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

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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{Introduction}
The traditional compartmental model of a neuron, in which the
behaviour of its complexly branched dendrites is described by the
solution of a family of ordinary differential equations, was
originally developed as a means of reducing the mathematical
complexity associated with the continuum description of a neuron
in terms of a family of connected partial differential equations
(Rall,\cite{Rall64}). The question posed in this project is
whether or not the accuracy of the traditional compartmental model
of a dendrite can be significantly improved without additional
computational effort. Toward this end, a generalised compartmental
model is developed and its computational properties examined with
reference to the behaviour of a test neuron with known exact
solution for given input. To facilitate this development, the
terminology used to describe the morphology of a neuron and its
input/output structure is now set out.

\begin{figure}[!h]
%THE FULLY IDEALISED NEURON
\centerline{\begin{mfpic}[0.8][0.8]{10}{210}{-80}{270}
\pen{1pt}
% Draws the dendrite
\shade\cyclic{(80,14),(85,15),(93,10),(100,7),(110,5),(110,3),(100,-3),(90,-5),(80,14)}
\curve{(80,14),(85,15),(93,10),(100,7),(110,5),(110,3),(100,-3),(90,-5),(80,14)}
\gfill\cyclic{(90,-5),(100,-2),(90,3),(85,1),(90,-5)}
\curve{(110,4),(115,10),(120,15),(125,19),(130,22),
(135,24),(140,25),(145,24),(150,22),(155,19)}
\curve{(135,24),(140,27),(145,32),(150,39),(155,48)}
\curve{(125,19),(120,30),(123,50),(125,80),(128,100),(126,120),(124,140),(128,160),(136,180),(148,200)}
\curve{(125,80),(130,85),(135,95),(140,110),(145,130)}
\curve{(124,140),(122,150),(120,158),(118,164)}
\curve{(80,14),(75,15),(70,17),(65,19.5),(60,23),(55,30)}
\curve{(80,14),(85,30),(90,50),(80,70),(70,80),(60,90),(50,110),(40,140),(5,190)}
\curve{(50,110),(60,125),(70,160),(80,195),(115,235)}
\curve{(80,70),(85,80),(90,100),(95,140)}
\curve{(85,30),(75,35),(65,45),(50,50),(40,65)}
% Draw axon
\pen{2pt}
\curve{(90,-5),(95,-20),(93,-30),(92,-40)}
\pen{1pt}
\lines{(118,-7),(118,-10),(143,-10),(143,-7)}
\tlabel[cc](130,-18){\footnotesize $100\;\mu$m}
\headlen7pt
% Deals with nodes
\tlabel[bc](180,170){\footnotesize branch}
\tlabel[bc](180,160){\footnotesize point}
\lines{(180,155),(165,140)}
\arrow\lines{(165,140),(129,140)}
\tlabel[bc](180,110){\footnotesize branch}
\tlabel[bc](180,100){\footnotesize point}
\lines{(180,95),(165,80)}
\arrow\lines{(165,80),(130,80)}
% Deals with terminal
\tlabel[bc](190,215){\footnotesize terminals}
\lines{(190,210),(180,200)}
\lines{(190,225),(180,235)}
\arrow\lines{(180,235),(120,235)}
\arrow\lines{(180,200),(153,200)}
% Deals with segments
\tlabel[bc](190,25){\footnotesize sections}
\lines{(190,35),(170,55)}
\arrow\lines{(170,55),(130,55)}
\lines{(190,20),(180,10)}
\arrow\lines{(180,10),(120,10)}
% Deals with Soma
