### Abstract

The number of solutions of a nonlinear equation x = sgn(Wx) is discussed. The equation is derived for the determination of equilibrium points of a kind of Hopfield neural networks. We impose some conditions on W. The conditions correspond to the case where a Hopfield neural network has n neurons arranged on a ring, each neuron has connections only from k preceding neurons and the magnitude of k connections decrease as the distance between two neurons increases. We show that the maximum number of solutions for the above case is extremely few and is independent of the number of neurons, n, if k is less than or equal to 4. We also show that the number of solutions generally increases exponentially with n by considering the case where k = n - 1.

Original language | English |
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Pages (from-to) | 1299-1305 |

Number of pages | 7 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E78-A |

Issue number | 10 |

Publication status | Published - Oct 1 1995 |

Externally published | Yes |

### ASJC Scopus subject areas

- Signal Processing
- Computer Graphics and Computer-Aided Design
- Electrical and Electronic Engineering
- Applied Mathematics