 Hebbian memory weights for a Hopfield network

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 $m_{\mu }=M^{-1}{\sum }_j{\xi }_j^{\mu }{x}_j$ givesfor $P$ 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.