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;
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Gene(s):
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Simulation Environment: NEURON; C or C++ program;
Model Concept(s): Methods;
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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
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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
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NC4.tex
NC5.tex
NC6.tex
NCFig2.eps *
NCFig3.eps *
NCFig4.eps *
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NCPics.tex
NeuronDriver.hoc
NewComExactSoln.c
NewComp.pdf
NewComp.ps
NewComp.tex
NewComp.toc
NewComp1.tex
NewComp2.tex
NewComp3.tex
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NewCompFig1.eps
NewCompFig2.eps *
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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
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TestSim080.hoc
TestSim090.hoc
TestSim1.hoc
TestSim100.hoc
TestSim200.hoc
TestSim300.hoc
TestSim400.hoc
TestSim500
TestSim500.hoc
                            
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\title{\bf A generalised compartmental model - increased
accuracy and precision of the traditional compartmental model
without increased computational effort}
\author{\Large\bf K.A. Lindsay\\
Department of Mathematics, University Gardens, University of Glasgow,\\
Glasgow G12 8QQ\\[10pt]
\Large\bf A.E. Lindsay\\
Department of Mathematics, Kings Buildings, University of Edinburgh,\\
Edinburgh EH9 3JZ\\[10pt]
\Large\bf J.R. Rosenberg \\
Division of Neuroscience and Biomedical Systems,\\
University of Glasgow, Glasgow G12 8QQ}

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\begin{center}
\begin{tabular}{p{5.2in}}
\multicolumn{1}{c}{\textbf{Abstract}}\\[10pt]

Compartmental models of complex branching dendrites are the most
widely used tool for investigating the behaviour of these
structures. This report demonstrates that both the accuracy and
precision of traditional compartmental models can be significantly
improved by relaxing the basic assumptions of these models,
namely, that compartments are iso-potential regions and that all
input to a compartment occurs at a designated node. The selective
relaxation of these assumptions is explored in this report and
leads to the development of the \emph{generalised compartmental
model} which achieves significantly more accuracy than the
traditional compartmental model without any increase in
computational effort beyond that already required by the
traditional compartmental model.
\end{tabular}
\end{center}

\vfil

\pagebreak[4]

\tableofcontents

\pagebreak[4]

\input gen1.tex
\input gen2.tex
\input gen3.tex
\input gen4.tex
\input gen5.tex
\input gen6.tex

\closegraphsfile

\end{document}


\section*{Percentage Mean and Standard deviation of Gen error}
\begin{tabular}{r|cccccccccc}
\hline&&&&&&&&&&\\[-8pt]
Nodes & t=1 & t=2 & t=3 & t=4 & t=5 & t=6 & t=7 & t=8 & t=9 & t=10\\[2pt]
\hline&&&&&&&&&&\\[-8pt]
  21 &-2.787 & -3.449 & -2.396 & -1.800 & -1.437 & -1.200 & -1.036 &  -0.917 & -0.829 & -0.760\\[2pt]
  34 & 1.323 & -0.510 & -0.430 & -0.341 & -0.279 & -0.237 & -0.207 &  -0.185 & -0.168 & -0.153\\[2pt]
  42 & 1.021 & -0.262 & -0.251 & -0.205 & -0.170 & -0.145 & -0.127 &  -0.113 & -0.103 & -0.093\\[2pt]
  54 & 0.521 & -0.150 & -0.139 & -0.113 & -0.092 & -0.078 & -0.067 &  -0.060 & -0.054 & -0.048\\[2pt]
  67 & 0.295 & -0.165 & -0.140 & -0.112 & -0.092 & -0.078 & -0.068 &  -0.060 & -0.055 & -0.050\\[2pt]
  75 & 0.228 & -0.094 & -0.082 & -0.066 & -0.054 & -0.046 & -0.040 &  -0.035 & -0.032 & -0.029\\[2pt]
  82 & 0.217 & -0.067 & -0.063 & -0.051 & -0.042 & -0.035 & -0.031 &  -0.027 & -0.025 & -0.022\\[2pt]
  93 & 0.125 & -0.070 & -0.059 & -0.047 & -0.038 & -0.032 & -0.028 &  -0.025 & -0.023 & -0.020\\[2pt]
 119 & 0.067 & -0.045 & -0.037 & -0.030 & -0.024 & -0.020 & -0.018 &  -0.016 & -0.014 & -0.013\\[2pt]
 142 & 0.072 & -0.022 & -0.020 & -0.017 & -0.014 & -0.011 & -0.010 &  -0.009 & -0.008 & -0.007\\[2pt]
 169 & 0.054 & -0.014 & -0.013 & -0.011 & -0.009 & -0.008 & -0.007 &  -0.006 & -0.005 & -0.005\\[2pt]
 193 & 0.033 & -0.014 & -0.012 & -0.010 & -0.008 & -0.006 & -0.006 &  -0.005 & -0.004 & -0.004\\[2pt]
 244 & 0.020 & -0.009 & -0.007 & -0.006 & -0.005 & -0.004 & -0.003 &  -0.003 & -0.003 & -0.002\\[2pt]
 293 & 0.011 & -0.006 & -0.005 & -0.004 & -0.003 & -0.003 & -0.002 &  -0.002 & -0.002 & -0.002\\[2pt]
 391 & 0.010 & -0.002 & -0.002 & -0.002 & -0.001 & -0.001 & -0.001 &  -0.001 & -0.001 & -0.000\\[2pt]
 495 & 0.005 & -0.001 & -0.001 & -0.001 & -0.001 & -0.001 & -0.000 &  -0.000 & -0.000 & -0.000\\[2pt]
 992 & 0.001 & -0.000 & -0.000 & -0.000 & -0.000 & -0.000 & -0.000 &  -0.000 & -0.000 & -0.000\\[2pt]
\hline
\end{tabular}

