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Learning and energy-entropy dependence in some nonlinear functional-differential systems

**Author(s):** Grossberg, S. |

**Year:** 1969

**Citation:** Bulletin of the American Mathematical Society, 75, 1238-1242

**Abstract:** 1. Introduction. This note describes limiting and oscillatory fea-tures of some nonlinear functional-differential systems having appli-cations in learning and nonstationary prediction theory. The mainresults discuss systems of the form

(1) z,(t) = A(W,, t)xt(t) + E Bk(W,, t)zkti(t) + C.(t)kEJand(2) i i i (t) = Di(Wg, t)zi:(t) + Ej(W,, t)x:(t),where i E I, j E J, and I and J are finite sets of indices such that eitherI = J or It'1 J = J. The coefficients are continuous functions of t,dependent perhaps on the 111 (1 + I JI) dimensional vector functionW=(x;,z;;: iEI, jEJ) evaluated at times no later than t, and onknown functions of t. All coefficients B; and E; are also nonnegative,and the initial data and inputs C{ are nonnegative and continuous.The main results discuss the probabilities y;; (t) =z;;(t) [ Ekâ‚¬rz;k(t) ]-'and Xi(t) =x;(t) [ Ekerxk(t) ]-1 defined for iEI and jEJ, given choicesof initial data and coefficient functionals for which (1) and (2) has aunique bounded solution.These results apply for example to systems of the form [. . .]

**Topics: **
Mathematical Foundations of Neural Networks,
**Models: **
Other,