Editing Hopfield network (section)

===Hebbian learning rule for Hopfield networks===
[[Hebbian theory]] was introduced by Donald Hebb in 1949 in order to explain "associative learning", in which simultaneous activation of neuron cells leads to pronounced increases in synaptic strength between those cells.<ref>{{harvnb|Hebb|1949}}</ref> It is often summarized as "Neurons that fire together wire together. Neurons that fire out of sync fail to link".

The Hebbian rule is both local and incremental. For the Hopfield networks, it is implemented in the following manner when learning <math>n</math>
binary patterns:

<math> w_{ij}=\frac{1}{n}\sum_{\mu=1}^{n}\epsilon_{i}^\mu \epsilon_{j}^\mu </math>

where <math>\epsilon_i^\mu</math> represents bit i from pattern <math>\mu</math>.

If the bits corresponding to neurons i and j are equal in pattern <math>\mu</math>, then the product  <math> \epsilon_{i}^\mu \epsilon_{j}^\mu </math> will be positive. This would, in turn, have a positive effect on the weight <math>w_{ij} </math> and the values of i and j will tend to become equal. The opposite happens if the bits corresponding to neurons i and j are different.