### 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;
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\section{Construction of compartmental models}
identical partitioning of a neuron into segments which interact
with the extracellular region (usually taken to be zero potential)
by means of transmembrane current driven by the potential of the
segment. In the mathematical model, each segment becomes a
compartment with electrical properties chosen to reflect the
biophysical properties of the segment it represents. Input to the
segment becomes input to the compartment, and shapes its potential
through a series of mathematical equations based on Kirchhoff's
circuit laws, and the fact that compartments only interact with
their nearest neighbours.

The traditional compartmental model treats each segment as an
iso-potential region of dendrite (\emph{e.g.} Segev and Burke,
\cite{Segev98}), and therefore all spatially distributed input
falling onto that segment experiences the same potential in the
model. Consequently, the electrical effect of spatial distributed
input on a segment is lost, and in practice all input to a segment
is assumed to act at a single point, conventionally taken to be
the centre of the segment.

In the generalised compartmental model of a neuron the membrane
potential of a segment is allowed to vary along its length
allowing  the solution for the membrane potential in the
generalised model to more accurately reflect the influence of
spatially distributed input. While this might appear to be a minor
difference between both models, and of no great consequence, it
will be shown by simulation that this modification improves both
the accuracy and precision of the compartmental model by an order
of magnitude. Furthermore, this improvement will be achieved
without any significant increase in computational effort. To
appreciate the consequences of a non-constant compartmental
potential, the steps in the construction of the traditional
compartmental model are reviewed briefly before a detailed
description of the generalised model is presented.

\subsection{Geometrical construction of dendritic segments}
Compartmental models of the neuron usually regard the dendritic
section as a sequence of contiguous tapered circular
cylinders\footnote{ One widely used implementation of
compartmental models is the NEURON simulator developed by Hines
and Carnevale (\cite{Hines97}).}. Any point at which a noticeable
change in taper occurs is a natural candidate for the designated
node of a dendritic segment, by this we mean the node at which the
membrane potential of that segment is recorded. In practice, the
designated nodes of a segment will correspond to the points at
which morphological information is available. Figure \ref{2d}
shows a dendritic segment
$\mathcal{C}=[x_\mathrm{P},x_\mathrm{Q}]$ where $x_\mathrm{P}$ and
$x_\mathrm{Q}$ are the respective distances of the left hand and
right hand endpoints of the segment from the somal end of the
dendritic section.

\begin{figure}[!h]
