### Increased computational accuracy in multi-compartmental cable models (Lindsay et al. 2005)

Accession:129149
Compartmental models of dendrites are the most widely used tool for investigating their electrical behaviour. Traditional models assign a single potential to a compartment. This potential is associated with the membrane potential at the centre of the segment represented by the compartment. All input to that segment, independent of its location on the segment, is assumed to act at the centre of the segment with the potential of the compartment. By contrast, the compartmental model introduced in this article assigns a potential to each end of a segment, and takes into account the location of input to a segment on the model solution by partitioning the effect of this input between the axial currents at the proximal and distal boundaries of segments. For a given neuron, the new and traditional approaches to compartmental modelling use the same number of locations at which the membrane potential is to be determined, and lead to ordinary differential equations that are structurally identical. However, the solution achieved by the new approach gives an order of magnitude better accuracy and precision than that achieved by the latter in the presence of point process input.
Reference:
1 . Lindsay AE, Lindsay KA, Rosenberg JR (2005) Increased computational accuracy in multi-compartmental cable models by a novel approach for precise point process localization. J Comput Neurosci 19:21-38 [PubMed]
Model Information (Click on a link to find other models with that property)
 Model Type: Neuron or other electrically excitable cell; Brain Region(s)/Organism: Cell Type(s): Channel(s): I Na,t; I K; Gap Junctions: Receptor(s): Gene(s): Transmitter(s): Simulation Environment: NEURON; C or C++ program; Model Concept(s): Methods; Implementer(s):
Search NeuronDB for information about:  I Na,t; I K;
 / LindsayEtAl2005 readme.txt 03-192.pdf AnalyseResults.c BitsAndPieces.c CellData.dat CompareSpikeTrain.c Ed04.tex ExactSolution.dat GammaCode Gen.tex Gen1.tex Gen2.tex Gen3.tex Gen4.tex Gen5.tex Gen6.tex GenCom.c GenCom1.c GenCom2.c GenComExactSoln.c GenerateInput.c GenerateInputText.c GenRan.ran GetNodeNumbers.c Info100.dat Info20.dat Info200.dat Info30.dat Info300.dat Info40.dat Info400.dat Info50.dat Info500.dat Info60.dat Info70.dat Info80.dat Info90.dat InputCurrents.dat InputDendrite.dat JaySpikeTrain.c JayTest1.dat JayTest100.dat KenSpikeTrain.c KenTest1.dat * KenTest10.dat KenTest100.dat * KenTest10p.dat KenTest1p.dat * KenTest2.dat KenTest2p.dat KenTest3.dat KenTest3p.dat KenTest4.dat KenTest4p.dat KenTest5.dat KenTest5p.dat KenTest6.dat KenTest6p.dat KenTest7.dat KenTest7p.dat KenTest8.dat KenTest8p.dat KenTest9.dat KenTest9p.dat LU.c Mean50.dat Mean500.dat mosinit.hoc NC.pdf NC.tex NC1.tex NC2.tex NC3.tex NC4.tex NC5.tex NC6.tex NCFig2.eps * NCFig3.eps * NCFig4.eps * NCFig5a.eps * NCFig5b.eps * NCFig6.eps * NCPics.tex NeuronDriver.hoc NewComExactSoln.c NewComp.pdf NewComp.ps NewComp.tex NewComp.toc NewComp1.tex NewComp2.tex NewComp3.tex NewComp4.tex NewComp5.tex NewComp6.tex NewCompFig1.eps NewCompFig2.eps * NewCompFig3.eps * NewCompFig4.eps * NewCompFig5a.eps * NewCompFig5b.eps * NewCompFig6.eps * NewCompPics.tex NewComSpikeTrain.c NewRes.dat NewRes60.dat NewRes70.dat NewRes80.dat NewSynRes40.dat NewTestCell.d3 NResults.res OldComExactSoln.c out.res principles_01.tex rand Ratio.dat RelErr.dat ReviewOfSpines.pdf SpikeTimes.dat TestCell.d3 TestCell1.d3 TestCell2.d3 TestCell3.d3 TestCell4.d3 testcellnew2.hoc TestCGS.c TestGen1.c TestSim.hoc TestSim020.hoc TestSim030.hoc TestSim040.hoc TestSim050.hoc TestSim060.hoc TestSim070.hoc TestSim080.hoc TestSim090.hoc TestSim1.hoc TestSim100.hoc TestSim200.hoc TestSim300.hoc TestSim400.hoc TestSim500 TestSim500.hoc
