\documentclass{seminar}
\includegraphics{synapticScaling2.eps}
\includegraphics{synapticRedist.eps}
\dot{w_{ij}} = \nu \cdot (rate_i \cdot rate_j)
Synaptic strength increases in proportion to the correlation between two inputs (the product of their firing rates).
If you measure firing rate w/r/t some baseline, you get:
\dot{w_{ij}} = \nu \cdot ((rate_i - baseline_i) \cdot (rate_j - baseline_j))
\dot{w_{ij}} = k \cdot (rate_i \cdot rate_j - \textrm{badness}(w))
\dot{w_{ij}} = k \cdot (rate_i \cdot rate_j - \textrm{badness}(w)*w_{ij})
For example, maybe
\begin{align*} \textrm{badness}(w) = \left\{ \begin{array}{ll} 0 & \textrm{when } (\sum_{i,j} w_{i,j}) < 10
(\sum_{i,j} w_{i,j}) - 10 & \textrm{otherwise} \end{array} \right. \end{align*}
\scriptsize{Popular paper: Miller, K. D. and MacKay?, D. J. C. The role of constraints in Hebbian learning. Neural Comput. 6, 100 126 (1994).}
(note: the second defn has correlation in place of causation. STDP, by requiring a pre-before-post, gets closer to the original causality requirement)
\dot{w}_{pre, post} = k \cdot (f(rate_{post}, \left< rate_{post} \right>) \cdot rate_{pre})
$f(x)$ might be something like $rate_{post} - \left< rate_{post} \right>$ \scriptsize{(I think)}
(orig paper had multiplicative normalization, too)
\scriptsize{Orig paper: Bienenstock, E. L., Cooper, L. N., and Munroe, P. W. (1982). Theory of the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex. J. Neurosci., 2:32-48. }
\scriptsize{Book: Theory Of Cortical Plasticity, Leon Cooper, Nathan Intrator, Brian S Blais, Harel Z Shouval. Some chapters online at http://nba.uth.tmc.edu/homepage/shouval/teaching.htm}
\dot{w}_{pre, post} = \nu \cdot (rate_{pre} \cdot rate_{post} - rate_{post}^2 * w_{ij}
\begin{itemize} \item Similar to the "multiplicative normalization" example given above \item Does PCA \end{itemize}
\scriptsize{Orig paper: Oja, E. (1982). A simplified neuron model as a principal component analyzer. J. Math. Biol., 15:267-273.}
\includegraphics[scale=.4]{PCAintro1.eps}
\includegraphics[scale=.4]{PCAintro2.eps}
\bullet direction of greatest variance
\scriptsize{Good PCA intro: http://www.okstate.edu/artsci/botany/ordinate/PCA.htm. See also http://purl.net/net/AIWiki/PCA.}