The idea that Rall discovered is that if the dendrites were related in
a particular fashion, then the whole thing could be collapsed to a
single cylindrical cable. This is called the *equivalent
cylinder.* Consider the tree shown in the figure 4. Suppose
that the branches, 0,1 and 2 have the same membrane resistivities,
*R*_{M} and *R*_{A}. Assume that the daughter branches, 1 and 2, have the
same electrotonic length, that is, their physical length divided by
their space constants (which of course depend on their diameters) are
all the same. (For example, if both have equal diameters and are the
same physical length.) Also, assume that the two have the same end
conditions, eg sealed. We want to know if it is possible to combine
the branches of the dendrite into a single equivalent cylinder. The
key is that we must avoid impedence mismatches. Thus, to combine the
dendrites, 1 and 2 with 0, we require:

- 1.
- All the ends are the same conditions, sealed.
- 2.
- The electrotonic lengths of 1 and 2 are the same
- 3.
- The diameters match as follows:

Example

In the above figure, we depict a dendritic tree consisting of several branches with their lengths and diameters in microns. (a) Can they be reduced to an equivalent cylinder (b) What is the electrotonic length (c) What is the input conductance. Assume sealed ends for all terminal dendrites and assume that and that

**Answer.**

*d*_{a}^{3/2} + *d*_{b}^{3/2}+*d*_{c}^{3/2} = 1+1+1 = 3 = 2.08^{3/2}=*d*_{d}^{3/2}

*d*_{d}^{3/2}+*d*_{e}^{3/2} = 3+3 = 6 = 3.3^{3/2} = *d*_{f}^{3/2}