diff --git a/docs/formulation.tex b/docs/formulation.tex index 93183fc00f847ed50e0526de711fe97ec1837e89..56dccf12c621506ffd1c68beb94ba9cfc6337135 100644 --- a/docs/formulation.tex +++ b/docs/formulation.tex @@ -7,8 +7,53 @@ The cable equation is a nonlinear parabolic PDE that can be written in the form where \begin{itemize} \item $V$ is the potential relative to the ECM $[mV]$ - \item $a$ is the cable radius \todo{units} + \item $a$ is the cable radius \todo{units} $[mm]$? \item $c_m$ is the {specific membrane capacitance}, approximately the same for all neurons $\approx 10~nF/mm^2$. Related to \emph{membrane capacitance} $C_m$ by the relationship $C_m=c_{m}A$, where $A$ is the surface area of the cell. - \item $i_m$ is the membrane current \todo{units} + \item $i_m$ is the membrane current \todo{units}. The total contribution from ion and synaptic channels is expressed as a the product of current per unit area $i_m$ and the surface area. + \item $i_e$ is the electrode current flowing into the cell, divided by surface area, i.e. $i_e=I_e/A$. \item $r_L$ is intracellular resistivity, typical value $1~k\Omega$ \end{itemize} + +Note that the standard convention is followed, whereby membrane and synapse currents ($i_m$) are positive when outward, and electrod currents ($i_e$) are positive inward. + +\begin{table*}[htp!] + \begin{center} + + \begin{tabular}{llll} + \hline + quality & symbol & unit & notes \\ + \hline + energy & $J$ & joule $j$ & work to push 1 $N$ through 1 $m$ \\ + charge & $q$ & coulomb $C$ & $6.25\cdot10^{18}$ electrons, $[A\cdot s]$ \\ + current & $I$ & ampere $A$ & $[C\cdot s^{-1}]$, $A$ is SI base unit\\ + voltage & $V$ & volt $V$ & potential work per unit charge \\ + resistance & $R$ & ohm $\Omega$ & recall Ohm's law $V=IR$ \\ + capacitance& $C$ & farad $F$ & $C=\frac{q}{V}$, $[J\cdot C^{2}]$\\ + \hline + \end{tabular} + + \begin{tabular}{llll} + \hline + symbol & unit & equivalents & SI base \\ + \hline + $J$ & $j$ & $J\cdot s^{-1}$, $V\cdot A$ & + $kg\cdot m^{2}\cdot s^{-2}$ \\ + + $q$ & $C$ & $s\cdot A$ & + $s\cdot A$ \\ + + $I$ & $A$ & & \\ + + $V$ & $V$ & & + $kg\cdot m^{2}\cdot s^{-3}\cdot A^{-1}$ \\ + + $R$ & $\Omega$ & & \\ + $C$ & $F$ & & \\ + \hline + \end{tabular} + + \end{center} + \caption{Symbols and quantities.} +\end{table*} + +