The length constant, time constant and velocity of a propagating action potential

Length and time constants are foundational to the study of conduction in neurons but are defined only for passive membranes. Here we define length and time constants for uniformly propagating action potentials. The derivation exploits transmembrane current transition (TCT) instants during action potential conduction when the net transmembrane ionic current is zero, but axial current remains non-zero. At these instants, we define a rate coefficient, {kappa}, and from it define {lambda}AP=1/{surd}({kappa}racm) and {tau}AP=1/{kappa}. We show that action potential propagation velocity is exactly {lambda}AP/{tau}AP. We establish that {lambda}AP equals the ratio of axial to local capacitive current throughout the waveform, sets the exponential spatial weighting of ionic current, and is the local length scale of the dV/dt field at a TCT. It also approximates the length scale of the leading edge to within a fractional error of {tau}foot/{tau}m. These identities are independent of channel gating formalism. We relate the new constants exactly to the passive DC, AC and AP foot constants and demonstrate the membrane and channel properties that determine {kappa}. Finally, we suggest how {kappa} may in principle be determined from experimental voltage waveforms.