Necessary and sufficient condition for multistability of neural networks evolving on a closed hypercube

Abstract

The paper considers nonsmooth neural networks described by a class of differential inclusions termed differential variational inequalities (DVIs). The DVIs include the relevant class of neural networks, introduced by Li, Michel and Porod, described by linear systems evolving in a closed hypercube of R(n). The main result in the paper is a necessary and sufficient condition for multistability of DVIs with nonsymmetric and cooperative (nonnegative) interconnections between neurons. The condition is easily checkable and provides a sharp bound between DVIs that can store multiple patterns, as asymptotically stable equilibria, and those for which this is not possible. Numerical examples and simulations are presented to confirm and illustrate the theoretic findings.

Keywords: Cooperative dynamical systems; Differential variational inequalities; Multistability; Neural networks; Nonsmooth dynamical systems.