A simple analytical method for determining the steady-state potential in models of geometrically complex neurons

Abstract

A method is presented for solving the cable equation for a spiking neuron below firing threshold or a nonspiking neuron of arbitrary geometry under constant stimulation. The neuron structure is considered as a tree composed of a set of cylinder cables of three types (terminal, intermediate and branching) characterized by their lengths, diameters and linear membrane properties. The stimulation can result from either a uniform conductance-change over a whole cable segment or a point injection of a current. Other special segments are considered (synapses, voltage clamp, lumped soma). Equations are given for replacing any segment with its Thévenin equivalent, i.e. resistance and electromotive force. The step by step use of these elementary equations allows one to find the Thévenin equivalent of the whole neuron and to determine the steady-state membrane potential at any point.