Solutions for transients in arbitrarily branching cables: I. Voltage recording with a somatic shunt

An analytical solution is derived for voltage transients in an arbitrarily branching passive cable neurone model with a soma and somatic shunt. The response to injected currents can be represented as an infinite series of exponentially decaying components with different time constants and amplitudes. The time constants of a given model, obtained from the roots of a recursive transcendental equation, are independent of the stimulating and recording positions. Each amplitude is the product of three factors dependent on the corresponding root: one constant over the cell, one varying with the input site, and one with the recording site. The amplitudes are not altered by interchanging these sites. The solution reveals explicitly some of the parameter dependencies of the responses. An efficient recursive root-finding algorithm is described. Certain regular geometries lead to "lost" roots; difficulties associated with these can be avoided by making small changes to the lengths of affected segments. Complicated cells, such as a CA1 pyramid, produce many closely spaced time constants in the range of interest. Models with large somatic shunts and dendrites of unequal electrotonic lengths can produce large amplitude waveform components with surprisingly slow time constants. This analytic solution should complement existing passive neurone modeling techniques.