= Hebbian memory weights for a Hopfield network
{c}
{title2=$w_{ij}=M^{-1}\sum_\mu\xi_i^\mu\xi_j^\mu,\quad i\ne j$}
For $M$ binary units and desired patterns $\xi^\mu$, use these symmetric weights and set $w_{ii}=0$. For one pattern, the field at that pattern is $(M-1)\xi_i/M$, so it is stable. Several patterns add cross-talk terms; storage by this rule alone does not guarantee stability of every arbitrary collection of patterns.
Writing $m_\mu=M^{-1}\sum_j\xi_j^\mu x_j$ gives
$$
\sum_{j\ne i}w_{ij}x_j=\sum_\mu\xi_i^\mu m_\mu-\frac{P}{M}x_i,
$$
for $P$ stored patterns. The subtraction removes the forbidden self-couplings. It makes explicit how the desired association competes with all other stored patterns. An unintended stable state can therefore arise even though every update obeys <asynchronous Hopfield energy descent>.
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