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Commit 8dbdde31 authored by Sam Yates's avatar Sam Yates
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...@@ -54,16 +54,16 @@ is the reversal potential. ...@@ -54,16 +54,16 @@ is the reversal potential.
\begin{table}[ht] \begin{table}[ht]
\centering \centering
\begin{tabular}{lSl} \begin{tabular}{lSl}
\toprule \toprule
Term & {Value} & Property\\ Term & {Value} & Property\\
\midrule \midrule
$d$ & \SI{1.0}{\um} & cable diameter \\ $d$ & \SI{1.0}{\um} & cable diameter \\
$L$ & \SI{1.0}{\mm} & cable length \\ $L$ & \SI{1.0}{\mm} & cable length \\
$R_A$ & \SI{1.0}{\ohm\m} & bulk axial resistivity \\ $R_A$ & \SI{1.0}{\ohm\m} & bulk axial resistivity \\
$R_M$ & \SI{4.0}{\ohm\m\squared} & areal membrane resistivity \\ $R_M$ & \SI{4.0}{\ohm\m\squared} & areal membrane resistivity \\
$C_M$ & \SI{0.01}{\F\per\m\squared} & areal membrane capacitance \\ $C_M$ & \SI{0.01}{\F\per\m\squared} & areal membrane capacitance \\
$E_M$ & \SI{-65.0}{\mV} & membrane reversal potential \\ $E_M$ & \SI{-65.0}{\mV} & membrane reversal potential \\
\bottomrule \bottomrule
\end{tabular} \end{tabular}
\caption{Cable properties for the Rallpack 1 model.} \caption{Cable properties for the Rallpack 1 model.}
\label{tbl:rallpack1} \label{tbl:rallpack1}
...@@ -87,9 +87,9 @@ and the linear membrane capacitance $c$. These determine $\lambda$ and $\tau$ by ...@@ -87,9 +87,9 @@ and the linear membrane capacitance $c$. These determine $\lambda$ and $\tau$ by
With the model boundary conditions, With the model boundary conditions,
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
v(x, 0) &= E, \\ v(x, 0) &= E, \\
\left.\frac{\partial v}{\partial x}\right\vert_{x=0} & = -Ir, \\ \left.\frac{\partial v}{\partial x}\right\vert_{x=0} & = -Ir, \\
\left.\frac{\partial v}{\partial x}\right\vert_{x=L} & = 0, \left.\frac{\partial v}{\partial x}\right\vert_{x=L} & = 0,
\end{align} \end{align}
\end{subequations} \end{subequations}
where $I$ is the injected current and $L$ is the cable length. where $I$ is the injected current and $L$ is the cable length.
...@@ -98,18 +98,18 @@ The solution $v(x, t)$ can be expressed in terms of the solution $g(x, t; L)$ ...@@ -98,18 +98,18 @@ The solution $v(x, t)$ can be expressed in terms of the solution $g(x, t; L)$
to a normalized version of the cable equation, to a normalized version of the cable equation,
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:normcable} \label{eq:normcable}
\frac{\partial^2 g}{\partial x^2} & = \frac{\partial^2 g}{\partial x^2} & =
\frac{\partial g}{\partial t} + g, \frac{\partial g}{\partial t} + g,
\\ \\
\label{eq:normcableinitial} \label{eq:normcableinitial}
g(x, 0) &= 0, g(x, 0) &= 0,
\\ \\
\label{eq:normcableleft} \label{eq:normcableleft}
\left.\frac{\partial g}{\partial x}\right\vert_{x=0} & = 1, \left.\frac{\partial g}{\partial x}\right\vert_{x=0} & = 1,
\\ \\
\label{eq:normcableright} \label{eq:normcableright}
\left.\frac{\partial g}{\partial x}\right\vert_{x=L} & = 0 \left.\frac{\partial g}{\partial x}\right\vert_{x=L} & = 0
\end{align} \end{align}
\end{subequations} \end{subequations}
by by
...@@ -158,7 +158,7 @@ and thus ...@@ -158,7 +158,7 @@ and thus
Consequently, Consequently,
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
G(x, s) &= \frac{1}{ms}\cdot\frac{e^{mx}+e^{2mL-mx}}{1-e^{2mL}}\\ G(x, s) &= \frac{1}{ms}\cdot\frac{e^{mx}+e^{2mL-mx}}{1-e^{2mL}}\\
&= - \frac{1}{ms}\cdot\frac{\cosh m(L-x)}{\sinh mL}. &= - \frac{1}{ms}\cdot\frac{\cosh m(L-x)}{\sinh mL}.
