Advertisement

Nonlinear Dynamics

, Volume 94, Issue 4, pp 2807–2825 | Cite as

Transition dynamics and adaptive synchronization of time-delay interconnected corticothalamic systems via nonlinear control

  • Denggui Fan
  • Liyuan Zhang
  • Qingyun Wang
Original Paper
  • 94 Downloads

Abstract

A modified corticothalamic (MCT) system with multiple delays for epileptic absence seizures is taken as a study object. Synchronization transition dynamics of two time-delay interconnected MCT systems via nonlinear control is investigated in this paper. Both the intrinsic delays in a single MCT system and the coupling delays between two MCT systems are considered. When there is no control, it is found that the occurrences of synchronization are dependent on the specific deviations of intrinsic delays and also correlated with the rhythmic periods of oscillations for the two MCT subsystems. To obtain the synchronous state transitions irrespective of the intrinsic delays, based on the Lyapunov stability theory, we propose a nonlinear adaptive control schemes with respect to the coupling delays. The numerical simulation results show the effectiveness of the designed control method.

Keywords

Corticothalamic model Time delay Transition dynamics Adaptive synchronization Nonlinear control 

Notes

Acknowledgements

The authors gratefully acknowledge helpful comments by the potential reviewers and acknowledge support from the National Natural Science Foundation of China (Grant Nos. 11702018 and 11772019), the National Key R&D Program of China (Grant No. 2017YFF0207401), the Project funded by China Postdoctoral Science Foundation (Grant Nos. 2016M600037 and 2018T110043) and the Fundamental Research Funds for the Central Universities (FRF-TP-16-068A1).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Gloor, P.: Neurophysiological bases of generalized seizures termed centrencephalic. In: Gaustaut, H., Jasper, H.H., Bancaud, J., Waltregny, A. (eds.) The Physiopathogenesis of the Epilepsies, pp. 209–236. Charles C. Thomas, Springfield (1969)Google Scholar
  2. 2.
    Kostopoulos, G.K.: Spike-and-wave discharges of absence seizures as a transformation of sleep spindles: the continuing development of a hypothesis. Clin. Neurophysiol. 2, S27–S38 (2000)CrossRefGoogle Scholar
  3. 3.
    Sitnikova, E.: Thalamo-cortical mechanisms of sleep spindles and spike-wave discharges in rat model of absence epilepsy (a review). Epilepsy Res. 89(1), 17–26 (2010)CrossRefGoogle Scholar
  4. 4.
    Meeren, H.K., Pijn, J.P., van Luijtelaar, E.L., Coenen, A.M., Lopes da Silva, F.H.: Cortical focus drives widespread corticothalamic networks during spontaneous absence seizures in rats. J. Neurosci. 22, 1480–1495 (2002)CrossRefGoogle Scholar
  5. 5.
    Meeren, H., van Luijtelaar, G., Lopes da Silva, F., Coenen, A.: Evolving concepts on the pathophysiology of absence seizures: the cortical focus theory. Arch. Neurol. 62, 371–376 (2005)CrossRefGoogle Scholar
  6. 6.
    Steriade, M.: Thalamic origin of sleep spindles: Morison and Bassett (1945). J. Neurophysiol. 73, 921–922 (1995)CrossRefGoogle Scholar
  7. 7.
    Crunelli, V., Hughes, S.W.: The slow (\(<\)1 Hz) rhythm of non-REM sleep: a dialogue between three cardinal oscillators. Nat. Neurosci. 13, 9–17 (2010)Google Scholar
  8. 8.
    Traub, R.D., Contreras, D., Cunningham, M.O., Murray, H., Lebeau, F.E.N., Roopun, A., et al.: Single-column thalamocortical network model exhibiting gamma oscillations, sleep spindles, and epileptogenic bursts. J. Neurophysiol. 93(4), 2194–2232 (2005)CrossRefGoogle Scholar
  9. 9.
    van Luijtelaar, E.L.: Spike-wave discharges and sleep spindles in rats. Acta. Neurobiol. Exp. 57(2), 113–121 (1997)Google Scholar
  10. 10.
    Mayville, C., Fakhoury, T., Abou-Khalil, B.: Absence seizures with evolution into generalized tonic-clonic activity: clinical and EEG features. Epilepsia 41, 391–394 (2000)CrossRefGoogle Scholar
  11. 11.
