<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\chapter{Procesoj Stokastikaj}
La stokastika proceso a\u u signalo estas nedetermina proceso $g$, ofte konsiderita sole per la tempa variablo $g=g(t)$, de kiu oni scias sole sia statistikaj ecoj. La stokastikaj procesoj povas esti {\bf stacionaraj} se la statistikaj ecoj ne variigas en la tempo kaj povas esti {\bf ergodecaj} se \v ciuj procesoj en la sama tempo evolvigas simile. Oni povas priskribi ian proceson per sia \ero{Probabla Denseco} kiu estas kiu estas difinita per
\begin{mat}{Asioma Difino de la Probabla Mizuro} La Funkcio $\Pro(t)$ estas \ldots
\end{mat} 
Kun la probabla denseco oni povas difini la momantojn de la proceso. La \ero{Probabla Momanto} estas la sumo de \v ciuj valoroj al la $n$ potenco per la probabla denseco en tiu punkto. \dm{m_{\nat n} \EDef \int\di x x^{\nat n}\Pro(x) \stackrel{a\breve u}= \frac1N\sum_i^N x_i^n=\sum_i^Mx_i\Pro_i} kir $\Pro_i$ estas la fekvenco de apero de la $M$ diversaj variabloj $x_i$ en la kampiono. La unua momanto estas la medio kun kiu oni povas konstrui centrajn momantojn \dm{\bar x \EDef \int \di x\Pro(x) \stackrel{a\breve u}=\frac1N\sum_i^Nx \qquad m^c_n \EDef \int\di x (x - \bar x)^n\Pro(x) \stackrel{a\breve u}= \sum_i^M (x_i - \bar x)^n \Pro_i} La pli grava centra momanto estas la dua anka\u u nomita \ero{Varianco}. 
\section{Distribuoj}
\subsection{Difinoj}
{\bf Probableco} \\
{\bf Medio} $&lt;x&gt; \EDef \med x \EDef \int_{-\infty}^{\infty}x\Pro(x) \di x$ \\
{\bf Varianco} $\sigma \EDef \big&lt;x-&lt;x&gt;{\big&gt;}^2 = &lt;x^2&gt; - &lt;x&gt;^2$ \\
{\bf Probableco Kumulita} $P(a&lt;x&lt;b) \EDef \int_a^b \Pro(x)\di x$ \noto{$P(-\infty&lt;x&lt;\infty) \stackrel ! = 1$}\\
{\bf Maksimumo, Miminumo} $q_0 \EDef \mathrm{min}(x_i)$, $q_4 \EDef \mathrm{max}(x_i)$ \\
{\bf Kvartilo} $q_1 | P_{q_1}\EDef P(-\infty&lt;x&lt;q_1) \stackrel ! = 1/4$, $q_2 | P_{q_2} \stackrel ! = 1/2$, $q_3 | P_{q_3} \stackrel ! = 3/4$ \\
{\bf Mediano} $  q_2\EDef\mu | P_\mu \stackrel ! = 1/2$ 
\subsection{Gauss-a, normala}
\dm{\Pro(x;\med x,\sigma) = \frac1{\sqrt{2\pi \sigma^2}} e^{\frac{-(\med x-x)^2 }{2\sigma^2} } }
\subsection{Lorentz, Cauchy, Breit-Wigner, -a}
\dm{\Pro(x;\med x,\sigma) = \frac1{\pi \sigma}\frac1{1+\Big(\frac{x-\med x}\sigma\Big)^2} }
Kie $\med x$ estas la mediano kaj $\sigma$ la medialteca largeco. Nenio momento eksistas.
\subsection{Binomiala}
\dm{\Pro(x=k) = \left(\begin{array}{c} n \\ k \end{array}\right)p^k(1-p)^{n-k} }
\subsection{Poisson-a}
La probabla distribuada funkcio la\u u la \homo{Poisson}-a distribuado estas \dm{\Pro(x) = \frac{\lambda^xe^{-\lambda}}{x!} }
\begin{legxo}{Centra Limito} Se la numero de procesoj estas granda la distribuado etendas a la normala\ldots
\end{legxo}
De la statistikaj procesoj oni povas anka\u u priskribi la media potenco de la signalo \dm{\Pot(t_0,T) \EDef \frac 1T\int_{t_0-T/2}^{t_0+T/2}|g(t)|^2\di t \qquad E_\infty = \int|g(t)|^2\di t} dume $E_\infty$ estas la energio de la signalo tempa limita. La spektro potenca estas anstata\u ue la potenco media sur \v ciu frekvencoj \dm{P_\infty=\int |\med g(\omega)|^2\di\omega \qquad S(\omega) \EDef |\med g(\omega)|^2 = |\int g(t)e^{-\imath\omega t}\di t|^2}
\begin{legxo}{Parseval}
