Difference between revisions of "Hebb's rule"

From Eyewire
Jump to: navigation, search
 
Line 1: Line 1:
 +
<translate>
 +
 
'''Hebb's Rule''' or Hebb's postulate attempts to explain "associative learning", in which simultaneous activation of cells leads to pronounced increases in [[Synapse|synaptic]] strength between those cells. Hebb stated:
 
'''Hebb's Rule''' or Hebb's postulate attempts to explain "associative learning", in which simultaneous activation of cells leads to pronounced increases in [[Synapse|synaptic]] strength between those cells. Hebb stated:
  
Line 49: Line 51:
  
 
[[Category: Neural computational models]]
 
[[Category: Neural computational models]]
 +
 +
</translate>

Latest revision as of 03:15, 24 June 2016

Hebb's Rule or Hebb's postulate attempts to explain "associative learning", in which simultaneous activation of cells leads to pronounced increases in synaptic strength between those cells. Hebb stated:

Let us assume that the persistence or repetition of a reverberatory activity (or "trace") tends to induce lasting cellular changes that add to its stability.… When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A's efficiency, as one of the cells firing B, is increased.[1]

Model

Model of a neuron. j is the index of the neuron when there is more than one neuron. For a linear neuron, the activation function is not present (or simply the identity function).

Given a set of k-dimensional inputs represented as a column vector:

生成缩略图出错:无法将缩略图保存到目标地点

and a linear neuron with (initially random, uniformly distributed between -1 and 1) synaptic weights from the inputs:

生成缩略图出错:无法将缩略图保存到目标地点

then the output the neuron is defined as follows:

生成缩略图出错:无法将缩略图保存到目标地点

Hebb's rule gives the update rule which is applied after an input pattern is presented:

生成缩略图出错:无法将缩略图保存到目标地点

where η is some small fixed learning rate.

It should be clear that given the same input applied over and over, the weights will continue to grow without bound. One solution is to limit the size of the weights. Another solution is to normalize the weights after every presentation:

生成缩略图出错:无法将缩略图保存到目标地点

Normalizing the weights leads to Oja's rule.

Hebb's rule and correlation

Instead of updating the weights after each input pattern, we can also update the weights after all input patterns. Suppose that there are N input patterns. If we set the learning rate η equal to 1/N, then the update rule becomes

生成缩略图出错:无法将缩略图保存到目标地点
where n is the pattern number, and
生成缩略图出错:无法将缩略图保存到目标地点
is the average over N input patterns. This is convenient, because we can now substitute
生成缩略图出错:无法将缩略图保存到目标地点
:
生成缩略图出错:无法将缩略图保存到目标地点
C is the correlation matrix for
生成缩略图出错:无法将缩略图保存到目标地点
, provided that
生成缩略图出错:无法将缩略图保存到目标地点
has mean zero and variance one. This means that strong correlation between elements of
生成缩略图出错:无法将缩略图保存到目标地点
will result in a large increase in the weights from those elements, which is what Hebb's rule is all about. Note that if
生成缩略图出错:无法将缩略图保存到目标地点
does not have mean zero and variance one, then the relationship holds up to a factor. Similarly, if the learning rate is not equal to 1/N, then the relationship is still true up to a factor.

References

  1. Hebb, D. O. (1949). The Organization of Behavior: A Neuropsychological Theory ISBN 978-0805843002.