Boundary Pointwise Control for Diffusion Hopfield Neural Network

Boundary Pointwise Control for Diffusion Hopfield Neural Network

Quan-Fang Wang (The Chinese University of Hong Kong, China)
Copyright: © 2010 |Pages: 17
DOI: 10.4018/jnmc.2010010102

Abstract

For a close to practical neural network in biology field, in this paper the author address the diffusion Hopfield neural network (HNN) with boundary pointwise control. In the framework of variational method at Hilbert space, the theoretical study finds and characterizes the boundary optimal control solution. Furthermore, with the numerical approach consist of finite element method (FEM) and conjugate gradient method (CGM), computational demonstration is performed for three neurons in two dimensions case. This approach adequately interpreted the effectiveness and feasibility of the control process in a realistic sense.
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Introduction

Neurobiology and neuronphysiology provides a great deal information on individual neuron as well as function of nuclei and other gross of neural structure. Inspired by the function of biological neurons, neural network is well utilized in a variety of fields with its featured structure. Especially, in industry and engineering areas, it easily find the extensive applications, such as combinatorial optimization, signal processing, image processing, pattern recognition and associative memories and so forth. Many of these designs become far from biological reality, and lost all interest in simulating neurology. They are more interested in using it as new tool to solve real problems arising in the world. Theory and computation of neural network can be highly mathematical, even some existing networks are entirely as mathematical models. The communications between theoreticians and neurophysiologists, the effort involved in analyzing its properties would be definitely hopeful. One feature of neural network is simple, finite speed to switch and transmit of signal, its input/output (I/O) layers can be easily constructed. Even its hidden layers connections (e.g., nodes) are linear and can be calculated quickly. Without regarding of its usefulness, what should be emphasized is controlling of large number neurons. In realistic case, how to achieve such progress in practical meaning, it would be a fairly highlighted topic in the relevant areas. Our previous researches (Wang, 2007; Wang, 2009) are attempting works in according to this direction. However, distributed and initial control in Wang (2007) is incredible for real neutrons, even pointwise control in the interior of neural network (Wang, 2009) is also impossible now. Although expect above controls can be realized some day. In fact, at present medical equipments and existing technology level, the most reasonable control which could be performed at neural network, that is external (i.e., boundary) control, particularly at finite points, namely “boundary pointwise control”. A lot kinds of neural networks are reported, for instance spiking neuron model and pulsed neural network and so on. It is well known, J. J. Hopfield proposed Hopfield neural network (HNN) since 1980s (Hopfield, 1982, 1984, 1986), the famous HNN is tremendously applied in a great deal researches (Fitz-Hugh, 1955; Hodgkin, 1952; Nagumo,1962; Nakagiri, 2002; Kunz, 1991; Wilde, 1997; Kaslik, 2007; Litinskii, 1999). It’s convenient to show Hopfield neural network consist of three neurons in Figure 1.

Figure 1.

3-neurons HNN

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