\begin{tabular}{r|cccccccccc}
\hline&&&&&&&&&&\\[-8pt]
Nodes & t=1 & t=2 & t=3 & t=4 & t=5 & t=6 & t=7 & t=8 & t=9 & t=10\\[2pt]
\hline&&&&&&&&&&\\[-8pt]
  21 & 37.21 & 10.48 & 6.241 & 4.536 & 3.619 & 3.048 & 2.660 & 2.380 & 2.170 & 2.007\\[2pt]
  34 & 8.319 & 2.417 & 1.420 & 1.021 & 0.810 & 0.680 & 0.593 & 0.531 & 0.484 & 0.449\\[2pt]
  42 & 4.983 & 1.432 & 0.831 & 0.592 & 0.466 & 0.388 & 0.336 & 0.300 & 0.272 & 0.252\\[2pt]
  54 & 2.721 & 0.791 & 0.457 & 0.324 & 0.254 & 0.211 & 0.182 & 0.162 & 0.147 & 0.137\\[2pt]
  67 & 2.235 & 0.667 & 0.393 & 0.283 & 0.223 & 0.187 & 0.162 & 0.145 & 0.132 & 0.122\\[2pt]
  75 & 1.427 & 0.434 & 0.255 & 0.182 & 0.144 & 0.120 & 0.104 & 0.093 & 0.084 & 0.078\\[2pt]
  82 & 1.123 & 0.341 & 0.201 & 0.143 & 0.113 & 0.094 & 0.081 & 0.072 & 0.065 & 0.061\\[2pt]
  93 & 0.887 & 0.277 & 0.166 & 0.120 & 0.095 & 0.080 & 0.069 & 0.062 & 0.056 & 0.053\\[2pt]
 119 & 0.542 & 0.168 & 0.100 & 0.072 & 0.057 & 0.047 & 0.041 & 0.037 & 0.033 & 0.031\\[2pt]
 142 & 0.386 & 0.117 & 0.068 & 0.049 & 0.038 & 0.032 & 0.027 & 0.024 & 0.022 & 0.021\\[2pt]
 169 & 0.253 & 0.077 & 0.045 & 0.032 & 0.025 & 0.021 & 0.018 & 0.016 & 0.015 & 0.014\\[2pt]
 193 & 0.210 & 0.065 & 0.038 & 0.027 & 0.021 & 0.018 & 0.015 & 0.014 & 0.012 & 0.012\\[2pt]
 244 & 0.124 & 0.038 & 0.022 & 0.016 & 0.012 & 0.010 & 0.009 & 0.008 & 0.007 & 0.007\\[2pt]
 293 & 0.088 & 0.027 & 0.016 & 0.011 & 0.009 & 0.007 & 0.006 & 0.005 & 0.005 & 0.005\\[2pt]
 391 & 0.051 & 0.015 & 0.008 & 0.006 & 0.005 & 0.004 & 0.003 & 0.003 & 0.002 & 0.002\\[2pt]
 495 & 0.032 & 0.009 & 0.005 & 0.003 & 0.003 & 0.002 & 0.002 & 0.002 & 0.001 & 0.001\\[2pt]
 992 & 0.007 & 0.002 & 0.001 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000\\[2pt]
\hline
\end{tabular}