    Caplan, R., Siddarth, P., Stahl, L., Lanphier, E., Vona, P., Gurbani, S., et al.: Childhood absence epilepsy: behavioral, cognitive, and linguistic comorbidities. Epilepsia 49(11), 1838–1846 (2008)CrossRefGoogle Scholar
  12. 12.
    Kostopoulos, G.K.: Spike-and-wave discharges of absence seizures as a transformation of sleep spindles: the continuing development of a hypothesis. Clin. Neurophysiol. 111(s2), S27–S38 (2000)CrossRefGoogle Scholar
  13. 13.
    Kostopoulos, G., Gloor, P., Pellegrini, A., Siatitsas, I.: A study of the transition from spindles to spike and wave discharge in feline generalized penicillin epilepsy: EEG features. Exp. Neurol. 73(1), 43–54 (1981)CrossRefGoogle Scholar
  14. 14.
    Wang, Z., Wang, Q.: Eliminating absence seizures through the deep brain stimulation to thalamus reticular nucleus. Front. Comput. Neurosci. 11, 22 (2017)Google Scholar
  15. 15.
    Liu, S., Wang, Q.: Transition dynamics of generalized multiple epileptic seizures associated with thalamic reticular nucleus excitability: a computational study. Commun. Nonlinear Sci. 52, 203–213 (2017)CrossRefGoogle Scholar
  16. 16.
    Zhang, H., Su, J., Wang, Q., Liu, Y., Good, L., Pascual, J.: Predicting seizure by modeling synaptic plasticity based on eeg signals-a case study of inherited epilepsy. Commun. Nonlinear Sci. 56, 330–343 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Chen, M., Guo, D., Li, M., Ma, T., Wu, S., Ma, J.: Critical roles of the direct gabaergic pallido-cortical pathway in controlling absence seizures. PLoS Comput. Biol. 11(10), e1004539 (2015)CrossRefGoogle Scholar
  18. 18.
    Costa, M.S., Weigenand, A., Ngo, H.V.V., Marshall, L., Born, J., Martinetz, T.: A thalamocortical neural mass model of the EEG during NREM Sleep and its response to auditory stimulation. PLoS Comput. Biol. 12(9), e1005022 (2016)CrossRefGoogle Scholar
  19. 19.
    Coombes, S.: Waves, bumps, and patterns in neural field theories. Biol. Cybern. 93(2), 91–108 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Deco, G., Jirsa, V.K., Robinson, P.A., Breakspear, M., Friston, K.J.: The dynamic brain: from spiking neurons to neural masses and cortical fields. PLoS Comput. Biol. 4(8), e1000092 (2008)CrossRefGoogle Scholar
  21. 21.
    Goodfellow, M., Schindler, K., Baier, G.: Intermittent spike-wave dynamics in a heterogeneous, spatially extended neural mass model. Neuroimage 55(3), 920–932 (2011)CrossRefGoogle Scholar
  22. 22.
    Taylor, P.N., Baier, G.: A spatially extended model for macroscopic spike-wave discharges. J. Comput. Neurosci. 31(3), 679–684 (2011)CrossRefGoogle Scholar
  23. 23.
    Fan, D., Wang, Q., Perc, M.: Disinhibition-induced transitions between absence and tonic-clonic epileptic seizures. Sci. Rep. 5, 12618 (2015)CrossRefGoogle Scholar
  24. 24.
    Breakspear, M., Roberts, J.A., Terry, J.R., Rodrigues, S., Mahant, N., Robinson, P.A.: A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. Cereb. Cortex 16(9), 1296–1313 (2005)CrossRefGoogle Scholar
  25. 25.
    Taylor, P.N., Wang, Y., Goodfellow, M., Dauwels, J., Moeller, F., Stephani, U.: A computational study of stimulus driven epileptic seizure abatement. PLoS ONE 9(12), e114316 (2014)CrossRefGoogle Scholar
  26. 26.
    Fan, D., Liu, S., Wang, Q.: Stimulus-induced epileptic spike-wave discharges in thalamocortical model with disinhibition. Sci. Rep. 6, 37703 (2016)CrossRefGoogle Scholar
  27. 27.
    Fan, D., Wang, Q., Su, J., Xi, H.: Stimulus-induced transitions between spike-wave discharges and spindles with the modulation of thalamic reticular nucleus. J. Comput. Neurosci. 43(3), 203–225 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Fan, D., Liao, F., Wang, Q.: The pacemaker role of thalamic reticular nucleus in controlling spike-wave discharges and spindles. Chaos 27(7), 073103 (2017)CrossRefGoogle Scholar
  29. 29.
    Chen, M., Guo, D., Wang, T., Jing, W., Xia, Y., Xu, P.: Bidirectional control of absence seizures by the basal ganglia: a computational evidence. PLoS Comput. Biol. 10(3), e1003495 (2014)CrossRefGoogle Scholar
