Neurodynamics and nonlinear integrable systems of Lax type
type: Item
instance of: scholarly article
title: Neurodynamics and nonlinear integrable systems of Lax type
zbMATH Open document ID: 0822.92003
DOI: 10.1007/BF03167210
author: Yoshimasa Nakamura
published in: Japan Journal of Industrial and Applied Mathematics
publication date: 1994-05-24T00:00:00Z
review text: We consider neurodynamics as a new topic of applied analysis of nonlinear integrable systems. We here take the position that we regard the problem of learning as a challenging problem in nonlinear dynamical systems. First we discuss a stochastic learning equation of Hebb type with a nonlinear damping term. Averaging the learning equation we derive a nonlinear dynamical system which is a gradient system for a potential function. Stable equilibrium points of the averaged learning equation are discussed. It is shown that the averaged learning equation also can be regarded as a constrained harmonic motion having a Lax pair representation. A competition learning rule appears as a first integral of the dynamical system. A relationship to the finite nonperiodic Toda lattice equation which is a typical nonlinear integrable system is discussed. It is also shown that a Lax type equation also emerges in the Hopfield model.
Mathematics Subject Classification ID: 92B20
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 68T05
zbMATH DE Number: 571991
zbMATH Keywords: neurodynamics
zbMATH Keywords: nonlinear integrable systems
zbMATH Keywords: stochastic learning equation of Hebb type
zbMATH Keywords: gradient system
zbMATH Keywords: averaged learning equation
zbMATH Keywords: constrained harmonic motion
zbMATH Keywords: Lax pair representation
zbMATH Keywords: finite nonperiodic Toda lattice equation
zbMATH Keywords: Lax type equation
zbMATH Keywords: Hopfield model
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03167210
OpenAlex ID: W2045520233
description: scientific article; zbMATH DE number 571991