Networks and Computations of Artificial Neurons

A Hopfield network is a completely connected directed graph of I nodes (“neurons”), i.e., a graph of I nodes, denoted {1, 2, . . . , I}, in which there is a directed edge from each node to every other node (there are no self directed loops). Each node i has an associated output value xi. For simplicity, we consider xi ∈ {−1, 1}. From a biological perspective, such an binary assignment of node values corresponds to a neuron not firing (xi = −1) and a neuron firing at maximum rate (xi = 1), respectively [1]. Finally, there is a weight assigned to each edge: wij is the weight of the directed edge from node j to node i. The value of every node represents the state or output of the Hopfield network (i.e., the network of the state is a vector x ∈ {−1, 1} .