### 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
\subsection{First simulation study}\label{sim1}

In this study, the performance of a traditional and the new
compartmental model is compared by assessing the accuracy with
which both models determine the time course of the somal potential
of the test neuron (Figure \ref{TestNeuron}) when the neuron is
subjected to large scale exogenous point input. Each simulation
distributes 75 point inputs at random over the dendritic tree of
the test neuron, where each input has strength
$2\times10^{-5}\,\mu$A. These inputs are then mapped to positions
on the Rall equivalent cylinder at the same electrotonic distance
from the soma (assumed to be a sphere of diameter $40\,\mu$m). The
time course of the potential at the soma of the equivalent
cylinder due to the combined effect of these inputs is determined
analytically and taken to be the reference potential against which
error in both compartmental models is measured. The difference
between a computed potential and its exact value is determined at
one millisecond intervals in the first 10 milliseconds of the
simulation, and each difference is divided by the exact potential
at that time to get a relative measure of error. The simulation
procedure is repeated 2000 times to determine the statistics of
the relative error for each of 13 different levels of spatial
discretisation (number of compartments).

\subsubsection{Results}

The results for this study are set out in Table \ref{simex1}. This
table shows the common logarithms of the mean value of the modulus
of the relative error and the standard deviation of that error
estimated ten milliseconds after the initiation of the stimulus.
Similar results (not shown) hold for all times at which the errors
were estimated.

\begin{table}[!h]
$\begin{array}{c|cc|cc} \hline \mbox{\begin{tabular}{c} Compartments \\[-5pt] (log_{10}(\mbox{Compartments})) \end{tabular}} & \multicolumn{2}{|c|}{\mbox{\begin{tabular}{cc} Traditional & New Model \\[-5pt] \multicolumn{2}{c}{\log_{10}(Mean)} \end{tabular}}} & \multicolumn{2}{|c}{\mbox{\begin{tabular}{cc} Traditional & New Model \\[-5pt] \multicolumn{2}{c}{\log_{10}(Standard Dev.)} \end{tabular}}}\\ \hline \phantom{0}17 \quad (1.2305) & -2.41151 & -2.71945 & -2.62290 & -3.19338 \\ \phantom{0}21 \quad (1.3222) & -2.47233 & -2.77674 & -2.69851 & -3.24583 \\ \phantom{0}34 \quad (1.5314) & -2.94299 & -3.41196 & -3.06731 & -3.88820 \\ \phantom{0}41 \quad (1.6127) & -3.04729 & -3.62138 & -3.17081 & -4.14997 \\ \phantom{0}54 \quad (1.7323) & -3.21258 & -3.89150 & -3.34889 & -4.41251 \\ \phantom{0}61 \quad (1.7853) & -3.24692 & -3.91268 & -3.37653 & -4.45051 \\ \phantom{0}75 \quad (1.8750) & -3.35180 & -4.12056 & -3.46881 & -4.65463 \\ \phantom{0}82 \quad (1.9138) & -3.39846 & -4.23567 & -3.51591 & -4.76498 \\ \phantom{0}93 \quad (1.9684) & -3.45602 & -4.30636 & -3.57633 & -4.82045 \\ 193 \quad (2.2855) & -3.77417 & -4.94731 & -3.89829 & -5.47886 \\ 293 \quad (2.4668) & -3.94409 & -5.31876 & -4.07811 & -5.84771 \\ 390 \quad (2.5910) & -4.08234 & -5.57349 & -4.20025 & -6.10791 \\ 495 \quad (2.6946) & -4.15996 & -5.78252 & -4.28525 & -6.32790 \\ \hline \end{array}$
\centering
\parbox{5in}{\caption{\label{simex1} The result of 2000 simulations
for each of 13 different levels of discretisation used in the
implementation of a traditional and new compartmental model. The
common logarithms of the mean value of the modulus of the relative
error and the standard deviation of that error are estimated at
ten milliseconds after the initiation of the stimulus.}}
\end{table}