\tlabel[bc](25,15){\footnotesize Soma showing}
\tlabel[bc](25,5){\footnotesize trigger zone}
\lines{(25,0),(30,-5)}
\arrow\lines{(30,-5),(80,-5)}
% Deals with Axon
\tlabel[cr](60,-25){\footnotesize Axon}
\arrow\lines{(65,-25),(90,-25)}
% Somal spike train
\curve{(30,-70),(39.2,-70),(40,-68),(40.2,-66),(40.4,-64),
(40.6,-62),(40.8,-60),(41,-58),(41.2,-56),(41.4,-54),
(41.6,-52),(41.8,-50),(42,-49),(43,-60),(44,-71),(45,-70)}
\lines{(45,-70),(49.2,-70)}
\curve{(49.2,-70),(50,-68),(50.2,-66),(50.4,-64),(50.6,-62),(50.8,-60),
(51,-58),(51.2,-56),(51.4,-54),(51.6,-52), (51.8,-50),(52,-49),(53,-60),
(54,-71),(55,-70)}
\lines{(55,-70),(69.2,-70)}
\curve{(69.2,-70),(70,-68),(70.2,-66),(70.4,-64),(70.6,-62),(70.8,-60),
(71,-58),(71.2,-56),(71.4,-54),(71.6,-52), (71.8,-50),(72,-49),(73,-60),
(74,-71),(75,-70)}
\lines{(75,-70),(83.2,-70)}
\curve{(83.2,-70),(84,-68),(84.2,-66),(84.4,-64),(84.6,-62),(84.8,-60),
(85,-58),(85.2,-56),(85.4,-54),(85.6,-52), (85.8,-50),(85,-49),(86,-60),
(87,-71),(88,-70)}
\lines{(88,-70),(114.2,-70)}
\curve{(114.2,-70),(115,-68),(115.2,-66),(115.4,-64),(115.6,-62),
(115.8,-60),(116,-58),(116.2,-56),(116.4,-54),(116.6,-52),
(116.8,-50),(117,-49),(118,-60),(119,-71),(120,-70)}
\lines{(120,-70),(130,-70)}
% Input spike train
\curve{(10,250),(19.2,250),(20,252),(20.2,254),(20.4,256),(20.6,258),
(20.8,260),(21,262),(21.2,264),(21.4,266),(21.6,268), (21.8,270),
(22,271),(23,260),(24,249),(25,250)}
\curve{(25,250),(29.2,250),(30,252),(30.2,254),(30.4,256),(30.6,258),
(30.8,260),(31,262),(31.2,264),(31.4,266),(31.6,268),(31.8,270),
(32,271),(33,260),(34,249),(35,250)}
\curve{(35,250),(39.2,250),(40,252),(40.2,254),(40.4,256),(40.6,258),
(40.8,260),(41,262),(41.2,264),(41.4,266),(41.6,268), (41.8,270),(42,271),(43,260),(44,249),(45,250)}
\lines{(45,250),(49.2,250)}
\curve{(49.2,250),(50,252),(50.2,254),(50.4,256),(50.6,258),
(50.8,260),(51,262),(51.2,264),(51.4,266),(51.6,268), (51.8,270),(52,271),(53,260),(54,249),(55,250)}
\lines{(55,250),(69.2,250)}
\curve{(69.2,250),(70,252),(70.2,254),(70.4,256),(70.6,258),
(70.8,260),(71,262),(71.2,264),(71.4,266),(71.6,268), (71.8,270),(72,271),(73,260),(74,249),(75,250)}
\lines{(75,250),(83.2,250)}
\curve{(83.2,250),(84,252),(84.2,254),(84.4,256),(84.6,258),
(84.8,260),(85,262),(85.2,264),(85.4,266),(85.6,268), (85.8,270),(85,271),(86,260),(87,249),(88,250)}
\lines{(88,250),(99.2,250)}
\curve{(99.2,250),(100,252),(100.2,254),(100.4,256),(100.6,258),
(100.8,260),(101,262),(101.2,264),(101.4,266),(101.6,268),
(101.8,270),(102,271),(103,260),(104,249),(105,250)}
\lines{(105,250),(114.2,250)}
\curve{(114.2,250),(115,252),(115.2,254),(115.4,256),(115.6,258),
(115.8,260),(116,262),(116.2,264),(116.4,266),(116.6,268),
(116.8,270),(117,271),(118,260),(119,249),(120,250)}
\lines{(120,250),(130,250)}
\tlabel[cc](180,265){\footnotesize An Input}
\tlabel[cc](180,255){\footnotesize Spike Train}
\arrow\lines{(145,260),(130,260)}
%  Heading
\tlabel[bc](180,-30){\footnotesize Output Spike}
\tlabel[bc](180,-40){\footnotesize Train}
