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\title{Fundamentals of Computational Neuroscience 2e}
%\author{Thomas Trappenberg}


\begin{document}

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\titlepage

\date{\today}

\begin{center} {Chapter 4: Associators and synaptic plasticity} \end{center}


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\begin{frame}
\frametitle{\large Types of plasticity}
\begin{itemize}
\item {\bf Structural plasticity} is the mechanism describing the generation of new connections and thereby redefining the topology of the network.
\item {\bf Functional plasticity} is the mechanism of changing the strength values of existing connections.
\end{itemize}

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\begin{frame}
\frametitle{\large Hebbian plasticity}

\begin{quotation}
  {\normalsize \emph{"When an axon of a cell A is near enough to excite cell B or repeatedly
  or persistently takes part in firing it, some growth or
  metabolic change takes place in both cells such that A's
  efficiency, as one of the cells firing B, is increased." }}
\end{quotation}

Donald O.~Hebb, \emph{The organization of behavior}, 1949

See also Sigmund Freud, \emph{Law of association by simultaneity}, 1888

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\begin{frame}
\frametitle{\large Association}

\begin{center}
\resizebox{6cm}{!}{\includegraphics{figures/Chapter4/associative_node.pdf}}
\end{center}

\emph{Neuron model:} In each time step the model neurons fires if $\sum_i w_i r^{\rm in}_i >1.5$ 

  \emph{Learning rule:} Increase the strength
of the synapses by a value $\Delta w=0.1$ if a presynaptic firing
is paired with a postsynaptic firing.





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\begin{frame}
\frametitle{\large }

\begin{center}
\resizebox{9cm}{!}{\includegraphics{figures/Chapter4/associative_learning.pdf}}
\end{center}


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\begin{frame}
\frametitle{\large Features of associators and Hebbian learning}

\begin{itemize}
\item Pattern completion and generalization
\item Prototypes and extraction of central tendencies
\item Graceful degradation and fault tolerance
\end{itemize}


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\begin{frame}
\frametitle{\large Classical LTP and LTD}

\begin{center}
\resizebox{8cm}{!}{\includegraphics{figures/plasticity1slides.pdf}} 
\end{center}

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\begin{frame}
\frametitle{\large Synaptic neurotransmitter release probability}

\begin{center}
\resizebox{8cm}{!}{\includegraphics{figures/fine.pdf}} 
\end{center}

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\begin{frame}
\frametitle{\large Spike timing dependent plasticity}


\begin{center}
\resizebox{7cm}{!}{\includegraphics{figures/Chapter4/hebb_stdp.pdf}}
\end{center}


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\begin{frame}
\frametitle{\large The calcium hypothesis and modelling chemical pathways}

\begin{center}
\resizebox{5cm}{!}{\includegraphics{figures/Chapter4/calcium.pdf}}
\hspace*{1cm}
\resizebox{4cm}{!}{\includegraphics{figures/Chapter4/signaling.pdf}}
\end{center}

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\begin{frame}
\frametitle{\large Mathematical formulation of Hebbian plasticity}

\begin{equation} \nonumber 
w_{ij}(t+\Delta t)=w_{ij}(t)+\Delta w_{ij}(t^{\rm f}_i,t^{\rm f}_j,\Delta t; w_{ij}).
\end{equation}

\begin{equation}\nonumber
\Delta w_{ij}^\pm = \epsilon^{\pm}(w)  {\rm e}^{\mp \frac{t^{\rm
post}-t^{\rm pre}}{\tau^\pm}} \;\; \Theta(\pm[t^{\rm post}-t^{\rm
pre}]).
 \end{equation}

Additive rule with hard (absorbing) boundaries:
\begin{equation} \nonumber
\epsilon^\pm = \left\{ \begin{array}{cc}
  a^\pm & \;\;\; \mbox{for} \;\; w_{ij}^{\rm min} \leq w_{ij} \leq
w_{ij}^{\rm max} \\
  0 & \;\;\; \mbox{otherwise}
\end{array} \right. ,
\end{equation}

Multiplicative rule (soft boundaries):
\begin{eqnarray}\nonumber
 \epsilon^+ &=& a^+ (w^{\rm max} - w_{ij}) \\
 \epsilon^- &=& a^- (w_{ij} - w^{\rm min}) .
\end{eqnarray}

