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Deriving theoretical phase locking values of a coupled cortico-thalamic neural mass model using center manifold reduction

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Abstract

Cognitive functions such as sensory processing and memory processes lead to phase synchronization in the electroencephalogram or local field potential between different brain regions. There are a lot of computational researches deriving phase locking values (PLVs), which are an index of phase synchronization intensity, from neural models. However, these researches derive PLVs numerically. To the best of our knowledge, there have been no reports on the derivation of a theoretical PLV. In this study, we propose an analytical method for deriving theoretical PLVs from a cortico-thalamic neural mass model described by a delay differential equation. First, the model for generating neural signals is transformed into a normal form of the Hopf bifurcation using center manifold reduction. Second, the normal form is transformed into a phase model that is suitable for analyzing synchronization phenomena. Third, the Fokker–Planck equation of the phase model is derived and the phase difference distribution is obtained. Finally, the PLVs are calculated from the stationary distribution of the phase difference. The validity of the proposed method is confirmed via numerical simulations. Furthermore, we apply the proposed method to a working memory process, and discuss the neurophysiological basis behind the phase synchronization phenomenon. The results demonstrate the importance of decreasing the intensity of independent noise during the working memory process. The proposed method will be of great use in various experimental studies and simulations relevant to phase synchronization, because it enables the effect of neurophysiological changes on PLVs to be analyzed from a mathematical perspective.

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References

  • Baddeley, A. (1992a). Working memory. Science, 255(5044), 556–559.

    Article  CAS  PubMed  Google Scholar 

  • Baddeley, A. (1992b). Working memory - an introduction. International Journal of Psychology, 27(3–4), 96–97.

    Google Scholar 

  • Buzsaki, G., & Draguhn, A. (2004). Neuronal oscillations in cortical networks. Science, 304(5679), 1926–1929. doi:10.1126/science.1099745.

    Article  CAS  PubMed  Google Scholar 

  • Cui, D., Li, X. Z., Wan, Y., & Li, X. L. (2010). Estimation of genuine and random synchronization in multivariate neural series. Neural Networks, 23(6), 698–704. doi:10.1016/j.neunet.2010.04.003.

    Article  PubMed  Google Scholar 

  • Dasilva, F. L. (1991). Neural mechanisms underlying brain waves - from neural membranes to networks. Electroencephalography and Clinical Neurophysiology, 79(2), 81–93.

    Article  Google Scholar 

  • David, O., & Friston, K. J. (2003). A neural mass model for MEG/EEG: coupling and neuronal dynamics. NeuroImage, 20(3), 1743–1755. doi:10.1016/j.neuroimaging.2003.07.015.

    Article  PubMed  Google Scholar 

  • David, O., Cosmelli, D., & Friston, K. J. (2004). Evaluation of different measures of functional connectivity using a neural mass model. NeuroImage, 21(2), 659–673. doi:10.1016/j.neuroimage.2003.10.006.

    Article  PubMed  Google Scholar 

  • Deco, G., Jirsa, V. K., Robinson, P. A., Breakspear, M., & Friston, K. (2008). The dynamic brain: from spiking neurons to neural masses and cortical fields. PLoS Computational Biology, 4(8). doi:10.1371/journal.pcbi.1000092.

  • Engel, A. K., Fries, P., & Singer, W. (2001). Dynamic predictions: oscillations and synchrony in top-down processing. Nature Reviews Neuroscience, 2(10), 704–716. doi:10.1038/35094565.

    Article  CAS  PubMed  Google Scholar 

  • Ermentrout, B. (1996). Type I membranes, phase resetting curves, and synchrony. Neural Computation, 8(5), 979–1001. doi:10.1162/neco.1996.8.5.979.

    Article  CAS  PubMed  Google Scholar 

  • Fell, J., & Axmacher, N. (2011). The role of phase synchronization in memory processes. Nature Reviews Neuroscience, 12(2), 105–U1500. doi:10.1038/Nrn2979.

    Article  CAS  PubMed  Google Scholar 

  • Gardiner, C. W. (1994). Handbook of stochastic methods for physics, chemistry, and the natural sciences, Springer series in synergetics (Vol. 13, 2nd ed.). Berlin, New York: Springer-Verlag.

    Google Scholar 

  • Glass, L., & Mackey, M. C. (1988). From clocks to chaos : the rhythms of life (Princeton paperbacks). Princeton, N.J.: Princeton University Press.

  • Goldobin, D. S., Teramae, J. N., Nakao, H., & Ermentrout, G. B. (2010). Dynamics of limit-cycle oscillators subject to general noise. Physical Review Letters, 105(15). doi:10.1103/Physrevlett.105.154101.

  • Halanay, A. (1966). Differential equations; stability, oscillations, time lags, Mathematics in science and engineering (Vol. 23). New York: Academic Press.

    Book  Google Scholar 

  • Hale, J. K. (1977). Theory of functional differential equations, Applied mathematical sciences (Vol. 3, 2d ed.). New York: Springer-Verlag.