\section{The model neuron}
Central to the comparison of the accuracy of the traditional and
generalised compartmental models is the construction of a typical
branched neuron for which the mathematical model has a closed form
expression for the membrane potential in response to input. This
solution then stands as a reference against which the performance
of the traditional and generalised compartmental models may be
measured. The most effective way to construct a branched model
neuron with a closed form solution for the membrane potential is
to choose the radii and lengths of its sections such that the Rall
conditions for an equivalent cylinder are satisfied (Rall,
\cite{Rall64}). The Rall conditions require that at any branch
point the sum of the three-halves power of the diameters of the
child limbs is equal to the three-halves power of the parent limb,
and that the total electrotonic length from a branch point to
dendritic tip is independent of path. In particular, the
electrotonic distance from soma-to-tip is independent of path. The
model neuron illustrated in Figure \ref{TestNeuron} satisfies
these conditions. When the Rall conditions are satisfied, the
effect at the soma of any configuration of input on the branched
model of the neuron is identical to the effect at the soma of the
unbranched equivalent cylinder with biophysical properties and
configuration of input determined uniquely from those of the
original branched neuron (Lindsay \emph{et al.},
\cite{Lindsay03}).

\begin{figure}[!h]
$\begin{array}{c} \begin{mfpic}[1][1]{0}{220}{-20}{220} \pen{2pt} \dotsize=1pt \dotspace=3pt \lines{(-5,100),(5,110),(15,100),(5,90),(-5 ,100)} % Upper dendrite % Root branch \dotted\lines{(5,115),(15,170),(20,170)} \lines{(20.0,160),(36.7,160)} \tlabel[tc](28.4,150){\textsf{(a)}} % Level 1 \lines{(50.0,190),(88.3,190)} \tlabel[bc](75,200){\textsf{(c)}} \lines{(50.0,130),(91.0,130)} \tlabel[tc](75,120){\textsf{(d)}} \dotted\lines{(36.7,160),(45,200),(55,200)} \dotted\lines{(36.7,160),(45,120),(55,120)} % Level 2 \lines{(100.0,210),(153.2,210)} \lines{(100.0,190),(153.2,190)} \lines{(100.0,170),(153.2,170)} \tlabel[cl](160,210){\textsf{(g)}} \tlabel[cl](160,190){\textsf{(g)}} \tlabel[cl](160,170){\textsf{(g)}} \dotted\lines{(88.3,190),(95,220),(105,220)} \dotted\lines{(88.3,190),(95,160),(105,160)} \lines{(100.0,140),(165.1,140)} \lines{(100.0,120),(165.1,120)} \dotted\lines{(91.0,130),(95,150),(105,150)} \dotted\lines{(91.0,130),(95,110),(105,110)} \tlabel[cl](175,140){\textsf{(h)}} \tlabel[cl](175,120){\textsf{(h)}} % % Lower dendrite % Root branch \lines{(20.0,40),(58.0,40)} \dotted\lines{(5,85),(15,30),(25,30)} \tlabel[bc](39,50){\textsf{(b)}} % Level 1 \lines{(70.0,70),(133.1,70)} \lines{(70.0,10),(127.1,10)} \dotted\lines{(58,40),(66.5,80),(76.5,80)} \dotted\lines{(58,40),(66.5,0),(76.5,0)} \tlabel[bc](105,80){\textsf{(e)}} \tlabel[tc](105,0){\textsf{(f)}} % Level 2 \lines{(145,80),(195.1,80)} \lines{(145,60),(195.1,60)} \dotted\lines{(133.1,70),(140,90),(150,90)} \dotted\lines{(133.1,70),(140,50),(150,50)} \tlabel[cl](205,80){\textsf{(i)}} \tlabel[cl](205,60){\textsf{(i)}} \lines{(140,30),(179.6,30)} \lines{(140,10),(179.6,10)} \lines{(140,-10),(179.6,-10)} \dotted\lines{(127.1,10),(134,40),(144,40)} \dotted\lines{(127.1,10),(134,-20),(144,-20)} \tlabel[cl](190,30){\textsf{(j)}} \tlabel[cl](190,10){\textsf{(j)}} \tlabel[cl](190,-10){\textsf{(j)}} \end{mfpic} \end{array}\qquad \begin{array}{ccc} \hline \mbox{Section} & \mbox{Length }\mu\mbox{m} & \mbox{Diameter }\mu\mbox{m}\\[2pt] \hline (a) & 166.809245 & 7.089751 \\ (b) & 379.828386 & 9.189790 \\ (c) & 383.337494 & 4.160168 \\ (d) & 410.137845 & 4.762203 \\ (e) & 631.448520 & 6.345604 \\ (f) & 571.445800 & 5.200210 \\ (g) & 531.582750 & 2.000000 \\ (h) & 651.053246 & 3.000000 \\ (i) & 501.181023 & 4.000000 \\ (j) & 396.218388 & 2.500000 \\ \hline \end{array}$
\centering
\parbox{5in}{\caption{\label{TestNeuron} A branched neuron
satisfying the Rall conditions. The radii and lengths of the
dendritic section are given in the right hand panel of the figure.
A each branch point, $l/\sqrt{r}$ is fixed for all children of the
branch point.}}
\end{figure}

To guarantee that any apparent errors between the closed form
solution and the numerical solution are not due to the lack of
precision with which the branched dendrite is represented as an
equivalent cylinder, a high degree of accuracy is used in the
specification of dendritic radii and section lengths in the model
neuron. The model neuron illustrated in Figure \ref{TestNeuron}
will be assumed to have a dendritic membrane of specific
conductance $g_\mathrm{M}=0.091$ mS/cm$^2$ and specific
capacitance $1.0\mu$F/cm$^2$, and an intracellular medium of
conductance $g_\mathrm{A}=14.286$ mS/cm. With these biophysical
properties, the equivalent cylinder has length one electrotonic
unit. The soma of the test dendrite is assumed to have a membrane
of area $A_\mathrm{S}$, specific conductance $g_\mathrm{S}=g_\mathrm{M}$ and
specific capacitance $c_\mathrm{S}=c_\mathrm{M}$.

\subsection{Closed form solution for the equivalent cylinder}
The first step in the assessment of the performance of both
compartmental models requires the construction of the closed form
solution for the Rall equivalent cylinder under the action of
input on the cylinder and at its soma.

Consider a uniform cylindrical dendrite of radius $r$ and length
$l$ attached to a soma of area $A_\mathrm{S}$ at its left hand end
which is taken to be the point $x=0$ of a coordinate system with
axis oriented along the length of the dendrite. If $V(x,t)$ is the
deviation of the transmembrane potential from its resting value at
point $x$ and time $t>0$, then $V(x,t)$ satisfies the cable equation
$$\label{es1} \ds 2\pi r\Big(c_\mathrm{M}\frac{\partial V}{\partial t} +g_\mathrm{M}V\Big)=\pi r^2 g_\mathrm{M}\frac{\partial^2V} {\partial x^2}-I(x,t)\,,\quad (x,t)\in(0,l)\times(0,\infty)$$
where $c_\mathrm{M}$, $g_\mathrm{M}$ and $g_\mathrm{A}$ have their
usual meanings and $I(x,t)$ is the linear density of exogenous
current along the dendrite. A solution of equation (\ref{es1}) is
sought satisfying the initial condition $V(x,0)=0$ and the
boundary conditions
$$\label{es2} A_\mathrm{S}\Big(c_\mathrm{M}\,\frac{\partial V(0,t)}{\partial t} +g_\mathrm{M}V(0,t)\Big)=\pi r^2g_\mathrm{A}\, \frac{\partial V(0,t)}{\partial x}-I_\mathrm{S}(t)\,, \qquad\frac{\partial V(l,t)}{\partial x}=0\,,$$
in which it has been recognised that the somal and dendritic
membranes have identical specific capacitances and conductances.