\lines{(180,-45),(165,-60)}
\arrow\lines{(165,-60),(130,-60)}
% Synaptic inputs
\pen{0.5pt}
\curve{(76,199),(75,200),(74,201),(73,203),(72,207),(71,213),(70,223),(69,240)}
\lines{(77,194),(76,199),(80,200)}
\curve{(91,220),(90,222),(89,223),(86,225),(81,226),(73,225),(69,225)}
\lines{(91,215),(91,220),(96,220)}
\lines{(17,180),(23,180),(23,174)}
\curve{(23,180),(30,183),(40,186),(50,188),(60,189),(65,190),(66,191),(67,193),
(68,196),(69,201),(70,209),(71,213)}
\lines{(37,153),(41,151),(41,145)}
\curve{(41,151),(45,155),(50,160),(55,166),(61,174),(64,185),(65,190)}
\tlabel[cc](30,235){\footnotesize Synaptic}
\tlabel[cc](30,225){\footnotesize inputs}
\arrow\lines{(30,215),(50,195)}
% Deal with tree heading
\tlabel[bc](20,110){\footnotesize An}
\tlabel[bc](20,100){\footnotesize Idealised}
\tlabel[bc](20,90){\footnotesize Dendritic}
\tlabel[bc](20,80){\footnotesize  Tree}
\end{mfpic}}
\centering
\parbox{2.2in}{\caption{\label{neuron} A stylised neuron with
some common nomenclature.}}
\end{figure}

Figure \ref{neuron} illustrates a typical neuron with two
dendrites or branching structures emanating from the cell body
(soma) of the neuron. The length of dendrite connecting one branch
point to a neighbouring branch point, the soma or a dendritic
terminal is called a dendritic section. In practice, dendritic
sections are divided into shorter units called segments. Following
Segev and Burke (\cite{Segev98}), a segment is assumed to be a
uniform cylinder and a section is represented by a series of such
cylinders as illustrated in Figure \ref{ld}.

\pagebreak[4]

\begin{figure}[!h]
\centerline{\begin{mfpic}[0.8][1]{0}{350}{160}{320}
\headlen7pt
\pen{1pt}
\dotspace=4pt
\dotsize=1.5pt
%
% Partial cylinder on right
\ellipse{(350,240),12,16}
\lines{(350,256),(380,256)}
\dashed\lines{(380,256),(400,256)}
\lines{(350,224),(380,224)}
\dashed\lines{(380,224),(400,224)}
%
% RH cylinder
\ellipse{(350,240),18,24}
\lines{(250,264),(350,264)}
\lines{(250,216),(350,216)}
\dotted\parafcn[s]{0,180,5}{(250+18*sind(t),240+24*cosd(t))}
\parafcn[s]{0,180,5}{(250-18*sind(t),240+24*cosd(t))}
%
% Central cylinder
\dotted\parafcn[s]{0,180,5}{(250+24*sind(t),240+32*cosd(t))}
\parafcn[s]{-30,210,5}{(250-24*sind(t),240+32*cosd(t))}
\lines{(100,272),(250,272)}
\lines{(100,208),(250,208)}
\dotted\parafcn[s]{0,180,5}{(100+24*sind(t),240+32*cosd(t))}
\parafcn[s]{0,180,5}{(100-24*sind(t),240+32*cosd(t))}
%
% LH cylinder
\dotted\parafcn[s]{0,180,5}{(100+30*sind(t),240+40*cosd(t))}
\parafcn[s]{-30,210,5}{(100-30*sind(t),240+40*cosd(t))}
\lines{(0,280),(100,280)}
\lines{(0,200),(100,200)}
\dotted\parafcn[s]{0,180,5}{(30*sind(t),240+40*cosd(t))}
\parafcn[s]{0,180,5}{(-30*sind(t),240+40*cosd(t))}
%
% Partial cylinder on left
\dotted\parafcn[s]{0,180,5}{(36*sind(t),240+48*cosd(t))}
\parafcn[s]{-30,210,5}{(-36*sind(t),240+48*cosd(t))}
\lines{(-30,288),(0,288)}
\lines{(-30,192),(0,192)}
\dashed\lines{(-50,288),(-30,288)}
\dashed\lines{(-50,192),(-30,192)}
%
% Delimit cylinders
\arrow\lines{(-50,300),(-5,300)}