$\begin{array}{c} \begin{mfpic}[1.1][1.1]{-30}{150}{155}{320} \pen{1pt} \headlen4pt \arrow\lines{(15,300),(5,300)} \arrow\lines{(80,300),(90,300)} \headlen7pt \dotspace=4pt \dotsize=1.5pt % % Central cylinder \lines{(100,272),(130,272)} \lines{(100,208),(130,208)} % % LH cylinder \dotted\parafcn[s]{0,180,5}{(100+24*sind(t),240+32*cosd(t))} \parafcn[s]{0,180,5}{(100-24*sind(t),240+32*cosd(t))} \lines{(0,288),(100,272)} \lines{(0,192),(100,208)} % % Partial cylinder on left \dotted\parafcn[s]{0,180,5}{(36*sind(t),240+48*cosd(t))} \parafcn[s]{0,180,5}{(-36*sind(t),240+48*cosd(t))} \lines{(-30,288),(0,288)} \lines{(-30,192),(0,192)} % % Annotation of LH cylinder \arrow\lines{(30,240),(60,240)} \tlabel[bl](55,250){I_\mathrm{PQ}} \tlabel[bc](0,250){V_\mathrm{P}} \tlabel[cc](0,300){\large x_\mathrm{P}} \tlabel[cc](0,240){\large \bullet} \arrow\lines{(0,230),(0,170)} \tlabel[tc](0,165){I^{(m)}_\mathrm{P}} % % Annotation of Central cylinder \dotted\parafcn[s]{0,180,5}{(50+30*sind(t),240+40*cosd(t))} \parafcn[s]{0,180,5}{(50-30*sind(t),240+40*cosd(t))} \tlabel[bc](100,250){V_\mathrm{Q}} \tlabel[cc](100,300){\large x_\mathrm{Q}} \tlabel[cc](100,240){\large \bullet} \arrow\lines{(100,230),(100,170)} \tlabel[tc](100,165){I^{(m)}_\mathrm{Q}} % % Form compartment \tlabel[cc](50,300){\textsf{Compartment}} \end{mfpic} \end{array}\qquad \begin{tabular}{p{1.9in}} \caption{\label{2d} The compartment occupying [x_\mathrm{P},x_\mathrm{Q}] is represented. Currents I^{(m)}_\mathrm{P} and I^{(m)}_\mathrm{Q} flow across the membrane at x_\mathrm{P} and x_\mathrm{Q} respectively and axial current I_\mathrm{PQ} flows from x_\mathrm{P} to x_\mathrm{Q} through the resistive dendritic core.} \end{tabular}$
\end{figure}

Let $r_\mathrm{P}$ and $r_\mathrm{Q}$ be the respective radii of
the dendrite at the nodes $x_\mathrm{P}$, and $x_\mathrm{Q}$. When
the dendritic membrane is formed by rotating the straight line
segment $[x_\mathrm{P},x_\mathrm{Q}]$ about the axis of the
dendrite, as illustrated in Figure \ref{2d}, the membrane of the
$$\label{car2} r(x) = \frac{r_\mathrm{P}(x_\mathrm{Q}-x) +r_\mathrm{Q}(x-x_\mathrm{P})}{x_\mathrm{Q}-x_\mathrm{P}}\,, \qquad x\in\mathcal{C}\,.$$
Moreover, it is straight forward Calculus to verify that
$$\label{dc1} \int_\mathcal{C}\,r(x)\,dx= \frac{1}{2}\,\big(x_\mathrm{Q}-x_\mathrm{P}\big) \big(r_\mathrm{P}+r_\mathrm{Q}\big)\,.$$

\subsection{Intracellular resistance of dendritic segments}
Assuming that the intracellular medium of the dendrite has
constant conductance $g_\mathrm{A}$ (mS/cm), the general
expressions for the axial resistance of the segment illustrated in
Figure \ref{2d} is
$$\label{car1} R_\mathrm{PQ} = \frac{1}{g_\mathrm{A}}\, \int_{\mathcal{C}}\,\frac{dx}{\pi r^2(x)}= \frac{x_\mathrm{Q}-x_\mathrm{P}} {\pi g_\mathrm{A} r_\mathrm{P}r_\mathrm{Q}}$$
where $r(x)$ is the radius of the dendritic section at position
$x$, and the value stated in (\ref{car1}) is that for the
piecewise tapered dendritic segment with radius given by
(\ref{car2}). If $V_\mathrm{L}$ and $V_\mathrm{Q}$ are the
respective potentials at the endpoints $x_\mathrm{P}$ and
$x_\mathrm{Q}$ of the segment $\mathcal{C}$ then, in the absence
of transmembrane current across the segment, the currents
$I_\mathrm{PQ}$ appearing in Figure \ref{2d} is determined by
Ohm's law and has value
$$\label{car4} I_\mathrm{PQ} = \ds\frac{(V_\mathrm{P}-V_\mathrm{Q})} {R_\mathrm{PQ}} = \frac{\pi g_\mathrm{A}r_\mathrm{P}r_\mathrm{Q}} {x_\mathrm{Q}-x_\mathrm{P}}\;\big(V_\mathrm{P}-V_\mathrm{Q}\big)\,.$$
