1.3 looks great ! Thanks! I found a bug though. in
The attached file was written in 1.2.3 and worked fine there, but gives latex errors in 1.3 (also after lyx2lyx) Mark -- ==== [EMAIL PROTECTED] ===== http://homepages.inf.ed.ac.uk/mvanross Dr. Mark van Rossum, Lecturer, ANC, University of Edinburgh. Rm C4, 5 Forrest Hill, Edinburgh, EH1 2QL, UK phone: +44-0131-650.3088 fax: +44-0131-650.6899 Home: 1-7 Dunbar's Close, 137 Canongate, Edinburgh EH8 8BW; +44-0131-557.1014
#LyX 1.3 created this file. For more info see http://www.lyx.org/ \lyxformat 221 \textclass book \begin_preamble \newcommand{\mvrbox}[1]{\fbox{$ #1 $}} %from chemistry.tex \def\longrharpup{\relbar\joinrel\rightharpoonup} \def\longlharpdn{\leftharpoondown\joinrel\relbar} \def\rlPOON#1{\vcenter{\hbox{\ooalign{\raise2.3pt \hbox{$#1\longrharpup$}\crcr $#1\longlharpdn$}}}} \def\eqbm{\mathrel{\mathpalette\rlPOON{}}\rm} \def\eqbmlab#1_#2{\mathrel{\mathop{\eqbm}\limits~{#1}_{#2}}\rm} \def\yields{\longrightarrow\rm} \def\yieldslab~#1{\mathrel{\mathop{\longrightarrow}\limits~{#1}} \rm} \def\expect#1{\langle{#1}\rangle} \usepackage{comment} %\includecomment{comment} \usepackage{namedplus} \newcommand{\eexp}[1]{{\rm e}^{#1}} \end_preamble \language british \inputencoding auto \fontscheme default \graphics default \paperfontsize default \spacing single \papersize a4paper \paperpackage widemarginsa4 \use_geometry 0 \use_amsmath 0 \use_natbib 0 \use_numerical_citations 0 \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \quotes_times 2 \papercolumns 1 \papersides 1 \paperpagestyle headings \layout Title Neural Computation \layout Author Mark van Rossum \layout Standard Microscopically, the gates are like little binary switches that switch on and off depending on the membrane voltage. The Na channel has 3 switches labelled \begin_inset Formula $m$ \end_inset and one labelled \begin_inset Formula $h$ \end_inset . In order for the sodium channel to conduct all three \begin_inset Formula $m$ \end_inset and the \begin_inset Formula $h$ \end_inset have to be switched on. The gating variables describe the probability that the gate is in the on or off state. Note that the gating variables depend both on time and voltage; their values range between 0 and 1. The gating variables evolve as \begin_inset Formula \begin{eqnarray} \frac{dm(V,t)}{dt} & = & \alpha_{m}(V)(1-m)-\beta_{m}(V)m\label{eq:m}\\ \frac{dh(V,t)}{dt} & = & \alpha_{h}(V)(1-h)-\beta_{h}(V)h\nonumber \end{eqnarray} \end_inset \layout Paragraph* \family sans \size small Intermezzo \layout Standard \family sans \size small Consider a simple reversible chemical reaction in which substance A is turned into substance B. \layout Standard \family sans \size small \begin_inset Formula \[ [A]\mathop\rightleftharpoons_{\alpha}^{\beta}[B]\] \end_inset the rate equation for reaction is: \begin_inset Formula $d[A]/dt=-\beta[A]+\alpha[B]$ \end_inset . Normalising \begin_inset Formula $[A]+[B]=1,$ \end_inset it is: \begin_inset Formula $d[A]/dt=\alpha(1-[A])-\beta[A]$ \end_inset , which is very similar to what we have for the gating variables. The solution to this differential equation is much like for the RC circuit. If at time 0, the concentration of \begin_inset Formula $A$ \end_inset is \begin_inset Formula $[A]_{0}$ \end_inset , it will settle to \begin_inset Formula \[ [A](t)=[A]_{\infty}+\left([A]_{0}-[A]_{\infty}\right)\exp(-t/\tau)\] \end_inset where the final concentration \begin_inset Formula $[A]_{\infty}=\alpha/(\alpha+\beta)$ \end_inset , and the time-constant \begin_inset Formula $\tau=1/(\alpha+\beta)$ \end_inset . (Check for yourself) \newline \layout Standard The interesting part for the voltage gated channel is that the rate constants depend on the voltage across the membrane. Therefore, as the voltage changes, the equilibrium shifts and the gating variables will try to establish a new equilibrium. The equilibrium value of the gating variable is \begin_inset Formula \[ m_{\infty}=\frac{\alpha_{m}}{\alpha_{m}+\beta_{m}}\] \end_inset and the time-constant is \begin_inset Formula \[ \tau_{m}=\frac{1}{\alpha_{m}+\beta_{m}}\] \end_inset \layout Section Markov description of channels \layout Standard Like the voltage gated channels, the synaptic channels can be written with state diagrams. These state, or Markov diagrams, are a convenient to simulate the channels. But the diagrams can also be used to calculate the power spectrum of the channel openings and the autocorrelation of the openings. Consider a simple state diagram \begin_inset Formula \[ C\mathop\rightleftharpoons_{\alpha}^{kT}C'\mathop\rightleftharpoons_{\alpha'}^{k'T}O\] \end_inset The first binding of transmitter \begin_inset Formula $T$ \end_inset move the channel from closed state \begin_inset Formula $C$ \end_inset to state \begin_inset Formula $C'$ \end_inset ; binding of another \begin_inset Formula $T$ \end_inset opens the channel. We can write the state of the channel as a vector \begin_inset Formula $\vec{s}(t)=(C,C',O)$ \end_inset . The dynamics of the channel can then be written as \begin_inset Formula \[ \frac{d\vec{s}}{dt}=W.\vec{{s}}\] \end_inset where \begin_inset Formula $W$ \end_inset is a transition matrix between the different states. (This formalism is also called ma \the_end