\end{frame}
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\begin{frame}
\frametitle{\large Hebbian learning in population and rate models}

\vspace*{0.3cm}

\emph{General:} $\Delta w_{ij}= \epsilon(t,w)  [f_{\rm post}(r_i) f_{\rm pre}(r_j) - f(r_i,r_j,w)] $
\vspace*{0.2cm}

\emph{Mnemonic equation (Caianiello): }
$\Delta w_{ij}= \epsilon(w)  [r_i r_j - f(w)] $
\vspace*{0.2cm}

\emph{Basic Hebb:} $ \Delta w_{ij}= \epsilon r_i  r_j $
\vspace*{0.2cm}

\emph{Covariance rule:} $  \Delta w_{ij}= \epsilon (r_i-\langle r_i
\rangle)(r_j-\langle r_j \rangle)$
\vspace*{0.2cm}

\emph{BCM theory:}
$ \Delta w_{ij}= \epsilon (f^{\rm BCM}(r_i;\theta^M)(r_j) - f(w))$
\vspace*{0.2cm}

\emph{ABS rule:} $  \Delta w_{ij}= \epsilon (f_{\rm ABS}(r_i; \theta^-,\theta^+) {\rm sign}(r_j-\theta^{\rm pre}))$
\vspace*{0.5cm}


\begin{center}
\vspace*{-0.5cm}\resizebox{3.9cm}{!}{\includegraphics{figures/Chapter4/f_prepost.pdf}}
\vspace*{0.cm} \resizebox{5cm}{!}{\includegraphics{figures/f_prepost2slides.pdf}}
\end{center}


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\begin{frame}
\frametitle{\large Synaptic scaling and weight distributions}

\begin{center}
\resizebox{10cm}{!}{\includegraphics{figures/Chapter4/if_STDP_Song.pdf}}
\end{center}

{\tiny after Song, Miller and Abbott 2000}
\end{frame}
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\begin{frame}
\frametitle{\large Hebbian rate rules on random pattern}

$$ w_{ij} = \frac{1}{\sqrt{N_p}} \sum_\mu (r_i^\mu-\langle r_i \rangle) (r_j^\mu-\langle r_j \rangle).
$$

\begin{center}
\resizebox{5cm}{!}{\includegraphics{figures/Chapter4/hebb1_distribution.pdf}}
\end{center}



\end{frame}
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\begin{frame}
\frametitle{\large Synaptic scaling and PCA}

\emph{Explicit normalization:} $w_{ij} \leftarrow \frac{w_{ij}}{\sum_j w_{ij}}$
\vspace*{0.2cm}

\emph{Basic decay:} $\Delta w_{ij} = r_i r_j - c  w_{ij} $
\vspace*{0.2cm}

\emph{Willshaw rule:} $ \Delta w_{ij} = (r_i - w_{ij}) r_j $
\vspace*{0.2cm}

\emph{Oja rule:} $  \Delta w_{ij}=r_i r_j - (r_i)^2 w_{ij} $
\vspace*{0.2cm}

\begin{center}
\resizebox{5cm}{!}{\includegraphics{figures/Chapter4/oja.pdf}}
\end{center}



\end{frame}

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\begin{frame}
\frametitle{\large Further Readings} 

\begin{list}{}{\setlength\itemindent{-.3in}  \setlength\rightmargin{.1in} \setlength\itemsep{0.05in}}
\small

\item Laurence F.~Abbott and Sacha B.~Nelson (2000), \emph{Synaptic
plasticity: taming the beast}, in \emph{Nature Neurosci.~(suppl.)}, 3: 1178--83.

\item Alain Artola and Wolf Singer (1993), \emph{Long-term depression of excitatory synaptic transmission and
its relationship to long-term potentiation}, in \emph{Trends in Neuroscience} 16: 480--487.

\item Mark C.~W.~van Rossum, Guo-chiang Bi, and Gina G.~Turrigiano (2000)
\emph{Stable Hebbian learning from spike timing-dependent plasticity},
in \emph{J.~Neuroscience}~20(23): 8812--21


\end{list} 

\end{frame}

\end{document}







