title: 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.
authors.affiliation: Department of Information Engineering and Mathematical Sciences, University of Siena, 53100 Siena, Italy. Electronic address: dimarco@dii.unisi.it.
authors.forename: Mauro
authors.initials: M
authors.lastname: Di Marco
authors.affiliation: Department of Information Engineering and Mathematical Sciences, University of Siena, 53100 Siena, Italy. Electronic address: forti@dii.unisi.it.
authors.forename: Mauro
authors.initials: M
authors.lastname: Forti
authors.affiliation: Department of Information Engineering and Mathematical Sciences, University of Siena, 53100 Siena, Italy. Electronic address: grazzini@dii.unisi.it.
authors.forename: Massimo
authors.initials: M
authors.lastname: Grazzini
authors.affiliation: Department of Information Engineering and Mathematical Sciences, University of Siena, 53100 Siena, Italy. Electronic address: pancioni@dii.unisi.it.
authors.forename: Luca
authors.initials: L
authors.lastname: Pancioni
mesh_terms: D000465:Algorithms; D008962:Models, Theoretical; D016571:Neural Networks, Computer; D009474:Neurons
keywords: Cooperative dynamical systems; Differential variational inequalities; Multistability; Neural networks; Nonsmooth dynamical systems