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Hopfield networks are a kind of recurrent neural network with binary threshold nodes.

## Definition

Nodes have indexes \(i \in \{1, \cdots, n\}\) and are in state \(s_i \in \{-1, 1\}\). Nodes have connections between them, characterized by a weight \(w_{ij}\). Each node also has an associated threshold \(\theta_i\) such that

\begin{equation*} s_i \leftarrow \begin{cases} +1 & \text{if}\ \sum_j w_{ij} s_j \geq \theta_i, \\\

-1 & \text{otherwise}. \end{cases} \end{equation*}

## Energy

A Hopfield network has an associated energy value \[ E = - \frac{1}{2} \sum_{i,j} w_{ij} s_i s_j + \sum_i \theta_i s_i \] which makes it part of the Ising models.

## Hopfield networks and attention

See Hopfield Networks is All You Need (Ramsauer et al. 2020)