\end{aligned} \end{aligned}
\end{equation} \end{equation}
...@@ -199,10 +199,10 @@ arising from $m=\sqrt{1+s}$, as letting $m=-\sqrt{1+s}$ leaves $G$ unchanged. ...@@ -199,10 +199,10 @@ arising from $m=\sqrt{1+s}$, as letting $m=-\sqrt{1+s}$ leaves $G$ unchanged.
For $|s+1|>\epsilon$, For $|s+1|>\epsilon$,
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
|s^{3/2}G(x,s)|^2 |s^{3/2}G(x,s)|^2
& \leq (1+\epsilon)^{-1} \left| \frac{\cosh m(L-x)}{\sinh mL} \right|^2 & \leq (1+\epsilon)^{-1} \left| \frac{\cosh m(L-x)}{\sinh mL} \right|^2
\\ \\
& \leq (1+\epsilon)^{-1} (1+|\coth mL|)^2 & \leq (1+\epsilon)^{-1} (1+|\coth mL|)^2
\label{eq:gbounds} \label{eq:gbounds}
\end{aligned} \end{aligned}
\end{equation} \end{equation}
...@@ -218,8 +218,8 @@ Recalling $m=\sqrt{1+s}$, $m$ and $\sinh mL$ are non-zero in ...@@ -218,8 +218,8 @@ Recalling $m=\sqrt{1+s}$, $m$ and $\sinh mL$ are non-zero in
a neighbourhood of $s=0$, and so the pole is simple and a neighbourhood of $s=0$, and so the pole is simple and
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
\Res(G; 0) & = - \frac{1}{m}\cdot\left.\frac{\cosh m(L-x)}{\sinh mL}\right|_{s=0}\\ \Res(G; 0) & = - \frac{1}{m}\cdot\left.\frac{\cosh m(L-x)}{\sinh mL}\right|_{s=0}\\
& = - \frac{\cosh (L-x)}{\sinh L}. & = - \frac{\cosh (L-x)}{\sinh L}.
\end{aligned} \end{aligned}
\end{equation} \end{equation}
...@@ -235,9 +235,9 @@ Let $G(x,s)=f(x,s)/h(s)$, where ...@@ -235,9 +235,9 @@ Let $G(x,s)=f(x,s)/h(s)$, where
Noting that $dm/ds = \frac{1}{2}m^{-1}$, Noting that $dm/ds = \frac{1}{2}m^{-1}$,
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
h'(s) &= \frac{1}{2}m^{-1}\sinh mL + \frac{1}{2}L\cosh mL \\ h'(s) &= \frac{1}{2}m^{-1}\sinh mL + \frac{1}{2}L\cosh mL \\
&= \frac{1}{2}L + \frac{1}{2}L + O(m^2) \quad(m\to 0) \\ &= \frac{1}{2}L + \frac{1}{2}L + O(m^2) \quad(m\to 0) \\
&= L + O(s+1) \quad(s\to -1). &= L + O(s+1) \quad(s\to -1).
