Sigmoidal Approximations of a Nonautonomous Neural Network with Infinite Delay and Heaviside Function
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Sigmoidal Approximations of a Nonautonomous Neural Network with Infinite Delay and Heaviside Function Peter E. Kloeden1 · Víctor M. Villarragut2 Dedicated to the memory of Russell A. Johnson Received: 4 June 2020 / Revised: 11 September 2020 / Accepted: 21 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we approximate a nonautonomous neural network with infinite delay and a Heaviside signal function by neural networks with sigmoidal signal functions. We show that the solutions of the sigmoidal models converge to those of the Heaviside inclusion as the sigmoidal parameter vanishes. In addition, we prove the existence of pullback attractors in both cases, and the convergence of the attractors of the sigmoidal models to those of the Heaviside inclusion. Keywords Neural networks · Differential inclusion · Nonautonomous set-valued dynamical system · Pullback attractor · Upper semi convergence Mathematics Subject Classification 34K09 · 34D45 · 37B55
1 Introduction Neural networks arise in many contexts, ranging from biological modeling to engineering applications. The study of the dynamics of neural networks has been of paramount importance in the process of establishing a theoretical framework for this fast-paced field. Extensive efforts have been devoted to the development of such theory (see e.g. Wu [17] and the
Both authors were partially supported by MICIIN/FEDER under Project RTI2018-096523-B-100.
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Víctor M. Villarragut [email protected] Peter E. Kloeden [email protected]
1
Mathematics Department, University of Tübingen, 72076 Tübingen, Germany
2
Departamento de Matemática Aplicada a la Ingeniería Industrial, Universidad Politécnica de Madrid, Calle de José Gutiérrez Abascal 2, 28006 Madrid, Spain
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Journal of Dynamics and Differential Equations
references therein). Specifically, in [17], the following system of differential equations with atomic delay was proposed: xi (t) = −Ai (xi (t)) xi (t) +
n (z ki (t) − cki ) f k (xk (t − τki ) − Γk ) + Ii (t) , k=1 k =i
z i j (t) = −Bi j (z i j (t)) z i j (t) + di j f i (xi (t − τi j ) − Γi ) [x j (t)]+ , where i, j ∈ {1, . . . , n} with i = j. The parameters in this model are thoroughly defined in [17], but we will give a few representative definitions for the reader’s convenience. Namely, xi (t) represents the deviation of the ith neuron’s potential from its equilibrium and z i j (t) is the neurotransmitter average release rate per unit axon signal frequency, usually referred to as synaptic coupling strength. We assume further that the neuron’s potential decays exponentially to its equilibrium in the absence of external processes, which is regulated by Ai . Also, the kth neuron sends a signal f k , which depends on the past values of xk and is subject to a threshold Γk , along the axon to the ith neuron. That signal affects the target neuron in an additive manner proportionally to the coupling strength z ki (t), giving rise to the term k=i z ki (t) f k (x k (t − τki ) − Γk
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