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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 2: Neurons and conductance-based model} \end{center}


\end{frame}
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\begin{frame}
\frametitle{\large Biological background}

\begin{figure}[ht]
\begin{center}
\resizebox{10cm}{!}{\includegraphics{figures/Chapter2/neuron.pdf}} 
\end{center}
\end{figure}

\end{frame}

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\begin{frame}

\frametitle{\large Ion channels} 

\begin{figure}[ht]
\begin{center}
\resizebox{!}{6cm}{\includegraphics{figures/Chapter2/ion_channels.pdf}}
\end{center}
\end{figure}



\end{frame}
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\begin{frame}
\frametitle{\large Synapse}

\begin{figure}[ht]
\begin{center}
\resizebox{5cm}{!}{\includegraphics{figures/Chapter2/synapse1.pdf}}
\hspace*{1cm}
\resizebox{3cm}{!}{\includegraphics{figures/Chapter2/synapse2.pdf}}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{\large non-NMDA: GABA, AMPA}

\begin{equation} \label{eq_epsp1} \nonumber
 \Delta V_{\rm m}^{\rm non-NMDA} \propto  t \; {\rm e}^{ - t/t^{\rm peak}}
\end{equation}

\begin{figure}[ht]
\begin{center}
\resizebox{5cm}{!}{\includegraphics{figures/nonNMDAfunction.pdf}}
\end{center}
\end{figure}


\end{frame}
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\begin{frame}
\frametitle{\large Conductance-based models}

\begin{equation}\label{eq_kirchhoff_syn}
c_m \frac{{\rm d}V(t)}{dt}=- I
\end{equation}

\begin{equation}\label{syn current}
I(t) = g_0 V(t) - g(t) (V(t) - E_{\rm syn})
\end{equation}


\begin{equation} \label{syn conductance}
\tau_{\rm syn} \frac{{\rm d}g(t)}{{\rm d}t} = - g(t) + \delta(t-t_{\rm pre} - t_{\rm delay})
\end{equation}

\begin{figure}[ht]
\begin{center}
\resizebox{9cm}{!}{\includegraphics{figures/Chapter2/circuit_syn.pdf}}
\end{center}
\end{figure}

\end{frame}
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\frame[containsverbatim] { \frametitle{\large MATLAB Program}
\small
\begin{verbatim}
 1   %% Synaptic conductance model to simulate an EPSP
 2    clear; clf; hold on;
 3   
 4   %% Setting some constants and initial values
 5    c_m=1; g_L=1; tau_syn=1; E_syn=10; delta_t=0.01;  
 6    g_syn(1)=0; I_syn(1)=0; v_m(1)=0; t(1)=0; 
 7   
 8   %% Numerical integration using Euler scheme
 9    for step=2:10/delta_t
10      t(step)=t(step-1)+delta_t;
11      if abs(t(step)-1)<0.001; g_syn(step-1)=1; end
12      g_syn(step)= (1-delta_t/tau_syn) * g_syn(step-1);
13      I_syn(step)= g_syn(step) * (v_m(step-1)-E_syn);
14      v_m(step) = (1-delta_t/c_m*g_L) * v_m(step-1) ...
15                      - delta_t/c_m * I_syn(step);
16    end
17
18   %% Plotting results
19    plot(t,v_m); plot(t,g_syn*5,'r--'); plot(t,I_syn/5,'k:') 
\end{verbatim}
}


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\begin{frame}
\frametitle{\large Hodgkin--Huxley model}

\begin{figure}[ht]
\begin{center}
\resizebox{8cm}{!}{\includegraphics{figures/Chapter2/action_potential.pdf}}
\caption{Typical form of an action potential; redrawn from an
oscilloscope picture from Hodgkin and Huxley (1939).
\label{fig_action_potential}}
\end{center}
\end{figure}

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\begin{frame}
\frametitle{\large The minimal mechanisms}

\begin{figure}[ht]
\begin{center}
\resizebox{7cm}{!}{\includegraphics{figures/Chapter2/hh_ion.pdf}}
\end{center}
\end{figure}


\end{frame}

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\begin{frame}
\frametitle{\large Hodgkin--Huxley equations and simulation}

\small
\begin{eqnarray}\label{eq_hh} \nonumber
  C\frac{{\rm d}V}{{\rm d}t} &=&  - g_{\rm K} n^4 (V - E_{\rm K} ) -
  g_{\rm Na} m^3 h(V - E_{\rm Na} ) - g_{\rm L} (V - E_{\rm L} ) +
  I(t) \nonumber \\  \nonumber
 {\tau_{\rm n} }(V) \frac{{\rm d}n}{{\rm d}t} &=&  - [n - n_0 (V)]  \\  \nonumber
 {\tau_{\rm m} }(V) \frac{{\rm d}m}{{\rm d}t} &=&  - [m - m_0 (V)] \\  \nonumber
 {\tau_{\rm h} }(V) \frac{{\rm d}h}{{\rm d}t} &=&  - [h - h_0 (V)]
\end{eqnarray}

\begin{figure}[ht]
\begin{center}
\resizebox{4cm}{!}{\includegraphics{figures/Chapter2/hh2A.pdf}}
\hspace*{1cm}
\resizebox{4cm}{!}{\includegraphics{figures/Chapter2/hh2B.pdf}}
\end{center}
\end{figure}


\end{frame}

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\begin{frame}
\frametitle{\large Compartmental models}

\begin{figure}[ht]
\begin{center}
\resizebox{9cm}{!}{\includegraphics{figures/Chapter2/compartment.pdf}}
\end{center}
\end{figure}


\end{frame}

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\begin{frame}
\frametitle{\large Simulators}

\begin{figure}[ht]
\begin{center}
\resizebox{9cm}{!}{\includegraphics{figures/Chapter2/neuron_sim2.pdf}}
\end{center}
\end{figure}


\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  Mark F.~Bear, Barry W.~Connors, and Michael A.~Paradiso (2006), \emph{Neuroscience: exploring the brain},
Lippincott Williams \& Wilkins , 3rd edition.

\item  Eric R.~Kandel, James H.~Schwartz, and Thomas M.~Jessell (2000), \emph{Principles of
neural science}, McGraw-Hill, 4th edition

\item Gordon M.~Shepherd (1994), \emph{Neurobiology}, Oxford University Press, 3rd edition.

\item Christof Koch (1999), \emph{Biophysics of computation; information processing in single neurons}, Oxford
University Press

\item Christof Koch and Idan Segev (eds.) (1998), \emph{Methods in
neural modelling}, MIT Press, 2nd edition.

\item C.~T. Tuckwell (1988), \emph{Introduction
to theoretical neurobiology}, Cambridge University Press.

\item Hugh R.~Wilson (1999) \emph{Spikes, decisions and actions:
dynamical foundations of neuroscience}, Oxford University Press. See also his paper in J.~Theor.~Biol.~200: 375--88, 1999.
\end{list}


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\end{document}