La kvar modulo de la signalo egalas la kvar modulo de sia \homo{Fourier}-a trasforma\v jo. \ldots \end{legxo}
\subsection*{Korelacioj}
La korelacio estas la produkto de du diversaj signaloj en siaj tempa variacio \dm{ K_{12} = \lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}g_1(t)g_2(t-\tau)\di t} La memkorelacio estas la korelacio de la signalo kun \v gi memo en diversaj tempoj \dm{K_{11}(\tau) = \lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}g_1(t)g_1(t-\tau)\di t =\lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}g(t)g(t+\tau)\di t}
\begin{legxo}{Wiener Khintchin} La spektra denseco de ia signalo estas akirebla trasforminte la\u ue \homo{Fourier} la memkorelacio de la signalo kaj inverse \dm{&lt;g(t)g(t')&gt;:=\int\frac{\di\omega}{(2\pi)}S_\phi(\omega)e^{\imath\omega(t-t')} } \end{legxo}
\section*{Parametraj Pruvoj}
La parametraj Provoj devas sub dri azuntoj validi:
\begin{itemize}
\item[-] {\bf Nedipendeco grupa sampla }\par
  \v ciu subjekto populacio povi in \v ciu samplo distribuita esti.
\item[-] {\bf Normaleco Distribuada}
\item[-] {\bf Varianca Homogeneco}
\end{itemize}
\subsection*{T Student-a}
\subsection*{F Fisher-a}
\subsection*{Regreso}
\section*{Neparametraj Pruvoj, Frekvenco}
Stata\u u la parametraj ili estas populacia distribuado nedepena, estas mapli zorga sed utilizema kiam la distribuadoj estas ne normalaj. 
\subsection*{Chi kvadrata}
Estas pruvo pri frekvencoj bazita kiu komparas la observitan distribuadon kun la atenditan (teoretikan). Sub la \ero{Nulo Hipotezo} $\mathbf {H_0}$ la empirikaj frekvencoj egalas la teorikajn $n_i = \med n_i$. \dm{\chi_c^2 = \sum_i\frac{(n_i-\med n_i)^2}{\med n_i} } a\u u du empirikaj aroj interne de la eraro egala estas. Se $\chi_c^2 &lt; \chi_\alpha^2$ la nulo hipoteso estas konsiderita kun erara probableco $\alpha$. La vertoj de $\chi_a^2$ estas tabulitaj. Alimaniere la hipotezo $\mathbf{H_1}=\bar{\mathbf{H_0}}$ estas konsiderita.
\section*{Neparametraj Pruvoj, Rango}
La \ero{Rango de la obeservadoj} konsideras la ordan numeron de la samplo, prefere la indeco $i$ per la ordenata serio $\{x_0&lt;\ldots&lt;x_i&lt;\ldots&lt;x_N\}$. Oni uzas tiu metodojn kiam la distribuado ne estas normala a\u u la la datoj estas tro malmalutaj a\u u la datoj referas al la sama variablo in rangoj disigitaj. 
\subsection{Mann-Withney Pruvo, sumo de la rangoj}
Kontrolas kiam du samploj apartenas al la sama distribuo. $\mathbf{H_0}: \med x_A = \med x_B$ \noto{$n_A$,$n_B$ estas la numero datoj en la samplo} \noto{$R$ Rango} \dm{U =  \mathrm{min}\left(\sum_i R_A,\sum_jR_B\right) \qquad \mu_u = \frac{n_A(n_A+n_B+1)}2 \\ \sigma_u = \sqrt{\frac{n_An_B(n_A+n_B+1)}{12}} \qquad Z = \frac{U-\mu_u\pm0.5}{\sigma_u} } 
prefere: \dm{U\EDef n_An_B+ \frac{n_A(n_A+1)}2-R }
\subsection{Mediana Pruvo}
\begin{center}
\begin{tabular}{c |c c| c}
&amp; A &amp; B &amp;\\
\hline
$&gt;\mu$ &amp; $x_{1A}$ &amp; $x_{1B}$ &amp; $x_1$\\
$\leq\mu$ &amp; $x_{2A}$ &amp; $x_{2B}$ &amp; $x_2$\\
\hline
&amp; $n_A$ &amp; $n_B$ &amp; $n$\\
\end{tabular}
\end{center}
\dm{\chi^2= \frac{N(|x_{1A}x_{2B}-x_{2A}x_{1B}|-n/2)^2}{x_1x_2n_an_B} }
\begin{komp}
Sur {\it R} oni kontrolas se la aroj A kaj B estas: $\mathbf{H_0}: p_{A1} \leq p_{A}p_{B}$, $\mathbf{H_1}: p_{A1} &gt; p_{A}p_{B}$ montrinte ke: $\chi_c^2 &gt; \chi_\alpha^2$ kaj $x_{A1}x_{B2} &gt; x_{A2}x_{B1}$
