Editing Hopfield network (section)

==Updating==

Updating one unit (node in the graph simulating the artificial neuron) in the Hopfield network is performed using the following rule:

<math>s_i \leftarrow \left\{\begin{array}{ll} +1 & \text{if }\sum_{j}{w_{ij}s_j}\geq\theta_i, \\
 -1 & \text{otherwise.}\end{array}\right.</math>

where:
* <math>w_{ij}</math> is the strength of the connection weight from unit j to unit i (the weight of the connection).
* <math>s_i</math> is the state of unit i.
* <math>\theta_i</math> 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 <math>w_{ij}</math> between two neurons i and j. If <math>w_{ij} > 0 </math>, the updating rule implies that:
* when <math>s_j = 1</math>, the contribution of ''j'' in the weighted sum is positive. Thus, <math>s_{i}</math> is pulled by ''j'' towards its value <math>s_{i} = 1</math>
* when <math>s_j = -1</math>, the contribution of ''j'' in the weighted sum is negative. Then again, <math>s_i</math> is pushed by ''j'' towards its value <math>s_i = -1</math>

Thus, the values of neurons ''i'' and ''j'' will converge if the weight between them is positive. Similarly, they will diverge if the weight is negative.