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3.2 Spatial discretization in a biological context: sections and segments

Every nerve simulation program solves for the longitudinal spread of voltage and current by approximating the cable equation as a series of compartments connected by resistors (Fig. 3.4 and Eq. 2). The sum of all the compartment areas is the total membrane area of the whole nerve. Unfortunately, it is usually not clear at the outset how many compartments should be used. Both the accuracy of the approximation and the computation time increase as the number of compartments used to represent the cable increases. When the cable is "short," a single compartment can be made to adequately represent the entire cable. For long cables or highly branched structures, it may be necessary to use a large number of compartments.
This raises the question of how best to manage all the parameters that exist within these compartments. Consider membrane capacitance, which has a different value in each compartment. Rather than specify the capacitance of each compartment individually, it is better to deal in terms of a single specific membrane capacitance which is constant over the entire cell and have the program compute the values of the individual capacitances from the areas of the compartments. Other parameters such as diameter or channel density may vary widely over short distances, so the graininess of their representation may have little to do with numerically adequate compartmentalization.
Although NEURON is a compartmental modeling program, the specification of biological properties (neuron shape and physiology) has been separated from the numerical issue of compartment size. What makes this possible is the notion of a section, which is a continuous length of unbranched cable. Although each section is ultimately discretized into compartments, values that can vary with position along the length of a section are specified in terms of a continuous parameter that ranges from 0 to 1 (normalized distance). In this way, section properties are discussed without regard to the number of segments used to represent it. This makes it easy to trade off between accuracy and speed, and enables convenient verification of the numerical correctness of simulations.

Figure 3.2. Top: cartoon of a neuron indicating the approximate boundaries between biologically significant structures.. The left hand side of the cell body (Soma) is attached to an axon hillock (AH) that drives a myelinated axon (myelinated internodes Ii alternating with nodes of Ranvier Ni). From the right hand side of the cell body originates a branched dendritic tree (Di). Bottom: how sections would be employed in a NEURON model to represent these structures.

Sections are connected together to form any kind of branched tree structure. Fig. 3.2 illustrates how sections are used to represent biologically significant anatomical features. The top of this figure is a cartoon of a neuron with a soma that gives rise to a branched dendritic tree and an axon hillock connected to a myelinated axon. Each biologically significant component of this cell has its counterpart in one of the sections of the NEURON model, as shown in the bottom of Fig. 3.2: the cell body (Soma), axon hillock (AH), myelinated internodes (Ii), nodes of Ranvier (Ni), and dendrites (Di). Sections allow this kind of functional/anatomical parcellation of the cell to remain foremost in the mind of the person who constructs and uses a NEURON model.
To accommodate requirements for numerical accuracy, NEURON represents each section by one or more segments of equal length (Figs. 3.3 and 3.4). The number of segments is specified by the parameter nseg, which can have a different value for each section.
At the center of each segment is a node, the location where the internal voltage of the segment is defined. The transmembrane currents over the entire surface area of a segment are associated with its node. The nodes of adjacent segments are connected by resistors.
It is crucial to realize that the location of the second order correct voltage is not at the edge of a segment but rather at its center, i.e. at its node. This is the discretization method employed by NEURON. To allow branching and injection of current at the precise ends of a section while maintaining second order correctness, extra voltage nodes that represent compartments with 0 area are defined at the section ends. It is possible to achieve second order accuracy with sections whose end nodes have nonzero area compartments. However, the areas of these terminal compartments would have to be exactly half that of the internal compartments, and extra complexity would be imposed on administration of channel density at branch points.

Figure 3.3 A. Cartoon of an unbranched neurite (thick lines) that is to be represented by a section in a NEURON model. Computer-assisted morphometry generates a file that stores successive measurements of diameter (thin circles) centered at x, y, z coordinates (thin crosses).
B. Each adjacent pair of diameter measurements (thick circles) becomes the parallel faces of a truncated cone or frustum, the height of which is the distance between the measurement locations. The outline of each frustum is shown with thin lines, and a thin centerline passes through the central axis of the chain of solids.

Based on the position of the nodes, NEURON calculates the values of internal model parameters such as the average diameter, axial resistance, and compartment area that are assigned to each segment. Figs. 3.3 and 3.4 show how an unbranched portion of a neuron, called a neurite (Fig. 3.3A), is represented by a section with one or more segments. Morphometric analysis generates a series of diameter measurements whose centers lie on the midline of the neurite (thin axial line in Fig. 3.3B). These measurements and the path lengths between their centers are the dimensions of the section, which can be regarded as a chain of truncated cones or frusta (Fig. 3.3C).

Figure 3.3 C. The centerline has been straightened so the faces of adjacent frusta are flush with each other. The scale underneath the figure shows the distance along the midline of the section in terms of the normalized position parameter x. The vertical dashed line at x = 0.5 divides the section into two halves of equal length.
D. Electrical equivalent circuit of the section as represented by a single segment (nseg = 1). The open rectangle includes all mechanisms for ionic (non-capacitive) transmembrane currents.

Distance along the length of a section is discussed in terms of the normalized position parameter x. That is, one end of the section corresponds to x = 0 and the other end to x = 1. In Fig. 3.3C these locations are depicted as being on the left and right hand ends of the section. The locations of the nodes and the boundaries between segments are conveniently specified in terms of this normalized position parameter. In general, a section has nseg segments that are demarcated by evenly spaced boundaries at intervals of 1 / nseg. The nodes at the centers of these segments are located at x = (2 i - 1) / 2 nseg where i is an integer in the range [1, nseg]. As we shall see later, x is also used in specifying model parameters or retrieving state variables that are a function of position along a section (see 4.4 Range variables).
The special importance of x and nseg lies in the fact that they free the user from having to keep track of the correspondence between segment number and position on the nerve. In early versions of NEURON, all nerve properties were stored in vector variables where the vector index was the segment number. Changing the number of segments was an error prone and laborious process that demanded a remapping of the relationship between the user's mental image of the biologically important features of the model, on the one hand, and the implementation of this model in a digital computer, on the other. The use of x and nseg insulates the user from the most inconvenient aspects of such low-level details.
When nseg = 1 the entire section is lumped into a single compartment. This compartment has only one node, which is located midway along its length, i.e. at x = 0.5 (Fig. 3.3C and D). The integral of the surface area over the entire length of the section (0 <= x <= 1) is used to calculate the membrane properties associated with this node. The values of the axial resistors are determined by integrating the cytoplasmic resistivity along the paths from the ends of the section to its midpoint (dashed line in Fig. 3.3C). The left and right hand axial resistances of Fig. 3.3D are evaluated over the x intervals [0, 0.5] and [0.5, 1], respectively.

Figure 3.4. How the neurite of Fig. 3.3 would be represented by a section with two segments (nseg = 2). Now the electrical equivalent circuit (bottom) has two nodes. The membrane properties attached to the first and second nodes are based on neurite dimensions and biophysical parameters over the x intervals [0, 0.5] and [0.5, 1], respectively. The three axial resistors are computed by integrating cytoplasmic resistance over the intervals shown.

Fig. 3.4 shows what happens when nseg = 2. Now NEURON breaks the section into two segments of equal length that correspond to x intervals [0, 0.5] and [0.5, 1]. The membrane properties over these intervals are attached to the nodes at 0.25 and 0.75, respectively. The three axial resistors Ri1, Ri2 and Ri3 are computed from the cytoplasmic resistivity and neurite dimensions over the x intervals [0, 0.25], [0.25, 0.75], and [0.75, 1].

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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.
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