\begin{verbatim}
R&gt; data &lt;- matrix(c(xA1, xB1, xA2, xB2), ncol = 2)
R&gt; line.names &lt;- c("A", "B")
R&gt; col.names &lt;- c("1", "2")
R&gt; dimnames(data) &lt;- list(line.names, col.names)
R&gt; alpha &lt;- 0.05;
R&gt; data.chi &lt;- chisq.test(data, correct = FALSE);
R&gt; data.chi$statistic &gt; qchisq(p = alpha, 
   df = data.chi$parameter, lower.tail = FALSE);
R&gt; xA1*xB2 &gt; xA2*xB1;
\end{verbatim}
\end{komp}
\subsection{Wilcoxon Pruvo, signa pruvo}
Montras la grandecon kaj direcion de du depenaj samploj. $\mathbf{H_0}:\med x_A=\med x_B$. 
\dm{T^+ = \sum R^+ \qquad \mu_T = \frac{n_A(n_A+1)}4 \qquad d=T-\mu_T \\ \sigma_T=\sqrt{\frac{n_A(n_A+1)(2n_A+1)}{24}} \qquad Z=\frac{T-\mu_T}{\sigma_T} } Per malgranda numeroj $n&lt;20$, $\mathbf{H_0}$ konsiderita se $P(T^+ &gt; \alpha)$ per grandaj numeroj $n&gt;20$, $\mathbf{H_0}$ konsiderita se $Z&lt;Z_\alpha$ kaj $Z&gt;-Z_\alpha$. 
\section{Bruoj}
\subsection*{Blanka} La blanka bruo estas karakterizita havi la sama frekvenca spektro per \v ciuj frekvencoj. \dm{&lt;\xi(t)&gt; = 0 \qquad &lt;\xi(t)\xi(t_0)&gt; = \sigma\delta(t-t_0)} En elektroniko estas frekventa la \eroh{Johnson}{Bruo} kiu depenas de la temperaturo kaj de la resistenco \ldots
\subsection*{Pafa}A\u u \fremd{shot noise}
\subsection*{1/f}
\section{Probableco evoluo}
Poste malgranda tempo la probableco evolvas inter la statoj $y_3-\toy_2$ kiel 
\dm{\Pro_\tau(y_3|y_2) = \delta(y_3-y_2) + W(y_3|y_2)\tau + O(\tau^2)}
estas la transiro procento $W(y_3|y_2)=\di_t\Pro(y_3|y_2)$. La probableco devas esti normalita: $\alpha(y_2) = \int W(y_3|y_2)\di y_3$}
  \blu{2}{\dm{\Pro_\tau(y_3|y_2) &amp;=&amp; (1-\tau\alpha(y_2))\delta(y_3-y_2) + W(y_3|y_2)\tau \\ &amp;=&amp; \delta(y_3-y_2) + W(y_3|y_2)\tau - W(y_2|y_3)\tau + O(\tau^2)}}
  \sogno{3}{A Chemical reaction: $R_1+R_2\rightleftharpoons P_1 + P_2$}
  \sogno{3}{\homo{Fermi}'s Rule: $\lambda_{if}=\frac{2\pi}{\hbar}|M_{if}|^2\rho_f$}
  \only&lt;3&gt;{
    \begin{figure}[!h]
      \psset{unit=3cm}
      \psset{linecolor=blue}
      \psset{linestyle=solid}
      \psset{linewidth=.5pt}
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        \pnode(0.15,.65){B1}
        \rput[ur](0,.4){\rnode{R2}{$R_2$}}
        \pnode(.15,.55){B2}
        \rput[bl](.4,.8){\rnode{P1}{$P_1$}}
        \pnode(.25,.65){B3}
        \rput[bl](.4,.4){\rnode{P2}{$P_2$}}
        \pnode(.25,.55){B4}
        % \fnode[framesize=3ex](.2,.6){B}
        \ncarc{&lt;-&gt;}{R1}{B1}     
        \ncarc{&lt;-&gt;}{R2}{B2}
        \ncarc{&lt;-&gt;}{B3}{P1}     
        \ncarc{&lt;-&gt;}{B4}{P2}
        \ncline{-}{B1}{B2}     
        \ncline{-}{B2}{B4}
        \ncline{-}{B4}{B3}     
        \ncline{-}{B3}{B1}
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        {\psset{linecolor=green,linewidth=2pt}\psline(0,.6)(.2,.6)}
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        \pnode(.2,.6){L1}
        \pnode(.4,.5){L2}
        \rput[bl](0,.6){$|n&gt;$}
        \rput[bl](.4,.5){$|m&gt;$}
        \nccurve[ArrowInside=-&gt;,ArrowInsidePos=0.75,angle=-30,angleB=180]{L1}{L2}
        % \psline(.6,.6)(.8,.6)
      \end{pspicture}
    \end{figure}
  }
\end{frame}

