Physics of the Propagating Action Potential

Abstract

We analyze the Rosenthal--Bezanilla experimental action potential waveforms using a charge-conserving phase-space formulation of the cable equation for a steadily propagated action potential. Each measured waveform determines the axial Ohmic current $J_z$, its geometry-converted divergence $J_m$, the capacitive current, and the total ionic current without requiring voltage-clamp currents as independent inputs. The current balance is expressed branchwise through $V$, $\Phi (V)=dV/dt$, and $d\Phi (V)/dV$; laboratory time remains recoverable from $dt=dV/\Phi (V)$. At points where $\Phi (V)\ne 0$, the total ionic current vanishes when $d\Phi /dV=k$, producing one zero crossing on each branch. Both crossings are present in the Rosenthal--Bezanilla data. The reconstructed currents remain continuous through the action-potential peak, while their state-dependent decomposition changes, supporting a continuous $M\to H$ transition between sodium-channel regimes. The second crossing occurs shortly after the peak on the recovery branch, after which the total ionic current is outward. Its effective reversal potential, channel-completion kinetics, and temperature dependence support a sodium-dominated interpretation. Quasilinear phase-space regions yield effective conductances, reversal potentials, and characteristic time rates. Channel completion follows a modified Avrami relation with nearly temperature-independent exponents and Arrhenius time rates. The fitted processes share an approximately invariant dimensionless prefactor, numerically close to the fine-structure constant, whose physical origin remains undetermined. This framework provides a unified charge-conserving, clock-free relational description of propagated action potentials and reveals current and channel-state structure not resolved by stationary voltage-clamp measurements.

47 pages, 27 figures