diff --git a/docs/bibliography.bib b/docs/bibliography.bib
new file mode 100644
index 0000000000000000000000000000000000000000..b3206ba47fde7e114969fd693bdf3f1c21ef9827
--- /dev/null
+++ b/docs/bibliography.bib
@@ -0,0 +1,20 @@
+@article{lindsay_2004,
+title = "From Maxwell's equations to the cable equation and beyond ",
+journal = "Progress in Biophysics and Molecular Biology ",
+volume = "85",
+number = "1",
+pages = "71 - 116",
+year = "2004",
+note = "",
+issn = "0079-6107",
+doi = "http://dx.doi.org/10.1016/j.pbiomolbio.2003.08.001",
+url = "http://www.sciencedirect.com/science/article/pii/S0079610703000786",
+author = "K.A. Lindsay and J.R. Rosenberg and G. Tucker",
+keywords = "Maxwell's equations",
+keywords = "Cable equation",
+keywords = "Finite elements",
+keywords = "Neuronal models",
+keywords = "Tapered dendrites",
+keywords = "Regular perturbation expansions "
+}
+
diff --git a/docs/formulation.tex b/docs/formulation.tex
index 7609dcbf56c585abf004852227c9dbdd300f4991..3812398dcff54a9852f34fe59ded3d8446fd6aea 100644
--- a/docs/formulation.tex
+++ b/docs/formulation.tex
@@ -3,70 +3,65 @@
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,
+ c_m \pder{V}{t} = \frac{1}{2\pi a r_{L}} \pder{}{x} \left( a^2 \pder{V}{x} \right) - i_m + i_e,
\end{equation}
where
\begin{itemize}
\item $V$ is the potential relative to the ECM $[mV]$
- \item $a$ is the cable radius \todo{units} $[mm]$?
+ \item $a$ is the cable radius $(mm)$, and can vary with $x$
\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}. 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 $i_m$ is the membrane current $[A]$. 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.
-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}
-
- \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
- $J$ & $j$ & $J\cdot s^{-1}$, $V\cdot A$ &
- $kg\cdot m^{2}\cdot s^{-2}$ \\
-
- $q$ & $C$ & $s\cdot A$ &
- $s\cdot A$ \\
-
- $I$ & $A$ & $C\cdot s^{-1}$ &
- $A$ \\
+The PDE in (\ref{eq:cable}) is derived from the following mass balance expression for a cable segment:
+\begin{align}
+ \int_{\Omega}{c_m \pder{V}{t} } \deriv{v} =
+ & - \int_{\Gamma_{\text{left}}} \left( \frac{1}{r_L}\pder{V}{x} \right) \deriv{s}
+ + \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}
+\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 $\Gamma$.
+The surface of the cable segment is sub-divided into the left, right and external parts of the surface.
- $V$ & $V$ & $W\cdot A$ &
- $kg\cdot m^{2}\cdot s^{-3}\cdot A^{-1}$ \\
+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.
- $R$ & $\Omega$ & $V\cdot A^{-1}$ &
- $kg\cdot m^{2}\cdot s^{-3}\cdot A^{-2}$ \\
+The left and right surface are the interface between the cable segment and its neighbour.
- $C$ & $F$ & $C\cdot V^{-1}$ &
- $kg^{-1}\cdot m^{-2}\cdot s^{4}\cdot A^{2}$ \\
- \hline
- \end{tabular}
+\subsection{Assumptions of the cable equation}
+See \cite{lindsay_2004} for a detailed derivation of the cable equation, and extensions to the one-dimensional model that account for radial variation of potential.
- \end{center}
- \caption{Symbols and quantities.}
-\end{table*}
+The formulation in equations~\eq{eq:cable} and~\eq{eq:cable_balance} is based on the following expression in three dimensions (based on Maxwell's equations adapted for neurological modelling)
+\begin{equation}
+ \nabla \cdot \vv{J} = 0,
+\end{equation}
+where $\vv{J}$ is current density (units $A/m^2$).
+Current density is in turn defined in terms of electric field $\vv{E}$ (units $V/m$)
+\begin{equation}
+ \vv{J} = \sigma \vv{E},
+\end{equation}
+where $\sigma$ is the specific electrical conductivity of intra-cellular fluid (typically 3.3 $S/m$).
+The derivation of the cable equation is based on two assumptions:
+\begin{enumerate}
+ \item that charge disperion is effectively instantaneous for the purposes of dendritic modelling.
+ \item that diffusion of magnetic field is instant, i.e. it behaves quasi-statically in the sense that it is determined by the electric field through the Maxwell equations.
+\end{enumerate}
+Under these conditions, $\vv{E}$ is conservative, and as such can be expressed in terms of a potential field
+\begin{equation}
+ \vv{E} = \nabla \phi,
+\end{equation}
+where the extra/intra-cellular potential field $\phi$ has units $mV$.
+The derivation of the one-dimensional conservation equation \eq{eq:cable_balance} is based on the assumption that the intra-cellular potential (i.e. inside the cell) does not vary radially.
+That is, potential is a function of the axial distance $x$ alone
+\begin{equation}
+ \vv{E} = \nabla \phi = \pder{V}{x}.
+\end{equation}
+This is not really true, because a potential field that is a variable of $x$ and $t$ alone can't support the axial gradients required to drive the potential difference over the cell membrane.
diff --git a/docs/report.bbl b/docs/report.bbl
new file mode 100644
index 0000000000000000000000000000000000000000..bc1039fa6704025f2927c47d36620acf270bb044
--- /dev/null
+++ b/docs/report.bbl
@@ -0,0 +1,9 @@
+\begin{thebibliography}{1}
+
+\bibitem{lindsay_2004}
+K.~Lindsay, J.~Rosenberg, and G.~Tucker.