The dependence of $V_\mathrm{P}$ and $V_\mathrm{Q}$ on time $t$
has been suppressed in equation (\ref{car4}) for representational
simplicity, but is will be understood henceforth that all
potentials at segment endpoints are functions of time although the
dependence on $t$ is not made explicit. Most importantly, the
calculation which leads to expressions (\ref{car1}) and
(\ref{car4}) for the compartment resistance and axial current also
implies that the potential distribution within the segment is
$$\label{tc1} V(x,t) = V_\mathrm{P}-I_\mathrm{PQ}\,\int^x_{x_\mathrm{P}} \;\frac{ds}{\pi g_\mathrm{A}\,r^2(s)}\,,\qquad x\in\mathcal{C}\,.$$
For a dendritic section that is uniformly tapered this
distribution is
$$\label{tc2} V(x,t) = \frac{V_\mathrm{P}\,r_\mathrm{P}\,(x_\mathrm{Q}-x)+ V_\mathrm{Q}\,r_\mathrm{Q}(x-x_\mathrm{P})}{r(x)\, (x_\mathrm{Q}-x_\mathrm{P})}\,,\qquad x\in\mathcal{C}\,.$$
Furthermore, the average potential of the compartment as measured
in terms of the charge carrying capacity of its membrane is
defined by
$$\label{tc3} V_\mathcal{C}=\frac{\ds\int_\mathcal{C}\,r(x)V(x,t)\,dx} {\ds\int_\mathcal{C}\,r(x)\,dx}= \frac{V_\mathrm{P}\,r_\mathrm{P}+V_\mathrm{Q}\,r_\mathrm{Q}} {r_\mathrm{P}+r_\mathrm{Q}}\,.$$

Both the traditional and generalised compartmental models are
based on Kirchhoff's current law which asserts that $I_\mathrm{LC} = I^{(m)}_\mathrm{C}+I_\mathrm{CR}$. Thus
$$\label{car5} \frac{V_\mathrm{L}}{R_\mathrm{LC}} -\Big(\,\frac{V_\mathrm{C}}{R_\mathrm{LC}} +\frac{V_\mathrm{C}}{R_\mathrm{CR}}\,\Big) +\frac{V_\mathrm{R}}{R_\mathrm{CR}} =I^{(m)}_\mathrm{C}\,,$$
or in terms of the dendritic radii,
$$\label{car6} \Big(\frac{\pi g_\mathrm{A}r_\mathrm{L}r_\mathrm{C}} {x_\mathrm{C}-x_\mathrm{L}}\Big)\;V_\mathrm{L} -\Big(\frac{\pi g_\mathrm{A}r_\mathrm{L}r_\mathrm{C}} {x_\mathrm{C}-x_\mathrm{L}} +\frac{\pi g_\mathrm{A}r_\mathrm{C}r_\mathrm{R}} {x_\mathrm{R}-x_\mathrm{C}}\Big)\;V_\mathrm{C} +\Big(\frac{\pi g_\mathrm{A}r_\mathrm{C}r_\mathrm{R}} {x_\mathrm{R}-x_\mathrm{C}}\Big)\;V_\mathrm{R}=I^{(m)}_\mathrm{C}\,.$$

\subsection{Specification of transmembrane current}
The current crossing the membrane of the dendritic segment in
Figure \ref{2d} has general expression
$$\label{tc0} \begin{array}{rcl} I^{(m)} & = & \ds 2\pi c_\mathrm{M}\,\frac{d}{dt}\, \int_\mathcal{C}\, r(x)\,V(x,t)\,dx +2\pi\int_\mathcal{C}\, r(x)J_\mathrm{IVDC}(V(x,t))\,dx\$12pt] &&\quad\ds+\;\sum_\mathcal{C}\,J_\mathrm{SYN}(V(x,t)) +\sum_\mathcal{C}\,I_\mathrm{EX}(x,t) \end{array}$$ where c_\mathrm{M} (\muF/cm^2) is the specific capacitance (assumed constant) of the segment membrane, V(x,t) is the membrane potential of the dendritic section at time t (ms) and distance x from its somal end, J_\mathrm{IVDC}(x,t) is the density of transmembrane current (\muA/cm^2) due to intrinsic voltage-dependent channel activity, J_\mathrm{SYN}(x,t) is the linear density of synaptic current (\muA/cm) due to synaptic activity falling on the segment and I_\mathrm{EX}(x,t) is the linear density of exogenous current (\muA/cm). The difference between the traditional and generalised compartmental models lies in the mathematical representation