Electrotonic properties of neurons: steady-state compartmental model

Abstract

If a neuron is represented by a network of resistively coupled isopotential regions, the passive flow of current in its dendritic structure and soma is described by a matrix differential equation. The matrix elements are defined in terms of membrane resistances and capacitances and of coupling resistances between adjoining regions. A uniform cylindrical dendrite can be represented by a chain of identical regions. In this case, a closed-form mathematical expression is derived for the voltage attenuation factor of the dendrite at steady state in terms of the ratio of membrane resistance to coupling resistance. A numerical method is given to determine the coupling resistances, which in turn yield a specified attenuation factor. Related expressions are given for a dendrite coupled to a soma. Formulas are also derived for the input resistance in these configurations. For more complicated neuronal structures, matrix manipulations are described which yield values for input resistances in all regions, attenuation factors between all pairs of regions and values of applied voltages necessary to attain specified steady-state potentials. Dynamic solutions to the differential equation provide voltage transients (PSP [post synaptic potentials]). Comparison of the shape parameters of these transients with those of experimental or cable-theoretical PSP establishes the number of regions necessary to achieve a given degree of approximation to the transients predicted by cable theory.