Stability, bifurcation, and synchronization of delay-coupled ring neural networks

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• 作者：Xiaochen Mao ; Zaihua Wang
• 关键词：Time delay ; Coupled systems ; Neural networks ; Synchronization ; Nonlinear dynamics
• 刊名：Nonlinear Dynamics
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：84
• 期：2
• 页码：1063-1078
• 全文大小：1,525 KB
• 参考文献：1.Yamaguchi, S., Isejima, H., Matsuo, T., Okura, R., Yagita, K., Kobayashi, M., Okamura, H.: Synchronization of cellular clocks in the suprachiasmatic nucleus. Science 302(5649), 1408–1412 (2003)CrossRef
  2.Nijmeijer, H., Rodriguez-Angeles, A.: Synchronization of mechanical systems. World Scientific Publishing, Singapore (2003)MATH
  3.Mao, X.C.: Stability switches, bifurcation, and multi-stability of coupled networks with time delays. Appl. Math. Comput. 218(11), 6263–6274 (2012)MathSciNet CrossRef MATH
  4.Murguia, C., Fey, R.H.B., Nijmeijer, H.: Network synchronization of time-delayed coupled nonlinear systems using predictor-based diffusive dynamic couplings. Chaos 25(2), 023108 (2015)MathSciNet CrossRef MATH
  5.Emelianova, Y.P., Emelyanov, V.V., Ryskin, N.M.: Synchronization of two coupled multimode oscillators with time-delayed feedback. Commun. Nonlinear Sci. Numer. Simulat 19(10), 3778–3791 (2014)MathSciNet CrossRef
  6.Louzada, V.H.P., Araujo, N.A.M., Andrade, J.S., Herrmann, H.J.: Breathing synchronization in interconnected networks. Sci. Rep. 3, 3289 (2013)CrossRef
  7.Weicker, L., Erneux, T., Keuninckx, L., Danckaert, J.: Analytical and experimental study of two delay-coupled excitable units. Phys. Rev. E 89(1), 012908 (2014)CrossRef
  8.Flunkert, V., Fischer, I., Fischer, I.: Dynamics, control and information in delay-coupled systems: an overview. Philos. Trans. R. Soc. A 371(1999), 20120465 (2013)MathSciNet CrossRef
  9.Hu, H.Y., Wang, Z.H.: Dynamics of controlled mechanical systems with delayed feedback. Springer, Heidelberg (2002)CrossRef MATH
  10.Sipahi, R., Niculescu, S.-I., Abdallah, C.T., Michiels, W., Gu, K.: Stability and stabilization of systems with time delay: limitations and opportunities. IEEE Control Syst. Mag. 31(1), 38–65 (2011)MathSciNet CrossRef
  11.Orosz, G., Wilson, R.E., Stepan, G.: Traffic jams: dynamics and control. Philos. Trans. R. Soc. A 368(1928), 4455–4479 (2010)MathSciNet CrossRef MATH
  12.Shepherd, G.M.: Neurobiology. Oxford University Press, New York (1983)
  13.Murray, J.D.: Mathematical Biology. Springer, New York (1990)MATH
  14.Marcus, C.M., Westervelt, R.M.: Stability of analog neural network with delay. Phys. Rev. A 39, 347–359 (1989)MathSciNet CrossRef
  15.Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984)CrossRef
  16.Timme, M., Wolf, F., Geisel, T.: Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators. Phys. Rev. Lett. 89, 258701 (2002)CrossRef
  17.Punetha, N., Prasad, A., Ramaswamy, R.: Phase-locked regimes in delay-coupled oscillator networks. Chaos 24(4), 043111 (2014)MathSciNet CrossRef
  18.Popovych, O.V., Yanchuk, S., Tass, P.A.: Delay- and coupling-induced firing patterns in oscillatory neural loops. Phys. Rev. Lett. 107(22), 228102 (2011)CrossRef
  19.Soriano, M.C., Flunkert, V., Fischer, I.: Relation between delayed feedback and delay-coupled systems and its application to chaotic lasers. Chaos 23(4), 043133 (2013)CrossRef MATH
  20.Sadeghi, S., Valizadeh, A.: Synchronization of delayed coupled neurons in presence of inhomogeneity. J. Comput. Neurosci. 36(1), 55–66 (2014)MathSciNet CrossRef
  21.Ge, J.H., Xu, J.: Computation of synchronized periodic solution in a BAM network with two delays. IEEE Trans. Neural Netw. 21(3), 439–450 (2010)MathSciNet CrossRef
  22.Ying, J., Guo, S., He, Y.: Multiple periodic solutions in a delay-coupled system of neural oscillators. Nonlinear Anal. Real World Appl. 12(5), 2767–2783 (2011)MathSciNet CrossRef MATH
  23.Song, Y.L.: Hopf bifurcation and spatio-temporal patterns in delay-coupled van der Pol oscillators. Nonlinear Dyn. 63(1–2), 223–237 (2011)MathSciNet CrossRef MATH
  24.Wirkus, S., Rand, R.: The dynamics of two coupled van der Pol oscillators with delay coupling. Nonlinear Dyn. 30(3), 205–221 (2002)MathSciNet CrossRef MATH
  25.Correa, D.P.F., Wulff, C., Piqueira, J.R.C.: Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 793–820 (2015)MathSciNet CrossRef MATH