\section*{Percentage Mean and Standard deviation of Mod error}
\begin{tabular}{r|cccccccccc}
\hline&&&&&&&&&&\\[-8pt]
Nodes & t=1 & t=2 & t=3 & t=4 & t=5 & t=6 & t=7 & t=8 & t=9 & t=10\\[2pt]
\hline&&&&&&&&&&\\[-8pt]
  21 & -42.97 & -9.952 & -4.695 & -2.896 & -2.070 & -1.622 & -1.351 & -1.171 & -1.045 & -0.950\\[2pt]
  34 & -15.44 & -3.114 & -1.294 & -0.732 & -0.498 & -0.381 & -0.315 & -0.272 & -0.243 & -0.220\\[2pt]
  42 & -9.796 & -1.927 & -0.803 & -0.458 & -0.314 & -0.241 & -0.199 & -0.172 & -0.154 & -0.138\\[2pt]
  54 & -5.498 & -1.070 & -0.443 & -0.251 & -0.170 & -0.130 & -0.107 & -0.092 & -0.082 & -0.073\\[2pt]
  67 & -4.514 & -0.904 & -0.387 & -0.225 & -0.157 & -0.121 & -0.100 & -0.087 & -0.078 & -0.070\\[2pt]
  75 & -2.983 & -0.589 & -0.247 & -0.141 & -0.097 & -0.074 & -0.061 & -0.053 & -0.047 & -0.042\\[2pt]
  82 & -2.406 & -0.469 & -0.196 & -0.112 & -0.077 & -0.059 & -0.048 & -0.042 & -0.037 & -0.033\\[2pt]
  93 & -1.944 & -0.387 & -0.164 & -0.095 & -0.066 & -0.051 & -0.042 & -0.036 & -0.032 & -0.029\\[2pt]
 119 & -1.187 & -0.237 & -0.102 & -0.059 & -0.041 & -0.032 & -0.026 & -0.023 & -0.020 & -0.018\\[2pt]
 142 & -0.794 & -0.154 & -0.065 & -0.037 & -0.025 & -0.019 & -0.016 & -0.014 & -0.012 & -0.010\\[2pt]
 169 & -0.570 & -0.110 & -0.045 & -0.026 & -0.017 & -0.013 & -0.011 & -0.009 & -0.008 & -0.007\\[2pt]
 193 & -0.443 & -0.087 & -0.036 & -0.021 & -0.014 & -0.011 & -0.009 & -0.008 & -0.007 & -0.006\\[2pt]
 244 & -0.276 & -0.054 & -0.023 & -0.013 & -0.009 & -0.007 & -0.005 & -0.005 & -0.004 & -0.003\\[2pt]
 293 & -0.191 & -0.038 & -0.016 & -0.009 & -0.006 & -0.005 & -0.004 & -0.003 & -0.003 & -0.002\\[2pt]
 391 & -0.104 & -0.020 & -0.008 & -0.004 & -0.003 & -0.002 & -0.002 & -0.001 & -0.001 & -0.001\\[2pt]
 495 & -0.065 & -0.012 & -0.005 & -0.003 & -0.002 & -0.001 & -0.001 & -0.001 & -0.001 & -0.000\\[2pt]
 992 & -0.016 & -0.003 & -0.001 & -0.000 & -0.000 & -0.000 & -0.000 & -0.000 & -0.000 & -0.000\\[2pt]
\hline
\end{tabular}

