Neuronal integrative analysis of the "Dumbbell" model for passive neurons

We analyze the so called "Dumbbell" model of Jackson (J. Neurophysiol. 69 (1993) pp. 464) for a single neuron consisting of a patch-clamped cell body attached to dendritic cable of finite length terminating in an oblique derivative ("natural termination") boundary condition representing a dendritic swelling or a natural ending sealed by a continuous surface of the cell membrane. The model is solved analytically via the Green's function method. Large and small time asymptotic behavior of the membrane potential is developed when there is a somatic voltage-clamp imposed. We discuss the difference in the voltage distribution if a sealed-end (Neumann) termination is used instead of the natural termination boundary condition. If the access resistance is large the differences between the potentials corresponding to the two boundary conditions are small at the soma, but can vary significantly near the dendritic termination. This discrepancy is amplified at the soma if there is a synaptic stimulus introduced between the soma and dendritic tip.