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Exponential Convergence Analysis of DCNNs having Unbounded Activations and Inhibitory Self-Connections

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Aspects of Mathematical Modelling

Part of the book series: Mathematics and Biosciences in Interaction ((MBI))

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Abstract

We investigate the exponential convergence characteristics of an equilibrium state of a delayed cellular neural network (DCNN) whose state variables are governed by a system of nonlinear integrodifferential equations with delays distributed continuously over unbounded intervals. The network is designed in such a way that the self-connections are inhibitory and instantaneous, and the activation functions are globally Lipschitz continuous and they are not necessarily bounded and differentiable. While monotonicity is generally not required for the activation functions, it is however needed for the activations of the inhibitory and instantaneous self-connections. By applying a Young inequality to an appropriate form of Lyapunov functionals, we establish the exponential convergence of the network towards a unique equilibrium state under a set of easily verifiable and delay independent sufficient conditions. It is shown that the restriction holding between the neural parameter values can be relaxed by the presence of the inhibitory and instantaneous self-connections and the corresponding monotonically increasing activation functions. The global exponential stability results obtained in this article will improve and extend the existing results which have been published in the literature on neural networks.

Results in this paper were presented at the International Conference on Mathematical Modelling and Computation held at the University of Brunei Darussalam during 5–8 June 2006. in conjunction with the 20th anniversary celebration of the foundation of the university.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Universiti Brunei Darussalam, Gadong, BE1410, Brunei Darussalam

    Sannay Mohamad

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  1. Sannay Mohamad
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Editors and Affiliations

  1. Dept. Mathematics, Universiti Brunei Darussalam, BE1410, Gadong, Brunei

    Roger J. Hosking

  2. Dipto. Matematica, UniversitĂ  di Torino, Via Carlo Alberto 10, 10123, Torino, Italy

    Ezio Venturino

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Mohamad, S. (2008). Exponential Convergence Analysis of DCNNs having Unbounded Activations and Inhibitory Self-Connections. In: Hosking, R.J., Venturino, E. (eds) Aspects of Mathematical Modelling. Mathematics and Biosciences in Interaction. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8591-0_14

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