<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[10pt,handout,
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dvipsnames,table]{beamer}
\usetheme{Padova}
\usepackage{Vorlage}
\newcommand{\Pro}{\mathcal{P}}%Probableco
\title[Fokker-Planck]{Fokker-Planck}
\subtitle{}
\author[G. Marelli]{Giovanni Marelli}
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% \PSforPDF{\usepackage{hyperref,pstricks,pst-plot,psfrag,pst-node,pstricks-add}}
\begin{document}

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  \begin{center}
    \colorbox{Titolo}{
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      {\Large \color{white}{Fokker Planck and Master equation}}
      \hspace{\stretch{1}}
    }{}\\ \vspace{4ex}
    {\large Giovanni Marelli}\\ \vspace{4ex}
  \end{center}
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  % \rput[lt](0,0){\includegraphics[clip=true,width=\textwidth]{unipd_logo}}
  \begin{figure}[!h]
    \normalcolor
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  \vspace{\stretch{1}}
  \begin{flushright}
    {\small G\"ottingen 27.01.10} 
  \end{flushright}
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\section{Introduction}
\begin{frame}[label=Contents]
  \frametitle{Contents}
  \begin{center}
    {\fontfamily{phv}\selectfont 
      \color{Rosso}{Connections}
    }
  \end{center}
  \begin{itemize}
  \item Temporal evolution of the probability
  \item Hypotesis for the \homo{Chapman-Kolmogorov} equation
  \item Meaning of the master equation
  \item The master equation is equivalent to a stochastic differential equation
  \item Derivation of the \homo{Fokker-Plank} from a \homo{Langevin} equation
  \end{itemize}


\end{frame}

\section{Derivation}
\begin{frame}[label=Trans]
  \frametitle{Transition Probability}
  \blu{1}{Transition probability at a time $\tau$ between two states $y_3\to y_2$ in a $\tau$ exspansion}
  \blu{1}{\dm{\Pro_\tau(y_3|y_2) = \delta(y_3-y_2) + W(y_3|y_2)\tau + O(\tau^2)} }\\
  \blu{1}{$W(y_3|y_2)=\di_t\Pro(y_3|y_2)$ transition rate}\\
  \blu{2}{Normalizing: $\alpha(y_2) = \int W(y_3|y_2)\di y_3$}
  \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]
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        \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}
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        \rput[bl](.4,.5){$|m&gt;$}
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        % \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}
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      \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}
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        \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)
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    \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}
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    \aput(0.5){Noise distribution}
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    \aput(0.7){Continuous}
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    \aput(0.7){Two moments}
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    \ncput*{Dissipative}
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    \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}


\end{document}
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