\documentclass{beamer}
\usetheme{Copenhagen}
\title{Decision Making Models}
\author{Shawn Gu}
\date{July XX, 2013}

\begin{document}

 \begin{frame}
  \titlepage
 \end{frame}

 \section{Introduction}
 \begin{frame}
  \begin{itemize}
   \pause \item Models for making decisions between two options.
   \pause \item In general, models work by integrating evidence.
   \pause \item Depending on the model, a decision is made when the evidence reaches a certain value, or the sign of
   the accumulator after a certain amount of time.
  \end{itemize}
 \end{frame}

 \begin{frame}
   \begin{figure}
    \includegraphics[width=3in]{ExampleHypotheses.eps} \footnote{\scriptsize Figure from ``The Neural Basis of Decision Making'' Gold, Shadlen}
    \caption{}
   \end{figure}
  \begin{itemize}
   \item If the evidence accumulated reaches either threshold, a decision is made corresponding to the threshold.
  \end{itemize}
 \end{frame}

 \begin{frame}
   \begin{figure}
    \includegraphics[width=2.5in]{otherDMaking.eps} \footnote{\scriptsize Figure slightly modified from ``Rats and Humans can Optimally Accumulate Evidence for Decision Making'' Brunton}
    \caption{Red and green arrows represent evidence.}
   \end{figure}
  \begin{itemize}
   \item After a certain amount of time, a decision is made based on if the evidence accumulated is positive or negative.
  \end{itemize}
 \end{frame}

 \section{Drift Diffusion}
 \begin{frame}
  \begin{itemize}
   \pause \item Modeled by
   \begin{align}
    dx &= Adt + cdW
   \end{align}
   \pause \item $Adt$ represents average increase in evidence supporting correct choice per time unit. Sign of A depends on which hypothesis is correct.
   \pause \item $cdW$ represents white noise, modeled by a Wiener process (also called Brownian motion)
   \begin{figure}
    \includegraphics[height=1.5in]{randomBrownian.eps}
    \caption{\scriptsize 5 randomly generated traces of a Wiener process (correcsponding to $dW$)}
   \end{figure}
  \end{itemize}
 \end{frame}

 \subsection{Stochastic Differential Equations}
 \begin{frame}
  \begin{itemize}
   \pause \item Differential equation with one or more terms as a stochastic process, meaning the solution is a stochastic process as well.
   \pause \item Solved numerically using the Euler-Maruyama method
   \begin{enumerate}
    \pause \item Extention of Euler's method for normal differential equations.
    \pause \item For an equation of the form
    \begin{align}
     dX(t) &= f(X(t))dt + g(X(t))dW(t), X(0) = X_0, 0 \le t \le T
    \end{align}
    EM takes the form
    \begin{align}
     X_j &= X_{j-1} + f(X_{j-1})\Delta t + g(X_{j-1})(W_j-W_{j-1}), j = 1,2,..,L
    \end{align}
    where $\Delta t$ is $T / L$ for some positive integer L. Note that if $g = 0$, we get the normal Euler method (corresponding to a normal differential equation).
   \end{enumerate}
  \end{itemize}
 \end{frame}

 \subsection{DDM}
 \begin{frame}
  \begin{figure}
   \includegraphics[height=2in]{5examplesEM.eps}
   \caption{\scriptsize 5 realizations of $dx = dt + 2dW$ using EM.}
  \end{figure}
 \end{frame}

 \begin{frame}
  \begin{itemize}
   \pause \item From this, we can see a few things to consider.
   \pause \item In particular, first passage times of the model ie how long it takes to make a decision.
   \pause \item If we make the barrier higher, we can be more sure of our decision. However, it will take longer to get there.
   \pause \item Is there some threshold value that maximizes the reward rate (defined as proportion of correct trials over average time between decisions)?
  \end{itemize}
 \end{frame}

 \begin{frame}
  \begin{figure}
   \includegraphics[height=2in]{RRvsThresh.eps}
   \caption{\scriptsize Reward Rate vs. Threshold for $dx = dt + 2dW$.}
  \end{figure}
  \begin{itemize}
   \pause \item As you can see, there is a threshold value that maximizes the reward rate.
   %include Bogacz as well?
  \end{itemize}
 \end{frame}

 \section{Ornstein-Uhlenbeck Process}
 \begin{frame}
  \begin{itemize}
   \pause \item Simple drift diffusion may not be optimal for describing decision making, as it has ``no destination''.
   \pause \item Instead, we can use the Ornstein Uhlenbeck process, which is ``mean-reverting''.
  \end{itemize}
 \end{frame}

 \begin{frame}
  \begin{itemize}
   \pause \item Modeled by
   \begin{align}
    dx &= (\lambda x + A)dt + cdW, x(0) = 0
   \end{align}
   \pause \item Addition of the $\lambda x$ term .
   \pause \item $A$ determines where the mean ``reverts'' to. $c$ determines how noisy the process is around the mean.
  \end{itemize}
 \end{frame}

 \begin{frame}
  \begin{figure}
   \includegraphics[height=2in]{exampleOUTrace.eps}
   \caption{\scriptsize For $dx = (-x - 4)dt + dW$.}
  \end{figure}
 \end{frame}


\end{document}