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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 10: The cognitive brain} \end{center}


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\frametitle{\large Hierarchical maps and attentive vision}

\begin{center}
\resizebox{9cm}{!}{\includegraphics{figures/Chapter10/visnet.pdf}}
\end{center}

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\frametitle{\large Attention in visual search and object recognition}

\begin{center}
\resizebox{10cm}{!}{\includegraphics{figures/Chapter10/deco_where_what.pdf}}
\end{center}

{\small Gustavo Deco}

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\begin{frame}
\frametitle{\large Model}

\begin{center}
\resizebox{11cm}{!}{\includegraphics{figures/Chapter10/deco_model.pdf}}
\end{center}

\end{frame}

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\begin{frame}
\frametitle{\large Example results}

\begin{center}
\resizebox{9cm}{!}{\includegraphics{figures/Chapter10/deco_res.pdf}}
\end{center}

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\frametitle{\large The interconnecting workspace hypothesis}

\begin{center}
\resizebox{6cm}{!}{\includegraphics{figures/Chapter10/dehaene.pdf}}
\end{center}

{\tiny Stanislas Dehaene, M. Kergsberg, and J.P. Changeux, PNAS 1998}
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\begin{frame}
\frametitle{\large Stroop task modelling}

\begin{center}
\resizebox{8cm}{!}{\includegraphics{figures/Chapter10/dehaene2.pdf}}
\end{center}

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\begin{frame}
\frametitle{\large The anticipating brain}

\begin{enumerate}
\item The brain can develop a model of the world, which can be used to anticipate or predict the environment.
\item The inverse of the model can be used to recognize causes by evoking internal concepts.
\item Hierarchical representations are essential to capture the richness of the world.
\item Internal concepts are learned through matching the brain's hypotheses with input from the world.
\item An agent can learn actively by testing hypothesis through actions.
\item The temporal domain is an important degree of freedom.
\end{enumerate}

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\frametitle{\large Outline}

\begin{center}
\resizebox{10cm}{!}{\includegraphics{figures/Chapter10/FristonTT.pdf}}
\end{center}

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\frametitle{\large Recurrent networks with hidden nodes}

\emph{The Boltzmann machine:}

\begin{center}
\resizebox{4cm}{!}{\includegraphics{figures/Chapter10/Boltzmann1.pdf}}
\end{center}

Energy: $ H^{nm}=-\frac{1}{2} \sum_{ij} w_{ij} s_i^n s_j^m$
\vspace*{.3cm}

Probabilistic update: $p(s_i^{n}=+1)=\frac{1}{1+\exp(-\beta \sum_j w_{ij} s_j^n)}$
\vspace*{.3cm}

Boltzmann-Gibbs distribution: $p(\mathbf{s}^v; \mathbf{w})=\frac{1}{Z} \sum_{m\in h} \exp(-\beta H^{vm})$

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\frametitle{\large Training Boltzmann machine}

\emph{Kulbach-Leibler divergence}
\begin{eqnarray}
{\rm KL}(p(\mathbf{s}^v),p(\mathbf{s}^v;\mathbf{w}))&=&\sum_\mathbf{s}^v p(\mathbf{s}^v) \log \frac{p(\mathbf{s}^v)}{p(\mathbf{s}^v;\mathbf{w})} \nonumber \\
&=&\sum_\mathbf{s}^v p(\mathbf{s}^v) \log p(\mathbf{s}^v) - \sum_\mathbf{s}^v p(\mathbf{s}^v) \log p(\mathbf{s}^v;\mathbf{w}) \nonumber 
\end{eqnarray}

Minimizing KL is equivalent to maximizing the average log-likelihood function
$$
l(\mathbf{w})=\sum_\mathbf{s}^v p(\mathbf{s}^v) \log p(\mathbf{s}^v;\mathbf{w})=\langle \log p(\mathbf{s}^v;\mathbf{w})\rangle.
$$

\emph{Gradient decent  $\rightarrow$ Boltzmann Learning}

\vspace*{0.3cm}
$\Delta w_{ij} = \eta \frac{\partial{l}}{\partial w_{ij}} = \eta \frac{\beta}{2} \left( \langle s_i s_j \rangle_{\rm clamped} - \langle s_i s_j \rangle_{\rm free} \right).
$



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\frametitle{\large The restricted Boltzmann machine}

\begin{center}
\resizebox{6cm}{!}{\includegraphics{figures/Chapter10/Boltzmann2.pdf}}
\end{center}

\vspace*{1cm}

\emph{Contrastive Hebbian learning: Alternating Gibbs sampling} 

\begin{center}
\resizebox{6cm}{!}{\includegraphics{figures/Chapter10/Boltzmann3.pdf}}
\end{center}

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\frametitle{\large Deep generative models}

\begin{center}
\resizebox{10cm}{!}{\includegraphics{figures/Chapter10/HintonColour.pdf}}
\end{center}

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\frametitle{\large Adaptive Resonance Theory (ART)}

\begin{center}
\resizebox{8cm}{!}{\includegraphics{figures/Chapter10/ART.pdf}}
\end{center}

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%%%%%%%%%%%% Slide with references %%%%%%%%%%%
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\frametitle{\large Further Readings} 

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

\item Edmund T.~Rolls and Gustavo Deco (2001), \emph{Computational neuroscience of vision}, Oxford University Press.

\item Karl Friston (2005), \emph{A theory of cortical responses}, in \emph{Philosophical Transactions of the Royal Society} B 360, 815--36.

\item Jeff Hawkins with Sandra Blakeslee (2004), \emph{On intelligence}, Henry Holt and Company.

\item Robert Rosen (1985), \emph{Anticipatory systems: Philosophical, mathematical and methodological foundations}, Pergamon Press.

\item Geoffrey E. Hinton (2007), \emph{Learning Multiple Layers of Representation}, in 
\emph{Trends in Cognitive Sciences} 11: 428--434.

\item Stephen Grossberg (1976), \emph{Adaptive pattern classification and universal recoding: Feedback, expectation, olfaction, and illusions}, in \emph{Biological Cybernetics} 23: 187--202.

\item Gail Carpenter and Stephen Grossberg (1987), \emph{A massively parallel architecture for a self-organizing neural pattern recognition machine} in \emph{Computer Vision, Graphics and Image Processing} 37: 54--115.

\item Daniel S. Levine (2000), \emph{Introduction to neural and cognitive modeling}, Lawrence Erlbaum, 2nd edition.

\item James A. Freeman (1994), \emph{Simulating neural networks with Mathematica}, Addison-Wesley.




\end{list} 

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