diff --git a/docs/appendix.tex b/docs/appendix.tex
new file mode 100644
index 0000000000000000000000000000000000000000..81a04377c2f6db62dc7f51e03705b21a6d657a9a
--- /dev/null
+++ b/docs/appendix.tex
@@ -0,0 +1,26 @@
+%-----------------------------------
+\subsection{The conic frustrum}
+%-----------------------------------
+The derivation of the surface area of conic frustrum.
+The edge length $l$ is defined
+\begin{equation}
+ l = \sqrt{(x_r - x_l)^2 + (a_r - a_l)^2} = \sqrt{\Delta x^2 + \Delta a^2}.
+\end{equation}
+The lateral area of the surface is found by integrating along surface of rotation:
+\begin{align}
+ \sigma_{\text{lateral}}
+ &= \int_{0}^{l} {2\pi a(s)} \deriv{s} \nonumber \\
+ &= 2\pi \int_{0}^{l} {a_{\ell} + \frac{s}{l}\left( a_r - a_\ell \right)} \deriv{s} \nonumber \\
+ &= 2\pi \left[ a_{\ell}s + \frac{s^2}{2l}\left( a_r - a_\ell \right) \right]_0^l \nonumber \\
+ &= \pi l \left( a_{\ell} + a_r \right) \nonumber \\
+ &= \pi \left( a_{\ell} + a_r \right) \sqrt{\Delta x^2 + \Delta a^2}. \label{eq:frustrum_area}
+\end{align}
+
+There are two degenerate cases of interest. The first is the \emph{cylinder}, for which the radii at each end are euqal, i.e. $a_\ell = a_r = a$. In this case the lateral area of the surface is
+\begin{equation}
+ \sigma_{\text{lateral}} = 2\pi a \Delta x.
+\end{equation}
+The second is a cone, for which $a_\ell=0$ and $a_r=a$:
+\begin{equation}
+ \sigma_{\text{lateral}} = \pi a \sqrt{\Delta x^2 + a^2}.
+\end{equation}
diff --git a/docs/formulation.tex b/docs/formulation.tex
index bc8254764a6cc8305ae48d36b1a2de1829a63da8..755817f70323b0eb0901490e7d0c05cf57a3d245 100644
--- a/docs/formulation.tex
+++ b/docs/formulation.tex
@@ -54,29 +54,32 @@ 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 $i$:
-\begin{align}
- \int_{\Omega}{c_m \pder{V}{t} } \deriv{v} =
- & - \sum_{j\in\mathcal{N}_i} {\int_{\Gamma_{i,j}} \frac{1}{r_L}\pder{V}{x} n_{i,j} \deriv{s} } \nonumber \\
- & - \int_{\Gamma_{ext}} {(i_m - i_e)} \deriv{s}
+%\begin{align}
+ %\int_{\Omega_i}{c_m \pder{V}{t} } \deriv{v} =
+ %& + \sum_{j\in\mathcal{N}_i} {\int_{\Gamma_{i,j}} \frac{1}{r_L}\pder{V}{x} n_{i,j} \deriv{s} } \nonumber \\
+ %& + \int_{\Gamma_{ext}} {(i_m - i_e)} \deriv{s}
+ %\label{eq:cable_balance}
+%\end{align}
+\begin{equation}
+ \int_{\Omega_i}{c_m \pder{V}{t} } \deriv{v} =
+ + \sum_{j\in\mathcal{N}_i} {\int_{\Gamma_{i,j}} q_{i,j} \deriv{s} }
+ + \int_{\Gamma_{ext}} {q_i} \deriv{s}
\label{eq:cable_balance}
-\end{align}
-where $\int_\Omega \cdot \deriv{v}$ is shorthand for the volume integral over the segment $\Omega$, and $\int_\Gamma \cdot \deriv{s}$ is shorthand for the surface integral over the surface $\Gamma$.
+\end{equation}
+where
+\begin{itemize}
+ \item $\int_\Omega \cdot \deriv{v}$ is shorthand for the volume integral over the segment $\Omega_i$
+ \item $\int_\Gamma \cdot \deriv{s}$ is shorthand for the surface integral over the surface $\Gamma$
+ \item $q_{i,j}=-\frac{1}{r_L}\pder{V}{x} n_{i,j}$ is the flux per unit area of current \emph{from segment $i$ to segment $j$} over the interface $\Gamma_{i,j}$ between the two segments.
+ \item $q_i=i_m - i_e$ is the flux per unit area over the cell membrane $\Gamma_i$ due to ion channels, synapses and electrodes.
