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Activity-dependent strengthening associated with coactivation of an input and its postsynaptic output. The idealized update $\Delta w_i\propto y{x}_i$ requires normalization, competition or other constraints to prevent indefinite weight growth.

In Hebbian memory weights for a Hopfield network, the local update $\Delta w_{ij}=\eta {\xi }_i{\xi }_j$ records a pairwise association, where $\eta >0$ and ${\xi }_i,{\xi }_j\in \{-1,1\}$. Equal signs strengthen a positive coupling; opposite signs favor a negative coupling. Summing such updates gives a symmetric matrix of pairwise associations. This is a storage prescription: it does not by itself compare the generated probability distribution with the training data, as Boltzmann machine learning does. Bipolar activity also makes joint inactivity contribute positively; this encoding is a mathematical idealization, not a literal statement about inactive biological cells.

Store only $\xi =(1,-1,1,-1,1)$ with $M=5$, zero thresholds, and Hebbian memory weights for a Hopfield network. If $d$ bits differ from $\xi$, their Hamming distance is $d$ and ${\sum }_i{\xi }_i{x}_i=M-2d$. Squaring this sum and subtracting the $M$ diagonal terms yields the displayed energy. Thus $d=0,1,2,3,4,5$ gives $E=-2,-2/5,2/5,2/5,-2/5,-2$.

For the cue $x=(-1,1,1,-1,1)$, the first unit has input $h_1=2/5$ and flips to $1$. The second then has $h_2=-4/5$ and flips to $-1$, completing recall. The energy drops $2/5\to -2/5\to -2$. Every cue with at most two wrong bits is corrected under a fair asynchronous schedule: a wrong bit sees the correct majority among the other four, while a correct bit either sees agreement or retains its value at a tie. The opposite pattern $-\xi$ is also stored implicitly, so cues with at least three errors retrieve it. This is a limitation of this symmetric one-pattern construction.

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.

A standard binary Hopfield network contains $M$ recurrently connected units with activities $x_i\in \{-1,+1\}$. Choose symmetric weights $w_{ij}=w_{ji}$, omit self-couplings by setting $w_{ii}=0$, and use zero thresholds for the energy in the next part. To store desired binary patterns ${\xi }^{\mu }$, a simple choice is Hebbian memory weights for a Hopfield network:Correlated activity strengthens connections between units agreeing across the training patterns and makes disagreement contribute negatively. This creates attraction toward the stored configurations when cross-talk is sufficiently small. For one stored pattern, its local field is $(M-1){\xi }_i/M$, so every bit is stable for $M>1$. With multiple patterns, the other patterns add cross-talk; the rule does not guarantee exact retrieval for an arbitrary collection.

Start retrieval from a partial or corrupted cue and update one unit at a time using its current fieldA cyclic or random fair asynchronous schedule lets the network relax toward a fixed point, completing a memory when the cue lies in its basin. This is content-addressable retrieval: the initial pattern selects the memory without an explicit address. Nonzero thresholds can be included in a more general model, but their corresponding linear terms must then be included in the energy.