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Fundamentals
The goal here is to design a network that stores a specific set of equilibrium points such that, when an initial condition is provided, the network eventually comes to rest at such a design point. The network is recursive in that the output is fed back as the input, once the network is in operation. Hopefully, the network output will settle on one of the original design points
The design method that we present is not perfect in that the designed network may have undesired spurious equilibrium points in addition to the desired ones. However, the number of these undesired points is made as small as possible by the design method. Further, the domain of attraction of the designed equilibrium points is as large as possible.
The design method is based on a system of first-order linear ordinary differential equations that are defined on a closed hypercube of the state space. The solutions exist on the boundary of the hypercube. These systems have the basic structure of the Hopfield model, but are easier to understand and design than the Hopfield model.
The material in this section is based on the following paper: Jian-Hua Li, Anthony N. Michel and Wolfgang Porod, "Analysis and synthesis of a class of neural networks: linear systems operating on a closed hypercube," IEEE Trans. on Circuits and Systems vol 36, no. 11, pp. 1405-22, November 1989.
For further information on Hopfield networks, read Chapter 18 of the Hopfield Network [HDB96].
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