+ \item the set $\mathcal{N}_i$ is the set of segments that are neighbours of $\Omega_i$
+\end{itemize}
The surface of the cable segment is sub-divided into the internal and external surfaces.
The external surface $\Gamma_{ext}$ is the cell membrane at the interface between the extra-cellular and intra-cellular regions.
The current, which is the conserved quantity in our conservation law, over the surface is composed of the synapse and ion channel contributions.
This is derived from a thin film approximation to the cell membrane, whereby the membrane is treated as an infinitesimally thin interface between the intra and extra cellular regions.
-The internal surfaces are the interface between the cable segment and its neighbour segments which are denoted by the set $\mathcal{N}_i$.
-Equation~\eq{eq:cable_balance} handles the general case where a cable might lie at a branch, and can be simplified for a one dimensional segment:
-\begin{align}
- \int_{\Omega}{c_m \pder{V}{t} } \deriv{v} =
- \quad & \int_{\Gamma_{\text{left}}} \left( \frac{1}{r_L}\pder{V}{x} \right) \deriv{s} \nonumber \\
- -& \int_{\Gamma_{\text{right}}} \left( \frac{1}{r_L}\pder{V}{x} \right) \deriv{s} \nonumber \\
- -& \int_{\Gamma_{ext}} {(i_m - i_e)} \deriv{s}
- \label{eq:cable_balance_1D}
-\end{align}
-
Note that some information is lost when going from a three-dimensional description of a neuron to a system of branching one-dimensional cable segments.
If the cell is represented by cylinders or frustrums\footnote{a frustrum is a truncated cone, where the truncation plane is parallel to the base of the cone.}, the three-dimensional values for volume and surface area at branch points can't be retrieved from the one-dimensional description.
@@ -104,7 +107,7 @@ The finite volume method is a natural choice for the solution of the conservatio
%-------------------------------------------------------------------------------
We proceed by defining the \emph{volume average} of a quantity $\varphi$ as follows:
\begin{equation}
- \bar{\varphi}_i = \frac{1}{v_i} \int_{\Omega_i}{\varphi}\deriv{v},
+ \bar{\varphi}_i = \frac{1}{\Delta_i} \int_{\Omega_i}{\varphi}\deriv{v},
\end{equation}
where $\Omega_i$ is the control volume with the point $x_i$ at its centroid illustrated in \fig{fig:segment}, and $v_i$ is the volume of the control volume $\Omega_i$.
The integral one the left hand side of~\eq{eq:cable_balance} can be expressed in terms of the volume average of $V$
@@ -149,7 +152,7 @@ where $\sigma_{i,j}=\pi a_{i,j}^2$ is the area of the surface $\Gamma_{i,j}$, wh
Some symmetries
\begin{itemize}
\item $\sigma_{i,j}=\sigma_{j,i}$ : surface area of $\Gamma_{i,j}$
- \item $\Delta_{i,j}=\Delta_{j,i}$ : distance between $x_i$ and $x_j$
+ \item $\Delta x_{i,j}=\Delta x_{j,i}$ : distance between $x_i$ and $x_j$
\item $n_{i,j}=-n_{j,i}$ : surface ``norm''/orientation
\item $q_{i,j}=n_{j,i}q_{i,j}=-q_{j,i}$ : charge flux over $\Gamma_{i,j}$
\end{itemize}
@@ -173,23 +176,53 @@ The current terms are an average per unit area, therefore the total flux
where $\sigma_i$ is the surface area the of the exterior of the cable segment, i.e. the surface corresponding to the cell membrane.
Each cable segment is a conical frustrum, as illustrated in \fig{fig:segment}.
-The lateral surface area of a frustrum with height $\Delta x_i$ and radii of $a_{i,\ell}$ and $a_{i,r}$ is
+The lateral surface area of a frustrum with height $\Delta x_i$ and radii of is
+The area of the external surface $\Gamma_{i}$ is
\begin{equation}
- \sigma_i = \pi (a_{i,\ell} + a_{i,r})+\sqrt{\Delta x_i^2 + (a_{i,\ell} - a_{i,r})^2}.
- \label{eq:frustrum_volume}
+ \sigma_i = \pi (a_{i,\ell} + a_{i,r}) \sqrt{\Delta x_i^2 + (a_{i,\ell} - a_{i,r})^2},
+ \label{eq:cv_volume}
\end{equation}
-
+where $a_{i,\ell}$ and $a_{i,r}$ are the radii of at the left and right end of the segment respectively (see~\eq{eq:frustrum_area} for derivation of this formula).