\label{eq:hprime} \label{eq:hprime}
\end{aligned} \end{aligned}
\end{equation} \end{equation}
...@@ -255,11 +255,11 @@ $m_k$ is non-zero for $k\geq 1$ and ...@@ -255,11 +255,11 @@ $m_k$ is non-zero for $k\geq 1$ and
Consequently the pole is simple and Consequently the pole is simple and
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
\Res(G; s_k) \Res(G; s_k)
& = f(x, s_k)/h'(s_k)\\ & = f(x, s_k)/h'(s_k)\\
& = -\frac{2}{s_k L}\frac{\cosh m_k(L-x)}{\cosh m_kL} \\ & = -\frac{2}{s_k L}\frac{\cosh m_k(L-x)}{\cosh m_kL} \\
& = -\frac{2}{s_k L}\frac{\cosh m_kL\cosh m_kx-\sinh m_kL\sinh m_kL}{\cosh m_kL} \\ & = -\frac{2}{s_k L}\frac{\cosh m_kL\cosh m_kx-\sinh m_kL\sinh m_kL}{\cosh m_kL} \\
& = -\frac{2}{s_k L}\cosh m_k x, & = -\frac{2}{s_k L}\cosh m_k x,
\end{aligned} \end{aligned}
\end{equation} \end{equation}
as $\sinh m_k=0$. as $\sinh m_k=0$.
...@@ -272,7 +272,7 @@ In terms of $a_k$, ...@@ -272,7 +272,7 @@ In terms of $a_k$,
The series exapnsion for $g(x, t)$ therefore is The series exapnsion for $g(x, t)$ therefore is
\begin{equation} \begin{equation}
g(x, t) = -\frac{\cosh(L-x)}{\sinh L} + \frac{1}{L}e^{-t}\left\{ g(x, t) = -\frac{\cosh(L-x)}{\sinh L} + \frac{1}{L}e^{-t}\left\{
1+2\sum_{k=1}^\infty \frac{e^{-ta_k^2}}{1+a_k^2}\cos a_k x\right\}. 1+2\sum_{k=1}^\infty \frac{e^{-ta_k^2}}{1+a_k^2}\cos a_k x\right\}.
\label{eq:theg} \label{eq:theg}
\end{equation} \end{equation}
...@@ -285,16 +285,16 @@ of stopping criteria for a given tolerance. ...@@ -285,16 +285,16 @@ of stopping criteria for a given tolerance.
Let $g_n$ be the partial sum Let $g_n$ be the partial sum
\begin{equation} \begin{equation}
g_n(x, t) = -\frac{\cosh(L-x)}{\sinh L} + \frac{1}{L}e^{-t}\left\{ g_n(x, t) = -\frac{\cosh(L-x)}{\sinh L} + \frac{1}{L}e^{-t}\left\{
1+2\sum_{k=1}^n \frac{e^{-ta_k^2}}{1+a_k^2}\cos a_k x\right\}. 1+2\sum_{k=1}^n \frac{e^{-ta_k^2}}{1+a_k^2}\cos a_k x\right\}.
\end{equation} \end{equation}
so that $g(x, t) =\lim_{n\to\infty} g_n(x,t)$. Let $\bar{g}_n = |g-g_n|$ be the so that $g(x, t) =\lim_{n\to\infty} g_n(x,t)$. Let $\bar{g}_n = |g-g_n|$ be the
residual. The $a_k$ form an increasing sequence, so residual. The $a_k$ form an increasing sequence, so
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
\bar{g}_n(x,t) \bar{g}_n(x,t)
& \leq \frac{2}{L}e^{-t}\sum_{n+1}^\infty\frac{e^{-ta_k^2}}{1+a_k^2}\\ & \leq \frac{2}{L}e^{-t}\sum_{n+1}^\infty\frac{e^{-ta_k^2}}{1+a_k^2}\\
& \leq \frac{2}{L}e^{-t}\int_{a_n}^\infty \frac{e^{-tu^2}}{1+u^2}\,du\\ & \leq \frac{2}{L}e^{-t}\int_{a_n}^\infty \frac{e^{-tu^2}}{1+u^2}\,du\\
& < \frac{2}{L}e^{-t}\int_{a_n}^\infty \frac{e^{-tu^2}}{u^2}\,du. & < \frac{2}{L}e^{-t}\int_{a_n}^\infty \frac{e^{-tu^2}}{u^2}\,du.
