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

x1 = (1, 1, 1, 1)T , x2 = (1,−1, 1,−1)T , x3 = (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)T on the Hopfield network.

Use your own set of random orders in (ii) and (iii).

[10]

(b) Derive a suitable Lyapunov function for the recurrent Hopfield network modeled using the differential equations [10]

˙ x = −x +

2

π

tan−1

γπx 2

+

2

π

tan−1

γπy 2

+ 6,

˙ y = −y +

2

π

tan−1

γπx 2

+ 4 2

π

tan−1

γπy 2

+ 10.