C++ Neural Networks and Fuzzy Logic: Preface

Figure 4.1

  Layout of a Hopfield Network

The two patterns we want the network to have stable recall for are A = (1, 0, 1, 0) and B = (0, 1, 0, 1). The

weight matrix W is given as follows:

               0    −3     3     −3

     W     =  −3     0    −3      3

               3    −3     0     −3

              −3     3    −3      0

NOTE:  The positive links (values with positive signs) tend to encourage agreement in a

stable configuration, whereas negative links (values with negative signs) tend to discourage

agreement in a stable configuration.

We need a threshold function also, and we define it using a threshold value, [theta], as follows:

                  1  if t >= [theta]

     f(t) =  

{

                  0  if t < [theta]

C++ Neural Networks and Fuzzy Logic:Preface

Chapter 4 Constructing a Neural Network

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The threshold value [theta] is used as a cut−off value for the activation of a neuron to enable it to fire. The

activation should equal or exceed the threshold value for the neuron to fire, meaning to have output 1. For our

Hopfield network, [theta] is taken as 0. There are four neurons in the only layer in this network. The first

node’s output is the output of the threshold function. The argument for the threshold function is the

activation of the node. And the activation of the node is the dot product of the input vector and the first

column of the weight matrix. So if the input vector is A, the dot product becomes 3, and f(3) = 1. And the dot

products of the second, third, and fourth nodes become –6, 3, and –6, respectively. The corresponding outputs

therefore are 0, 1, and 0. This means that the output of the network is the vector (1, 0, 1, 0), which is the same

as the input pattern. Therefore, the network has recalled the pattern as presented. When B is presented, the dot

product obtained at the first node is –6 and the output is 0. The activations of all the four nodes together with

the threshold function give (0, 1, 0, 1) as output from the network, which means that the network recalled B

as well. The weight matrix worked well with both input patterns, and we do not need to modify it.