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**Introduction to neural dynamics and signal transmission delay.**
*(English)*
Zbl 0977.34069

de Gruyter Series in Nonlinear Analysis and Applications. 6. Berlin: Walter de Gruyter. x, 182 p. (2001).

This book deals with mathematical modeling and analysis of neural networks from the viewpoint of dynamical systems. It consists of five chapters. In Chapter 1 the author gives an introductory description of the structure of a single neuron and a neural circuit. The electrical and chemical mechanism of neural signal transmission is discussed as well. Basic dynamic models of neural networks are considered in Chapter 2. Typical signal functions as step function, piecewise linear function, sigmoid function, are presented for the main neural architectures. Chapter 3 deals with some simple neural networks models which have application in pattern learning, pattern recognition and noise suppression. The author shows how the choice of signal function affects the limiting patterns. Signal weights are determined as well. In Chapter 4 the dynamics of neural networks is studied in the sense of stability of the equilibrium points. The author presents several approaches towards the global analysis of neural networks, such as Lyapunov (or energy) function and monotone dynamical systems theory. Chapter 5 deals with signal transmission delays which occur in biological neural networks. It is shown that the signal delays change the stability of equilibria, causing nonlinear oscillations and inducing periodic solutions. A description of the topological and geometric structure of the attractor is made. A summary of recent results on dynamics of delay differential equations arising from neural networks is presented in this chapter, too.

Reviewer: Angela Slavova (Sofia)

### MSC:

34K20 | Stability theory of functional-differential equations |

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

34K13 | Periodic solutions to functional-differential equations |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

92-02 | Research exposition (monographs, survey articles) pertaining to biology |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

94-02 | Research exposition (monographs, survey articles) pertaining to information and communication theory |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37N25 | Dynamical systems in biology |

34K11 | Oscillation theory of functional-differential equations |