Neurodynamics and nonlinear integrable systems of Lax type
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.
stochastic learning equation of Hebb type
finite nonperiodic Toda lattice equation
scientific article; zbMATH DE number 571991