\end{aligned} \end{aligned}
\label{eq:gbar} \label{eq:gbar}
\end{equation} \end{equation}
...@@ -313,9 +313,9 @@ For real $\alpha<1$ and $z>0$, \textcite[][Theorem 2.3]{borwein2009} give the up ...@@ -313,9 +313,9 @@ For real $\alpha<1$ and $z>0$, \textcite[][Theorem 2.3]{borwein2009} give the up
Substituting into \eqref{eq:gbar} gives Substituting into \eqref{eq:gbar} gives
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
\bar{g}_n(x,t) \bar{g}_n(x,t)
& < \frac{1}{L}e^{-t}\sqrt{t}\,\Gamma(-\frac{1}{2},a_n^2 t) \\ & < \frac{1}{L}e^{-t}\sqrt{t}\,\Gamma(-\frac{1}{2},a_n^2 t) \\
& \leq \frac{1}{L}e^{-t}\sqrt{t}\,(a_n^2 t)^{-\frac{3}{2}}e^{-a_n^2t} \\ & \leq \frac{1}{L}e^{-t}\sqrt{t}\,(a_n^2 t)^{-\frac{3}{2}}e^{-a_n^2t} \\
& = \frac{e^{-t(1+a_n^2)}}{L t a_n^3}. & = \frac{e^{-t(1+a_n^2)}}{L t a_n^3}.
\end{aligned} \end{aligned}
\end{equation} \end{equation}
......
...@@ -38,7 +38,7 @@ foreach(target ${TARGETS}) ...@@ -38,7 +38,7 @@ foreach(target ${TARGETS})
) )
if(BUILD_VALIDATION_DATA) if(BUILD_VALIDATION_DATA)
add_dependencies(${target} validation_data) add_dependencies(${target} validation_data)
endif() endif()
endforeach() endforeach()
...@@ -31,10 +31,10 @@ function(add_validation_data) ...@@ -31,10 +31,10 @@ function(add_validation_data)
set(out "${VALIDATION_DATA_DIR}/${ADD_VALIDATION_DATA_OUTPUT}") set(out "${VALIDATION_DATA_DIR}/${ADD_VALIDATION_DATA_OUTPUT}")
string(REGEX REPLACE "([^;]+)" "${CMAKE_CURRENT_SOURCE_DIR}/\\1" deps "${ADD_VALIDATION_DATA_DEPENDS}") string(REGEX REPLACE "([^;]+)" "${CMAKE_CURRENT_SOURCE_DIR}/\\1" deps "${ADD_VALIDATION_DATA_DEPENDS}")
add_custom_command( add_custom_command(
OUTPUT "${out}" OUTPUT "${out}"
DEPENDS ${deps} DEPENDS ${deps}
WORKING_DIRECTORY "${CMAKE_CURRENT_SOURCE_DIR}" WORKING_DIRECTORY "${CMAKE_CURRENT_SOURCE_DIR}"
COMMAND ${ADD_VALIDATION_DATA_COMMAND} > "${out}") COMMAND ${ADD_VALIDATION_DATA_COMMAND} > "${out}")
# Cmake, why can't we just write add_dependencies(validation_data "${out}")?! # Cmake, why can't we just write add_dependencies(validation_data "${out}")?!
make_unique_target_name(ffs_cmake "${out}") make_unique_target_name(ffs_cmake "${out}")
......
...@@ -11,7 +11,7 @@ set(models ...@@ -11,7 +11,7 @@ set(models
foreach(model ${models}) foreach(model ${models})
set(script "${model}.py") set(script "${model}.py")
add_validation_data( add_validation_data(
OUTPUT "neuron_${model}.json" OUTPUT "neuron_${model}.json"
DEPENDS "${script}" "nrn_validation.py" DEPENDS "${script}" "nrn_validation.py"
COMMAND ${NRNIV_BIN} -nobanner -python "${script}") COMMAND ${NRNIV_BIN} -nobanner -python "${script}")
endforeach() endforeach()
......
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