Title : The action potential as a soliton: A geometric derivation from the de Broglie relation with anatomical and clinical implications
Abstract:
Objective: Derive the action potential from first principles using the de Broglie relation λp = h and the geometry of the Euler Complex Plane (ECP), and validate the resulting anatomical and clinical predictions.
Background: The Hodgkin-Huxley (HH) model describes the action potential empirically using voltage-gated ion channels but does not explain why the action potential is all-or-none, why it propagates at a fixed speed, or why it regenerates at nodes of Ranvier. Using de Broglie wave mechanics and phase harmony, we hypothesized that the action potential is a soliton—a localized, self-sustaining nonlinear wave—derived from the geometry of the ECP. Design/Methods: Mathematical derivation of the action potential from the Prince de Broglie nonlinear wave equation on the ECP. The soliton solution was obtained for r < 1, where the nonlinear term dominates. Anatomical predictions were derived from soliton geometry: larger axons require longer myelinated segments, Ranvier nodes are spaced further apart, and no closed circuit is required for propagation.
Results: Mathematical derivation yields the soliton solution:
Vsoliton(x, t) = V0 sech (x − vgt/∆ )
which reproduces all known properties of the action potential: all-or-none generation, fixed prop agation speed vg = 2βc, and a refractory period arising from a reverse soliton on the opposite face of the ECP. Myelination preserves the soliton; nodes of Ranvier regenerate it. Anatomical predictions are confirmed: larger axons have longer internodes and no closed circuit is required for propagation. The framework unifies quantum mechanics, circuit theory, and neurophysiology under a single geometric principle: λp = h.
Conclusions: The action potential is a soliton—a geometric consequence of the de Broglie relation on the ECP. The reverse soliton mechanism explains the refractory period and ensures ordered firing. The soliton model provides a first-principles explanation of myelination, saltatory conduction, and the anatomical structure of the axon. Understanding the action potential as 1 a geometric wave opens novel therapeutic targets: preserving soliton integrity in demyelinating diseases and restoring phase harmony in seizure disorders. This quantitative framework transforms neurophysiology from an empirical description to a geometric necessity.
