Evidence for quantized magnetic flux in an axon

Open access

Abstract

In December of 2018 I published my consolidated findings of a closed-form description of propagated signaling phenomena in the membrane of an axon [1]. Those results demonstrate how intracellular conductance, the thermodynamics of magnetization, and current modulation, function together in generating an action potential in a unified differential equation. At present, I report on a subsequent finding within this model. Namely, evidence of quantized magnetic flux Φ0 in an axon.

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 Φ0 [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:

[Vm(Δr)]21ΓVmu=0

is a novel, closed-form quantification of the membrane action potential, Vm. (1a) is in contrast to the Hodgkin-Huxley quantification of Vm [6] which requires numerically integrating four differential equations to solve for the membrane voltage. Here, u = (67.9 × 10–3–1, where:

Γ=ε0Δr(t0.5esinπt)(0.5π)tanh(4πμBkT)(GincoshπX)

In review [1] (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 [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 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 Vm=ΓE2m67.9×103V(p. 112), where Em is the membrane electric field (V/m).

From classical electrodynamics [10], Melendy showed that (p. 110) Em(πa2)ρm1(tsinnπt)=B2m(2πa/μ0)(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/m2), and μ0 is the vacuum permittivity of free space (4π x 10–7 H/m). From this relationship between Em and Bm is Melendy’s origianly-derived model (p. 109) but that includes identical numerical parameters as in (1b):

Vm=Bm2(2πaμ0)(t0.5esinπt)(0.5π)tanh(4πμBkT)(GincoshπX)67.9×103V

In other words, (1c) produces identical results to (4b) (p. 111) [1] 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 [1] 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 [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 Vm 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 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).

References

  • 1

    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

  • 2

    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

  • 3

    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

  • 4

    U. Schollwöck et al. Quantum Magnetism (Springer 2004).

  • 5

    Ş. Erkoç Fundamentals of Quantum Mechanics (Taylor and Francis 2006).

  • 6

    A.L. Hodgkin A.F. Huxley A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology 117 500-544 (1952). https://doi.org/10.1113/jphysiol.1952.sp004764

    • Crossref
    • Export Citation
  • 7

    W. Rall Core Conductor Theory and Cable Properties of Neurons: Handbook of Physiology the Nervous System Cellular Biology of Neurons (American Physiological Society 1977) pp. 39-93. https://doi.org/10.1002/cphy.cp010103

  • 8

    R. Hobbie Intermediate Physics for Medicine and Biology (AIP Press New York 1997).

  • 9

    W.T. Coffey Y.P. Kalmykov J.T. Waldron The Langevin Equation with Applications in Physics Chemistry and Electrical Engineering (World Scientific River Edge NJ 1996). https://doi.org/10.1142/2256

  • 10

    R.L. Armstrong J.D. King The Electromagnetic Interaction (Prentice Hall Englewood Cliffs NJ 1973).

  • 11

    The Nation Institute of Standards and Technology (NIST) Reference on Constants Unit and Uncertainty [Internet]. [Cited June 26 2019]; Available from https://physics.nist.gov/cgi-bin/cuu/Value?flxquhs2e

  • 12

    P. Hawkes (Ed.) Advances in Imaging and Electron Physics (Academic Press 2011).

  • 13

    L. Deecke J. Eccles and V. Mount castle (Eds.) From Neuron to Action: An Appraisal of Fundamental and Clinical Research (Springer-Verlag 1990).

  • 14

    The Nation Institute of Standards and Technology (NIST) New Measurement will Help Redefine International Unit of Mass [Internet]. [Cited June 26 2019]; Available from https://www.nist.gov/news-events/news/2017/06/new-measurement-will-help-redefine-international-unit-mass

  • 15

    The Nation Institute of Standards and Technology (NIST) Reference on Constants Unit and Uncertainty [Internet]. [Cited June 26 2019]; Available from https://physics.nist.gov/cgi-bin/cuu/Value?e

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • 1

    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

  • 2

    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

  • 3

    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

  • 4

    U. Schollwöck et al. Quantum Magnetism (Springer 2004).

  • 5

    Ş. Erkoç Fundamentals of Quantum Mechanics (Taylor and Francis 2006).

  • 6

    A.L. Hodgkin A.F. Huxley A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology 117 500-544 (1952). https://doi.org/10.1113/jphysiol.1952.sp004764

    • Crossref
    • Export Citation
  • 7

    W. Rall Core Conductor Theory and Cable Properties of Neurons: Handbook of Physiology the Nervous System Cellular Biology of Neurons (American Physiological Society 1977) pp. 39-93. https://doi.org/10.1002/cphy.cp010103

  • 8

    R. Hobbie Intermediate Physics for Medicine and Biology (AIP Press New York 1997).

  • 9

    W.T. Coffey Y.P. Kalmykov J.T. Waldron The Langevin Equation with Applications in Physics Chemistry and Electrical Engineering (World Scientific River Edge NJ 1996). https://doi.org/10.1142/2256

  • 10

    R.L. Armstrong J.D. King The Electromagnetic Interaction (Prentice Hall Englewood Cliffs NJ 1973).

  • 11

    The Nation Institute of Standards and Technology (NIST) Reference on Constants Unit and Uncertainty [Internet]. [Cited June 26 2019]; Available from https://physics.nist.gov/cgi-bin/cuu/Value?flxquhs2e

  • 12

    P. Hawkes (Ed.) Advances in Imaging and Electron Physics (Academic Press 2011).

  • 13

    L. Deecke J. Eccles and V. Mount castle (Eds.) From Neuron to Action: An Appraisal of Fundamental and Clinical Research (Springer-Verlag 1990).

  • 14

    The Nation Institute of Standards and Technology (NIST) New Measurement will Help Redefine International Unit of Mass [Internet]. [Cited June 26 2019]; Available from https://www.nist.gov/news-events/news/2017/06/new-measurement-will-help-redefine-international-unit-mass

  • 15

    The Nation Institute of Standards and Technology (NIST) Reference on Constants Unit and Uncertainty [Internet]. [Cited June 26 2019]; Available from https://physics.nist.gov/cgi-bin/cuu/Value?e

Search
Journal information
Impact Factor


CiteScore 2018: 0.83

SCImago Journal Rank (SJR) 2018: 0.138
Source Normalized Impact per Paper (SNIP) 2018: 0.649

Metrics
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 39 39 22
PDF Downloads 20 20 13