\begin{tabular}{r|cccccccccc}
\hline&&&&&&&&&&\\[-8pt]
Nodes & t=1 & t=2 & t=3 & t=4 & t=5 & t=6 & t=7 & t=8 & t=9 & t=10\\[2pt]
\hline&&&&&&&&&&\\[-8pt]
  21 & 55.50 & 14.95 & 8.130 & 5.572 & 4.280 & 3.516 & 3.018 & 2.671 & 2.417 & 2.226\\[2pt]
  34 & 17.36 & 3.986 & 1.947 & 1.268 & 0.955 & 0.779 & 0.669 & 0.593 & 0.538 & 0.498\\[2pt]
  42 & 10.94 & 2.426 & 1.165 & 0.750 & 0.559 & 0.453 & 0.386 & 0.340 & 0.308 & 0.285\\[2pt]
  54 & 6.200 & 1.353 & 0.644 & 0.412 & 0.306 & 0.246 & 0.210 & 0.185 & 0.167 & 0.155\\[2pt]
  67 & 5.229 & 1.164 & 0.565 & 0.366 & 0.274 & 0.222 & 0.189 & 0.167 & 0.151 & 0.140\\[2pt]
  75 & 3.479 & 0.769 & 0.369 & 0.237 & 0.176 & 0.142 & 0.121 & 0.107 & 0.096 & 0.090\\[2pt]
  82 & 2.768 & 0.607 & 0.290 & 0.185 & 0.137 & 0.110 & 0.094 & 0.082 & 0.074 & 0.070\\[2pt]
  93 & 2.325 & 0.515 & 0.248 & 0.160 & 0.119 & 0.096 & 0.082 & 0.072 & 0.065 & 0.061\\[2pt]
 119 & 1.396 & 0.309 & 0.149 & 0.095 & 0.071 & 0.057 & 0.049 & 0.043 & 0.039 & 0.036\\[2pt]
 142 & 0.940 & 0.205 & 0.098 & 0.063 & 0.047 & 0.038 & 0.032 & 0.028 & 0.025 & 0.024\\[2pt]
 169 & 0.669 & 0.145 & 0.069 & 0.043 & 0.032 & 0.026 & 0.022 & 0.019 & 0.017 & 0.016\\[2pt]
 193 & 0.526 & 0.116 & 0.055 & 0.036 & 0.026 & 0.021 & 0.018 & 0.016 & 0.014 & 0.014\\[2pt]
 244 & 0.320 & 0.069 & 0.033 & 0.021 & 0.015 & 0.012 & 0.010 & 0.009 & 0.008 & 0.008\\[2pt]
 293 & 0.223 & 0.049 & 0.023 & 0.015 & 0.011 & 0.009 & 0.007 & 0.006 & 0.006 & 0.005\\[2pt]
 391 & 0.122 & 0.026 & 0.012 & 0.008 & 0.006 & 0.004 & 0.004 & 0.003 & 0.003 & 0.003\\[2pt]
 495 & 0.076 & 0.016 & 0.008 & 0.005 & 0.003 & 0.003 & 0.002 & 0.002 & 0.002 & 0.002\\[2pt]
 992 & 0.018 & 0.004 & 0.001 & 0.001 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000\\[2pt]
\hline
\end{tabular}

\section*{Percentage Mean and Standard deviation of Old error}
\begin{tabular}{r|cccccccccc}
\hline&&&&&&&&&&\\[-8pt]
Nodes & t=1 & t=2 & t=3 & t=4 & t=5 & t=6 & t=7 & t=8 & t=9 & t=10\\[2pt]
\hline&&&&&&&&&&\\[-8pt]
  21 & -25.38 & -5.964 & -2.919 & -1.874 & -1.388 & -1.122 & -0.958 & -0.848 & -0.769 & -0.708\\[2pt]
  34 & -11.46 & -2.073 & -0.815 & -0.458 & -0.321 & -0.256 & -0.220 & -0.198 & -0.183 & -0.169\\[2pt]
  42 & -7.455 & -1.423 & -0.634 & -0.398 & -0.299 & -0.247 & -0.216 & -0.196 & -0.181 & -0.167\\[2pt]
  54 & -3.503 & -0.406 & -0.074 & -0.000 &  0.018 &  0.022 &  0.022 &  0.020 &  0.019 &  0.019\\[2pt]
  67 & -2.783 & -0.338 & -0.082 & -0.025 & -0.009 & -0.005 & -0.004 & -0.004 & -0.004 & -0.003\\[2pt]
  75 & -2.241 & -0.447 & -0.211 & -0.138 & -0.106 & -0.089 & -0.078 & -0.070 & -0.065 & -0.060\\[2pt]
  82 & -1.687 & -0.249 & -0.081 & -0.038 & -0.023 & -0.016 & -0.013 & -0.011 & -0.010 & -0.008\\[2pt]
  93 & -1.684 & -0.392 & -0.202 & -0.137 & -0.107 & -0.090 & -0.079 & -0.071 & -0.066 & -0.061\\[2pt]
 119 & -0.664 & -0.044 &  0.010 &  0.019 &  0.020 &  0.018 &  0.017 &  0.015 &  0.014 &  0.014\\[2pt]
 142 & -0.509 & -0.068 & -0.021 & -0.010 & -0.006 & -0.005 & -0.004 & -0.004 & -0.004 & -0.003\\[2pt]
 169 & -0.572 & -0.164 & -0.096 & -0.070 & -0.057 & -0.049 & -0.043 & -0.039 & -0.036 & -0.034\\[2pt]
 193 & -0.423 & -0.106 & -0.055 & -0.036 & -0.027 & -0.022 & -0.019 & -0.017 & -0.015 & -0.013\\[2pt]
 244 & -0.264 & -0.059 & -0.026 & -0.016 & -0.011 & -0.008 & -0.007 & -0.006 & -0.005 & -0.005\\[2pt]
 293 & -0.175 & -0.030 & -0.009 & -0.004 & -0.002 & -0.001 & -0.000 & -0.000 & -0.000 & -0.000\\[2pt]
 391 & -0.050 & -0.009 & -0.006 & -0.006 & -0.006 & -0.005 & -0.005 & -0.005 & -0.005 & -0.004\\[2pt]
 495 & -0.071 & -0.022 & -0.014 & -0.010 & -0.008 & -0.007 & -0.006 & -0.005 & -0.005 & -0.004\\[2pt]
 992 &  0.020 &  0.012 &  0.008 &  0.005 &  0.004 &  0.003 &  0.003 &  0.002 &  0.002 &  0.002\\[2pt]
\hline
\end{tabular}


