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Strategies for numerical solution of the equations that describe
chemical and electrical signaling in neurons have been discussed in many
places. Elsewhere we have briefly presented an intuitive rationale for the
most commonly used methods (Hines and Carnevale 1995). Here we start from this
base and proceed to address those aspects which are most pertinent to the
design and application of NEURON.
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The application of cable theory to the study of electrical signaling in
neurons has a long history, which is summarized elsewhere (Rall 1989).
The basic computational task is to numerically solve the cable equation
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(1) |
| which describes the relationship between current and voltage in a one-dimensional cable. The branched architecture typical of most neurons is incorporated by combining equations of this form with appropriate boundary conditions. |
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| Figure 3.1. The net current entering a region must equal zero. |
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Spatial discretization of this partial differential equation is equivalent to
reducing the spatially distributed neuron to a set of connected compartments.
The earliest example of a multicompartmental approach to the analysis of
dendritic electrotonus was provided by Rall (1964).
Spatial discretization produces a family of ordinary differential equations of
the form
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(2) |
Equation 2 is a statement of Kirchhoff's current law, which asserts that net
transmembrane current leaving the jth compartment must equal the sum of
axial currents entering this compartment from all sources (Fig. 3.1). The left
hand side of Eq. 2 is the total membrane current, which is the sum of
capacitive and ionic components. The capacitive component is
where cj is the membrane capacitance of the compartment. The
ionic component
includes all currents through ionic channel conductances. The right hand side
of Eq. 2 is the sum of axial currents that enter this compartment from its
adjacent neighbors. Currents injected through a microelectrode would be added
to the right hand side. The sign conventions for current are: outward
transmembrane current is positive; axial current flow into a region is
positive; positive injected current drives vj in a positive
direction.
Equation 2 involves two approximations. First, axial current is specified in
terms of the voltage drop between the centers of adjacent compartments. The
second approximation is that spatially varying membrane current is represented
by its value at the center of each compartment. This is much less drastic than
the often heard statement that a compartment is assumed to be "isopotential."
It is far better to picture the approximation in terms of voltage varying
linearly between the centers of adjacent compartments. Indeed, the linear
variation in voltage is implicit in the usual description of a cable in terms
of discrete electrical equivalent circuits.
If the compartments are of equal size, it is easy to use Taylor's series to
show that both of these approximations have errors proportional to the square
of compartment length. Thus replacing the second partial derivative by its
central difference approximation introduces errors proportional to
, and doubling the number of compartments reduces
the error by a factor of four.
It is often not convenient for the size of all compartments to be equal.Unequal compartment size might be expected to yield simulations that are only
first order accurate. However, comparison of simulations in which unequal
compartments are halved or quartered in size generally reveals a second-order
reduction of error. A rough rule of thumb is that simulation error is
proportional to the square of the size of the largest compartment.
The first of two special cases of Eq. 2 that we wish to discuss allows us to
recover the usual parabolic differential form of the cable equation. Consider
the interior of an unbranched cable with constant diameter. The axial current
consists of two terms involving compartments with the natural indices
j-1 and j+1, i.e.
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If the compartments have the same length
and diameter d,
then the capacitance of a compartment is
and the axial resistance is
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The specific capacitance of the membrane is Cm,
which is generally taken to be 1 uf / cm2.
Ra is the axial resistivity, which has different reported
values for different cell classes
(e.g. 35.4
cm for squid axon).
Eq. 2 then becomes
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where we have replaced the total ionic current
with the current density ij.
The right hand term, as
, is just
at the location of the now infinitesimal compartment j.
The second special case of Eq. 2 allows us to recover the boundary condition.
This is an important exercise since naive discretizations at the ends of the
cable have destroyed the second order accuracy of many simulations. Nerve
boundary conditions are that no axial current flows at the end of the cable,
i.e. the end is sealed. This is implicit in Eq. 2, where the right hand side
consists only of the single term
when compartment j lies at the end of an unbranched cable.
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Address questions and inquiries to Michael Hines or Ted Carnevale
Digital preprint of "The NEURON Simulation Environment" by M.L. Hines and N.T. Carnevale,
Neural Computation, Volume 9, Number 6 (August 15, 1997), pp. 1179-1209.
Copyright © 1997 by the Massachusetts Institute of Technology, all rights reserved. |