A model in a coupled system of simple neural oscillators with delays.

*(English)*Zbl 1176.34087A coupled system of simple neural oscillators with delays is studied. The authors prove that there is fully symmetric solution which loses stability as a parameter varies, and this loss of stability is due to the crossing of imaginary eigenvalues through the imaginary axis. So, Hopf bifurcation to periodic solutions appears. Moreover, spatio-temporal patterns appear as mirror-reflecting waves, standing waves, etc. which can be seen from the computer simulations.

Reviewer: Angela Slavova (Sofia)

##### MSC:

34K18 | Bifurcation theory of functional-differential equations |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

34K20 | Stability theory of functional-differential equations |

34K13 | Periodic solutions to functional-differential equations |

##### Keywords:

neural networks; coupled oscillator theory; delay differential equation; stability; Hopf bifurcation; periodic solution
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\textit{C. Zhang} et al., J. Comput. Appl. Math. 229, No. 1, 264--273 (2009; Zbl 1176.34087)

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##### References:

[1] | Atay, F.M., Oscillator death in coupled functional differential equations near Hopf bifurcation, Journal of differential equations, 221, 190-209, (2006) · Zbl 1099.34066 |

[2] | Dias, Ana Paula S.; Lamb, Jeroen S.W., Local bifurcation in symmetric coupled cell networks: linear theory, Physica D, 223, 1, 93-108, (2006) · Zbl 1112.34025 |

[3] | Golubitsky, M.; Stewart, I.N.; Schaeffer, D.G., () |

[4] | Guo, S.; Huang, L., Hopf bifurcating periodic orbits in a ring of neurons with delays, Physica D, 183, 19-44, (2003) · Zbl 1041.68079 |

[5] | Krawcewicz, W.; Wu, J., Theory and applications of Hopf bifurcations in symmetric functional differential equations, Nonlinear analysis, 35, 37, 845-870, (1999) · Zbl 0917.58027 |

[6] | Peng, M., Bifurcation and stability analysis of nonlinear waves in symmetric delay differential systems, J. differential equations, 232, 2, 521-543, (2007) · Zbl 1118.34067 |

[7] | Drubi, Fátima; Ibáñez, Santiago; Ángel Rodríguez, J., Coupling leads to chaos, J. differential equations, 239, 371-385, (2007) · Zbl 1133.34027 |

[8] | Tachikawa, M., Specific locking in populations dynamics: symmetry analysis for coupled heteroclinic cycles, Journal of computational and applied mathematics, 201, 374-380, (2007) · Zbl 1121.34044 |

[9] | Wu, J., Symmetric functional differential equations and neural networks with memory, Transactions of the American mathematical society, 350, 12, 4799-4838, (1998) · Zbl 0905.34034 |

[10] | Wang, L.; Zou, Xi., Hopf bifurcation in bidirectional associative memory neural networks with delays: analysis and computation, Journal of computational and applied mathematics, 167, 73-90, (2004) · Zbl 1054.65076 |

[11] | Yuri, A.K., Elements of applied bifurcation theory, (1995), Springer-Verlag New-York · Zbl 0829.58029 |

[12] | Zhang, C.; Zheng, B., Stability and bifurcation of a two-dimension discrete neural network model with multi-delays, Chaos, solitons and fractals, 31, 1232-1242, (2007) · Zbl 1141.39013 |

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