\arrow\lines{(50,300),(5,300)}
\arrow\lines{(50,300),(95,300)}
\arrow\lines{(175,300),(105,300)}
\arrow\lines{(175,300),(245,300)}
\arrow\lines{(300,300),(255,300)}
\arrow\lines{(300,300),(345,300)}
\arrow\lines{(400,300),(355,300)}
%
% Annotation of LH cylinder
\arrow\lines{(0,240),(20,240)}
\arrow\lines{(100,240),(150,240)}
\tlabel[br](112,250){$I_\mathrm{LC}$}
\tlabel[bc](50,250){$V_\mathrm{L}$}
\tlabel[tl](55,235){\large $x_\mathrm{L}$}
\tlabel[cc](50,240){\large $\bullet$}
\arrow\lines{(50,225),(50,180)}
\tlabel[tc](50,170){$I_\mathrm{L}$}
%
% Annotation of Central cylinder
\arrow\lines{(250,240),(290,240)}
\tlabel[br](265,250){$I_\mathrm{CR}$}
\tlabel[bc](175,250){$V_\mathrm{C}$}
\tlabel[tl](180,235){\large $x_\mathrm{C}$}
\tlabel[cc](175,240){\large $\bullet$}
\arrow\lines{(175,225),(175,180)}
\tlabel[tc](175,170){$I_\mathrm{C}$}
%
% Annotation of RH cylinder
\arrow\lines{(350,240),(390,240)}
\tlabel[bc](300,250){$V_\mathrm{R}$}
\tlabel[tl](305,235){\large $x_\mathrm{R}$}
\tlabel[cc](300,240){\large $\bullet$}
\arrow\lines{(300,225),(300,180)}
\tlabel[tc](300,170){$I_\mathrm{R}$}
\end{mfpic}}
\centering
\parbox{5.5in}{\caption{\label{ld}
Three consecutive segments of a dendritic section are illustrated.
Current $I_\mathrm{C}$ flows across the membrane at
$x_\mathrm{C}$, axial currents $I_\mathrm{LC}$ and $I_\mathrm{CR}$
flow from $x_\mathrm{L}$ to $x_\mathrm{C}$ and from $x_\mathrm{C}$
to $x_\mathrm{R}$ respectively through a resistive dendritic
core.}}
\end{figure}

In the traditional compartmental model, each segment is
represented by an elemental circuit with electrical properties
that incorporate the local biophysical and morphological
properties of the dendritic segment it represents. For example,
Figure \ref{circuit} illustrates how the three segments of Figure
\ref{ld} might be represented by three elemental circuits.

\begin{figure}[!h]
\centerline{\begin{mfpic}[1][0.8]{0}{400}{20}{260}
\headlen7pt
\pen{1pt}
% Build axonal resistances
\dashed\lines{(0,230),(27.5,230)}
\lines{(27.5,230),(30,235),(35,225),(40,235),(45,225),
(50,235),(55,225),(60,235),(65,225),(67.5,230),(80,230)}
\lines{(80,230),(92.5,230),(95,235),(100,225),(105,235),
(110,225),(115,235),(120,225),(125,235),(130,225),(135,235),
(140,225),(145,235),(150,225),(155,235),(160,225),(165,235),
(170,225),(175,235),(180,225),(185,235),(187.5,230),(200,230)}
\tlabel[tc](145,220){$R_\mathrm{LC}$}
\lines{(200,230),(212.5,230),(215,235),(220,225),(225,235),
(230,225),(235,235),(240,225),(245,235),(250,225),(255,235),
(260,225),(265,235),(270,225),(275,235),(280,225),(285,235),
(290,225),(295,235),(300,225),(305,235),(307.5,230),(320,230)}
\tlabel[tc](260,220){$R_\mathrm{CR}$}
\lines{(320,230),(332.5,230),(335,235),(340,225),(345,235),
(350,225),(355,235),(360,225),(365,235),(370,225),(372.5,230)}
\dashed\lines{(372.5,230),(400,230)}
\tlabel[cc](80,230){$\bullet$}
\tlabel[cc](200,230){$\bullet$}
\tlabel[cc](320,230){$\bullet$}
\arrow\lines{(100,245),(180,245)}
\arrow\lines{(220,245),(300,245)}
