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


\begin{document}

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\begin{frame}
\titlepage

\date{\today}

\begin{center} {Chapter 3: Simplified neuron and population models} \end{center}


\end{frame}
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\begin{frame}
\frametitle{\large \large The leaky integrate-and-fire neuron}

\begin{equation}\label{eq:leakint}
  \tau_{\rm m} \frac{{\rm d}v(t)}{{\rm d}t} =  - (v(t)-E_L) + R I(t),
\end{equation}

\begin{equation}
  v(t^{\rm f}) = \vartheta.
\end{equation}

\begin{equation}\label{eq:ureset}
  \lim_{\delta \rightarrow 0} v(t^{\rm f}+\delta) = v_{\rm res},
\end{equation}

\begin{center}
\resizebox{5cm}{!}{\includegraphics{figures/Chapter3/if_node.pdf}}
\end{center}

\end{frame}
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\begin{frame}
\frametitle{\large \large The leaky integrate-and-fire neuron (cont.)}

\begin{center}
\resizebox{10cm}{!}{\includegraphics{figures/Chapter3/leaky_if.pdf}}
\end{center}

\end{frame}
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\begin{frame}
\frametitle{\large \large The LIF-neuron (cont.): Gain function}

The inverse of the first passage time defines the \emph{firing rate}
\begin{equation} \label{eq_gain_if}
\bar{r}=(t^{\rm ref}-\tau_{\rm m} \ln \frac{\vartheta-RI}{v_{\rm res}-RI})^{-1}
\end{equation}

\begin{center}
\resizebox{6cm}{!}{\includegraphics{figures/Chapter3/mean_firing_if.pdf}}
\end{center}

\end{frame}

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\begin{frame}
\frametitle{\large \large The LIF-neuron (cont.): Noise}

\begin{center}
\resizebox{8cm}{!}{\includegraphics{figures/Chapter3/if_poisson.pdf}}
\end{center}

\end{frame}

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\begin{frame}
\frametitle{\large \large The Izhikevich neuron}

\begin{eqnarray}\label{eq:izhikevich} \nonumber
  \frac{{\rm d}v(t)}{{\rm d}t} &=&  0.04 v^2 + 5 v+ 140 -u + I(t)  \\ \nonumber
  \frac{{\rm d}u(t)}{{\rm d}t} &=&  a (bv-u)
\end{eqnarray}
\begin{equation}
   v(v>30) = c \;\; \mbox{and} \;\; u(v>30) = u+d \nonumber
\end{equation}

\begin{center}
\resizebox{10cm}{!}{\includegraphics{figures/Chapter3/izhikevich_neuron.pdf}}
\end{center}

\end{frame}

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\begin{frame}
\frametitle{\large \large McCulloch-Pitts neuron}

\begin{equation}
h = \sum_i x^{\rm in}_i \nonumber
\end{equation}

\begin{equation} \label{McCullochPittsThrehold}
x^{\rm out} = \left\{ \begin{array}{cc}
  1 & \mbox{if } h > \Theta \\
  0 & \mbox{otherwise}
\end{array} \right. \nonumber
\end{equation}

\vspace*{-3cm}
\begin{center}
\resizebox{10cm}{!}{\includegraphics{figures/McCullochPittsArticle.pdf}}
\end{center}



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\begin{frame}
\frametitle{\large \large The firing rate hypothesis}

\begin{center}
\resizebox{4cm}{!}{\includegraphics{figures/Chapter3/adrian.pdf}}
\hspace*{1cm}
\resizebox{4cm}{!}{\includegraphics{figures/Chapter3/tuning.pdf}}
\end{center}

\end{frame}

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\frametitle{\large \large Counter example: correlation code (?)}

\begin{center}
\resizebox{7cm}{!}{\includegraphics{figures/Chapter3/decharms_merzenich.pdf}}
\end{center}

{\tiny From DeCharms and Merzenich 1996}
\end{frame}

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\begin{frame}
\frametitle{\large \large Integrator or coincidence detector?}

\begin{center}
\resizebox{8cm}{!}{\includegraphics{figures/Chapter3/inte_coincidence.pdf}}
\end{center}
\begin{center}
\resizebox{8cm}{!}{\includegraphics{figures/Chapter3//buracas.pdf}}
\end{center}

{\tiny From Buracas et al.  1998}
\end{frame}

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\begin{frame}
\frametitle{\large \large Population model}

\begin{center}
\resizebox{5cm}{!}{\includegraphics{figures/Chapter3/rate_temporal.pdf}}
\hspace*{1cm}
\resizebox{3cm}{!}{\includegraphics{figures/Chapter3/rate_population.pdf}}
\end{center}

\end{frame}

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\begin{frame}
\frametitle{\large \large Population dynamics}

For slow varying input (adiabatic limit), when all nodes do practically the same, same input, etc (Wilson and Cowan, 1972):
\begin{equation}\label{eq:population_dynamics}
  \tau \frac{{\rm d}A(t)}{{\rm d}t} =  - A(t) + g(R I^{\rm ext}(t)).
\end{equation}

Gain function:
\begin{equation} \label{eq_gain_adiabatic}
g(x)=\frac{1}{t^{\rm ref}-\tau \log(1-\frac{1}{\tau x})},
\end{equation}



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

\end{frame}

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\frametitle{\large \large Other gain functions}

\begin{center}
\resizebox{10cm}{!}{\includegraphics{figures/Chapter3/table_transf.pdf}}
\end{center}

\end{frame}

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\frametitle{\large \large Fast population response!!!}

\begin{center}
\resizebox{7cm}{!}{\includegraphics{figures/Chapter3/if_fast.pdf}}
\end{center}

\end{frame}

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\frametitle{\large \large Further Readings} 


\begin{list}{}{\setlength\itemindent{-.3in}  \setlength\rightmargin{.1in} \setlength\itemsep{0.05in}}
\small
\item Wolfgang Maass and Christopher M.~Bishop (eds.) (1999), \emph{Pulsed neural networks}, MIT Press.

\item  Wulfram Gerstner (2000), \emph{Population dynamics of spiking neurons: fast transients, asynchronous states,
and locking}, in \emph{Neural Computation} 12: 43--89.

\item Eugene M. Izhikevich (2003), \emph{Simple Model of Spiking Neurons},  in \emph{IEEE Transactions on Neural Networks}, 14: 1569--1072.

\item Eugene M. Izhikevich (2004), \emph{Which model to use for cortical spiking neurons?},  in \emph{IEEE Transactions on Neural Networks}, 15: 1063--1070.

\item Warren McCulloch and Walter Pitts (1943) \emph{A logical calculus of the ideas immanent in nervous activity}, in\emph{Bulletin of Mathematical Biophysics} 7:115--133.

\item Huge R. Wilson and Jack D. Cowan (1972), \emph{Excitatory and inhibitory interactions in localized populations of model neurons}, in \emph{Biophys. J.} 12:1--24.
 
\item Nicolas Brunel and Xiao-Jing Wang, (2001), \emph{Effects of neuromodulation in a cortical network model of working memory dominated by recurrent inhibition},  in \emph{Journal of Computational Neuroscience} 11: 63--85.



\end{list}


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\end{document}







