Hopfield networks

A Hopfield network is a form of recurrent artificial neural network popularized by John Hopfield in 1982, but described earlier by Little in 1974. Hopfield nets serve as content-addressable ("associative") memory systems with binary threshold nodes. They are guaranteed to converge to a local minimum, but will sometimes converge to a false pattern (wrong local minimum) rather than the stored pattern (expected local minimum). Hopfield networks also provide a model for understanding human memory.

The units in Hopfield nets are binary threshold units, i.e. the units only take on two different values for their states and the value is determined by whether or not the units' input exceeds their threshold. Hopfield nets normally have units that take on values of 1 or -1, and this convention will be used throughout this page. However, other literature might use units that take values of 0 and 1.

Every pair of units i and j in a Hopfield network have a connection that is described by the connectivity weight ${\displaystyle w_{ij}}$. In this sense, the Hopfield network can be formally described as a complete undirected graph ${\displaystyle G=\langle V,f\rangle }$, where ${\displaystyle V}$ is a set of McCulloch-Pitts neurons and ${\displaystyle f:V^{2}\rightarrow R}$ is a function that links pairs of nodes to a real value, the connectivity weight.

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