The attenuation of passively propagating dendritic potentials in a motoneurone cable model

Abstract

1. The Rall model of the motoneurone, which consists of a lumped resistance and capacitance, representing the soma, in parallel with a number of distributed resistance-capacitance networks of finite and equal electrical length, representing equivalent dendritic cables, has been used to study the effects of varying electrical and geometrical parameters on the time course and amplitude of transients generated at different locations on the dendritic cables.2. An analytical solution has been obtained for the time course of the voltage transient generated at the point of current injection on the parallel combination of all dendritic cables, in terms of the distance from the soma to the current injection point, the electrotonic length of the equivalent dendritic cable, the dendritic to soma conductance ratio and the membrane time constant. The current applied is a current impulse, and the response to any synaptic current time course may be obtained from the analytical expression for the current impulse response. A smooth current time course of the form Te(-alphaT) has been used in computations.3. An analytical expression has been obtained for the early part of the voltage response at the point of current injection, when the current is applied to a fraction of the total dendritic cable. This response is in terms of all the cable parameters, and the assumed fraction of the dendritic cable which receives the synaptic current. Computations of this response have been carried out assuming a smooth time course of synaptic current.4. The computations of the peak amplitude of the voltage transient obtained from these expressions, together with similar computations for the peak amplitude of the voltage transient after propagation to the soma (Jack & Redman, 1971b), have been used to derive a set of attenuation curves for dendritic propagation. These curves give the ratio of the peak amplitude of the voltage transient at the synaptic location on the dendritic cable, and the peak amplitude after propagation to the soma, in terms of the electrotonic distance to the synaptic location, the time course of current injection, and the cable parameters for the motoneurone model.