+\newblock From maxwell's equations to the cable equation and beyond.
+\newblock {\em Progress in Biophysics and Molecular Biology}, 85(1):71 -- 116,
+ 2004.
+
+\end{thebibliography}
diff --git a/docs/report.blg b/docs/report.blg
new file mode 100644
index 0000000000000000000000000000000000000000..34a47a81b90faf6b3fae66b78548023ed1450d4f
--- /dev/null
+++ b/docs/report.blg
@@ -0,0 +1,46 @@
+This is BibTeX, Version 0.99d (TeX Live 2015/Arch Linux)
+Capacity: max_strings=35307, hash_size=35307, hash_prime=30011
+The top-level auxiliary file: report.aux
+The style file: abbrv.bst
+Database file #1: bibliography.bib
+You've used 1 entry,
+ 2118 wiz_defined-function locations,
+ 504 strings with 3885 characters,
+and the built_in function-call counts, 383 in all, are:
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diff --git a/docs/report.tex b/docs/report.tex
index fdc2cca75baaec1ca7702cb73d4f493bb96ac5d5..f087be39ee12d6ab5d09d43fd500112a6dd489b6 100644
--- a/docs/report.tex
+++ b/docs/report.tex
@@ -22,6 +22,7 @@
\usepackage{xspace}
\usepackage{color}
+\usepackage{bm} % bold math
%----------------------------------------------------------------------------------------
% COLUMNS
@@ -58,20 +59,6 @@
% CUSTOM COMMANDS
%----------------------------------------------------------------------------------------
-\newcommand{\HWtwelve}{HW12\xspace}
-\newcommand{\HWtwelves}{HW12$^*$\xspace}
-\newcommand{\HWeight}{HW8\xspace}
-\newcommand{\HWsix}{HW6\xspace}
-\newcommand{\HWfour}{HW4\xspace}
-\newcommand{\SBeight}{SB8\xspace}
-
-\newcommand{\HWtwelvecode}{E5-2690v3\xspace}
-\newcommand{\HWtwelvescode}{E5-2680v3\xspace}
-\newcommand{\HWeightcode}{E5-1660v3\xspace}
-\newcommand{\HWsixcode}{E5-1650v3\xspace}
-\newcommand{\HWfourcode}{E5-1620v3\xspace}
-\newcommand{\SBeightcode}{E5-2670\xspace}
-
\newcommand{\todo}[1]{\textbf{\textcolor{blue}{TODO: #1}}} % add a comment to the article
\newcommand{\tbl}[1]{\textbf{Table \ref{#1}}\xspace}
@@ -80,6 +67,8 @@
\newcommand{\ssec}[1]{\textbf{\S\ref{#1}}\xspace}
\newcommand{\pder}[2]{\frac{\partial{#1}}{\partial{#2}}}
+\newcommand{\deriv}[1]{~\text{d}{#1}}
+\newcommand{\vv}[1]{\bm{#1}\xspace}
%----------------------------------------------------------------------------------------
% ARTICLE INFORMATION
@@ -101,8 +90,16 @@
\thispagestyle{empty} % Removes page numbering from the first page
%------------------------------------------------
+\section{Formulation}
\input{formulation.tex}
+\section{Symbols and Units}
+\input{symbols.tex}
+
+%*************************************************
+\bibliographystyle{abbrv}
+\bibliography{bibliography}
+%*************************************************
\end{document}
diff --git a/docs/symbols.tex b/docs/symbols.tex
new file mode 100644
index 0000000000000000000000000000000000000000..814b7d4a4ea907f8118be91820e200a1e611e393
--- /dev/null
+++ b/docs/symbols.tex
@@ -0,0 +1,43 @@
+\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
+ $J$ & $j$ & $J\cdot s^{-1}$, $V\cdot A$ &
+ $kg\cdot m^{2}\cdot s^{-2}$ \\
+
+ $q$ & $C$ & $s\cdot A$ &
+ $s\cdot A$ \\
+
+ $I$ & $A$ & $C\cdot s^{-1}$ &
+ $A$ \\
+
+ $V$ & $V$ & $W\cdot A$ &
+ $kg\cdot m^{2}\cdot s^{-3}\cdot A^{-1}$ \\
+
+ $R$ & $\Omega$ & $V\cdot A^{-1}$ &
+ $kg\cdot m^{2}\cdot s^{-3}\cdot A^{-2}$ \\
+
+ $C$ & $F$ & $C\cdot V^{-1}$ &
+ $kg^{-1}\cdot m^{-2}\cdot s^{4}\cdot A^{2}$ \\
+ \hline
+ \end{tabular}
+
+ \end{center}
+ \caption{Symbols and quantities.}
+\end{table*}
diff --git a/tree.hpp b/tree.hpp
index b67244c721d97ad0084df57612d09c62e6f55f5f..772eb86c99d55ac65b076ee8ab164d1ae0c6c675 100644
--- a/tree.hpp
+++ b/tree.hpp
@@ -188,7 +188,6 @@ class tree {
new_tree.init(num_nodes());
// add the root node
- new_tree.data_(memory::all) = -std::numeric_limits<int_type>::min();
new_tree.parents_[0] = -1;
new_tree.child_index_[0] = 0;