diff --git a/docs/formulation.tex b/docs/formulation.tex index 56dccf12c621506ffd1c68beb94ba9cfc6337135..7609dcbf56c585abf004852227c9dbdd300f4991 100644 --- a/docs/formulation.tex +++ b/docs/formulation.tex @@ -2,6 +2,7 @@ The cable equation is a nonlinear parabolic PDE that can be written in the form \begin{equation} + \label{eq:cable} c_m \pder{V}{t} = \frac{1}{2ar_{L}} \pder{}{x} \left( a^2 \pder{V}{x} \right) - i_m + i_e, \end{equation} where @@ -16,6 +17,14 @@ where 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. +The PDE in (\ref{eq:cable}) is derived from the following mass balance expression for a cable segment +\begin{equation} + \label{eq:cable} + 2\pi a \Delta x c_m \pder{V}{t} = -\left. \left( \frac{\pi a^2}{r_L}\pder{V}{x} \right) \right|_\text{left} + +\left. \left( \frac{\pi a^2}{r_L}\pder{V}{x} \right) \right|_\text{right} + - 2\pi a \Delta x (i_m - i_e) +\end{equation} + \begin{table*}[htp!] \begin{center} @@ -42,13 +51,17 @@ Note that the standard convention is followed, whereby membrane and synapse curr $q$ & $C$ & $s\cdot A$ & $s\cdot A$ \\ - $I$ & $A$ & & \\ + $I$ & $A$ & $C\cdot s^{-1}$ & + $A$ \\ - $V$ & $V$ & & + $V$ & $V$ & $W\cdot A$ & $kg\cdot m^{2}\cdot s^{-3}\cdot A^{-1}$ \\ - $R$ & $\Omega$ & & \\ - $C$ & $F$ & & \\ + $R$ & $\Omega$ & $V\cdot A^{-1}$ & + $kg\cdot m^{2}\cdot s^{-3}\cdot A^{-2}$ \\ + + $C$ & $F$ & $C\cdot V^{-1}$ & + $kg^{-1}\cdot m^{-2}\cdot s^{4}\cdot A^{2}$ \\ \hline \end{tabular}