A portion of this work is reprinted from  R.F. Melendy, Resolving the biophysics of axon transmembrane polarization in a single closed-form description. Journal of Applied Physics, 118(24), Copyright © (2015); and  R.F. Melendy, A subsequent closed-form description of propagated signaling phenomena in the membrane of an axon. AIP Advances, 6(5), Copyright © (2016), with the permission of AIP Publishing. Said published works are copyright protected by Robert. F. Melendy, Ph.D. and the AIP journals in which these articles appear. Under §107 of the Copyright Act of 1976, allowance is made for “fair use” for purposes such as criticism, comment, news reporting, teaching, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Nonprofit, educational (i.e., teaching, scholarship, and research) or personal use tips the balance in favor of fair use.
I Scope of this Article
Melendy  demonstrated that the differential equation:
is a novel, closed-form quantification of the membrane action potential, Vm. (1a) is in contrast to the Hodgkin-Huxley quantification of Vm  which requires numerically integrating four differential equations to solve for the membrane voltage. Here, u = (67.9 × 10–3)Γ –1, where:
In review  (p. 107), Gin is the leaky cable input conductance along the longitudinal length of neuronal fiber (Ω–1) and Χ (Chi) is a normalized length (dimensionless) [7,8]. The hyperbolic tangent term (p. 108) is Langevin’s thermodynamic relation . The sine term (p. 108-109) is the current-modulation function and is a magnetic field-dependent current. On p. 110  the myelinated thickness of the axon is given as Δr and ε0 is the vacuum permittivity of free space (8.854 x 10–12 F/m). The relationship between Vm and Γ is given as
From classical electrodynamics , Melendy showed that (p. 110)
In other words, (1c) produces identical results to (4b) (p. 111)  for Bm = 2.964(74) × 10–5 Wb/m2 and a = 4.710(94) μm.
The laws of electrodynamics state that Bm = Φm/πa2, where Φm is the membrane magnetic flux (Wb). The axon cross-sectional area  is πa2 = 6.972(15) × 10–11 m2, resulting in a flux of Φm = 2.067(06) × 10–15 Wb. This is very nearly the value of the magnetic flux quantum, Φ0 = h/2e [11, 12, 13], where h = 6.626 069 934 x 10–34 J/Hz is the currently reported measured value of Planck’s constant  and e = 1.602 176 634 x 10–19 C is elementary charge .
The development of an original, quantitative model of the membrane (action) potential Vm was presented in . This is a conductance-based model rooted in cable theory. It is independent of the chemistry and physics behind the contribution of sodium, potassium, and leakage ions to the action potential cycle. In this article, the electric field model (1a) was re-stated in terms of the membrane magnetic field Bm and subsequently, the membrane magnetic flux, Φm. It was discovered that Φm = 2.067(06) × 10–15 Wb, which is very nearly the value of the magnetic flux quantum, Φ0 = h/2e. This is evidence for quantized magnetic flux in the membrane of an axon.
IV Ethical Approval
The conducted research reported in this article is not related to either human or animal use.
V Compliance with Ethical Standards
Conflict of Interests: The author Robert F. Melendy, Ph.D. declares that I have no conflict of interest(s).
R.F. Melendy A single differential equation description of membrane properties underlying the action potential and the axon electric field. Journal of Electrical Bioimpedance 9(1) (2018). https://doi.org/10.2478/joeb-2018-0015
R.F. Melendy Resolving the biophysics of axon transmembrane polarization in a single closed-form description. Journal of Applied Physics 118(24) (2015). https://doi.org/10.1063/1.4939278
R.F. Melendy A subsequent closed-form description of propagated signaling phenomena in the membrane of an axon. AIP Advances 6(5) (2016). https://doi.org/10.1063/1.5052550
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