The left hand panel of Figure \ref{mean} shows regression lines of
the common logarithms of the modulus of the mean relative error
(denoted by $\overline{RE\phantom{\vert\hskip-4pt}}\;$) for the
traditional (dashed line) and new (solid line) compartmental
models on the logarithm of the number of compartments (denoted by
$N$) used to represent the model neuron. These lines, based on the
data in Table \ref{simex1}, have equations
$$\label{mean1} \begin{array}{rcl} \log_{10}\overline{RE\phantom{\vert\hskip-4pt}}_\mathrm{\,\small traditional} & = & -1.09-1.17\log_{10}N\,, \$5pt] \log_{10}\overline{RE\phantom{\vert\hskip-4pt}}_\mathrm{\,\small new} & = & -0.17-2.10\log_{10}N \end{array}$$ in which the regressions are achieved with adjusted R^2 values\footnote{R^2 measures the proportion of the total variation of the dependent variable about its mean value that is explained by the regression, and necessarily takes a value between zero and one expressed as a percentage.} of 97.4\% and 99.5\% respectively. In view of the very high R^2 values for these regression equations, a number of conclusions can be drawn from this simulation study. For a fixed number of compartments, the error in the new compartmental model is always less than that of the traditional model. The regression equations (\ref{mean1}) support the argument made in Section \ref{assertion} that the error in a traditional compartmental model in the presence of exogenous point current input is approximately O(1/n), whereas the comparable error in the new compartmental model is approximately O(1/n^2). In practical terms, for example, the regression results (\ref{mean1}) suggest that the new compartmental model with 100 compartments achieves approximately the same level of accuracy as a traditional model with 500 compartments. \begin{figure}[!h] \centerline{\includegraphics[ ]{NCFig5a.eps} \quad\includegraphics[ ]{NCFig5b.eps}} \centering \parbox{5.2in}{\caption{\label{mean} The left panel shows the regression lines of the common logarithm of the mean relative errors in the new compartmental model (solid line) and a traditional compartmental model (dashed line) against the common logarithm of the number of compartments. All errors are measured ten milliseconds after initiation of the stimulus. The right panel shows the regression lines for the standard deviations of the mean relative errors for the new compartmental model (solid line) and for a traditional compartmental model (dashed line).}} \end{figure} %\begin{figure}[!h] %\centerline{\begin{mfpic}[56][24]{0.4}{3}{-7.5}{1} %\headlen7pt %\pen{1pt} %\dotspace=4pt %\dotsize=1.5pt %% %% x-axis %\tlabel[br](3.0,0.9){\textsf {\log_{10}(\mbox{No. Compartments})}} %\lines{(1.0,0),(3.0,0)} %\lines{(1.5,0),(1.5,-0.2)} %\lines{(2.0,0),(2.0,-0.2)} %\lines{(2.5,0),(2.5,-0.2)} %\lines{(3.0,0),(3.0,-0.2)} %\tlabel[bc](1.0,0.3){\textsf{1.0}} %\tlabel[bc](1.5,0.3){\textsf{1.5}} %\tlabel[bc](2.0,0.3){\textsf{2.0}} %\tlabel[bc](2.5,0.3){\textsf{2.5}} %\tlabel[bc](3.0,0.3){\textsf{3.0}} %% y-axis %\tlabel[bc](0.5,-6){\rotatebox{90}{\textsf{\log_{10}(\mbox{Mean relative error})}}} %\lines{(1,0),(1,-7)} %\lines{(1.0,-1.0),(1.05,-1.0)} %\lines{(1.0,-2.0),(1.05,-2.0)} %\lines{(1.0,-3.0),(1.05,-3.0)} %\lines{(1.0,-4.0),(1.05,-4.0)} %\lines{(1.0,-5.0),(1.05,-5.0)} %\lines{(1.0,-6.0),(1.05,-6.0)} %\lines{(1.0,-7.0),(1.05,-7.0)} %\tlabel[cr](0.95,-0.0){\textsf{0.0}} %\tlabel[cr](0.95,-1.0){\textsf{-1.0}} %\tlabel[cr](0.95,-2.0){\textsf{-2.0}} %\tlabel[cr](0.95,-3.0){\textsf{-3.0}} %\tlabel[cr](0.95,-4.0){\textsf{-4.0}} %\tlabel[cr](0.95,-5.0){\textsf{-5.0}} %\tlabel[cr](0.95,-6.0){\textsf{-6.0}} %\tlabel[cr](0.95,-7.0){\textsf{-7.0}} %% %% Mean values at t=10 %\dashed\lines{(1.2,-2.494),(3.0,-4.60)} %\lines{(1.2,-2.686),(3.0,-6.466)} %\end{mfpic} %\begin{mfpic}[56][24]{0}{3}{-7.5}{1} %\headlen7pt %\pen{1pt} %\dotspace=4pt %\dotsize=1.5pt %% %% x-axis %\tlabel[br](3.0,0.9){\textsf{\log_{10}(\mbox{No, Compartments})}} %\lines{(1.0,0),(3.0,0)} %\lines{(1.5,0),(1.5,-0.2)} %\lines{(2.0,0),(2.0,-0.2)} %\lines{(2.5,0),(2.5,-0.2)} %\lines{(3.0,0),(3.0,-0.2)} %\tlabel[bc](1.0,0.3){\textsf{1.0}} %\tlabel[bc](1.5,0.3){\textsf{1.5}} %\tlabel[bc](2.0,0.3){\textsf{2.0}} %\tlabel[bc](2.5,0.3){\textsf{2.5}} %\tlabel[bc](3.0,0.3){\textsf{3.0}} %% y-axis %\tlabel[bc](0.5,-6){\rotatebox{90}{\textsf{\log_{10}(\mbox{Standard Dev.