a new state is subjected to the interaction matrix, each neuron will change until it matches the original state (see the Updates section below). The connections in a Hopfield net typically have the following restrictions: (no unit has a connection with itself) (connections are symmetric) The constraint that weights are symmetric guarantees that the energy function decreases monotonically while following the activation rules. A network with asymmetric weights may exhibit some periodic or chaotic behaviour; however, Hopfield found that this behavior is confined to relatively small parts of the phase space and does not impair the network's ability to act as a content-addressable associative memory system. Hopfield also modeled neural nets for continuous values, in which the electric output of each neuron is not binary but some value between 0 and 1. He found that this type of network was also able to store and reproduce memorized states. Notice that every pair of units i and j in a Hopfield network has a connection that is described by the connectivity weight . In this sense, the Hopfield network can be formally described as a complete undirected graph , where is a set of McCulloch–Pitts neurons and is a function that links pairs of units to a real value, the connectivity weight. Updating Updating one unit (node in the graph simulating the artificial neuron) in the Hopfield network is performed using the following rule: where: is the strength of the connection weight from unit j to unit i (the weight of the connection). is the state of unit i. is the threshold of unit i. Updates in the Hopfield network can be performed in two different ways: Asynchronous: Only one unit is updated at a time. This unit can be picked at random, or a pre-defined order can be imposed from the very beginning. Synchronous: All units are updated at the same time. This requires a central clock to the system in order to maintain synchronization. This method is viewed by some as less realistic, based on an absence of observed global clock influencing analogous biological or physical systems of interest. Neurons "attract or repel each other" in state space The weight between two units has a powerful impact upon the values of the neurons. Consider the connection weight between two neurons i and j. If , the updating rule implies that: when , the contribution of j in the weighted