3. MATHEMATICAL BASIS

 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.

3.1 The cable equation
 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
 (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.

 Figure 3.1. The net current entering a region must equal zero.

 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
 (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.
 If the compartments have the same length and diameter d, then the capacitance of a compartment is and the axial resistance is . 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
 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.

 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. HTML formatting and graphics for page navigation copyright © 1997 by N.T. Carnevale amd M.L. Hines.