Convergence of Neural Networks with a Class of Real Memristors with Rectifying Characteristics

Mauro Di Marco (), Mauro Forti, Riccardo Moretti, Luca Pancioni and Alberto Tesi
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Mauro Di Marco: Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, Via Roma 56, 53100 Siena, Italy
Mauro Forti: Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, Via Roma 56, 53100 Siena, Italy
Riccardo Moretti: Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, Via Roma 56, 53100 Siena, Italy
Luca Pancioni: Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, Via Roma 56, 53100 Siena, Italy
Alberto Tesi: Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Firenze, Via S. Marta 3, 50139 Firenze, Italy

Mathematics, 2022, vol. 10, issue 21, 1-18

Abstract: The paper considers a neural network with a class of real extended memristors obtained via the parallel connection of an ideal memristor and a nonlinear resistor. The resistor has the same rectifying characteristic for the current as that used in relevant models in the literature to account for diode-like effects at the interface between the memristor metal and insulating material. The paper proves some fundamental results on the trajectory convergence of this class of real memristor neural networks under the assumption that the interconnection matrix satisfies some symmetry conditions. First of all, the paper shows that, while in the case of neural networks with ideal memristors, it is possible to explicitly find functions of the state variables that are invariants of motions, the same functions can be used as Lyapunov functions that decrease along the trajectories in the case of real memristors with rectifying characteristics. This fundamental property is then used to study convergence by means of a reduction-of-order technique in combination with a Lyapunov approach. The theoretical predictions are verified via numerical simulations, and the convergence results are illustrated via the applications of real memristor neural networks to the solution of some image processing tasks in real time.

Keywords: convergence; diode-like nonlinearities; flux–charge analysis method; image processing; invariants of motion; Lyapunov functions; memristor; neural networks (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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