    Book  Google Scholar 

  • Hassard, B. D., Kazarinoff, N. D., & Wan, Y. H. (1981). Theory and applications of Hopf bifurcation, London Mathematical Society lecture note series (Vol. 41). Cambridge, New York: Cambridge University Press.

    Google Scholar 

  • Hempel, C. M., Hartman, K. H., Wang, X. J., Turrigiano, G. G., & Nelson, S. B. (2000). Multiple forms of short-term plasticity at excitatory synapses in rat medial prefrontal cortex. Journal of Neurophysiology, 83(5), 3031–3041.

    CAS  PubMed  Google Scholar 

  • Hirata, A., & Castro-Alamancos, M. A. (2010). Neocortex network activation and deactivation states controlled by the thalamus. Journal of Neurophysiology, 103(3), 1147–1157. doi:10.1152/jn.00955.2009.

    Article  PubMed  PubMed Central  Google Scholar 

  • Hoppensteadt, F. C., & Izhikevich, E. M. (1997). Weakly connected neural networks, Applied mathematical sciences (Vol. 126). New York: Springer.

    Google Scholar 

  • Kim, J. W., & Robinson, P. A. (2007). Compact dynamical model of brain activity. Physical Review E, 75(3). doi:10.1103/Physreve.75.031907.

  • Kotani, K., Yamaguchi, I., Ogawa, Y., Jimbo, Y., Nakao, H., & Ermentrout, G. B. (2012). Adjoint method provides phase response functions for delay-induced oscillations. Physical Review Letters, 109(4). doi:10.1103/Physrevlett.109.044101.

  • Kuramoto, Y. (1984). Chemical oscillations, waves, and turbulence, Springer series in synergetics (Vol. 19). Berlin, New York: Springer-Verlag.

    Book  Google Scholar 

  • Kurebayashi, W., Shirasaka, S., & Nakao, H. (2013). Phase reduction method for strongly perturbed limit cycle oscillators. Physical Review Letters, 111(21). doi:10.1103/Physrevlett.111.214101.

  • Liebe, S., Hoerzer, G. M., Logothetis, N. K., & Rainer, G. (2012). Theta coupling between V4 and prefrontal cortex predicts visual short-term memory performance. Nature Neuroscience, 15(3), 456–U150. doi:10.1038/Nn.3038.

    Article  CAS  PubMed  Google Scholar 

  • Lumer, E. D., Edelman, G. M., & Tononi, G. (1997). Neural dynamics in a model of the thalamocortical system.1. Layers, loops and the emergence of fast synchronous rhythms. Cerebral Cortex, 7(3), 207–227.

    Article  CAS  PubMed  Google Scholar 

  • Ly, C., & Ermentrout, G. B. (2009). Synchronization dynamics of two coupled neural oscillators receiving shared and unshared noisy stimuli. Journal of Computational Neuroscience, 26(3), 425–443. doi:10.1007/s10827-008-0120-8.

    Article  PubMed  Google Scholar 

  • Nakao, H., Arai, K., & Kawamura, Y. (2007). Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators. Physical Review Letters, 98(18). doi:10.1103/Physrevlett.98.184101.

  • Naruse, Y., Matani, A., Miyawaki, Y., & Okada, M. (2010). Influence of coherence between multiple cortical columns on alpha rhythm: a computational modeling study. Human Brain Mapping, 31(5), 703–715. doi:10.1002/Hbm.20899.

    Article  PubMed  Google Scholar 

  • Palva, J. M., Monto, S., Kulashekhar, S., & Palva, S. (2010). Neuronal synchrony reveals working memory networks and predicts individual memory capacity. Proceedings of the National Academy of Sciences of the United States of America, 107(16), 7580–7585. doi:10.1073/pnas.0913113107.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Payne, L., & Kounios, J. (2009). Coherent oscillatory networks supporting short-term memory retention. Brain Research, 1247, 126–132. doi:10.1016/j.brainres.2008.09.095.

    Article  CAS  PubMed  Google Scholar 

  • Pikovsky, A., Rosenblum, M., & Kurths, J. (2001). Synchronization: a universal concept in nonlinear sciences, vol. 12 of Cambridge nonlinear science series. Cambridge, England.

  • Ponten, S. C., Daffertshofer, A., Hillebrand, A., & Stam, C. J. (2010). The relationship between structural and functional connectivity: graph theoretical analysis of an EEG neural mass model. NeuroImage, 52(3), 985–994. doi:10.1016/j.neuroimage.2009.10.049.

    Article  CAS  PubMed  Google Scholar 

  • 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). doi:10.1103/Physreve.65.041924.

  • Rodriguez, E., George, N., Lachaux, J. P., Martinerie, J., Renault, B., & Varela, F. J. (1999). Perception's shadow: long-distance synchronization of human brain activity. Nature, 397(6718), 430–433.