})}}} %\lines{(1,0),(1,-7)} %\lines{(1.0,-1.0),(1.05,-1.0)} %\lines{(1.0,-2.0),(1.05,-2.0)} %\lines{(1.0,-3.0),(1.05,-3.0)} %\lines{(1.0,-4.0),(1.05,-4.0)} %\lines{(1.0,-5.0),(1.05,-5.0)} %\lines{(1.0,-6.0),(1.05,-6.0)} %\lines{(1.0,-7.0),(1.05,-7.0)} %\tlabel[cr](0.95,-0.0){\textsf{0.0}} %\tlabel[cr](0.95,-1.0){\textsf{-1.0}} %\tlabel[cr](0.95,-2.0){\textsf{-2.0}} %\tlabel[cr](0.95,-3.0){\textsf{-3.0}} %\tlabel[cr](0.95,-4.0){\textsf{-4.0}} %\tlabel[cr](0.95,-5.0){\textsf{-5.0}} %\tlabel[cr](0.95,-6.0){\textsf{-6.0}} %\tlabel[cr](0.95,-7.0){\textsf{-7.0}} %% %% Standard deviations at t=10 %\dashed\lines{(1.2,-2.664),(3.0,-4.680)} %\lines{(1.2,-3.169),(3.0,-7.021)} %\end{mfpic}} %\centering %\parbox{5.8in}{\caption{\label{mean} The left panel shows the %regression lines of the mean relative errors in the new %compartmental model (solid line) and that of a traditional %compartmental model (NEURON - dashed line) against number of %compartments. All errors are measured ten milliseconds after %initiation of the stimulus. The right panel shows the regression %lines for the standard deviations of the mean relative errors for %the new compartmental model (solid line) and for a traditional %compartmental model (NEURON - dashed line).}} %\end{figure} The standard deviation (SD) of the modulus of the relative error can be regarded as an indicator of the reliability of a single application of the model. The right hand panel of Figure \ref{mean} shows regression lines of the common logarithms of the standard deviation of the modulus of the relative error for the traditional (dashed line) and new (solid line) compartmental models on the logarithm of the number of compartments used to represent the model neuron. These lines, based on the data in Table \ref{simex1}, have equations $$\label{mean2} \begin{array}{rcl} \log_{10}\,\mbox{SD}_\mathrm{\,\small traditional} & = & -1.32-1.12\log_{10}N\,, \\[5pt] \log_{10}\,\mbox{SD}_\mathrm{\,\small new} & = & -0.60-2.14\log_{10}N \end{array}$$ in which the regressions are achieved with adjusted R^2 values of 98.7\% and 99.4\% respectively. These regression lines show that the new compartmental model is more reliable than a traditional compartmental model. For example, a traditional compartmental model requires at least 100 compartments to give a standard deviation of the modulus of the relative error that is smaller than that of the new compartmental model using 40 compartments. \subsection{Second simulation study}\label{sim2} In the second simulation study 100 synapses are distributed at random over the dendritic tree of the test neuron illustrated in Figure \ref{TestNeuron}. Each synapse is activated independently of all other synapses, has a maximum conductance of 3\times10^{-5}\,\mbox{mS} and a rise time of 0.5 msec. Activation times for each synapse follow Poisson statistics with a mean rate of 30 pre-synaptic spikes per second. Spikes are generated by the soma of the test neuron using Hodgkin-Huxley kinetics. This study is based on 12 different levels of spatial discretisation (number of compartments) in which each simulation of the traditional and new compartmental models use identical synaptic firing times and identical numbers of compartments. \subsubsection{Results} Table \ref{spikerate} gives the spike rate of soma-generated action potentials based on 11 seconds of activity, the first second of which is ignored. \begin{table}[!h] \[ \begin{array}{c|c|c} \hline \mbox{\begin{tabular}{c} Compartments \\[-5pt] (log_{10}(\mbox{Compartments})) \end{tabular}} & \mbox{\begin{tabular}{c} Traditional Model \\[-5pt] Mean Firing Rate \end{tabular}} &\mbox{\begin{tabular}{c} New Model \\[-5pt] Mean Firing Rate \end{tabular}}\\ \hline \phantom{0}34 \quad (1.5314) & 31.5 & 27.6 \\ \phantom{0}41 \quad (1.6127) & 30.3 & 27.9 \\ \phantom{0}54 \quad (1.7323) & 30.5 & 27.5 \\ \phantom{0}61 \quad (1.7853) & 29.8 & 27.2 \\ \phantom{0}75 \quad (1.8750) & 29.2 & 27.0 \\ \phantom{0}82 \quad (1.9138) & 28.5 & 27.0 \\ \phantom{0}93 \quad (1.9684) & 28.3 & 26.8 \\ 193 \quad (2.2855) & 26.5 & 26.5 \\ 293 \quad (2.4668) & 25.9 & 26.2 \\ 390 \quad (2.5910) & 26.2 & 26.2 \\ 495 \quad (2.6946) & 26.7 & 26.2 \\ 992 \quad (2.9965) & 26.0 & 26.1 \\ \hline \end{array}$
\centering
\parbox{5in}{\caption{\label{simex2} The spike rate estimated from
a 10 second record of spike train activity obtained from a
traditional and the new compartmental model at 12 different levels
of spatial discretisation (number of compartments).}}
\end{table}

