Estimating space and time constants for active neuronal models from measurements of conduction speed

Space constants and time constants characterize the spatial and temporal behavior of the membrane potential of a neuronal membrane with constant conductance. However, more realistic models of membrane potential assume that membrane conductance depends on the membrane potential and its history, and therefore, it is not clear that space and time constants can be defined for membranes with this property. However, through a consideration of the properties of trains of action potentials treated as traveling waves, space and time constants for the total membrane current during a propagated action potential can be derived and estimated. We show that the formal definitions of the space and time constants for the membranes with constant conductance can be extended to membranes with voltage-dependent conductance in which the behavior of each type of membrane is distinguished through the choice of the representation of the membrane conductance used in these definitions. In the case of a membrane with voltage-dependent conductance, the conductance to be used in the definitions of the space and time constants can be estimated from the extracellularly determined conduction speed of the propagated action potential.