\begin{frame}
  \frametitle{Chapman-Kolmogorow equation}
  \blu{1}{The probability of a Markov process depends only on the probability of the last process}
  \blu{1}{\dm{\Pro(y_n|y_{n-1},,y_1) = \Pro(y_n|y_{n-1})}}\\
  \blu{2}{We assume a stationary ($\Pro(t) = \Pro(t+\tau)$) and homogenues process ($\Pro(t_1,t_2) = \Pro(t_1-t_2)$) (a more general case is the \homo{Boltzmann} equation)}\\
  \blu{3}{The conditional probability between the state $y_3$ and $y_1$ can be written:}
  \blu{3}{\dm{\Pro(y_3|y_1) = \int \Pro(y_3|y_2)\Pro(y_2|y_1)\di y_2}} \\
  \sogno{3}{One step process: Brownian motion, shot noise, dacay.}\\
\end{frame}

\begin{frame}
  \frametitle{Continous master equation}
  \blu{1}{We derivate the conditional probability with respect to the first order}\\
  \blu{1}{$\partial_\tau \Pro(y_3|y_2) = W(y_3|y_2) - W(y_2|y_3)$}\\
  \blu{2}{The derivative of the conditional probability is:}
  \blu{2}{\dm{\partial_\tau\Pro(y_3|y_1) = \int \Big( W(y_3|y_2)\Pro(y_2|y_1) - W(y_2|y_3)\Pro(y_3|y_1) \Big)\di y_2 } }
  \blu{2}{We remove the stationarity condition, multiply by $P(y_1,t)$ and integrate over $x_1$}
  \blu{2}{ \dm{\di_t\Pro(y,t) = \int W(y,y')\Pro(y',t) - W(y',y)\Pro(y,t)\di y'}}
  \sogno{3}{Einstein coefficients: Spontaneus emission $A_{mn}$, Absorption: $B_{nm}J$, Stimulated emission: $B_{mn}J$} 
\end{frame}