Figure \ref{spikerate} illustrates the data set out in Table
\ref{simex2} in which the spike rates for the traditional model
(dashed line) and new model (solid line) are plotted against the
common logarithm of $N$, the number of compartments used in each
simulation. As $N$ is increased, the spike rates generated by both
models approach a common limit. However, the spike rate generated
by the traditional model approaches this limit more slowly and
appears to oscillate as the limit is approached. The spike rate
obtained using the traditional model with 500 compartments is
achieved in the new model with only 100 compartments. These
differences in the number of compartments required to achieve the
same level of accuracy in both models are identical to those
observed in the first study.

\begin{figure}[!h]
\centering
\includegraphics[ ]{NCFig6.eps}
\vskip5pt
\parbox{5.5in}{\caption{\label{spikerate} The spike rate plotted
against the common logarithm of the number of compartments for a
traditional compartmental model (dashed line) and the new
compartmental model (solid line). The dotted line shows the
expected spike rate.}}
\end{figure}

%\begin{figure}[!h]
%\centerline{\begin{mfpic}[75][20]{0.3}{3}{-1}{8.5}
%\pen{1pt}
%\dotspace=4pt
%\dotsize=1.5pt
%%
%% x-axis
%\tlabel[tr](3.0,-1){\textsf {$\log_{10}(\mbox{No. Compartments})$}}
%\lines{(1.0,0),(3.0,0.0)}
%\lines{(1.0,0),(1.0,-0.3)}
%\lines{(1.5,0),(1.5,-0.3)}
%\lines{(2.0,0),(2.0,-0.3)}
%\lines{(2.5,0),(2.5,-0.3)}
%\lines{(3.0,0),(3.0,-0.3)}
%\tlabel[tc](1.0,-0.5){\textsf{1.0}}
%\tlabel[tc](1.5,-0.5){\textsf{1.5}}
%\tlabel[tc](2.0,-0.5){\textsf{2.0}}
%\tlabel[tc](2.5,-0.5){\textsf{2.5}}
%\tlabel[tc](3.0,-0.5){\textsf{3.0}}
%%
%% Expected spike rate
%\dotted\lines{(1.5,2.6),(3.2,2.6)}
%%
%% Traditional model (Modulo spike rate of 25)
%\dashed\lines{
%(1.531,8.0),(1.613,6.8),(1.732,7.0),(1.785,6.3),
%(1.875,5.7),(1.914,5.0),(1.968,4.8),(2.286,3.0),
%(2.467,2.4),(2.591,2.7),(2.696,3.2),(2.997,2.5)}
%%
%% New model (Modulo spike rate of 25)
%\lines{
%(1.531,4.1),(1.613,4.4),(1.732,4.0),(1.785,3.7),
%(1.875,3.6),(1.914,3.5),(1.968,3.3),(2.286,3.0),
%(2.467,2.7),(2.591,2.7),(2.695,2.7),(3.000,2.6)}
%% y-axis
%\lines{(1,0),(1,0.5)}
%\dashed\lines{(1,0.5),(1,2.0)}
%\lines{(1,2.0),(1,8.5)}
%\lines{(1.0,0.0),(0.95,0.0)}
%\lines{(1.0,2.5),(0.95,2.5)}
%\lines{(1.0,4.5),(0.95,4.5)}
%\lines{(1.0,6.5),(0.95,6.5)}
%\lines{(1.0,8.5),(0.95,8.5)}
%\tlabel[cr](0.9,0.0){\textsf{0.0}}
%\tlabel[cr](0.9,2.5){\textsf{26.0}}
%\tlabel[cr](0.9,4.5){\textsf{28.0}}
%\tlabel[cr](0.9,6.5){\textsf{30.0}}
%\tlabel[cr](0.9,8.5){\textsf{32.0}}
%\tlabel[tc](0.5,6.5){\rotatebox{90}{\textsf{Spikes per second}}}
%\end{mfpic}}
%\centering
%\vskip5pt
%\parbox{5.5in}{\caption{\label{spikerate} The spike rate plotted against
%the common logarithm of the number of compartments for a
%traditional compartmental model (dashed line) and the new
%compartmental model (solid line). The dotted line shows the
%expected spike rate.}}
%\end{figure}

