Asynchronous Hopfield energy descent ($\Delta E=-(x_k^{\prime }-x_k){\sum }_{j\ne k}{w}_{kj}{x}_j\le 0$)

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.