\begin{tabular}{r|cccccccccc}
\hline&&&&&&&&&&\\[-8pt]
Nodes & t=1 & t=2 & t=3 & t=4 & t=5 & t=6 & t=7 & t=8 & t=9 & t=10\\[2pt]
\hline&&&&&&&&&&\\[-8pt]
  21 & 54.93 & 27.29 & 18.97 & 14.58 & 11.90 & 10.13 & 8.887 & 7.984 & 7.304 & 6.774\\[2pt]
  34 & 31.70 & 15.50 & 10.65 & 8.194 & 6.726 & 5.762 & 5.089 & 4.597 & 4.225 & 3.933\\[2pt]
  42 & 24.44 & 12.36 & 8.509 & 6.553 & 5.386 & 4.622 & 4.089 & 3.699 & 3.404 & 3.171\\[2pt]
  54 & 18.11 & 9.264 & 6.324 & 4.842 & 3.965 & 3.394 & 2.998 & 2.709 & 2.491 & 2.319\\[2pt]
  67 & 16.27 & 8.326 & 5.673 & 4.339 & 3.552 & 3.040 & 2.684 & 2.426 & 2.230 & 2.076\\[2pt]
  75 & 13.08 & 6.724 & 4.584 & 3.509 & 2.874 & 2.461 & 2.173 & 1.964 & 1.806 & 1.681\\[2pt]
  82 & 11.73 & 6.057 & 4.137 & 3.172 & 2.601 & 2.230 & 1.971 & 1.783 & 1.640 & 1.528\\[2pt]
  93 & 10.64 & 5.500 & 3.747 & 2.866 & 2.346 & 2.008 & 1.773 & 1.602 & 1.473 & 1.371\\[2pt]
 119 & 8.315 & 4.318 & 2.945 & 2.256 & 1.849 & 1.584 & 1.399 & 1.265 & 1.164 & 1.084\\[2pt]
 142 & 6.574 & 3.412 & 2.320 & 1.772 & 1.449 & 1.239 & 1.094 & 0.988 & 0.908 & 0.845\\[2pt]
 169 & 5.535 & 2.876 & 1.961 & 1.502 & 1.232 & 1.055 & 0.933 & 0.844 & 0.776 & 0.723\\[2pt]
 193 & 4.978 & 2.580 & 1.754 & 1.340 & 1.097 & 0.938 & 0.828 & 0.748 & 0.688 & 0.640\\[2pt]
 244 & 3.938 & 2.052 & 1.396 & 1.067 & 0.873 & 0.747 & 0.660 & 0.596 & 0.548 & 0.510\\[2pt]
 293 & 3.321 & 1.721 & 1.169 & 0.893 & 0.730 & 0.625 & 0.552 & 0.499 & 0.459 & 0.427\\[2pt]
 391 & 2.395 & 1.252 & 0.854 & 0.654 & 0.535 & 0.459 & 0.405 & 0.366 & 0.337 & 0.314\\[2pt]
 495 & 1.928 & 1.007 & 0.686 & 0.525 & 0.430 & 0.369 & 0.326 & 0.294 & 0.271 & 0.252\\[2pt]
 992 & 0.946 & 0.497 & 0.339 & 0.260 & 0.213 & 0.182 & 0.161 & 0.146 & 0.134 & 0.125\\[2pt]
\hline
\end{tabular}


