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.

To obtain the displayed identity, subtract the two energies. Terms not involving $x_k$ cancel, and symmetry combines the two copies of each pair. With thresholds included the result is $\Delta E=-(x_k^{\prime }-x_k)h_k$. The zero-threshold expression above is its specialization. At $h_k=0$, keeping the old state rules out motion along equal-energy states. Finiteness then excludes infinitely many strict decreases, while fairness ensures that an unstable unit cannot be ignored forever. A local minimum here means no single flip lowers energy; equality directions can remain.

The asynchronous condition matters even for two units: with $w_{12}=w_{21}=1$ and zero thresholds, simultaneous updates alternate $(1,-1)$ and $(-1,1)$ forever. Either single-unit update instead reaches an aligned stable state.