\tlabel[bc](140,255){$I_\mathrm{LC}$}
\tlabel[bc](260,255){$I_\mathrm{CR}$}
\tlabel[bc](80,255){$V_\mathrm{L}$}
\tlabel[bc](200,255){$V_\mathrm{C}$}
\tlabel[bc](320,255){$V_\mathrm{R}$}
% Build Right hand circuit
\lines{(320,230),(320,180),(300,180)}
\lines{(300,142.5),(295,140),(305,135),(295,130),(305,125),
(295,120),(305,115),(295,110),(305,105),(295,100),
(300,97.5),(300,90),(340,90),(340,130)}
\lines{(350,130),(330,130)}
\lines{(350,140),(330,140)}
\tlabel[bl](345,150){$c^{(m)}_\mathrm{R}$}
\lines{(340,140),(340,180),(320,180)}
\lines{(320,90),(320,40)}
\lines{(310,40),(330,40)}
\lines{(312.5,35),(327.5,35)}
\lines{(315,30),(325,30)}
\lines{(317.5,25),(322.5,25)}
\dashed\lines{(300,180),(260,180)}
\dashed\lines{(300,90),(260,90)}
\lines{(300,180),(320,180)}
\lines{(300,180),(300,170)}
% Battery
\lines{(295,170),(305,170)}
\lines{(295,162),(305,162)}
\lines{(295,154),(305,154)}
\lines{(255,150),(265,150)}
\lines{(255,166),(265,166)}
\lines{(255,158),(265,158)}
\pen{2pt}
\lines{(297.5,150),(302.5,150)}
\lines{(297.5,166),(302.5,166)}
\lines{(297.5,158),(302.5,158)}
\lines{(257.5,170),(262.5,170)}
\lines{(257.5,162),(262.5,162)}
\lines{(257.5,154),(262.5,154)}
\pen{1pt}
\lines{(300,150),(300,142.5)}
\lines{(260,180),(260,170)}
\lines{(260,150),(260,142.5),(255,140),(265,135),(255,130),
(265,125),(255,120),(265,115),(255,110),(265,105),(255,100),
(260,97.5),(260,90)}
\arrow\lines{(290,135),(290,105)}
\arrow\lines{(310,80),(310,50)}
\tlabel[cr](305,65){$I^{(m)}_\mathrm{R}$}
% Build Left hand circuit
\lines{(80,230),(80,180),(60,180)}
\lines{(60,142.5),(55,140),(65,135),(55,130),(65,125),
(55,120),(65,115),(55,110),(65,105),(55,100),(60,97.5),
(60,90),(100,90),(100,130)}
\lines{(110,130),(90,130)}
\lines{(110,140),(90,140)}
\tlabel[bl](105,150){$c^{(m)}_\mathrm{L}$}
\lines{(100,140),(100,180),(80,180)}
\lines{(80,90),(80,40)}
\lines{(70,40),(90,40)}
\lines{(72.5,35),(87.5,35)}
\lines{(75,30),(85,30)}
\lines{(77.5,25),(82.5,25)}
\dashed\lines{(60,180),(20,180)}
\dashed\lines{(60,90),(20,90)}
\lines{(60,180),(80,180)}
\lines{(60,180),(60,170)}
% Battery
\lines{(55,170),(65,170)}
\lines{(55,162),(65,162)}
\lines{(55,154),(65,154)}
\lines{(15,150),(25,150)}
\lines{(15,166),(25,166)}
\lines{(15,158),(25,158)}
\pen{2pt}
\lines{(57.5,150),(62.5,150)}
\lines{(57.5,166),(62.5,166)}
\lines{(57.5,158),(62.5,158)}
\lines{(17.5,170),(22.5,170)}
\lines{(17.5,162),(22.5,162)}
\lines{(17.5,154),(22.5,154)}
\pen{1pt}
\lines{(60,150),(60,142.5)}
\lines{(20,180),(20,170)}
\lines{(20,150),(20,142.5),(15,140),(25,135),(15,130),
(25,125),(15,120),(25,115),(15,110),(25,105),(15,100),
(20,97.5),(20,90)}
\arrow\lines{(50,135),(50,105)}
\arrow\lines{(70,80),(70,50)}
\tlabel[cr](65,65){$I^{(m)}_\mathrm{L}$}
% Build Middle circuit
\lines{(200,230),(200,180),(180,180)}
\lines{(180,142.5),(175,140),(185,135),(175,130),(185,125),
(175,120),(185,115),(175,110),(185,105),(175,100),
(180,97.5),(180,90),(220,90),(220,130)}
\lines{(230,130),(210,130)}
\lines{(230,140),(210,140)}