\subsubsection{Comparison of model-generated spike trains}

It is clear from Figure \ref{spikerate} that the mean rate of the
spike train generated by the new compartmental model converges
more quickly to the theoretical mean spike rate than that
generated by a traditional compartmental model. One would
therefore infer from the behaviour of this summary statistic that
the spike train generated by the former is a more accurate
representation of the spiking behaviour of the test neuron in
response to synaptic activity than that generated by the latter.
To investigate the validity of this inference requires an accurate
comparison of the times of occurrence of the spikes in the spike
trains generated by each model with identical synaptic activity
applied to the test neuron. We take as our reference the times of
occurrence of the spikes generated in ten seconds using the new
compartmental model with 100 compartments (spike train
$\mathcal{N}_{100}$). These spike times are compared with those
generated by a traditional compartmental model with 100
compartments and with 500 compartments\footnote{All the
simulations were run on a PC with dual Athlon 1500MP processors.
The times required to simulate 10 seconds of spike train data were
61 minutes for the new compartmental model with 100 compartments,
41 minutes and 353 minutes for a traditional compartmental model
with 100 and 500 compartments respectively. In the presence of
point current input alone, the computational times for both models
are identical.} (spike trains $\mathcal{T}_{100}$ and
$\mathcal{T}_{500}$ respectively). The times of occurrence of
spikes in the spike trains to be compared are taken to be
identical if they occur within one millisecond of each other. The
comparison between $\mathcal{N}_{100}$ and $\mathcal{T}_{100}$
revealed 244 spikes common to both spike trains (\emph{i.e.}
occurring within one millisecond of each other). There were 24
spikes unique to $\mathcal{N}_{100}$ and 39 spikes unique to
$\mathcal{T}_{100}$. The comparison between $\mathcal{N}_{100}$
and $\mathcal{T}_{500}$ revealed 258 spikes common to both spike
trains with 10 spikes unique to $\mathcal{N}_{100}$ and 9 spikes
unique to $\mathcal{T}_{500}$. Since the reference spike train
$\mathcal{N}_{100}$ is common to both comparisons, it is clear
that as the number of compartments in a traditional model
increases, the spike train generated by that model will conform
more closely to that generated by the new compartmental model with
significantly fewer compartments.

\section{Concluding remarks}

We have demonstrated that it is possible to achieve a significant
increase in the accuracy and precision of compartmental models by
developing a new compartmental model in which compartments have
two potentials -- one at either end of the segment which the
compartment represents. The new compartment acts as fundamental
unit in the construction of a model of a branched dendrite. When
these compartments are connected by requiring continuity of
potential and conservation of current at segment boundaries, they
provide a new type of compartmental model with a mathematical form
identical to that of a traditional model in the sense that both
types of compartmental model involve only nearest neighbour
interactions. One demonstrated benefit of the new compartmental
model is that it provides a mechanism to take account of the exact
location of point process input by contrast with traditional
compartmental models which would assign this input to an accuracy
of half the length of a segment. We would anticipate that the
application of the new compartmental model would be most useful in
association with experiments in which the precise timing of spikes
is thought to be important (\emph{e.g.}, Oram \emph{et al}.,
\cite{Oram99} and the references therein) or in studies
investigating the influence of the location of synaptic input on
the mean rate of the spike train output (\emph{e.g.}, Poirazi
\emph{et al}., \cite{Poirazi03}).

\section*{Acknowledgement}

A.E. Lindsay would like to thank the Wellcome Trust for the award
of Vacation Scholarship (VS/03/GLA/8/SL/TH/FH).