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**Frustration, stability, and delay-induced oscillations in a neural network model.**
*(English)*
Zbl 0840.92003

Summary: The effect of time delays on the linear stability of equilibria in an artificial neural network of Hopfield type is analyzed. The possibility of delay-induced oscillations occurring is characterized in terms of properties of the (not necessarily symmetric) connection matrix of the network. Such oscillations are possible exactly when the network is frustrated, equivalently when the signed digraph of the matrix does not require the Perron property. Nonlinear analysis (centre manifold computation) of a three-unit frustrated network is presented, giving the nature of the bifurcations taking place. A supercritical Hopf bifurcation is shown to occur, and a codimension-two bifurcation is unfolded.

### MSC:

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

34K11 | Oscillation theory of functional-differential equations |

34K18 | Bifurcation theory of functional-differential equations |

15B48 | Positive matrices and their generalizations; cones of matrices |

05C20 | Directed graphs (digraphs), tournaments |

68T05 | Learning and adaptive systems in artificial intelligence |