\tlabel[bl](225,150){$c^{(m)}_\mathrm{C}$}
\lines{(220,140),(220,180),(200,180)}
\lines{(200,90),(200,40)}
\lines{(190,40),(210,40)}
\lines{(192.5,35),(207.5,35)}
\lines{(195,30),(205,30)}
\lines{(197.5,25),(202.5,25)}
\dashed\lines{(180,180),(140,180)}
\dashed\lines{(180,90),(140,90)}
\lines{(180,180),(200,180)}
\lines{(180,180),(180,170)}
% Battery
\lines{(175,170),(185,170)}
\lines{(175,162),(185,162)}
\lines{(175,154),(185,154)}
\lines{(135,150),(145,150)}
\lines{(135,166),(145,166)}
\lines{(135,158),(145,158)}
\pen{2pt}
\lines{(177.5,150),(182.5,150)}
\lines{(177.5,166),(182.5,166)}
\lines{(177.5,158),(182.5,158)}
\lines{(137.5,170),(142.5,170)}
\lines{(137.5,162),(142.5,162)}
\lines{(137.5,154),(142.5,154)}
\pen{1pt}
\lines{(180,150),(180,142.5)}
\lines{(140,180),(140,170)}
\lines{(140,150),(140,142.5),(135,140),(145,135),(135,130),
(145,125),(135,120),(145,115),(135,110),(145,105),(135,100),
(140,97.5),(140,90)}
\arrow\lines{(170,135),(170,105)}
\arrow\lines{(190,80),(190,50)}
\tlabel[cr](185,65){$I^{(m)}_\mathrm{C}$}
% Far right
\dashed\lines{(400,180),(380,180)}
\dashed\lines{(400,90),(380,90)}
\lines{(380,150),(380,142.5),(375,140),(385,135),(375,130),
(385,125),(375,120),(385,115),(375,110),(385,105),(375,100),
(380,97.5),(380,90)}
\arrow\lines{(50,135),(50,105)}
\arrow\lines{(70,80),(70,50)}
% Battery
\lines{(375,150),(385,150)}
\lines{(375,166),(385,166)}
\lines{(375,158),(385,158)}
\lines{(380,180),(380,170)}
\pen{2pt}
\lines{(377.5,170),(382.5,170)}
\lines{(377.5,162),(382.5,162)}
\lines{(377.5,154),(382.5,154)}
\pen{1pt}
\end{mfpic}}
\centering
\parbox{5.6in}{\caption{\label{circuit} A diagrammatic
representation of two elemental circuits used to construct the
compartmental model of a dendritic section. Axial current
$I_\mathrm{L}$ flows in the left hand compartment under the
influence of the potential difference
$(V_\mathrm{L}-V_\mathrm{C})$. Axial current $I_\mathrm{R}$ flows
in the right hand compartment under the influence of the potential
difference $(V_\mathrm{C}-V_\mathrm{R})$. Transmembrane current
flow from the left and right hand circuits to the extracellular
medium at the endpoints of the left and right hand compartments.}}
\end{figure}

The accuracy of the two compartmental models can be assessed by
comparing their response to deterministic input for situations in
which the potential distribution in the model dendrite is known
analytically. The objective of this report is to examine the error
in the potential at the soma of a branched neuron in response to a
sustained input randomly placed on the dendritic tree. This
procedure will be repeated 2000 times and because of the linearity
of the model, the results of the simulation study will mimic the
effect of large scale exogenous input on the branched neuron. The
accuracy and precision of the models are assessed by comparing
their responses to this input in situations in which the potential
at the soma is known analytically.

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