  26.Shen, Z., Zhang, C.: Double Hopf bifurcation of coupled dissipative Stuart–Landau oscillators with delay. Appl. Math. Comput. 227, 553–566 (2014)MathSciNet CrossRef
  27.Usacheva, S.A., Ryskin, N.M.: Phase locking of two limit cycle oscillators with delay coupling. Chaos 24(2), 023123 (2014)MathSciNet CrossRef
  28.Caceres, M.O.: Time-delayed coupled logistic capacity model in population dynamics. Phys. Rev. E 90(2), 022137 (2014)MathSciNet CrossRef
  29.Campbell, S.A., Edwards, R., Van Den Driessche, P.: Delayed coupling between two neural network loops. SIAM. J. Appl. Math. 65(1), 316–335 (2005)MathSciNet CrossRef MATH
  30.Cheng, C.: Induction of Hopf bifurcation and oscillation death by delays in coupled networks. Phys. Lett. A 374(2), 178–185 (2009)
  31.Song, Z.G., Xu, J.: Stability switches and multistability coexistence in a delay-coupled neural oscillators system. J. Theor. Biol. 313, 98–114 (2012)MathSciNet CrossRef
  32.Jiang, Y., Guo, S.: Linear stability and Hopf bifurcation in a delayed two-coupled oscillator with excitatory-to-inhibitory connection. Nonlinear Anal. Real World Appl. 11(3), 2001–2015 (2010)MathSciNet CrossRef MATH
  33.Peng, Y., Song, Y.: Stability switches and Hopf bifurcations in a pair of identical tri-neuron network loops. Phys. Lett. A 373(20), 1744–1749 (2009)CrossRef MATH
  34.Mao, X.C.: Stability and Hopf bifurcation analysis of a pair of three-neuron loops with time delays. Nonlinear Dynam. 68(1), 151–159 (2012)MathSciNet CrossRef MATH
  35.Song, Y., Tade, M.O., Zhang, T.: Bifurcation analysis and spatio-temporal patterns of nonlinear oscillations in a delayed neural network with unidirectional coupling. Nonlinearity 22(5), 975–1001 (2009)MathSciNet CrossRef MATH
  36.Kandel, E.R., Schwartz, J.H., Jessell, T.M.: Principles of neural science. McGraw-Hill, New York (2000)
  37.Graybiel, A.M.: Basal ganglia-input, neural activity, and relation to the cortex. Curr. Opin. Neurobiol. 1(4), 644–651 (1991)CrossRef
  38.Nana, B., Woafo, P.: Synchronization in a ring of four mutually coupled van der Pol oscillators: theory and experiment. Phys. Rev. E 74(4), 046213 (2006)CrossRef
  39.Hoppensteadt, F.C., Izhikevich, E.M.: Weakly connected neural networks. Springer, New York (1997)CrossRef MATH
  40.Hisi, A.N.S., Guimaraes, P.R., de Aguiar, M.A.M.: The role of predator overlap in the robustness and extinction of a four species predator–prey network. Phys. A 389(21), 4725–4733 (2010)CrossRef
  41.Xu, X.: Complicated dynamics of a ring neural network with time delays. J. Phys. A 41(3), 035102 (2008)MathSciNet CrossRef MATH
  42.Guo, S.J., Huang, L.H.: Hopf bifurcating periodic orbits in a ring of neurons with delays. Phys. D 183, 19–44 (2003)MathSciNet CrossRef MATH
  43.Campbell, S.A., Yuan, Y., Bungay, S.D.: Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling. Nonlinearity 18(6), 2827–2846 (2005)MathSciNet CrossRef MATH
  44.Burić, N., Grozdanović, I., Vasović, N.: Excitable systems with internal and coupling delays. Chaos Soliton Fractals 36(4), 853–861 (2008)MathSciNet CrossRef MATH
  45.Song, Y., Makarov, V.A., Velarde, M.G.: Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks. Biol. Cybern. 101(2), 147–167 (2009)MathSciNet CrossRef MATH
  46.Tass, P.A., Hauptmann, C.: Therapeutic modulation of synaptic connectivity with desynchronizing brain stimulation. Int. J. Psychophysiol. 64(1), 53–61 (2007)CrossRef
  
• 作者单位：Xiaochen Mao (1) (2)
  Zaihua Wang (2) (3)
  
  1. Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, 210098, China
  2. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
  3. Institute of Science, PLA University of Science and Technology, Nanjing, 211101, China
  
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X

文摘

This paper studies the nonlinear dynamics of coupled ring networks each with an arbitrary number of neurons. Different types of time delays are introduced into the internal connections and couplings. Local and global asymptotical stability of the coupled system is discussed, and sufficient conditions for the existence of different bifurcated oscillations are given. Numerical simulations are performed to validate the theoretical results, and interesting neuronal activities are observed, such as rest state, synchronous oscillations, asynchronous oscillations, and multiple switches of the rest states and different oscillations. It is shown that the number of neurons in the sub-networks plays an important role in the network characteristics.