\begin{frame}
  \frametitle{Discrete master equation}
  \blu{1}{A chemical reaction generation/ricombination (discrete, nonlinear) $X\stackrel[W^\dagger]{W}{\rightleftharpoons} 2X$:}
  \blu{1}{\dm{\dot \Pro_n = W^\dagger n(n+1)\Pro_{n+1}+W(n-1)\Pro_{n-1}-W^\dagger n\Pro_{n} - W n(n-1)\Pro_n}}
  \blu{1}{The master equation says that the probability of a transition in a time $t$ is the sum of the gain in changing from $m\to n$ minus the loss between $n\to m$}\\
  \onslide&lt;2&gt;{
    \begin{figure}[!h]
      \psset{unit=6cm}
      \psset{linecolor=green}
      \psset{linestyle=solid}
      \psset{linewidth=1.5pt}
      \begin{pspicture}(0,-.1)(1,.1)
        \psline(0,.0)(.2,.0)
        \rput[bl](0.,.0){$(n-1)$}
        \psline(.4,.0)(.6,.0)
        \rput[bl](.4,.0){$(n)$}
        \psline(.8,.0)(1.0,.0)
        \rput[bl](.8,.0){$(n+1)$}
        \pnode(.2,.0){P1}
        \pnode(.4,.0){P2}
        \pnode(.6,.0){P3}
        \pnode(.8,.0){P4}
        \psset{linewidth=.5pt}
        \psset{linecolor=blue}
        \nccurve[ArrowInside=&lt;-&gt;,ArrowInsidePos=0.75,angle=30,angleB=120]{P1}{P2}
        \aput*{:U}{$W$}
        \nccurve[ArrowInside=&lt;-&gt;,ArrowInsidePos=0.75,angle=30,angleB=120]{P3}{P4}
        \aput*{:U}{$W^\dagger$}
      \end{pspicture}
    \end{figure}
  }
  \sogno{2}{In the steady state the lhs of the master equation is zero $\sum_mW_{nm}\Pro_m = (\sum_m W_{mn})\Pro_n$}\\
  \sogno{2}{Detailed balance: $W_{nm}\Pro_m = W_{mn}\Pro_n$}\\
  \sogno{2}{Detailed balance is necessary but not sufficient for the equilibrium (microscopic reversibility)}
  % the solution of the master equation of discrete states gives the Poisson distribution
\end{frame}