Prior to describing the critical steps in the construction of the
exact mathematical solution to the initial boundary value problem
posed by equations (\ref{es1}) and (\ref{es2}), it is convenient
to introduce the well-known non-dimensional electrotonic length
$$\label{es3} L=l\,\sqrt{\ds\frac{2 g_\mathrm{M}}{r g_\mathrm{A}}}\,.$$
The required solution is now constructed by observing that the
series
$$\label{es4} V(x,t)=e^{-t/\tau}\,\Big[\,\phi_0(t)+\sum_\beta\;\phi_\beta(t) e^{-\beta^2 t/L^2\tau}\,\cos{\beta(1-x/l)}\,\Big]\,, \qquad\tau=\frac{c_\mathrm{M}}{g_\mathrm{M}}\,,$$
satisfies the gradient boundary condition at $x=l$ for all values
of $\beta$ and functions $\phi_0(t)$ and $\phi_\beta(t)$, and will
also satisfy the initial condition $V(x,0)=0$ provided
$\phi_0(0)=\phi_\beta(0)=0$. This series solution for $V(x,t)$
also satisfies the partial differential equation (\ref{es1})
provided
$$\label{es4} \ds \frac{d\phi_0}{dt}+\sum_\beta\,\frac{d\phi_\beta}{dt} \,e^{-\beta^2\,t/L^2\tau}\,\cos{\beta(1-x/l)} =-\frac{I(x,t)e^{t/\tau}}{2\pi r c_\mathrm{M}}$$
and the boundary condition at the soma provided
$$\label{es5} \frac{d\phi_0}{dt}+\sum_\beta\,\Big[\,\frac{d\phi_\beta}{dt}\, \cos\beta-\frac{\beta\cos\beta}{\gamma\tau\,L^2}\,\Big(\, \gamma\beta+\tan\beta\,\Big)\,\phi_\beta\,\Big] \,e^{-\beta^2\,t/L^2\tau}= -\frac{I_\mathrm{S}(t)}{A_\mathrm{S} c_\mathrm{M}}\; e^{t/\tau}$$
where $\gamma=A_\mathrm{S}/2\pi r l$, that is, $\gamma$ is the
ratio of the membrane surface area of the soma to the membrane
surface area of the dendrite. Equation (\ref{es5}) suggests that
the values of $\beta$ in expression (\ref{es4}) should be chosen
to be the zeros of the transcendental equation
$$\label{es6} \tan\beta+\gamma\beta = 0\,.$$
With this choice for the values of $\beta$, equations (\ref{es4})
and (\ref{es5}) take the simplified form
$$\label{es7} \begin{array}{rcl} \ds \frac{d\phi_0}{dt}+\sum_\beta\,\frac{d\phi_\beta}{dt} \,e^{-\beta^2\,t/L^2\tau}\,\cos{\beta(1-x/l)} & = & -\ds\frac{I(x,t)e^{t/\tau}}{2\pi r c_\mathrm{M}}\,, \\[12pt] \ds\frac{d\phi_0}{dt}+\sum_\beta\,\frac{d\phi_\beta}{dt}\, \cos\beta\,e^{-\beta^2\,t/L^2\tau} & = & - \ds\frac{I_\mathrm{S}(t)}{A_\mathrm{S}c_\mathrm{M}} \; e^{t/\tau}\,. \end{array}$$
The coefficients $\phi_0(t)$ and $\phi_\beta(t)$ are determined
from equations (\ref{es7}) by two different procedures. To find
$\phi_0(t)$, the first of equations (\ref{es7}) is integrated over
$(0,l)$ to obtain
$$\label{es8} l\,\frac{d\phi_0}{dt}-\gamma\,l\,\sum_\beta\,\frac{d\phi_\beta}{dt} \,e^{-\beta^2\,t/L^2\tau}\,\cos\beta = -\frac{e^{t/\tau}}{2\pi r c_\mathrm{M}}\,\int_0^l\,I(x,t)\,dx\,.$$
The summation in this expression is now eliminated using the
second of equations (\ref{es7}) to get
$$\label{es9} \frac{d\phi_0}{dt}= -\frac{e^{t/\tau}} {(2\pi r l+A_\mathrm{S})c_\mathrm{M}}\,\Big[\, I_\mathrm{S}(t)+\int_0^l\,I(x,t)\,dx\,\Big]\,.$$
Note that $(2\pi r l+A_\mathrm{S})c_\mathrm{M}$ is simply the
total membrane capacitance of the dendrite and soma. The
coefficient $\phi_0(t)$ is obtained by integrating equation
(\ref{es9}) with respect to time with the initial condition
$\phi_0(0)=0$ in the simulations to carried out here.

