Cable properties of a neuron model with non-uniform membrane resistivity

We propose an exact, self-closed and simple method for simultaneous estimation of the electrotonic length of the equivalent dendritic cylinder, and the ratio of dendritic to somatic input conductance in the Rall's motoneuron model (1969), from a voltage transient at the soma in response to a current step applied to the soma. We prove the theoretical constraint in the Rall's motoneuron model that one half of the ratio of amplitudes of the first two peeled exponentials in a membrane voltage transient caused by a current step, must be smaller than the ratio of the corresponding first two time constants. This theoretical prediction is not satisfied for several types of neurons, and our method to estimate cable parameters is not applicable to these neurons. By extending the Rall's neuron model, we develop a neuron model, which contains two membrane resistance per unit area; one for somatic membrane and one for dendritic membrane. In this model we obtain the transient solution of membrane potential at the soma in response to a current step applied to the soma. It is shown that the amplitude ratio can be larger than double of the time constant ratio when the somatic resistance is lower than the dendritic resistance. Moreover, we give a purely electrophysiological method to estimate cable parameters of the extended model from soma transient in response to a current step.