% \begin{frame}
%   \frametitle{Fokker Planck equation}
%   \blu{1}{If $W(x_3|x_2)\to 0$, small jumps $\delta_x = x_2-x_3$ approximation of continous paths. We develop in series of $\delta_x$}
%   \blu{1}{}



% \end{frame}

\begin{frame}
  \frametitle{Fokker-Planck equation}
  \blu{1}{The evolution of a single event $\Pro(t)$ is governed by the master equation which describes all statistical properties.}\\
  \blu{1}{We recast the integro equation of a master equation into the form of a \homo{Kramers-Moyal} expansion ($r=x-x'$, small)}
  \blu{1}{\dm{\partial_t \Pro(x,t) &amp;=&amp; \Pro(x,t)\int W(x|r)\Pro(x,t)-W(x|-r)\Pro(x,t)\di r \\ &amp;-&amp;\int r \partial_x(W(x|r)\Pro(x,t) )\di r + \frac12\int r^2\partial^2_x(W(x|r)\Pro(x,t))\di r \pm \ldots } }
  \blu{1}{or}
  \blu{1}{\dm{\dot \Pro_t(x) = \sum_n\frac {-^n}{n!}\partial_x^nD_{KM}^{(n)}\Pro_t(x) \quad D^{(n)}_{KM}(x,t)=\frac1{n!}\lim_{\tau\to 0}\frac 1\tau&lt;(x(t+\tau)-x(t))^n&gt;} }
  \blu{2}{From the \homo{Pawula} Theorem one can either use the first moment, the first and the second or every of them}\\
  \blu{2}{In Most of the case two moments are enough, (e.g. Gaussian noise): \homo{Fokker-Planck} equation}
  \blu{2}{\dm{\dot \Pro(x,t) = -\partial_x D^{(1)}\Pro(x,t) + \partial_x^2 D^{(2)}\Pro(x,t) } }
\end{frame}

\begin{frame}
  \frametitle{From master equation to stochastic differential equation}
  \blu{1}{We consider the equation:}
  \blu{1}{\dm{\dot x(t) = X(x) + \xi(t) }}
  \blu{1}{The moments of the noise $\xi$ are connected with the \homo{Kramers-Moyal} expansion} 
  \blu{1}{\dm{lim_{\Delta\to 0}\frac 1\Delta\int_\tau^{\tau\Delta}\di s &lt;\xi(s)|x(t)=x_0&gt; = D_{KM}^{(1)}-X(x) \\
      \lim_{\Delta\to 0}\frac 1\Delta\int_\tau^{\tau+\Delta}\di t_1\ldots\di t_n &lt;\xi(t_1)\ldots \xi(t_n)|x(t)=x_0&gt; = D_{KM}^{(n)}}
  }
  \blu{2}{The first moment is deterministic, the higher oder stochastic (mesoscopic description)}\\
  \blu{2}{Considering the same moments the following SDEq is equivalent to the previous MEq}\\
  \sogno{3}{A non zero mean of the noise contributes to a drift term}
  % multiplicative noise which leads to non-constant diffusion coefficient
  % for a continous Markov process the relavant moments are the first two
  \vspace{\stretch{1}}
  \begin{flushleft}
    \colorbox{Reference}{
      \scalebox{.7}{ \color{white}{P. Hanggi Z., Physik B, {\bf 43}, 269-273,(1981) }}
    }
  \end{flushleft}
\end{frame}


\begin{frame}
  \frametitle{Flux}
  \blu{1}{From the \homo{Fokker-Planck} equation we define a flux}
  \blu{1}{\dm{J := - D^{(1)}X(x)\Pro(x) + \frac12D^{(2)}\Pro(x)}}
  \blu{1}{The \homo{FP}Eq can be written as a conservation law:}
  \blu{1}{\dm{\partial_t \Pro(x) - \partial_x J(x) = 0}}
  \blu{2}{Tipical boundary condition} \\
  \begin{itemize}
  \item&lt;2-&gt; Natural (decay): $\int\Pro(x) = \mathrm{norm}$ 
  \item&lt;2-&gt; Reflecting (wall): $J(x=a,t)=0$
  \item&lt;2-&gt; Absorbing (first passage time): $P(x=a,t)=0$
  \end{itemize}
  
