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A binary Hopfield network has $N$ recurrently connected units $s_i\in \{-1,+1\}$, symmetric weights $w_{ij}=w_{ji}$, zero self-weights, and thresholds ${\theta }_i$. A distributed memory is a desired population pattern ${\xi }^{\mu }$. One Hebbian learning prescription isCoactive signs strengthen a connection, and opposite signs weaken it. This is Hebbian memory weights for a Hopfield network, rather than a separate address for each memory.

Initialize the units with the input cue, which can be a corrupted or partially specified pattern. Update one unit at a time by $s_i\leftarrow \mathrm{sign}({\sum }_j{w}_{ij}{s}_j-{\theta }_i)$, retaining its old sign when the field is exactly zero. A fair asynchronous schedule revisits every unit. The energychanges on a single update by $\Delta E=-(s_i^{\prime }-s_i)({\sum }_j{w}_{ij}{s}_j-{\theta }_i)\le 0$. An actual flip with the tie convention strictly reduces energy; finitely many states imply eventual arrival at a fixed point. This proves asynchronous Hopfield energy descent. Synchronous updating lacks this single-unit argument and can cycle.

For one stored pattern and zero thresholds, the field at $s=\xi$ is $(N-1){\xi }_i/N$, so the pattern is a fixed point. With multiple patterns, the desired signal is accompanied by cross-talk from other memories; not every arbitrary set is guaranteed to be stable. Retrieval is attraction from a cue to a stored-pattern fixed point, provided the cue lies in its basin of attraction. Spurious minima and limited basins are part of the model, not errors in its update algorithm.

For the symmetric zero-diagonal weights and asynchronous rule just specified, update only unit $k$. In the energy difference, terms involving neither $k$ nor its symmetric partner cancel. Thus asynchronous Hopfield energy descent givesIf the unit was already aligned, it does not change and $\Delta E=0$. If it flips into alignment, $x_k^{\prime }-x_k=2\mathrm{sgn}(h_k)$, soAt $h_k=0$ our tie rule retains the old activity. The network has only $2^M$ configurations. Strict descent on every actual change prevents revisiting a configuration and permits only finitely many changes. With a fair schedule every misaligned unit is eventually updated, so the terminal configuration is a fixed point. At that point each possible single-bit flip has energy cost $2x_k{h}_k\ge 0$. The network reaches a minimum against single-unit changes, not necessarily the global minimum. Its energy is a Lyapunov function for these asynchronous dynamics.

The qualifications are essential to the printed claim. Synchronous updates can cycle: for two units with $w_{12}=w_{21}=1$, the state $(1,-1)$ alternates with $(-1,1)$, keeping energy $+1$. Symmetry alone therefore does not prove convergence under simultaneous updates.

Nor does the valid descent proof guarantee a global minimum. Consider four units with $w_{12}=w_{34}=3$ and all four cross-group weights equal to $1$, symmetrically and with zero diagonal. The state $(1,1,-1,-1)$ has aligned fields $(1,1,-1,-1)$ and energy $-2$, so it is a strict single-flip local minimum. The all-positive state has energy $-10$. This explicit counterexample shows why local minima may trap retrieval even though the energy always descends under the stipulated rule.