36(2), 1083–1090.CrossRefPubMedGoogle Scholar
  37. Lee, J., Song, K., Lee, K., Hong, J., Lee, H., Chae, S., et al. (2013). Sleep spindles are generated in the absence of t-type calcium channel-mediated low-threshold burst firing of thalamocortical neurons. Proceedings of the National Academy of Sciences, 110(50), 20266.CrossRefGoogle Scholar
  38. Lewis, L.D., Jakob, V., Flores, F.J., Ian, S.L., Wilson, M.A., Halassa, M.M., et al. (2015). Thalamic reticular nucleus induces fast and local modulation of arousal state. Elife Sciences, 4, e08760.Google Scholar
  39. Liu, Z., Vergnes, M., Depaulis, A., & Marescaux, C. (1991). Evidence for a critical role of GABAergic transmission within the thalamus in the genesis and control of absence seizures in the rat. Brain Research, 545(1), 1–7.CrossRefPubMedGoogle Scholar
  40. Loring, D.W., & Meador, K.J. (2004). Cognitive side effects of antiepileptic drugs in children. Neurology, 62 (6), 872–877.CrossRefPubMedGoogle Scholar
  41. Meeren, H.K., Veening, J.G., Möderscheim, T.A., Coenen, A.M., & van Luijtelaar, G. (2009). Thalamic lesions in a genetic rat model of absence epilepsy: dissociation between spike-wave discharges and sleep spindles. Experimental Neurology, 217(1), 25– 37.CrossRefPubMedGoogle Scholar
  42. Moeller, F., Muthuraman, M., Stephani, U., Deuschl, G., Raethjen, J., & Siniatchkin, M. (2013). Representation and propagation of epileptic activity in absences and generalized photoparoxysmal responses. Human Brain Mapping, 34(8), 1896–1909.CrossRefPubMedGoogle Scholar
  43. Molle, M., Bergmann, T.O., Marshall, L., & Born, J. (2011). Fast and slow spindles during the sleep slow oscillation: Disparate coalescence and engagement in memory processing. Sleep, 34(10), 1411–1421.CrossRefPubMedPubMedCentralGoogle Scholar
  44. O’reilly, C., Godin, I., Montplaisir, J., & Nielsen, T. (2015). REM Sleep behaviour disorder is associated with lower fast and higher slow sleep spindle densities. Journal of Sleep Research, 24(6), 593–601.CrossRefPubMedGoogle Scholar
  45. Panayiotopoulos, C.P. (1997). Absence epilepsies. In Engel, J., & Pedley, T.A. (Eds.) Epilepsy: A comprehensive textbook (pp. 2327–2346). Philadelphia: Lippincott-Raven.Google Scholar
  46. Pinault, D., & O’Brien, T.J. (2005). Cellular and network mechanisms of genetically-determined absence seizures. Thalamus & Related Systems, 3(3), 181.CrossRefGoogle Scholar
  47. 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(4), 041924.CrossRefGoogle Scholar
  48. Rosanova, M., & Ulrich, D. (2005). Pattern-specific associative long-term potentiation induced by a sleep spindle-related spike train. Journal of Neuroscience, 25(41), 9398.CrossRefPubMedGoogle Scholar
  49. Salem, K.M., Goodger, L., Bowyer, K., Shafafy, M., & Grevitt, M.P. (2016). Does transcranial stimulation for motor evoked potentials (TcMEP) worsen seizures in epileptic patients following spinal deformity surgery?. European Spine Journal, 25(10), 3044–3048.CrossRefPubMedGoogle Scholar
  50. Sanchez-Vives, M.V., & Mccormick, D.A. (2000). Cellular and network mechanisms of rhythmic recurrent activity in neocortex. Nature Neuroscience, 3(10), 1027.CrossRefPubMedGoogle Scholar
  51. Sargsyan, A., Sitnikova, E., Melkonyan, A., Mkrtchian, H., & van Luijtelaar, G. (2007). Simulation of sleep spindles and spike and wave discharges using a novel method for the calculation of field potentials in rats. Journal of Neuroscience Methods, 164(1), 161–176.CrossRefPubMedGoogle Scholar
  52. Schiller, Y., & Bankirer, Y. (2007). Cellular mechanisms underlying antiepileptic effects of low- and high-frequency electrical stimulation in acute epilepsy in neocortical brain slices in vitro. Journal of Neurophysiology, 97(3), 1887–1902.CrossRefPubMedGoogle Scholar
  53. Shouse, M.N., Farber, P.R., & Staba, R.J. (2000). Physiological basis: How nrem sleep components can promote and rem sleep components can suppress seizure discharge propagation. Clinical Neurophysiology, 111(S2), S9–S18.CrossRefPubMedGoogle Scholar