\end{frame}
\section{Application}
\begin{frame}
  \frametitle{Neuron dynamics, role of noise}
  \color{Rosso}
  \psset{linewidth=.2pt}
  The dynamics of the voltage in a neuron is ruled by the equation:
  \dm{\dot V(t) = - \frac {V(t)} \tau + I(t) \qquad I(t)=\mu+\sigma_w\rnode[b]{White}{\eta(t)}+\sigma_w\frac{\beta}{\sqrt{2\tau_c}}\rnode[b]{Col}{z(t)} }
  \color{Blu}
  Where $\eta(t)$ is a white noise and $z(t)$ is an auxiliary colored noise, $V$ is a potential between $H$ and $\Theta$
  \dm{C(t,t') = &lt;(I(t) - &lt;I(t)&gt;)(I(t') - &lt;I(t')&gt;)&gt; = \sigma_w^2\rnode[u]{White1}{\delta}(t-t')+\frac{\Sigma_2}{2\tau_c}\rnode[u]{Col1}{e}^{-\frac{|t-t'|}{\tau_c}} }
  \ncarc{White}{White1}
  \ncarc{Col}{Col1}
  \color{Verde}
  The \homo{Fokker-Planck} equation is:
  \dm{\left(\partial_V\Big( f(V) - \mu+\frac{\sigma^2}2\partial_V\Big)+\frac 1 {\tau_c}\partial_z(z+\partial_z)-\sqrt{\frac{2\sigma^2\alpha^2}{\tau_c}}\partial_V\right)\Pro=-\delta(V-H)J(z)}
  The {\it firing rate} $\nu$ is the probability per unit time that the potential cross a threshold $\Theta$. $J(z)$ is the escape probability current. 
  
\end{frame}

\begin{frame}
  \frametitle{Firing rate}
  \color{Rosso}
  $\Pro(V,z)$ is the staedy state probability. We suppose $\tau_c&lt;\tau_{ref}$ (correlation, refractory). 
  \dm{J(z) = \frac{\nu_{out}}{\sqrt{2\pi}} e^{-x^2/2} }
  $z$ after a spike relaxes to the stationary distribution. The \homo{Fokker-Planck} has to be resolved with the normalisation condition
  \dm{\nu_{out}\tau_{ref}+\int_{-\infty}^\Theta\di V\int_{-\infty}^\infty\di z \Pro(V,z) = 1 }
  \color{Blu}
  Finally the output firing rate is give by
  \dm{\nu_{out} = \int \di z J(z) }
  
\end{frame}

\begin{frame}
  \frametitle{Summary}
  \psset{unit=8cm}
  \begin{pspicture}(-.3,0)(1,1)
    \psset{linecolor=red,fillstyle=solid}
    % % \psset{gradient,gradbegin=white,gradend=blue,gradmidpoint=1,gradangle=-45](0,0)
    % %   \psset{fillcolor=pink}
    \psset{linewidth=.9\pslinewidth}
    \rput[bt](0.,.9){\ovalnode{Transition}{Transition prob}}
    \rput[bt](.1,.7){\ovalnode{MEq}{MEq}}
    \rput(.6,.7){\ovalnode{Langevin}{Langevin}}
    \rput(.4,.5){\ovalnode{Kramers}{Kramers-Moyal}}
    \rput[b](.4,.3){\ovalnode{Fokker}{Fokker-Planck}}
    \rput(0.,.1){\rnode{Olstein}{\psframebox{Olstein-Uhlenbeck}{}} }
    \rput(.8,.1){\rnode{Neuro}{\psframebox{Neuronal model}{}} }
    \nccurve[ArrowInside=-&gt;,ArrowInsidePos=0.75,angle=-40,angleB=50]{Transition}{MEq}
    % \aput(0.5){Evolution}
    % \mput*{Evolution}
    \nccurve[ArrowInside=-&gt;,ArrowInsidePos=0.75,angle=30,angleB=160]{MEq}{Langevin}
    \aput(0.5){Stochastic}
    \nccurve[ArrowInside=-&gt;,ArrowInsidePos=0.75,angle=230,angleB=30]{Langevin}{Kramers}
    \aput(0.5){Noise distribution}
    \nccurve[ArrowInside=-&gt;,ArrowInsidePos=0.75,angle=-30,angleB=140]{MEq}{Kramers}
    \aput(0.7){Continuous}
    \nccurve[ArrowInside=-&gt;,ArrowInsidePos=0.75,angle=250,angleB=60]{Kramers}{Fokker}
    \aput(0.7){Two moments}
    \nccurve[ArrowInside=-&gt;,ArrowInsidePos=0.75,angle=230,angleB=20]{Fokker}{Olstein}
    \ncput*{Dissipative}
    \nccurve[ArrowInside=-&gt;,ArrowInsidePos=0.75,angle=-30,angleB=120]{Fokker}{Neuro}
    \ncput*{Non-gaussian noise}
  \end{pspicture}
\end{frame}

