A Hopfield network (or Ising model of a neural network or Ising–Lenz–Little model) is a form of recurrent artificial neural network and a type of spin glass system popularised by John Hopfield in 1982 as described earlier by Little in 1974 based on Ernst Ising's work with Wilhelm Lenz on the Ising model. Hopfield networks serve as content-addressable ("associative") memory systems with binary threshold nodes, or with continuous variables. Hopfield networks also provide a model for understanding human memory. Origins The Ising model of a neural network as a memory model was first proposed by William A. Little in 1974, which was acknowledged by Hopfield in his 1982 paper. Networks with continuous dynamics were developed by Hopfield in his 1984 paper. A major advance in memory storage capacity was developed by Krotov and Hopfield in 2016 through a change in network dynamics and energy function. This idea was further extended by Demircigil and collaborators in 2017. The continuous dynamics of large memory capacity models was developed in a series of papers between 2016 and 2020. Large memory storage capacity Hopfield Networks are now called Dense Associative Memories or modern Hopfield networks. Structure 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 unit's input exceeds its threshold . Discrete Hopfield nets describe relationships between binary (firing or not-firing) neurons . At a certain time, the state of the neural net is described by a vector , which records which neurons are firing in a binary word of bits. The interactions between neurons have units that usually take on values of 1 or −1, and this convention will be used throughout this article. However, other literature might use units that take values of 0 and 1. These interactions are "learned" via Hebb's law of association, such that, for a certain state but . (Note that the Hebbian learning rule takes the form when the units assume values in .) Once the network is trained, no longer evolve. If a new state of neurons is introduced to the neural network, the net acts on neurons such that if if where is the threshold value of the i'th neuron (often taken to be 0). In this way, Hopfield networks have the ability to "remember" states stored in the interaction matrix, because if