The procedure to find $\phi_\beta(t)$ begins by subtracting
equations (\ref{es7}) to get
$$\label{es10} \sum_\beta\,\frac{d\phi_\beta}{dt} \,e^{-\beta^2\,t/L^2\tau}\,\Big(\,\cos{\beta(1-x/l)} -\cos\beta\,\Big) = \frac{e^{t/\tau}}{2\pi r c_\mathrm{M}} \,\Big[\,\frac{I_\mathrm{S}(t)}{\gamma\,l}-I(x,t)\,\Big]\,.$$
Further progress is based on the observation that if $\alpha$ and
$\beta$ are solutions of equation (\ref{es6}) then
$$\label{es11} \int_0^l\,\cos\alpha\big(1-x/l\big)\big(\cos\beta\big(1-x/l\big) -\cos\beta\,\big)\,dx= \left[\begin{array}{cc} 0 & \alpha\ne\beta\,, \\[10pt] \ds\frac{l(1+\gamma\cos^2\alpha)}{2} & \alpha=\beta\,. \end{array}\right.$$
Equation (\ref{es10}) is multiplied by $\cos\alpha\big(1-x/l\big)$
and integrated over $[0,l]$ with respect to $x$ to obtain
$$\label{es12} \frac{d\phi_\alpha}{dt}=-\frac{2e^{(1+\alpha^2/L^2)/\tau}} {(2\pi r l+A_\mathrm{S}\cos^2\alpha)c_\mathrm{M}}\,\Big[\, \int_0^1\,I(x,t)\cos\alpha\big(1-x/l\big)\,dx +\cos\alpha\,I_\mathrm{S}(t)\,\Big]\,.$$
The coefficient $\phi_\alpha(t)$ is obtained by integrating
equation (\ref{es12}) with respect to time with the initial
condition $\phi_\alpha(0)=0$ in the simulations to be carried out
here. Once $\phi_0(t)$ and $\phi_\alpha(t)$ are known, the
potential is determined everywhere from expression (\ref{es4}).

\subsubsection{Computation of current on equivalent cylinder}
The special case in which the current $I(x,t)$ is constructed from
a series of point currents $I_1, \cdots,I_n$ at locations
$x_1,\cdots x_n$ from the soma of the equivalent cable is
particularly useful for the investigation of large scale synaptic
activity. In this case
$$\label{ec1} I(x,t)=\sum_{k=1}^n\;I_k(t)\delta(x-x_k)$$
and the corresponding coefficient functions $\phi_0$ and
$\phi_\alpha$ satisfy
$$\label{ec2} \begin{array}{rcl} \ds\frac{d\phi_0}{dt} & = &\ds -\frac{e^{t/\tau}} {(2\pi r l+A_\mathrm{S})c_\mathrm{M}}\,\Big[\, I_\mathrm{S}(t)+\sum_{k=1}^n\;I_k(t)\,\Big]\,,\\[10pt] \ds\frac{d\phi_\alpha}{dt} & = & \ds-\frac{2e^{(1+\alpha^2/L^2)/\tau}} {(2\pi r l+A_\mathrm{S}\cos^2\alpha)c_\mathrm{M}}\,\Big[\, \sum_{k=1}^n \;I_k(t)\cos\alpha\big(1-x_k/l\big) +\cos\alpha\,I_\mathrm{S}(t)\,\Big]\,. \end{array}$$
If $X_k$ is the electrotonic distance of the input $I_k(t)$ at
distance $x_k$ from the soma of the equivalent cylinder, then
$I_k(t)$ is the sum of the exogenous current inputs to the
branched neuron taken across all dendritic sections at
electrotonic distance $X_k$ from the soma.