= Asynchronous Hopfield energy descent
{title2=$\Delta E=-(x_k'-x_k)\sum_{j\ne k}w_{kj}x_j\leq0$}
For symmetric zero-diagonal weights and zero thresholds, update one binary unit to the sign of its current field, retaining it at a zero field. The displayed energy change is negative on every actual flip. There are finitely many states, so a fair asynchronous update schedule eventually reaches a fixed point that is a minimum against single-unit flips. Synchronous updates can instead cycle, and local minima need not be global minima.
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