### 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{Comparison of the traditional and new approaches to
compartmental modelling}

Two simulation studies are used to compare the performance of the
traditional and new compartmental models. These studies are based
on a branched model neuron with known expression for the somal
potential in response to large scale exogenous input (see Appendix
4). The first study examines the accuracy with which each type of
compartmental model estimates this somal potential, and uses the
NEURON simulator (Hines and Carnevale, \cite{Hines97}) as an
example of a traditional compartmental model. The second study
assesses the accuracy of the two types of models by comparing the
statistics of the spike train output generated by each model when
the test neuron is subjected to large scale synaptic input. Here a
traditional compartmental model developed by the authors is used.
This model gave results identical to those of NEURON in the first
study. Finally, a time step of one microsecond is used in the
numerical integration of each compartmental model to ensure that
errors in temporal integration make no significant contribution to
the error in the calculation of membrane potential.

\subsection{The test neuron}

One way to construct a branched test neuron with a closed form
solution for the somal 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}). These
conditions require that the sum of the three-halves power of the
diameters of the child limbs is equal to the three-halves power of
the diameter of the parent limb at any branch point, and that the
total electrotonic length from a branch point or the soma to a
dendritic tip is independent of path. The test 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]
$\begin{array}{c} \includegraphics[ ]{NCFig4.eps} \end{array}\qquad \begin{array}{ccc} \hline \mbox{Section} & \mbox{Length }\mu\mbox{m} & \mbox{Diameter }\mu\mbox{m}\\[2pt] \hline (a) & 166.809245 & 7.089751 \\ (b) & 379.828386 & 9.189790 \\ (c) & 383.337494 & 4.160168 \\ (d) & 410.137845 & 4.762203 \\ (e) & 631.448520 & 6.345604 \\ (f) & 571.445800 & 5.200210 \\ (g) & 531.582750 & 2.000000 \\ (h) & 651.053246 & 3.000000 \\ (i) & 501.181023 & 4.000000 \\ (j) & 396.218388 & 2.500000 \\ \hline \end{array}$
\centering
\parbox{5in}{\caption{\label{TestNeuron} A branched neuron
satisfying the Rall conditions. The diameters and lengths of the
dendritic sections are given in the right hand panel of the
figure. At each branch point, the ratio of the length of a section
to the square root of its radius is fixed for all children of the
branch point.}}
\end{figure}

%\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{5.5in}{\caption{\label{TestNeuron} A branched neuron
%satisfying the Rall conditions. The diameters and lengths of the
%dendritic sections are given in the right hand panel of the
%figure. At each branch point, the ratio of the length of a section
%to the square root of its radius is fixed for all children of the
%branch point.}}
%\end{figure}

The high degree of accuracy used in the specification of the
dendritic radii and section lengths of the test neuron is required
to ensure that the equivalent cylinder is an adequate
representation of the test neuron. The membrane of the test neuron
is assigned a specific conductance of $0.091\,$mS/cm$^2$
($g_\mathrm{M}$) and specific capacitance of $1.0\,\mu$F/cm$^2$
($c_\mathrm{M}$), and has axoplasm of conductance $14.286\,$mS/cm
($g_\mathrm{A}$). With these biophysical properties, the
equivalent cylinder has length one electrotonic unit. The soma of
the test neuron is assumed to have membrane area $A_\mathrm{S}$,
and specific conductance $g_\mathrm{S}$ and specific capacitance
$c_\mathrm{S}$ identical to that of the dendritic membrane. The
analytical expression for its somal potential is given in Appendix
4.