\section{The system of model differential equations}

\subsection{Designated node of an internal segment of a section}
If $x_\mathrm{C}$ is the designated node of an internal segment of
dendritic section then the equation contributed by this node has
form
\begin{equation}\label{is1}
\hskip-2pt
\begin{array}{l}
\ds\Big(\frac{\pi g_\mathrm{A}r_\mathrm{L}r_\mathrm{C}}
{x_\mathrm{C}-x_\mathrm{L}}\Big)\;V_\mathrm{L}
-\Big(\frac{\pi g_\mathrm{A}r_\mathrm{L}r_\mathrm{C}}
{x_\mathrm{C}-x_\mathrm{L}}+\frac{\pi g_\mathrm{A}
r_\mathrm{C}r_\mathrm{R}}
{x_\mathrm{R}-x_\mathrm{C}}\Big)\;V_\mathrm{C}
+\Big(\frac{\pi g_\mathrm{A}r_\mathrm{C}r_\mathrm{R}}
{x_\mathrm{R}-x_\mathrm{C}}\Big)\;V_\mathrm{R} = \\[12pt]
\ds\quad\frac{\pi c_\mathrm{M}}{2}\,\Big[\,
\big(x_\mathrm{C}-x_\mathrm{L}\big)r_\mathrm{L}\,
\frac{dV_\mathrm{L}}{dt}+3\big(x_\mathrm{R}-x_\mathrm{L}\big)
r_\mathrm{C}\,\frac{dV_\mathrm{C}}{dt}+
\big(x_\mathrm{R}-x_\mathrm{C}\big)r_\mathrm{R}\,
\frac{dV_\mathrm{R}}{dt}\,\Big]\\[12pt]
\ds\qquad+\;\frac{\pi}{2}\,\sum_\alpha\,\Big[\,
\big(x_\mathrm{C}-x_\mathrm{L}\big)r_\mathrm{L}
g_\alpha(V_\mathrm{L})(V_\mathrm{L}-E_\alpha)+
3\big(x_\mathrm{R}-x_\mathrm{L}\big)\,r_\mathrm{C}
g_\alpha(V_\mathrm{C})(V_\mathrm{C}-E_\alpha)\\[12pt]
\qquad\quad\ds+\;\big(x_\mathrm{R}-x_\mathrm{C}\big)\,
r_\mathrm{R} g_\alpha(V_\mathrm{R})(V_\mathrm{R}-E_\alpha)\,\Big]
+G_\mathrm{L}(t)\,V_\mathrm{L}+G_\mathrm{C}(t)\,V_\mathrm{C}
+G_\mathrm{R}(t)\,V_\mathrm{R}+I_\mathrm{C}(t)\,.
\end{array}
\end{equation}
in which $G_\mathrm{L}(t)$, $G_\mathrm{C}(t)$ and
$G_\mathrm{R}(t)$ are time dependent synaptic conductances taking
the form specified in equation (\ref{si5}). The current
$I_\mathrm{C}(t)$ is a function of time only and takes a value
which is constructed by combining an appropriate form of
expression (\ref{ei3}) for the contribution of pure exogenous
current with the current arising from synaptic reversal
potentials. Equation (\ref{is1}) has been constructed from
equation (\ref{car6}) by replacing in an appropriate way each
constituent of the membrane current. For example, the contribution
due to capacitative current is given by the right hand side of
expression (\ref{gcm2}) and the contribution due to intrinsic
voltage-dependent current is given by the right hand side of
expression (\ref{gcm4}).

It is clear that the equation arising from the designated node of
an internal segment of a dendritic section contains only
$V_\mathrm{C}$, the potential at the designated node of the
segment itself, and the potentials $V_\mathrm{L}$ and
$V_\mathrm{R}$ at the designated nodes of the segments connected
respectively to its somal and distil ends. The model equation will
be linear if $g_\alpha$ is a constant function of $V$ across the
region of dendrite occupying $[x_\mathrm{L},x_\mathrm{R}]$ for
each ionic species $\alpha$, otherwise the presence of intrinsic
voltage-dependent channels will lead to nonlinear
behaviour.

