\documentclass[10pt]{beamer}

% for themes, etc.
\mode<presentation>
{ \usetheme{default} }
%{ \usetheme{boxes} }

\usepackage{times}  % fonts are up to you
\usepackage{graphics}
\usepackage{amsmath}
\definecolor{red}{rgb}{0.5,0.0,0.0}
\renewcommand{\emph}[1]{\textcolor{red}{\em\bf #1}}

% these will be used later in the title page
\title{Fundamentals of Computational Neuroscience 2e}
%\author{Thomas Trappenberg}


\begin{document}

%%%%%%%%%%%% Slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\titlepage

\date{\today}

\begin{center} {Chapter 8: Recurrent associative networks and episodic memory} \end{center}


\end{frame}
%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Memory classification scheme (Squire)}

\begin{center}
\resizebox{11cm}{!}{\includegraphics{figures/Chapter8/memoryClassification.pdf}}
\end{center}

\end{frame}
%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Auto-associative network and hippocampus}

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

\end{frame} 

%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Point attractor neural network (ANN)}

\emph{Update rule:} ${\tau} \frac{{\rm d}u_i(t)}{{\rm d}t}=-u_i(t)+ \frac{1}{N} \sum_j w_{ij} r_j(t)+I^{\rm ext}_i(t) $
\vspace*{0.3cm}

\emph{Activation function} $r_i = g(u_i)$  (e.g. threshold functions)
\vspace*{0.3cm}

\emph{Learning rule} $w_{ij} =  \epsilon \sum_{\mu=1}^{N_p} (r_i^\mu-\langle r_i \rangle) (r_j^\mu-\langle r_j \rangle) - c_{i}$
\vspace*{0.3cm}

\emph{Training patterns:} Random binary states with components  \\
\hspace*{2cm} $s_i^\mu \in \{-1,1\}$,   $r_i=\frac{1}{2} (s_i+1)$
\vspace*{0.3cm}

\emph{Update equations for fixed-point model}( ${{\rm d}u_i}/{{\rm d}t}=0$)\emph{:} \\
\hspace*{2cm} 
$s_i(t+1) = \mbox{sign}\left(\sum_j w_{ij} s_j(t)\right)$

\end{frame} 


%%%%%%%%%%%% Slide %%%%%%%%%%%
\frame[containsverbatim] { \frametitle{\large  ann\_cont.m}
%\scriptsize
\footnotesize

\begin{verbatim}
 1   %% Continuous time ANN
 2    clear; clf; hold on;
 3    nn = 500; dx=1/nn; C=0;
 4
 5   %% Training weight matrix
 6    pat=floor(2*rand(nn,10))-0.5;
 7    w=pat*pat'; w=w/w(1,1); w=100*(w-C);
 8   %% Update with localised input
 9    tall = []; rall = [];
10    I_ext=pat(:,1)+0.5; I_ext(1:10)=1-I_ext(1:10);
11    [t,u]=ode45('rnn_ode_u',[0 10],zeros(1,nn),[],nn,dx,w,I_ext);
12    r=u>0.; tall=[tall;t]; rall=[rall;r];
13   %% Update without input
14    I_ext=zeros(nn,1);
15    [t,u]=ode45('rnn_ode_u',[10 20],u(size(u,1),:),[],nn,dx,w,I_ext);
16    r=u>0.; tall=[tall;t]; rall=[rall;r];
17   %% Plotting results
18    plot(tall,4*(rall-0.5)*pat/nn)

\end{verbatim}

}



%%%%%%%%%%%% Slide %%%%%%%%%%%
\frame[containsverbatim] { \frametitle{\large ann\_fixpoint.m}

\scriptsize

\begin{verbatim}
 1   pat=2*floor(2*rand(500,10))-1;             % Random binary pattern
 2   w=pat*pat';                                % Hebbian learning
 3   s=rand(500,1)-0.5;                         % Initialize network
 4   for t=2:10; s(:,t)=sign(w*s(:,t-1)); end   % Update network
 5   plot(s'*pat/500)
\end{verbatim}

\begin{center}
\resizebox{10cm}{!}{\includegraphics{figures/Chapter8/ann_sim_color.pdf}} 
\end{center}

}


%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Memory breakdown}

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

\end{frame} 

%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Probabilistic RNN}

\emph{Probabilistic update rule:} 
$$P(s_i(t)=+ 1)=\frac{1}{1+\exp(- 2 \sum_j w_{ij} s_j(t-1) / T)}$$

Recovers deterministic rule in $\lim_{T\rightarrow 0}$
$$ s_i(t)=\mbox{sign}(\sum_j w_{ij} s_j(t-1))$$

\end{frame} 

%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Phase diagram}

\begin{center}
\resizebox{9cm}{!}{\includegraphics{figures/Chapter8/GH_phasediagram.pdf}} 
\end{center}

\end{frame} 

%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Noisy weights and diluted attractor networks}

\begin{center}
\resizebox{11cm}{!}{\includegraphics{figures/Chapter8//ann_diluted.pdf}} 
\end{center}

\end{frame} 


%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large How to minimize interference between patterns?}

Associative memory in ANN is strongly influenced by interference between patter due to
\begin{itemize}
\item correlated patterns  
\item random overlap 
\end{itemize}

Storage capacity can be much enhanced through decorrelating patterns. Simplest approach is generating sparse representations with expansion re-coding.

