Given that W is symmetric, Hopfield proved stability for the network by showing that with the abovementioned equations (7 and 8), the network output v evolves so as to minimize the Liapunov function:
where denotes the network state. Therefore we
have an energy function defined throughout
and
dynamics which guarantee convergence to minima. Also, for a suitable
W, the minima of the first term lie at the hypercube vertices. The
second term is minimized at the hypercube center but has negligible
impact at low
.
In regard to updating the states of the net (or the output of the neurons), it is carried out either asynchronously or synchronously. From a simulation point of view, we found that asynchronous updating is a bit more efficient than synchronous updating since the later gave more limit cycles, hence, worse convergence. From a theoretical point of view, it makes not much of a difference on the operation of the net as to which updating mode is used.