  54. Sinha, N., Taylor, P.N., Dauwels, J., & Ruths, J. (2014). Development of optimal stimuli in a heterogeneous model of epileptic spike wave oscillations. IEEE International Conference on Systems, Man and Cybernetics, 2014, 3160–3165.Google Scholar
  55. Sitnikova, E. (2010). Thalamo-cortical mechanisms of sleep spindles and spikeCwave discharges in rat model of absence epilepsy (a review). Epilepsy Research, 89(1), 17–26.CrossRefPubMedGoogle Scholar
  56. Sitnikova, E., Hramov, A.E., Grubov, V., & Koronovsky, A.A. (2014a). Age-dependent increase of absence seizures and intrinsic frequency dynamics of sleep spindles in rats. Neuroscience Journal, 2014, 370764.Google Scholar
  57. Sitnikova, E., Hramov, A.E., Grubov, V., & Koronovsky, A.A. (2014b). Time-frequency characteristics and dynamics of sleep spindles in WAG/rij rats with absence epilepsy. Brain Research, 1543, 290–299.Google Scholar
  58. Sotero, R.C., Trujillo-Barreto, N.J., Iturria-Medina, Y., Carbonell, F., & Jimenez, J.C. (2007). Realistically coupled neural mass models can generate EEG rhythms. Neural Computation, 19(2), 478–512.CrossRefPubMedGoogle Scholar
  59. Steriade, M. (2003). The corticothalamic system in sleep. Frontiers in Bioscience A Journal & Virtual Library, 8(1-3), d878.CrossRefGoogle Scholar
  60. Steriade, M., Deschnes, M., Domich, L., & Mulle, C. (1985). Abolition of spindle oscillation in thalamic neurons disconnected from nucleus reticularis thalami. Journal of Neurophysiology, 54(6), 1473.PubMedGoogle Scholar
  61. Steriade, M., Domich, L., Oakson, G., & Deschenes, M. (1987). The deafferented reticular thalamic nucleus generates spindle rhythmicity. Journal of Neurophysiology, 57(1), 260–273.PubMedGoogle Scholar
  62. Steriade, M., Mccormick, D.A., & Sejnowski, T.J. (1993). Thalamocortical oscillations in the sleeping and aroused brain. Science, 262(5134), 679.CrossRefPubMedGoogle Scholar
  63. Steyn-Ross, D.A., Steyn-Ross, M.L., Sleigh, J.W., Wilson, M.T., Gillies, I.P., & Wright, J.J. (2005a). The sleep cycle modelled as a cortical phase transition. Journal of Biological Physics, 31(3-4), 547–569.Google Scholar
  64. Steyn-Ross, M.L., Steyn-Ross, D.A., Sleigh, J.W., Wilson, M.T., & Wilcocks, L.C. (2005b). Proposed mechanism for learning and memory erasure in a white-noise-driven sleeping cortex. Physical Review E, 72(6), 061910.Google Scholar
  65. Steyn-Ross, A., & Steyn-Ross, M. (2010). Modeling phase transitions in the brain. New York: Springer.CrossRefGoogle Scholar
  66. Su, Y., Radman, T., Vaynshteyn, J., Parra, L.C., & Bikson, M. (2008). Effects of high-frequency stimulation on epileptiform activity in vitro: ON/OFF control paradigm. Epilepsia, 49(9), 1586–1593.CrossRefPubMedGoogle Scholar
  67. Suffczynski, P., Kalitzin, S., & Silva, F.H.L.D. (2004). Dynamics of non-convulsive epileptic phenomena modeled by a bistable neuronal network. Neuroscience, 126(2), 467–484.CrossRefPubMedGoogle Scholar
  68. Tamaki, M., Matsuoka, T., Nittono, H., & Hori, T. (2008). Fast sleep spindle (13-15 hz) activity correlates with sleep-dependent improvement in visuomotor performance. Sleep, 31(2), 204.CrossRefPubMedPubMedCentralGoogle Scholar
  69. Taylor, P.N., & Baier, G. (2011). A spatially extended model for macroscopic spike-wave discharges. Journal of Computational Neuroscience, 31(3), 679–684.CrossRefPubMedGoogle Scholar
  70. Taylor, P.N., Baier, G., Cash, S.S., Dauwels, J., Slotine, J.J., & Wang, Y. (2013). A model of stimulus induced epileptic spike-wave discharges. IEEE Symposium on Computational Intelligence, Cognitive Algorithms, Mind, and Brain (CCMB), 9(5), 53–59.CrossRefGoogle Scholar
  71. Taylor, P.N., Thomas, J., Sinha, N., Dauwels, J., Kaiser, M., Thesen, T., & Ruths, J. (2015). Optimal control based seizure abatement using patient derived connectivity. Frontiers in Neuroscience, 1(9), 202.Google Scholar
