## Attribution

A portion of this work is reprinted from [2] 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 [3] 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

To present evidence that the natural constant magnetic flux quantum Φ_{0} [4,5] is built-into the fabric of the action potential as per my recently published model [1].

## II Method

Melendy [1] demonstrated that the differential equation:

is a novel, closed-form quantification of the membrane action potential, *V _{m}*. (1a) is in contrast to the Hodgkin-Huxley quantification of

*V*[6] which requires numerically integrating four differential equations to solve for the membrane voltage. Here,

_{m}*u*= (67.9 × 10

^{–3})Γ

^{–1}, where:

In review [1] (p. 107), *G _{in}* 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 [9]. The sine term (p. 108-109) is the current-modulation function and is a magnetic field-dependent current. On p. 110 [1] the myelinated thickness of the axon is given as Δ

*r*and

*ε*

_{0}is the vacuum permittivity of free space (8.854 x 10

^{–12}F/m). The relationship between

*V*and Γ is given as

_{m}*Em*is the membrane electric field (V/m).

From classical electrodynamics [10], Melendy showed that (p. 110) *t* sin *nπt*), where *a* is the axon radius (μm), *ρm* is the longitudinal membrane resistivity (Ω-m), *Bm* is the membrane magnetic field (T or Wb/m^{2}), and μ_{0} is the vacuum permittivity of free space (4π x 10^{–7} H/m). From this relationship between *E _{m}* and

*B*is Melendy’s origianly-derived model (p. 109) but that includes identical numerical parameters as in (1b):

_{m}In other words, (1c) produces identical results to (4b) (p. 111) [1] for *B _{m}* = 2.964(74) × 10

^{–5}Wb/m

^{2}and

*a*= 4.710(94) μm.

The laws of electrodynamics state that *B _{m}* = Φ

*/πa2, where Φ*

_{m}*is the membrane magnetic flux (Wb). The axon cross-sectional area [1] is*

_{m}*π*a

^{2}= 6.972(15) × 10

^{–11}m

^{2}, resulting in a flux of Φ

*= 2.067(06) × 10*

_{m}^{–15}Wb. This is very nearly the value of the magnetic flux quantum, Φ

_{0}=

*h*/2

*e*[11, 12, 13], where

*h*= 6.626 069 934 x 10

^{–34}J/Hz is the currently reported measured value of Planck’s constant [14] and

*e*= 1.602 176 634 x 10

^{–19}C is elementary charge [15].

## III Summary

The development of an original, quantitative model of the membrane (action) potential *V _{m}* was presented in [1]. 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

*B*and subsequently, the membrane magnetic flux, Φ

_{m}*. It was discovered that Φ*

_{m}*= 2.067(06) × 10*

_{m}^{–15}Wb, which is very nearly the value of the magnetic flux quantum, Φ

_{0}=

*h*/2

*e*. 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).

## References

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- 2↑
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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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