of I^{(m)}. \section{The traditional compartmental model} In the traditional compartmental model, the compartment is assumed to be iso-potential with membrane potential V_\mathcal{C}, the average potential of the compartment. With this assumption, the transmembrane current in the traditional compartmental model simplifies to $$\label{tc00} I^{(m)} = \pi\big(x_\mathrm{Q}-x_\mathrm{P}\big) \big(r_\mathrm{P}+r_\mathrm{Q}\big)\,\Big[\,c_\mathrm{M}\, \frac{dV_\mathcal{C}}{dt}\,+J_\mathrm{IVDC}(V_\mathrm{C})\,\Big] +\sum_\mathcal{C}\,J_\mathrm{SYN}(V_\mathrm{C}) +\sum_\mathcal{C}\,I_\mathrm{EX}(x,t)\,.$$ For segments constructed from piecewise tapered elements, the values of these integrals are given in equations (\ref{dc1}). \subsection{The model differential equations} To make explicit the essential features of the mathematical problem that must be solved when using the traditional compartmental model to describe neuronal behaviour, it is necessary to state how intrinsic voltage-dependent current and synaptic current are to be modelled. The most common description of intrinsic voltage-dependent current is due to Hodgkin and Huxley (\cite{Hodgkin52}) and assumes that the contribution to transmembrane current density from ionic channels of species \alpha is J_\mathrm{IVDC}=g_\alpha(V)(V-E_\alpha) where E_\alpha is the reversal potential for the species \alpha. The conductance g_\alpha is defined in terms of auxiliary variables which themselves satisfy differential equations with coefficients that are dependent on the local membrane potential. It is in this sense that the conductance g_\alpha(V) is dependent on the membrane potential. On the other hand, synaptic input due to ionic species \beta is modelled by the specification J_\mathrm{SYN}=g_\beta(t)(V-E_\beta) where E_\beta is the reversal potential for the species and g_\beta(t) is now the time course of the synaptic conductance. With these model representations of intrinsic voltage-dependent current and synaptic current, the transmembrane current at x_\mathrm{C} has generic form $$\label{mde1} \begin{array}{rcl} I^{(m)}_\mathrm{C} & = & \ds C_\mathrm{C}\, \frac{dV_\mathrm{C}}{dt}\,+\sum_\alpha\; G^\mathrm{IVDC}_\alpha(V_\mathrm{C})\big(\,V_\mathrm{C}-E_\alpha\,\big) +\sum_\beta\,G^\mathrm{SYN}_\beta(t) \big(\,V_\mathrm{C}-E_\beta\,\big)+I_\mathrm{C}(t) \end{array}$$ where C_\mathrm{C} (constant) is the total membrane capacitance of the segment, G^\mathrm{IVDC}_\alpha(V_\mathrm{C}) denotes the total intrinsic voltage-dependent conductance of the channels of ionic species \alpha associated with the segment, G^\mathrm{SYN}_\beta(t) is the total synaptic conductance at time t associated with channels of ionic species \beta falling on the segment and I_\mathrm{C}(t) plays the role of the total exogenous current input to the segment at time t. Suppose that the neuron is partitioned into m compartments where the membrane potential at the designated node of the k^{th} compartment is V_k(t) and let $$\label{mde2} V(t)=\big[\,V_1(t),V_1(t),\cdots,V_m(t)\,]^\mathrm{T}\,.