(a) Asix-neuron discrete Hopfield network is required to store the following fundamental memories:

x1 = (1, 1, 1, 1, 1, 1)T , x2 = (1,−1, 1,−1,−1, 1)T , x3 = (1,−1,−1, 1,−1, 1)T .

(i) Compute the synaptic weight matrix W.
(ii) Use asynchronous updating to show that the three fundamental memories are stable.
(iii) Test the vector (−1,−1,−1, 1, 1, 1)T on the Hopfield network.
Use your own set of random orders in (ii) and (iii).
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(b) Derive a suitable Lyapunov function for the recurrent Hopfield network modeled using the differential equations ˙ x = −x+2 2 π
tan−1 γπx 2 , ˙ y = −y+2 2 π
tan−1 γπy 2 .
Plot a vector field plot and Lyapunov function surface plot for γ = 0.5.