\begin{frame}
  \begin{center}
    {\fontfamily{phv}\selectfont 
      \color{Rosso}{Thank you for your attention}
    }
  \end{center}
  \vspace{\stretch{1}}
  \color{blue}{Aknowledgments}\nul\\\nul
  \blu{1}{Tatjana, Mirko (tutors)}

  \vspace{\stretch{1}}
  \begin{flushright}
    \color{blue}{References}\nul\\\nul
    \rosso{1}{Response of Spiking Neuros to Correlated Inputs}\nul\\\nul
    \blu{1}{R. Moreno, J. de la Rocha, A. Renart, N. Parga}\nul\\\nul
    \verde{1}{PRL {\bf 89} 288101  (2002)}\nul\\\nul
    \rosso{1}{How Spike Generation Mechanisms Determine the Neuronal Response to Fluctuating Inputs}\nul\\\nul
    \blu{1}{N. Foucard-Trocm\'e, D. Hansel, C. van Vreeswijk, N. Brunel}\nul\\\nul
    \verde{1}{J. Neuroscience {\bf 23} 11628}\nul\\\nul
  \end{flushright}
\end{frame}

\section{Appendix}

\begin{frame}
  \frametitle{Langevin equation}
  \blu{1}{\dm{\dot x = X(x) + \xi(t)}}
  \blu{1}{The probability distribution $P(y,t)$ is defined as:}
  \blu{1}{\dm{P(y,t) = &lt;\delta(y - x(t))&gt;_\xi}}
  \blu{2}{Advection (drift) term $D^{(1)}$: $\dot x = X(x)$}
  \blu{2}{\dm{\partial_t\Pro(y,t) &amp;=&amp; -\dot x \di_y \delta(y-x) = -X(x)\di_y\delta(y-x)=-\di_y(\delta(y-x)F(x))\\ &amp;=&amp;-\di_y(\delta(y-x)F(y))=-\di_y(F(y)\Pro(y,t))}}
  \blu{3}{Diffusive term $D^{(2)}$: $\dot x = \xi(t)$}
  \blu{3}{\dm{\Pro(x,t)=\frac{1}{\sqrt{2\pi\Delta t}}e^{-\frac{y^2}{2\Delta t}}\qquad \partial_t\Pro(x,t) = \frac{D^{(2)}}{2}\partial_x^2\Pro(x,t) \\
      \partial_t \Pro(x,t) = -D^{(1)}\partial_x\Pro(x,t) + D^{(2)}\partial_x^2\Pro(x,t) } }
\end{frame}


\begin{frame}
  \frametitle{Ornstein-Uhlenbeck process}
  We start from the dissipative \homo{Langevin} equation
  \dm{\Delta v(t) = -\gamma v(t) + \sigma\Delta \xi(t)}
  The \homo{KM} coefficients are $D^{(1)} = -\gamma v$, $D^{(2)} = \frac{\sigma^2}2$ \\
  The stationary solution is given by $\dot \Pro(x) \stackrel != 0$
  \dm{\Pro(v) = \sqrt{\frac{m}{2\pi k_BT}}e^{-\frac{v^2m}{2K_BT}} }
\end{frame}

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