$$ It follows immediately from the expression (\ref{mde1}) for the transmembrane current I^{(m)}_\mathrm{C} and (\ref{car5}) that V(t), the column vector of membrane potentials, satisfies the ordinary differential equations $$\label{mde3} D^\mathrm{C}\,\frac{dV}{dt}+D^\mathrm{IVDC}(V)\,V+D^\mathrm{SYN}(t)\,V +I(t)=AV$$ where D^\mathrm{C} is a constant diagonal matrix, D^\mathrm{IVDC}(V) is a diagonal matrix of intrinsic voltage-dependent conductances, D^\mathrm{SYN}(t) is a diagonal matrix of synaptic conductances and I(t) is a column vector of exogenous currents. The (j,k)^{th} entry of the matrix A, which is interpreted as a conductance matrix, is nonzero if the j^{th} and k^{th} designated nodes are neighbours, otherwise the entry is zero. The computational complexity of the final mathematical problem is determined by the structure of A provided all other matrices arising in the mathematical specification of the problem are not more complex than A. This is certainly true for the traditional compartmental model since all matrices other than A are diagonal. Integration of equation (\ref{mde3}) over the interval [t,t+h] yields $$\label{mde4} D^\mathrm{C}\,\big[\,V(t+h)-V(t)\,\big]+ \int_t^{t+h}\,\big[\,D^\mathrm{IVDC}(V)+ D^\mathrm{SYN}(t)\,\big]\,V(t)\,dt +\int_t^{t+h}\,I(t)=A\int_t^{t+h}\,V(t)\,dt\,.$$ The trapezoidal quadrature is used to replace each integral in equation (\ref{mde4}) with the exception of the integral of intrinsic voltage-dependent current, which is replaced by the midpoint quadrature. The result of this calculation is $$\label{mde5} \begin{array}{l} \ds D^\mathrm{C}\,\big[\,V(t+h)-V(t)\,\big]+ h\,D^\mathrm{IVDC}(V(t+h/2))\,V(t+h/2)\\[10pt] \quad\ds+\;\frac{h}{2}\,\Big[\,D^\mathrm{SYN}(t+h)V(t+h) +D^\mathrm{SYN}(t)V(t)\,\Big]+\frac{h}{2} \Big[\,I(t+h)+I(t)\,\Big]\\[10pt] \qquad\ds = \frac{h}{2}\,\Big[\,A V(t+h)+AV(t)\,\Big]+O(h^3)\,. \end{array}$$ On taking account of the fact that \[ V(t+h/2)=\frac{1}{2} \,\Big[\,V(t+h)+V(t)\,\Big]+O(h^2)\,,$
equation (\ref{mde5}) may be reorganised to give
$$\label{mde6} \begin{array}{l} \ds \Big[\,2 D^\mathrm{C}-hA+h\,D^\mathrm{SYN}(t+h) +h\,D^\mathrm{IVDC}(V(t+h/2))\,\Big]\,V(t+h) = \$10pt] \qquad\ds \Big[\,2 D^\mathrm{C}+hA+h D^\mathrm{SYN}(t) -h\,D^\mathrm{IVDC}(V(t+h/2))\,\Big]\,V(t) -h\Big[\,I(t+h)+I(t)\,\Big] \end{array}$$ when the error structure is ignored. The detailed computation of D^\mathrm{IVDC}(V(t+h/2)) is determined entirely by the structure of the auxiliary equations. In the case of Hodgkin-Huxley like channels, it is standard knowledge that D^\mathrm{IVDC}(V(t+h/2)) can be computed to adequate accuracy from V(t) and the differential satisfied by the auxiliary variables (Lindsay \emph{et al.}, \cite{Lindsay01a}). \section{The generalised compartmental model} Equations (\ref{tc1}) and (\ref{tc2}) provide the basis for the construction of the transmembrane current in the generalised compartmental model. Within the framework of this model, distributed transmembrane current and point sources of transmembrane current receive a different mathematical treatment. To appreciate why this is the case, consider a cylindrical dendritic segment of radius a, length L and with membrane of constant conductance g_\mathrm{M}. Suppose that the segment is filled with intracellular medium of conductance g_\mathrm{A} and that a potential difference V exists across its length L. The axial current flowing along the segment is I_\mathrm{A}=\pi a^2 g_\mathrm{A} V/L and the total transmembrane current is I_\mathrm{M}=2\pi a L g_\mathrm{M}\,(V/2). Thus $$\label{pc1} \frac{\mbox{Transmembrane current}}{\mbox{Axial current}} =\frac{I_\mathrm{M}}{I_\mathrm{A}}=\frac{\pi a L g_\mathrm{M}\,V} {\pi a^2 g_\mathrm{A}\,(V/L)}=\frac{L^2 g_\mathrm{M}} {a g_\mathrm{A}}=\Big(\frac{L}{a}\Big)^2\, \frac{a g_\mathrm{M}}{g_\mathrm{A}}\,.