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{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
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\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)}
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\curve{(80,14),(75,15),(70,17),(65,19.5),(60,23),(55,30)}
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\curve{(80,70),(85,80),(90,100),(95,140)}
\curve{(85,30),(75,35),(65,45),(50,50),(40,65)}
% Draw axon
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\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}
% 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)}
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% 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
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\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)}
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(51,-58),(51.2,-56),(51.4,-54),(51.6,-52), (51.8,-50),(52,-49),(53,-60),
(54,-71),(55,-70)}
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(87,-71),(88,-70)}
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(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),
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\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)}
\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}
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\lines{(77,194),(76,199),(80,200)}
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\lines{(91,215),(91,220),(96,220)}
\lines{(17,180),(23,180),(23,174)}
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(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)}
\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}
\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)}
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\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}
\pen{1pt}
% Build axonal resistances
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\tlabel[tc](145,220){$R_\mathrm{LC}$}
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\tlabel[cc](200,230){$\bullet$}
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\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)}
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% Battery
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\arrow\lines{(310,80),(310,50)}
\tlabel[cr](305,65){$I^{(m)}_\mathrm{R}$}
% Build Left hand circuit
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% Battery
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\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){$c^{(m)}_\mathrm{C}$}
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% Battery
\lines{(175,170),(185,170)}
\lines{(175,162),(185,162)}
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\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.