Unequal diameters and their effects on time-varying voltages in branched neurons

A theoretical method, developed in a previous paper, enables one to calculate analytical expressions for time-varying voltages at specific locations in branching dendritic systems in response to synaptic current inputs at other sites. Exact results were obtained for a number of dendritic trees that possessed certain symmetries: all branch lengths had to be integral multiples of one another, and all branch diameters had to be equal. Because the second of these conditions is unrealistic, the method has been generalized to treat dendritic trees whose branches differ in diameter. The method entails adding onto the symmetric results a sum of correction terms. It is found that the correction terms, as well as the symmetric results, can be expressed as combinations of two families of functions. These functions, generalizations of those found in our earlier paper, provide a precise formalism for analyzing how voltage transients depend on the geometrical structure of the dendritic tree. Examples are given that show how the correction terms affect the value of the voltage, and how variations in branch diameters alter the behavior of the propagated postsynaptic potential. The implications of these results for our understanding of neuronal functioning are discussed.