$$ For a typical dendrite a g_\mathrm{M}/g_\mathrm{A}\approx 10^{-5} which in turn suggest that membrane current losses are comparable to axial current only provided the segment is several orders of magnitude longer than its radius. Since the model only allows current to flow across the membrane at designated nodes, a well-structured compartmental model requires that the internodal distance does not become several orders of magnitude greater than the radius of the dendrite. By meeting this requirement, the effect of the transmembrane current on the axial current is locally negligible. Provided a compartment is not excessively long, the implication of this argument is that distributed transmembrane current has negligible impact on the local axial current flowing between designated nodes. Consequently, the effect of this distributed transmembrane current may be described in terms of the membrane potential computed from the axial current by ignoring the transmembrane current itself. On the other hand, point input of current due to synaptic activity or exogenous input necessarily causes a discontinuity in axial current irrespective of the size of the compartment or the strength of the input. Consequently, point current input must be treated separately from distributed input because it necessarily generates discontinuities in axial current and therefore affects the local transmembrane potential. The treatment of distributed transmembrane current takes advantage of the identity $$\label{gcm1} \begin{array}{rcl} \ds\int_{x_\mathrm{P}}^{(x_\mathrm{P}+x_\mathrm{Q})/2}\,r(x)\,V(x,t)\,dx & = & \ds\frac{1}{8}\big(x_\mathrm{Q}-x_\mathrm{P}\big) \big(\,3r_\mathrm{P} V_\mathrm{P}+r_\mathrm{Q} V_\mathrm{Q}\,\big)\,,\\[12pt] \ds\int_{(x_\mathrm{P}+x_\mathrm{Q})/2}^{x_\mathrm{Q}}\,r(x)\,V(x,t)\,dx & = & \ds\frac{1}{8}\big(x_\mathrm{Q}-x_\mathrm{P}\big) \big(r_\mathrm{P} V_\mathrm{P}+3r_\mathrm{Q} V_\mathrm{Q}\,\big)\,. \end{array}$$ These equations describe how transmembrane current falling on the segment should be partitioned between the membrane currents I^{(m)}_\mathrm{P} and I^{(m)}_\mathrm{Q} crossing the membrane at x_\mathrm{P} and x_\mathrm{Q}. \subsection{Capacitative current} It now follows immediately from the general expression (\ref{tc0}) for transmembrane current that the capacitative component of this current is $$\label{gcm2} \begin{array}{l} \ds 2\pi c_\mathrm{M}\,\frac{d}{dt}\, \int_\mathcal{L}\, r(x)\,V(x,t)\,dx+2\pi c_\mathrm{M}\, \frac{d}{dt}\,\int_\mathcal{R}\, r(x)\,V(x,t)\,dx = \\[12pt] \qquad\ds\frac{\pi c_\mathrm{M}\big(x_\mathrm{C}-x_\mathrm{L}\big)}{2} \Big[\,r_\mathrm{L}\frac{dV_\mathrm{L}}{dt}+3r_\mathrm{C} \frac{dV_\mathrm{C}}{dt}\,\Big]+ \frac{\pi c_\mathrm{M}\big(x_\mathrm{R}-x_\mathrm{C}\big)}{2} \Big[\,3r_\mathrm{C}\frac{dV_\mathrm{C}}{dt}+r_\mathrm{R} \frac{dV_\mathrm{R}}{dt}\,\Big]\,. \end{array}$$ \subsection{Intrinsic voltage-dependent current} A common specification of intrinsic voltage-dependent current describing the behaviour of channels of ionic species \alpha assumes that J_\mathrm{IVDC}(x,t)=g_\alpha(V)(V-E_\alpha) where V is the transmembrane potential, E_\alpha (mV) is the reversal potential for species \alpha and g_\alpha(V) (mS/cm^2) is a voltage-dependent membrane conductance (which may depend on a set of auxiliary variables such as the m, n and h appearing in the Hodgkin-Huxley (\cite{Hodgkin52}) model). The simplest case is the \emph{passive} membrane in which g_\alpha(V) is constant for each species \alpha, albeit a different constant for each species. It then follows immediately from identity (\ref{gcm1}) and the general expression (\ref{tc0}) that the contribution of intrinsic voltage-dependent current to the segment due to species \alpha in a passive membrane is $$\label{gcm3} \begin{array}{l} \ds 2\pi\int_\mathcal{L}\, r(x) g_\alpha(V-E_\alpha)\,dx +2\pi\int_\mathcal{R}\, r(x) g_\alpha(V-E_\alpha)\,dx = \\[12pt] \qquad\ds\frac{\pi\big(x_\mathrm{C}-x_\mathrm{L}\big)}{2} \Big[\,r_\mathrm{L} g_\alpha(V_\mathrm{L}-E_\alpha) +3r_\mathrm{C} g_\alpha(V_\mathrm{C}-E_\alpha)\,\Big]\\[12pt] \qquad\qquad\ds+\;\frac{\pi\big(x_\mathrm{R}-x_\mathrm{C}\big)}{2} \Big[\,3r_\mathrm{C} g_\alpha(V_\mathrm{C}-E_\alpha) +r_\mathrm{R} g_\alpha(V_\mathrm{R}-E_\alpha)\,\Big]\,. \end{array}$$ On the other hand, when g_\alpha(V) is a non-constant function of V as happens, for example, with a Hodgkin-Huxley membrane (\cite{Hodgkin52}), no analytical expression for the effect of intrinsic voltage transmembrane current exist. To resolve this impasse, one requires a generalised expression to describe the effect of intrinsic voltage-dependent transmembrane current. This expression must have the following properties. First, it must be tractable when g_\alpha(V) is a non-constant function of V in the sense that g_\alpha(V) is evaluated for values of V at designate nodes only. Second, the expression must incorporate the effect of changing membrane potential along a dendritic segment, and third, the generalised expression must reduce to expression (\ref{gcm3}) when g_\alpha(V) is a constant function of V. For the specification J_\mathrm{IVDC}(x,t)=g_\alpha(V)(V-E_\alpha), these three conditions are satisfied by replacing \[ \ds 2\pi\int_\mathcal{L}\, r(x) g_\alpha(V)(V-E_\alpha)\,dx +2\pi\int_\mathcal{R}\, r(x) g_\alpha(V)(V-E_\alpha)\,dx$
with the generalised expression
$$\label{gcm4} \begin{array}{l} \ds\frac{\pi\big(x_\mathrm{C}-x_\mathrm{L}\big)}{2} \Big[\,r_\mathrm{L} g_\alpha(V_\mathrm{L})(V_\mathrm{L}-E_\alpha) +3r_\mathrm{C} g_\alpha(V_\mathrm{C})(V_\mathrm{C}-E_\alpha)\,\Big]\\[12pt] \qquad\ds+\;\frac{\pi\big(x_\mathrm{R}-x_\mathrm{C}\big)}{2} \Big[\,3r_\mathrm{C} g_\alpha(V_\mathrm{C})(V_\mathrm{C}-E_\alpha) +r_\mathrm{R} g_\alpha(V_\mathrm{R})(V_\mathrm{R}-E_\alpha)\,\Big]\,. \end{array}$$
Moreover, it is clear that the specification of intrinsic
voltage-dependent current used in the traditional compartmental
model is recovered from (\ref{gcm4}) by replacing $V_\mathrm{L}$
and $V_\mathrm{R}$ with $V_\mathrm{C}$.