\subsection{Designated node of a terminal segment}
A terminal segment of a dentritic section occurs at a dendritic
tip. In this case, the designated node $x_\mathrm{C}$ is at the
dendritic tip and $V_\mathrm{C}$ is the membrane potential at the
tip. In these circumstances the model value assigned to
$I_\mathrm{CR}$ is determined by the nature of the dendritic tip.
Here it will be assumed that dendritic terminals leak no axial
current (a sealed terminal) so that $I_\mathrm{CR}=0$. However,
the cut terminal characterised by the condition
$V_\mathrm{C}=V_\mathrm{ext}$ and the leaky terminal characterised
by $I_\mathrm{CR}+\kappa ( V_\mathrm{C}- V_\mathrm{ext})=0$ are
other less common possibilities where $V_\mathrm{ext}$ denotes the
potential of the exterior region. If $x_\mathrm{C}$ is the
designated node of a terminal segment of dendritic section then
the equation contributed by this node has form
\begin{equation}\label{ts1}
\hskip-2pt
\begin{array}{l}
\ds\Big(\frac{\pi g_\mathrm{A}r_\mathrm{L}r_\mathrm{C}}
{x_\mathrm{C}-x_\mathrm{L}}\Big)\;V_\mathrm{L}
-\Big(\frac{\pi g_\mathrm{A}r_\mathrm{L}r_\mathrm{C}}
{x_\mathrm{C}-x_\mathrm{L}}\Big)\;V_\mathrm{C} =
\frac{\pi\big(x_\mathrm{C}-x_\mathrm{L}\big)
c_\mathrm{M}}{2}\,\Big[\,r_\mathrm{L}\,\frac{dV_\mathrm{L}}{dt}
+3r_\mathrm{C}\,\frac{dV_\mathrm{C}}{dt}\,\Big]\\[12pt]
\ds\qquad+\;\frac{\pi\big(x_\mathrm{C}-x_\mathrm{L}\big)}{2}\,
\sum_\alpha\,\Big[\,r_\mathrm{L}g_\alpha(V_\mathrm{L})
(V_\mathrm{L}-E_\alpha)+3r_\mathrm{C}
g_\alpha(V_\mathrm{C})(V_\mathrm{C}-E_\alpha)\,\Big]\\[12pt]
\qquad\quad\ds+\;G_\mathrm{L}(t)\,V_\mathrm{L}+G_\mathrm{C}(t)\,
V_\mathrm{C}+I_\mathrm{C}(t)
\end{array}
\end{equation}
in which $G_\mathrm{L}(t)$, $G_\mathrm{C}(t)$ and $I_\mathrm{C}$
are again functions of time only. Again it is clear that the
equation arising from the terminal node of a dendritic section
contains only $V_\mathrm{C}$, the potential at the designated node
of the segment itself and $V_\mathrm{L}$, the potential at the
designated node of the segment connected to its somal end. The
model equation contributed by $x_\mathrm{C}$ will be linear in
this case if $g_\alpha$ is a constant function of $V$ across the
region of dendrite occupying $[x_\mathrm{L},x_\mathrm{C}]$ for
each ionic species $\alpha$, otherwise the presence of intrinsic
voltage-dependent channels will lead to nonlinear behaviour.

\subsection{Designated node at soma}
Suppose that $x_\mathrm{C}$ is the designated node of the soma
(assumed to be lumped) to which are connected a number of
dendritic segments and let $I_\mathrm{Soma}$ be the axial current
supplied by the soma to these segments then conservation of
current requires that
\begin{equation}\label{ns1}
\begin{array}{rcl}
I_\mathrm{Soma} \hskip-5pt & = & \hskip-5pt \ds \sum_\mathcal{S}\,\Big[\,
\Big(\frac{\pi g_\mathrm{A}r_\mathrm{C}r_\mathrm{R}}
{x_\mathrm{R}-x_\mathrm{C}}\Big)\;V_\mathrm{C}
-\Big(\frac{\pi g_\mathrm{A}r_\mathrm{C}r_\mathrm{R}}
{x_\mathrm{R}-x_\mathrm{C}}\Big)\;V_\mathrm{R}+
\frac{\pi\big(x_\mathrm{R}-x_\mathrm{C}\big)
c_\mathrm{M}}{2}\,\Big[\,3r_\mathrm{C}\,\frac{dV_\mathrm{C}}{dt}+
r_\mathrm{R}\,\frac{dV_\mathrm{R}}{dt}\,\Big]\\[12pt]
&&\ds\quad+\;\frac{\pi\big(x_\mathrm{R}-x_\mathrm{C}\big)}{2}\,\sum_\alpha\,\Big[\,
3r_\mathrm{C}g_\alpha(V_\mathrm{C})(V_\mathrm{C}-E_\alpha)+
r_\mathrm{R} g_\alpha(V_\mathrm{R})(V_\mathrm{R}-E_\alpha)\,
\Big]\\[12pt]
&&\qquad\ds+\;G_\mathrm{C}(t)\,V_\mathrm{C}+G_\mathrm{R}(t)
+I_\mathrm{C}(t)\,\Big]
\end{array}
\end{equation}
where $\mathcal{S}$ indicates that the summation is taken over all
dendritic segments connected to the soma. In the summation
$V_\mathrm{C}$ is the potential of the soma and $V_\mathrm{R}$ is
the potential at the node $x_\mathrm{R}$ of a somal segment that
has the somal node as neighbour. As previously, the equation
arising from the designated node at the soma contains only
$V_\mathrm{C}$, the potential of the soma, and $V_\mathrm{R}$, the
potential at the designated node of each somal segment nearest to
$x_\mathrm{C}$.

