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We would like to understand and make some quantitative predictions about the attenuation of post-synaptic potentials as they propagate through a length of dendrite towards the soma. So far, we haven’t said anything about the length of the neural compartments that we use when we model a section of dendrite. When we approximate the continous dendrite with discrete compartments, how many should we use in order to have a relatively smooth variation in membrane potential from one compartment to the next?

If we consider a length of dendrite as a chain of linked compartments that are
represented by the generic compartment diagram, we can see that a PSP
propagating from one compartment to another must flow through the axial
resistance R_{a}, and can be reduced as current is shunted off to ground as it flows
through the membrane resistance R_{m}. (If there are conducting ion channels, the
conductance G_{k} can further affect the PSP.) So, we would expect that the
attenuation of a PSP will be least when R_{a} is small and R_{m} is large, and that it
should be possible to calculate the attentuation from a knowledge of these
quantities.

In 1855, Prof. William Thompson (Lord Kelvin) presented to the Royal Society a theoretical analysis of attentuation of signals in the transatlantic telephone cable that was then being planned. An undersea cable is similar to a nerve fiber. It has a conducting core covered with an insulating sheath, and is surrounded by sea water. As the insulation is not perfect, there will be a finite leakage resistance through the insulation. The main quantitative difference is that the core of the cable is made of copper, which is a much better conductor than the salt solution inside a neuron, and the cable covering is a much better insulator than the cell membrane. As a result, signals in the cable may travel for many miles without significant attenuation.

In the 1940’s, Hodgkin and others applied cable theory to nerve fibers. Many papers by Wilfred Rall in the 1960’s and 1970’s made further application of cable theory to the analysis of PSPs in dendrites.

So far, we haven’t said much about the units used to express the quantities R_{m},
R_{a}, C_{m}, V _{m}, etc. that appear in the neural compartment diagram (make link)
and the differential equation for V _{m} (make link).

Physicists and engineers like to use SI (MKS) units of ohms (Ω), farads (F), volts V , and meters (m) for describing resistance, capacitance, voltage, and length. Neurophysiologists are more likely to prefer kilohms (KΩ), microfarads (μF), millivolts (mV ), and either centimeters (cm) or micrometers (μm). Importantly, GENESIS uses SI units.

The problem with using any of these units for resistance and capacitance is
that R_{m}, C_{m}, and R_{a} will depend on the dimensions of the section of dendrite
that is represented by the neural compartment. In order to specify parameters
that are independent of the cell dimensions, specific units are used. For a
cylindrical compartment, the membrane resistance is inversely proportional to the
area of the cylinder, so we define a specific membrane resistance R_{M}, which has
units of ohms⋅m^{2}. The membrane capacitance is proportional to the area, so it is
expressed in terms of a specific membrane capacitance C_{M}, with units of
farads∕m^{2}. Compartments are connected to each other through their
axial resistances R_{a}. The axial resistance of a cylindrical compartment is
proportional to its length and inversely proportional to its cross-sectional
area. Therefore, we define the specific axial resistance R_{A} to have units of
ohms∕m.

For a piece of dendrite or a compartment of length l and diameter d we then have

Note the membrane time constant R_{m} ⋅C_{m} is also equal to R_{M} ⋅C_{M}, so that it
is independent of the dimensions of the membrane.

WARNING: Many treatments of the passive properties of neural tissue use the
symbols R_{m}, R_{a}, and C_{m} for the specific resistances and capacitance, instead of
this notation with R_{M}, R_{A}, and C_{M}. Also, many textbooks and journal papers
define the resistance and capacitance in terms of that for a unit length of cable
having a specified diameter, where

Although this notation is convenient and widely used, it obscures the fact that
r_{m} and r_{a} depend on the dendrite diameter. In your reading, you should be aware
of the units that are being used.

Linear cable theory assumes that the cable (or neural fiber) has a constant
diameter and leakage resistance. Thus, it applies to passive membrane properties,
as when PSPs propagate through a uniform section of dendrite that has no time
or voltage dependent conductances. When R_{m} is much greater than R_{a}, as is
typically the case, the current flow is essentially one-dimensional, along the
length of the cable. Under these conditions, there are exact mathematical
solutions for the voltage as a function of time and position along the cable.
With the additional simplifying assumptions that a constant, or very
slowly varying, voltage V _{0} is applied to one end of a very long cable, there
is a simple solution to the cable equation for the voltage at a distance
x:

The quantity “lambda” (λ) in this equation is called the “space constant” (or “length constant”), and represents the distance at which the voltage will have decreased to 1∕e, or about 37 %, of its original value.

The GENESIS Neuron tutorial simulation uses values of R_{M} = 5 KΩ⋅cm^{2} and
R_{A} = 0.025 KΩ/cm, with a dendrite diameter of 0.0002 cm (2 μm) and a length of
0.01 cm for each compartment. You should be able to show that the space
constant is then 0.1 cm and that and the attenuation over the length of seven
compartments (0.07 cm) is e^{-0.7} = 0.497.

Note the the space constant depends on the square root of the radius. Thus,
large diameter axons are best for propagation of action potentials over large
distances. The space constant also influences spatial summation. Spatially
separated synaptic inputs will combine differently in small diameter dendrites
than in ones with larger diameters. The giant axon of the squid has a diameter
of about 1 mm. How much larger would the space constant be for the
squid, if it had the same values of R_{M} and R_{A} as used in the Neuron
tutorial?

If the leakage resistance varies with time and voltage, as when there are active
channels present, then one must solve the equations for V _{m} numerically, dividing
the dendrite into finite compartments. Nevertheless, the space constant is a useful
quantity for giving a rough idea of the attenuation.

As the length of a compartment approaches zero, the behavior of this “lumped
parameter” model approaches that of a continous cable. In practice, we want to
have only “small” jumps in V _{m} from one compartment to the next, approximating
the smooth variation of a continous medium. But, we don’t want to make the
compartments any smaller than we have to in order to obtain reasonably accurate
results. Experience with compartmental simulations suggests that good results
can usually be obtained when the compartment length is less than 1/20 of the
space constant.

The help menus for the GENESIS Neuron and Cable tutorial simulations suggest some experiments and exercises involving passive propagation in dendrites. Chapters 5 and 6 of The Book of GENESIS provide further details and give additional theoretical background. Other chapters describe the process of creating simulations of multicompartmental neurons and of networks, such as the Traub model and Piriform Cortex model described in these lectures. You can read more about the application of cable theory to dendrites in:

Rall, W. and Agmon-Snir, H. (1998) Cable Theory for Dendritic Neurons, in C. Koch and I. Segev (eds.) Methods in Neuronal Modeling, second edn, MIT Press, Cambridge, MA, Chapter 2, pp. 27–92.

Jack, J. J. B., Noble, D. and Tsien, R. W. (1975) Electric Current Flow in Excitable Cells, Calderon Press, Oxford.

Rall, W. (1977) Cable Theory for Neurons, in E. R. Kandel, J. M. Brookhardt and V. B. Mountcastle (eds),

Handbook of Physiology: The Nervous System, Vol. 1, Williams and Wilkins, Baltimore, Chapter 3, pp. 39–98.