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5.12 Ho~field Networks 229

Figure 5.52. Binary image prototypes used in CAM experiments with the Hopfield
program: (a) "zero"; (b) "plus"; (c) "cross".

Three such "unknown" patterns are shown in Figure 5.53. The first one, Figure
5.53a, corresponds to a "plus" pattern with noise and converges in one cycle to the
right prototype.
The second one, Figure 5.53b, corresponds to a "cross" pattern with substantial
noise. It can either converge to a cross or to one of the spurious states shown in
Figures 5.54a and 5.54b. The type of convergence depends on the random
behaviour of the output neuron updating and, therefore, on the particular
"trajectory" followed in the space of the d-dimensional hypercube vertices. When
the spurious state is the one shown in Figure 5.54a, the matching prototype is the
"plus" pattern, therefore corresponding to an incorrect retrieval.
Finally, the "grid" pattern also exhibits convergence to a spurious state. An
interesting thing about this pattern is that the Hopfield net will not converge and,
instead, will oscillate between two states, the "grid" itself and its complement
shown in Figure 5.54c, if the algorithm performs a full parallel updating instead of
a random serial updating. The possibility of obtaining oscillations between two
states with full parallel updating was previously mentioned.

Figure 5.53. Three unknown binary images: (a) "plus" with noise; (b) a highly
corrupted "cross"; (c) "grid".