One common constitutive formula for $I_\mathrm{Soma}$ is
\begin{equation}\label{ns2}
I_\mathrm{Soma}=-r_\mathrm{S}\Big[c_\mathrm{S}\frac{dV_\mathrm{C}}
{dt}+\sum_\sigma\,g_\sigma(V_\mathrm{C})(V_\mathrm{C}-E_\sigma)
\Big]-I_\mathrm{S}
\end{equation}
where $r_\mathrm{S}$ is the membrane area of the soma,
$c_\mathrm{S}$ is the specific capacitance of the somal membrane,
$g_\sigma(V)$ is the membrane conductance of intrinsic
voltage-dependent channels on the soma of ionic species $\sigma$,
and $I_\mathrm{S}(t)$ is exogenous current. The model equation
contributed by $x_\mathrm{C}$ will be linear in this case if
$g_\alpha$ is a constant function of $V$ across the region of
dendrite occupying $[x_\mathrm{L},x_\mathrm{C}]$ for each ionic
species $\alpha$, otherwise the presence of intrinsic
voltage-dependent channels will lead to nonlinear behaviour.

\subsection{Designated node at a branch point}
The mathematical description of a dendritic branch point resembles
closely that of the soma except that the contribution of the soma
due to capacitance and membrane current is replaced by the
contribution from a parent dendrite. Suppose that node
$x_\mathrm{C}$ is a branch point connected to $m$ child sections.
The expression for the axial current leaving the end of the parent
section is
\begin{equation}\label{ld43}
\begin{array}{l}
\dfrac{V_\mathrm{L}}{r_\mathrm{L}}
-\dfrac{V_\mathrm{C}}{r_\mathrm{L}}
-c_\mathrm{M}\Big[\alpha_\mathrm{C}\;\dfrac{dV_\mathrm{L}}{dt}
+\xi_\mathrm{C}\;\dfrac{dV_\mathrm{C }}{dt}\Big]-
\ds{\sum_{\sigma,\;x_\mathrm{syn}}}\;g_\mathrm{syn}(t)\,
(V_\mathrm{syn}-E_\sigma)-I_\mathrm{injected}\\[12pt]
\qquad-\:\ds{\sum_\sigma}\;g_\sigma(V_\mathrm{C},t)\,
\Big[\alpha_\mathrm{C}\;(V_\mathrm{L}-E_\sigma)
+\xi_\mathrm{C}\;(V_\mathrm{C}-E_\sigma)\Big]
\end{array}
\end{equation}
where $\alpha_\mathrm{C}$, $\xi_\mathrm{C}$ and $r_\mathrm{L}$ are
defined by expressions (\ref{ld6}) and (\ref{ld2}) respectively.
Note that this expression is identical to that for $I_\mathrm{R}$
at a dendritic tip. In conclusion, the differential
equation contributed by the branch point is therefore
\begin{equation}\label{ld44}
\begin{array}{l}
c_\mathrm{M}\Big[\alpha_\mathrm{C}\;\dfrac{dV_\mathrm{L}}{dt}
+\xi_\mathrm{C}\;\dfrac{dV_\mathrm{C }}{dt}\Big]+
\ds{\sum_{\sigma,\;x_\mathrm{syn}}}\;g_\mathrm{syn}(t)\,
(V_\mathrm{syn}-E_\sigma)+I_\mathrm{injected}\\[12pt]
\qquad+\:\ds{\sum_\sigma}\;g_\sigma(V_\mathrm{C},t)\,
\Big[\alpha_\mathrm{C}\;(V_\mathrm{L}-E_\sigma)
+\xi_\mathrm{C}\;(V_\mathrm{C}-E_\sigma)\Big]+\dfrac{V_\mathrm{C}-V_\mathrm{L}}
{r_\mathrm{L}}+\sum_{k=1}^m\; I^{(k)}_\mathrm{L}=0\,.
\end{array}
\end{equation}

\section{The system of model differential equations}
Both the traditional and generalised compartmental models involve
the solution of a system of ordinary differential equations with
one equation arising from the description of the potential at each
designated node. Let
\begin{equation}\label{de1}
V(t)=\big[\,V_0(t),V_1(t),\cdots,V_n(t)\,]^\mathrm{T}
\end{equation}
be the column vector of dimension $(n+1)$ with $k$-th entry the
potential of the $k$-th designated node at time $t$. Each model
equation is based on conservation of axial current at the
designate node taking account of the connectivity of the node. The
construction of the model differential equations requires the
separate consideration of segments internal to a dendritic
section, terminal segments of a dendritic section, segments
connected at a branch point and segments attached to the soma.

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