  72. Taylor, P.N., Wang, Y., Goodfellow, M., Dauwels, J., Moeller, F., Stephani, U., & et al. (2014). A computational study of stimulus driven epileptic seizure abatement. PloS one, 9(12), e114316.CrossRefPubMedPubMedCentralGoogle Scholar
  73. Traub, R.D., Contreras, D., Cunningham, M.O., Murray, H., Lebeau, F.E.N., Roopun, A., & et al. (2005). Single-column thalamocortical network model exhibiting gamma oscillations, sleep spindles, and epileptogenic bursts. Journal of Neurophysiology, 93(4), 2194–2232.CrossRefPubMedGoogle Scholar
  74. Tsekou, H., Angelopoulos, E., Paparrigopoulos, T., Golemati, S., Soldatos, C.R., Papadimitriou, G.N., et al. (2015). Sleep eeg and spindle characteristics after combination treatment with clozapine in drug-resistant schizophrenia: a pilot study. Journal of Clinical Neurophysiology, 32(2), 159–63.CrossRefPubMedGoogle Scholar
  75. Ursino, M., Cona, F., & Zavaglia, M. (2010). The generation of rhythms within a cortical region: analysis of a neural mass model. NeuroImage, 52(3), 1080–1094.CrossRefPubMedGoogle Scholar
  76. van Luijtelaar, E.L. (1997). Spike-wave discharges and sleep spindles in rats. Acta Neurobiologiae Experimentalis, 57(2), 113–121.PubMedGoogle Scholar
  77. Veggiotti, P., Beccaria, F., Guerrini, R., Capovilla, G., & Lanzi, G. (1999). Continuous spike-and-wave activity during slow-wave sleep: syndrome or eeg pattern?. Epilepsia, 40(11), 1593– 1601.CrossRefPubMedGoogle Scholar
  78. Wamsley, E.J., Tucker, M.A., Shinn, A.K., Ono, K.E., Mckinley, S.K., Ely, A.V., et al. (2011). Reduced sleep spindles and spindle coherence in schizophrenia: mechanisms of impaired memory consolidation?. Biological Psychiatry, 71(2), 154–161.CrossRefPubMedPubMedCentralGoogle Scholar
  79. Wang, Q., Perc, M., Duan, Z., & Chen, G. (2009). Synchronization transitions on scale-free neuronal networks due to finite information transmission delays. Physical Review E, 80(2), 026206.CrossRefGoogle Scholar
  80. Wang, Y., Goodfellow, M., Taylor, P.N., & Baier, G. (2012). Phase space approach for modeling of epileptic dynamics. Physical Review E, 85(6), 061918.CrossRefGoogle Scholar
  81. Westmijse, I., Ossenblok, P., Gunning, B., & Van, L.G. (2009). Onset and propagation of spike and slow wave discharges in human absence epilepsy: a meg study. Epilepsia, 50(12), 2538–2548.Google Scholar
  82. Wilson, M.T., Steyn-Ross, M.L., Steyn-Ross, D.A., & Sleigh, J.W. (2005). Predictions and simulations of cortical dynamics during natural sleep using a continuum approach. Physical Review E, 72(5), 051910.CrossRefGoogle Scholar
  83. Yan, B., & Li, P. (2011). An integrative view of mechanisms underlying generalized spike-and-wave epileptic seizures and its implication on optimal therapeutic treatments. PloS one, 6(7), e22440.CrossRefPubMedPubMedCentralGoogle Scholar
  84. Yousif, N.A., & Denham, M. (2005). A population-based model of the nonlinear dynamics of the thalamocortical feedback network displays intrinsic oscillations in the spindling (7C14 Hz) range. European Journal of Neuroscience, 22(12), 3179–3187.CrossRefPubMedGoogle Scholar
  85. Zhao, X., Kim, J.W., & Robinson, P.A. (2015). Slow-wave oscillations in a corticothalamic model of sleep and wake. Journal of Theoretical Biology, 370, 93–102.CrossRefPubMedGoogle Scholar
  86. Zhao, X., & Robinson, P.A. (2015). Generalized seizures in a neural field model with bursting dynamics. Journal of Computational Neuroscience, 39(2), 197–216.CrossRefPubMedGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Denggui Fan
    • 1
  • Qingyun Wang
    • 2
    Email author
  • Jianzhong Su
    • 3
  • Hongguang Xi
    • 3
  1. 1.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingChina
  2. 2.Department of Dynamics and ControlBeihang UniversityBeijingChina
  3. 3.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA

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