Boundary Pointwise Control for Diffusion Hopfield Neural Network

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. DOI: 10.4018/jnmc.2010010102 14 International Journal of Nanotechnology and Molecular Computation, 2(1), 13-29, January-March 2010 Copyright © 2010, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. 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. As far as we know that most neurons communicate through punctate events (called spikes: a sharp upswing, then a restoring downswing). Spiking neurons connected with output ones, the synapse propagate (exchange) information by spikes once cell’s membrane voltage (firing rate) is going to peak and broken the thresholds, the whole event typically lasting 1~2 millisecond. Those sharp voltage transients travel down the output cables of the axons. As the spikes reach the axon terminal synapses, which connecting the neuron to further downstream neurons, they form the signal indicated that chemical neurotransmitters are released, thus communicating a signal to other neuron. Addtionally, the synaptic excitatory delay is 0.3~1 ms. Omit the slight delay in transmission due to random delays provide more robust (synaptic efficacies) network. For more rational consideration, suppose diffusion would be happen between neurons activities, the involved target control system will be HNN with diffusion term (Wang, 2004, 2005, 2006; Wang & Nakagiri, 2006; Wang, 2007, 2009). As an example and simulation purpose, three neurons in Figure 1 can be expressed by diffusion HNN.