%-------------------------------------------------------------------------------
\subsubsection{Putting it all together}
%-------------------------------------------------------------------------------
-By substituting the volume averaging of the temporal derivative in~\eq{eq:dvdt} approximations for the flux over the surfaces in~\eq{eq:q_ij} and~\eq{eq:frustrum_volume} respectively into the conservation equation~\eq{eq:cable_balance} we get the following ODE defined for each node in the cell
+By substituting the volume averaging of the temporal derivative in~\eq{eq:dvdt} approximations for the flux over the surfaces in~\eq{eq:q_ij} and~\eq{eq:cv_volume} respectively into the conservation equation~\eq{eq:cable_balance} we get the following ODE defined for each node in the cell
\begin{equation}
\frac{c_m}{\Delta_i} \dder{\bar{V}_i}{t}
= -\sum_{j\in\mathcal{N}_i} {\frac{\sigma_{i,j}}{r_L \Delta x_{i,j}} (V_i-V_j)} + \sigma_i\cdot(i_m(V_i) - i_e(x_i)),
\label{eq:ode}
\end{equation}
where
-\begin{itemize}
- \item $\sigma_{i,j}=\pi a_{i,j}^2$ is the area of the surface between two adjacent segments $i$ and $j$. Care must be taken if the neighbour is a child or parent of the other at a branch point. In this case, the radius is that of the child segment (which ever has the largest index in the minimum degree ordering used for numbering segments). \todo{what about if we balance the tree so that the parent-child relationship is swapped?, maybe the relationship ``farther from soma'' has to used to choose.}
- \item $\sigma_{i}=(a_{i,\ell} + a_{i,r})+\sqrt{\Delta x_i^2 + (a_{i,\ell} - a_{i,r})^2}.$ is the lateral area of the conical frustrum describing segment $i$. For this area, we take the value of the segment radius at the left and right of the frustrum iteself, regardless of whether either end lies at a branch point.
-\end{itemize}
+%\begin{itemize}
+% \item $\sigma_{i,j}=\pi a_{i,j}^2$ is the area of the surface between two adjacent segments $i$ and $j$.
+% \item $\sigma_{i}=\pi(a_{i,\ell} + a_{i,r})+\sqrt{\Delta x_i^2 + (a_{i,\ell} - a_{i,r})^2}.$ is the lateral area of the conical frustrum describing segment $i$.
+% \item $\Delta_{i}=\frac{\pi\Delta x_i}{2} \left( a_{i,l}^2 + a_{i,r}^2 \right)$ is the volume of the segment $\Omega_i$.
+%\end{itemize}
+\begin{equation}
+ \sigma_{i,j} = \pi a_{i,j}^2
+ \label{eq:sigma_ij}
+\end{equation}
+is the area of the surface between two adjacent segments $i$ and $j$, and
+\begin{equation}
+ \sigma_{i} = \pi(a_{i,\ell} + a_{i,r}) \sqrt{\Delta x_i^2 + (a_{i,\ell} - a_{i,r})^2},
+ \label{eq:sigma_i}
+\end{equation}
+is the lateral area of the conical frustrum describing segment $i$, and
+\begin{equation}
+ \Delta_{i} = \frac{\pi\Delta x_i}{2} \left( a_{i,l}^2 + a_{i,r}^2 \right)
+ \label{eq:delta_i}
+\end{equation}
+is the volume of the segment $\Omega_i$.
+
+%-------------------------------------------------------------------------------
+\subsubsection{Handling branches}
+%-------------------------------------------------------------------------------
+The value of the lateral area and volume, $\sigma_i$ and $\Delta_i$ in~\eq{eq:sigma_i} and~\eq{eq:delta_i} respetively, must include contributions from each branch at branch points.
+
+\todo{a picture of a branching point to illustrate}
+
+\todo{a picture of a soma to illustrate the ball and stick model with a sphere for the soma and sticks for the dendrites branching off the soma.}
+
+\begin{equation}
+ \sigma_i = \sum_{j\in\mathcal{N}_i} {}
+\end{equation}
diff --git a/docs/report.tex b/docs/report.tex
index 4702f766a0a7b0bb30f80e939baff2a123930241..24d7f7657a0ac00703eede3999f132c79d108caa 100644
--- a/docs/report.tex
+++ b/docs/report.tex
@@ -94,6 +94,9 @@
\section{Formulation}
\input{formulation.tex}
+\section{Apendix}
+\input{appendix.tex}
+
\section{Symbols and Units}
\input{symbols.tex}