Rallian Equivalent Cylinders Reconsidered: Comparisons with Literal Compartments

In Rall\u27s equivalent cylinder morphological-to-electrical transformation, neuronal arborizations are reduced to single unbranched core-conductors. The conventional assumption that such an equivalent reconstructs the electrical properties of the fibers it represents was tested directly; electrical properties and responses of equivalent cylinders were compared with those of their literal branch constituents for fibers with a single symmetrical bifurcation. The numerical solution methods were validated independently by their accurate reconstruction of the responses of an analog circuit configured with compartmental architecture to solve the cable equation for passive fibers with a symmetrical bifurcation. In passive fibers, equivalent cylinders misestimated the spatial distribution of voltage amplitudes and steady-state input resistance, partly due to the lack of axial current bifurcation. In active fibers with a single propagating action potential, the spatial distributions of point-to-point conduction velocity values (measured in meters/second) for a literal branch point differed significantly from those of their equivalent cylinders. Equivalent cylinders also underestimated the diameter-dependent delay in propagation through the branch point and branches, due to the larger equivalent diameter. Corrections to the equivalent cylinder did not reconcile differences between equivalent and literal models. However, equivalent and literal branch fibers had the same (a) steady-state resistance looking into an isolated symmetrical branch point and (b) geometry-independent point-to-point propagation velocity when measured in space constants per millisecond except within ±1 space constant from the geometrical inhomogeneity. In summary, Rall\u27s equivalent cylinders did not accurately reconstruct all passive or active electrophysiological properties and responses of their literal compartments. For the modeling of individual neurons, the requirement of single-branch resolution is discussed