  30. 30.
    Drover, J.D., Schiff, N.D., Victor, J.D.: Dynamics of coupled thalamocortical modules. J. Comput. Neurosci. 28(3), 605–616 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Taylor, P.N., Thomas, J., Sinha, N., Dauwels, J., Kaiser, M., Thesen, T., et al.: Optimal control based seizure abatement using patient derived connectivity. Front. Neurosci. 1(9), 202 (2015)Google Scholar
  32. 32.
    Jirsa, V.K.: Neural field dynamics with local and global connectivity and time delay. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 367(1891), 1131–1143 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Nakagawa, T.T., Woolrich, M., Luckhoo, H., Joensson, M., Mohseni, H., Kringelbach, M.L.: How delays matter in an oscillatory whole-brain spiking-neuron network model for MEG alpha-rhythms at rest. Neuroimage 87, 383–394 (2014)CrossRefGoogle Scholar
  34. 34.
    Destexhe, A., Mainen, Z.F., Sejnowski, T.J.: Fast kinetic models for simulating AMPA, NMDA, GABAA and GABAB receptors. In: The Neurobiology of Computation. Springer, Boston, pp. 9–14 (1995)CrossRefGoogle Scholar
  35. 35.
    Rodrigues, S., Barton, D., Szalai, R., Benjamin, O., Richardson, M.P., Terry, J.R.: Transitions to spike-wave oscillations and epileptic dynamics in a human cortico-thalamic mean-field model. J. Comput. Neurosci. 27(3), 507–526 (2009)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Rodrigues, S., Goncalves, J., Terry, J.R.: Existence and stability of limit cycles in a macroscopic neuronal population model. Physica D 233(1), 39–65 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Egghe, L., Leydesdorff, L.: The relation between pearson’s correlation coefficient R, and salton’s cosine measure. J. Am. Soc. Inf. Sci. Technol. 60(5), 1027–1036 (2009)CrossRefGoogle Scholar
  38. 38.
    Nguyen, L.H., Hong, K.: Adaptive synchronization of two coupled chaotic Hindmarsh–Rose neurons by controlling the membrane potential of a slave neuron. Appl. Math. Model. 37(4), 2460–2468 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Shi, X., Wang, Z.: Adaptive synchronization of time delay Hindmarsh–Rose neuron system via self-feedback. Nonlinear Dyn. 69(4), 2147–2153 (2012)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Hettiarachchi, I.T., Lakshmanan, S.: Chaotic synchronization of time-delay coupled Hindmarsh–Rose neurons via nonlinear control. Nonlinear Dyn. 86(2), 1249–1262 (2016)zbMATHCrossRefGoogle Scholar
  41. 41.
    Achermann, P., Borbely, A.: Low-frequency (\(<\)1 Hz) oscillations in the human sleep electroencephalogram. Neuroscience 81(1), 213–222 (1997)Google Scholar
  42. 42.
    Amzica, F., Steriade, M.: The K-complex: its slow (\(<\)1-Hz) rhythmicity and relation to delta waves. Neurology 49(4), 952–959 (1997)Google Scholar
  43. 43.
    Jin, X., Wang, S., Yang, G., Ye, D.: Robust adaptive hierarchical insensitive tracking control of a class of leader-follower agents. Inf. Sci. 406, 234–247 (2017)CrossRefGoogle Scholar
  44. 44.
    Jin, X., Wang, S., Qin, J., Zheng, W., Kang, Y.: Adaptive fault-tolerant consensus for a class of uncertain nonlinear second-order multi-agent systems with circuit implementation. IEEE Trans. Circuits Syst. I Regul. Pap. 65(7), 2243–2255 (2018)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001). (ISBN 0521592852) zbMATHCrossRefGoogle Scholar
  46. 46.
    Frank, T., Richardson, M.: On a test statistic for the Kuramoto order parameter of synchronization: an illustration for group synchronization during rocking chairs. Phys. D Nonlinear Phenom. 239(23–24), 2084–2092 (2010)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingPeople’s Republic of China
  2. 2.Beijing Key Laboratory of Knowledge Engineering for Materials ScienceUniversity of Science and Technology BeijingBeijingPeople’s Republic of China
  3. 3.Department of Dynamics and ControlBeihang UniversityBeijingPeople’s Republic of China

Personalised recommendations