\emph{Storage capacity:}  $\alpha_{\rm c} \approx \frac{k}{a \ln(1/a)}$  (Rolls \& Treves) 


\end{frame} 
%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Expansion re-coding (e.g. in dentate gyrus)}


\begin{center}
\resizebox{9cm}{!}{\includegraphics{figures/Chapter8/expansion_recoding.pdf}} 
\end{center}

\end{frame} 

%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Sparse pattern and inhibition}

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

\end{frame} 

%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Three nodes with $\sum$ and $\sum-\prod$ coupling (Lorenz attractor):}
\vspace*{0.3cm}

$$\frac{{\rm d}x_i}{{\rm d}t} = \sum_j w^1_{ij} x_i + \sum_{jk}
w^2_{ijk} x_j x_k
$$
\vspace*{0.3cm}

$ \small
\mathbf{w}^1=\left( \begin{array}{ccc}
  -1 & a & 0 \\
  b & -1 & 0 \\
  0 & 0 & -c
\end{array} \right) \;\;\;  \mbox{and} \;\;\;
\mathbf{w}^2=\left\{ \begin{array}{ccc}
  w^2_{213} = -1 \\
  w^2_{312} = -1 \\
  0 \;\;\; \mbox{otherwise}
\end{array} \right. 
$
\vspace*{0.3cm}

\begin{center}
\resizebox{5cm}{!}{\includegraphics{figures/Chapter8/lorenz.pdf}} 
\end{center}

\end{frame} 

%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Cohen--Grossberg theorem}

\index{}

Dynamical system of the form $\frac{dx_i}{dt} = -a_i(x_i) \left( b_i(x_i) - \sum_{j=1}^{N}(w_{ij} g_j(x_j)) \right) $

Has a Lyapunov (Energy) function, which guaranties point attractors, under the conditions that
\begin{enumerate}
\item{\bf Positivity $a_i \geq 0$:} The dynamics must be a leaky
integrator rather than an amplifying integrator.
\item{\bf Symmetry $w_{ij}=w_{ji}$:} The influence of one node on another
 has to be the same as the reverse influence.
\item{\bf Monotonicity} $\mbox{sign}({\rm d}g(x)/{\rm d}x)={\rm const}${\bf :} The
activation function has to be a monotonic function.
\end{enumerate}

\emph{$\rightarrow$ more general dynamics possible with:}
\begin{itemize} 
\item{Non-symmetric weight matrix}

\item{Non-monotone activation functions (tuning curves)}

\item{Networks with hidden nodes}
\end{itemize}

\end{frame}

%%%%%%%%%%%% Slide %%%%%%%%%%%
\begin{frame}
\frametitle{\large Recurrent networks with non-symmetric weights}

\begin{center}
\resizebox{11cm}{!}{\includegraphics{figures/Chapter8/ann_asym.pdf}} 
\end{center}

\emph{$\rightarrow$ strong asymmetry is necessary to abolish point attractors}


\end{frame} 

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

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

\item  Daniel J.~Amit (1989), \emph{Modelling brain function: the world of attractor neural networks}, Cambridge University Press.

\item John Hertz, Anders Krogh, and Richard G.~Palmer (1991), \emph{Introduction
to the theory of neural computation}, Addison-Wesley.

\item Edmund T.~Rolls and Alessandro Treves (1998), \emph{Neural
networks and brain function}, Oxford University Press

\item Eduardo R.~Caianello (1961), \emph{Outline of a theory of thought-proccess and thinking machines}, in \emph{Journal of Theoretical Biology} 2: 204--235.

\item John J.~Hopfield (1982), \emph{Neural networks
and physical systems with emergent collective computational abilities}, in
 \emph{Proc.~Nat.~Acad.~Sci., USA} 79: 2554--8.

\item Michael A.~Cohen and Steven Grossberg (1983), \emph{Absolute stability
of global pattern formation and parallel memory storage
 by competitive neural networks}, in
 \emph{IEEE Trans. on Systems, Man and Cybernetics}, SMC-13: 815--26.
 
\item MWenlian Lu and Tianping Chen, \emph{New Conditions on Global Stability of Cohen-Grossberg Neural Networks}, in \emph{Neural Computation}, 15: 1173--1189.
 
\item Masahiko Morita (1993), \emph{Associative memory with nonmonotone dynamics}, in \emph{Neural Networks} 6: 115--26.

\item Michael E.~Hasselmo and Christiane Linster (1999), \emph{Neuromodulation
and memory function}, in \emph{ Beyond neurotransmission:
neuromodulation and its importance for information processing},
Paul S. Katz (ed.), Oxford University Press.

\item Pablo Alvarez and Larry R.~Squire (1991), \emph{Memory consolidation and the medial temporal lobe: a simple network model}, in \emph{Proc Natl Acad Sci} 15: 7041-–7045.


\end{list} 

\end{frame}

\end{document}