    Article  CAS  PubMed  Google Scholar 

  • Sawaguchi, T., & Goldmanrakic, P. S. (1994). The role of D1-dopamine receptor in working-memory - local injections of dopamine antagonists into the prefrontal cortex of rhesus-monkeys performing an Oculomotor delayed-response task. Journal of Neurophysiology, 71(2), 515–528.

    CAS  PubMed  Google Scholar 

  • Seamans, J. K., & Yang, C. R. (2004). The principal features and mechanisms of dopamine modulation in the prefrontal cortex. Progress in Neurobiology, 74(1), 1–57. doi:10.1016/j.pneurobio.2004.05.006.

    Article  CAS  PubMed  Google Scholar 

  • Shayer, L. P., & Campbell, S. A. (2000). Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays. SIAM Journal on Applied Mathematics, 61(2), 673–700. doi:10.1137/S0036139998344015.

    Article  Google Scholar 

  • Sherman, S. M., & Guillery, R. W. (1996). Functional organization of thalamocortical relays. Journal of Neurophysiology, 76(3), 1367–1395.

    CAS  PubMed  Google Scholar 

  • Simmendinger, C., Wunderlin, A., & Pelster, A. (1999). Analytical approach for the Floquet theory of delay differential equations. Physical Review E, 59(5), 5344–5353. doi:10.1103/PhysRevE.59.5344.

    Article  CAS  Google Scholar 

  • Stam, C. J., van der Made, Y., Pijnenburg, Y. A. L., & Scheltens, P. (2003). EEG synchronization in mild cognitive impairment and Alzheimer's disease. Acta Neurologica Scandinavica, 108(2), 90–96.

    Article  CAS  PubMed  Google Scholar 

  • Steriade, M. (2001). Impact of network activities on neuronal properties in corticothalamic systems. Journal of Neurophysiology, 86(1), 1–39.

    CAS  PubMed  Google Scholar 

  • Takeda, Y. (2011). Influence of connection type on phase synchrony: analysis of a neural mass model. Biological Cybernetics, 105(5–6), 349–354. doi:10.1007/s00422-011-0470-6.

    Article  PubMed  Google Scholar 

  • Ton, R., Deco, G., & Daffertshofer, A. (2014). Structure-function discrepancy: inhomogeneity and delays in synchronized neural networks. PLoS Computational Biology, 10(7). doi:10.1371/journal.pcbi.1003736.

  • Varela, F., Lachaux, J. P., Rodriguez, E., & Martinerie, J. (2001). The brainweb: Phase synchronization and large-scale integration. Nature Reviews Neuroscience, 2(4), 229–239. doi:10.1038/35067550.

    Article  CAS  PubMed  Google Scholar 

  • Victor, J. D., Drover, J. D., Conte, M. M., & Schiff, N. D. (2011). Mean-field modeling of thalamocortical dynamics and a model-driven approach to EEG analysis. Proceedings of the National Academy of Sciences of the United States of America, 108, 15631–15638. doi:10.1073/pnas.1012168108.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Winfree, A. T. (1980). The geometry of biological time, Biomathematics v 8. New York: Springer Verlag.

    Book  Google Scholar 

  • Yamaguchi, I., Ogawa, Y., Jimbo, Y., Nakao, H., & Kotani, K. (2011). Reduction theories elucidate the origins of complex biological rhythms generated by interacting delay-induced oscillations. PloS One, 6(11), e26497. doi:10.1371/journal.pone.0026497.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Zavaglia, M., Astolfi, L., Babiloni, F., & Ursino, M. (2008). The effect of connectivity on EEG rhythms, power spectral density and coherence among coupled neural populations: Analysis, with a neural mass model. IEEE Transactions on Biomedical Engineering, 55(1), 69–77. doi:10.1109/Tbme.2007.897814.

    Article  PubMed  Google Scholar 

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Acknowledgements

This study was supported in part by KAKENHI (15H05324) and SCOPE (142103017).

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  1. Yutaro Ogawa

    Present address: Research Center for Advanced Science and Technology, The University of Tokyo, Tokyo, Japan

Authors and Affiliations

  1. Research Center for Advanced Science and Technology, The University of Tokyo, Tokyo, Japan

    Kiyoshi Kotani

  2. Graduate School of Education, The University of Tokyo, Tokyo, Japan

    Ikuhiro Yamaguchi

  3. PRESTO, Japan Science and Technology Agency (JST), Kawaguchi, Japan

    Kiyoshi Kotani

  4. Graduate School of Engineering, The University of Tokyo, Tokyo, Japan

    Yasuhiko Jimbo

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  1. Yutaro Ogawa
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  2. Ikuhiro Yamaguchi
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Correspondence to Yutaro Ogawa.

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Ogawa, Y., Yamaguchi, I., Kotani, K. et al. Deriving theoretical phase locking values of a coupled cortico-thalamic neural mass model using center manifold reduction. J Comput Neurosci 42, 231–243 (2017). https://doi.org/10.1007/s10827-017-0638-8

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