title: The propagation of the nerve impulse.
abstract: A partial differential equation for the propagated action potential is derived using symmetry, charge conservation, and Ohm's law. Charge conservation analysis explicitly includes the gating charge when applied in the laboratory frame. When applied in the system of reference in which capacitive currents are zero, it yields a relation between orthogonal components of the ionic current allowing us to express the nonlinear ionic current in terms of the voltage-dependent membrane capacitance C(V) and the axial current that satisfies Ohm's law. The ionic current is shown to behave as C(V)V[C(V)V2]' at the foot of the action potential while the gating current behaves as C(V)V[Cg(V)V]' where Cg(V) is the capacitance associated with gating. Improved knowledge of the nonlinear current makes it possible to describe the propagated action potential in an approximated way with quasilinear partial differential equations. These equations have analytical solutions that travel with constant velocity, retain their shape, and account for other properties of the action potential. Furthermore, the quasilinear approximation is shown to be equivalent to the FitzHugh-Nagumo equation without recovery making apparent its physical content.
mesh_terms: D000200:Action Potentials; D000818:Animals; D008433:Mathematics; D008959:Models, Neurological; D009431:Neural Conduction; D009474:Neurons / Q000502:physiology*