<abstract xmlns="http://www.w3.org/1999/xhtml">In December of 2018 I published my consolidated findings of a closed-form description of propagated signaling phenomena in the membrane of an axon [<xref ref-type="bibr" rid="j_joeb-2019-0009_ref_001_w2aab3b7d147b1b6b1ab2ab1Aa">1</xref>]. 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.</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.

<div xmlns="http://www.w3.org/1999/xhtml"><sec id="j_joeb-2019-0009_s_001_w2aab3b7d147b1b6b1ab1aAa"><div>Attribution</div>A portion of this work is reprinted from [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_002_w2aab3b7d147b1b6b1ab2ab2Aa">2</a>] R.F. Melendy, <italic>Resolving the biophysics of axon transmembrane polarization in a single closed-form description</italic>. Journal of Applied Physics, 118(24), Copyright © (2015); and [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_003_w2aab3b7d147b1b6b1ab2ab3Aa">3</a>] R.F. Melendy, <italic>A subsequent closed-form description of propagated signaling phenomena in the membrane of an axon</italic>. 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.</sec><sec id="j_joeb-2019-0009_s_002_w2aab3b7d147b1b6b1ab1b1Aa"><label>I</label><div>Scope of this Article</div>To present evidence that the natural constant magnetic flux quantum Φ0 [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_004_w2aab3b7d147b1b6b1ab2ab4Aa">4</a>,<a ref-type="bibr" href="#j_joeb-2019-0009_ref_005_w2aab3b7d147b1b6b1ab2ab5Aa">5</a>] is built-into the fabric of the action potential as per my recently published model [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_001_w2aab3b7d147b1b6b1ab2ab1Aa">1</a>].</sec><sec id="j_joeb-2019-0009_s_003_w2aab3b7d147b1b6b1ab1b2Aa"><label>II</label><div>Method</div>Melendy [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_001_w2aab3b7d147b1b6b1ab2ab1Aa">1</a>] demonstrated that the differential equation:<disp-formula id="j_joeb-2019-0009_eq_001_w2aab3b7d147b1b6b1ab1b2b3Aa"><label>(1a)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_joeb-2019-0009_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mrow><mrow><mo>[</mo><mrow><mfrac><mrow><mo>∂</mo><msub><mi>V</mi><mi>m</mi></msub></mrow><mrow><mo>∂</mo><mrow><mo>(</mo><mrow><mi>Δ</mi><mi>r</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>]</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mfrac><mn>1</mn><mi>Γ</mi></mfrac><msub><mi>V</mi><mi>m</mi></msub><mo>−</mo><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math><tex-math>$${{\left[ \frac{\partial {{V}_{m}}}{\partial \left( \Delta r \right)} \right]}^{2}}-\frac{1}{\Gamma }{{V}_{m}}-u=0$$</tex-math></alternatives></disp-formula>is a novel, closed-form quantification of the membrane action potential, <italic>Vm</italic>. (1a) is in contrast to the Hodgkin-Huxley quantification of <italic>Vm</italic> [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_006_w2aab3b7d147b1b6b1ab2ab6Aa">6</a>] which requires numerically integrating four differential equations to solve for the membrane voltage. Here, <italic>u</italic> = (67.9 × 10–3)Γ –1, where:<disp-formula id="j_joeb-2019-0009_eq_002_w2aab3b7d147b1b6b1ab1b2b5Aa"><label>(1b)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_joeb-2019-0009_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>Γ</mi><mo>=</mo><mfrac><mrow><msub><mi>ε</mi><mn>0</mn></msub><mi>Δ</mi><mi>r</mi><mrow><mo>(</mo><mrow><msup><mi>t</mi><mrow><mn>0.5</mn><mi>e</mi></mrow></msup><mi>sin</mi><mi>π</mi><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>0.5</mn><mi>π</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi>tanh</mi><mrow><mo>(</mo><mrow><mn>4</mn><mi>π</mi><mfrac><mrow><mi>μ</mi><mi>B</mi></mrow><mrow><mi>k</mi><mi>T</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></msup><mrow><mo>(</mo><mrow><msub><mi>G</mi><mrow><mi>i</mi><mi>n</mi></mrow></msub><mi>cosh</mi><mi>π</mi><mtext>X</mtext></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math><tex-math>$$\Gamma =\frac{{{\varepsilon }_{0}}\Delta r\left( {{t}^{0.5e}}\sin \pi t \right)}{{{\left( 0.5\pi \right)}^{\tanh \left( 4\pi \frac{\mu B}{kT} \right)}}\left( {{G}_{in}}\cosh \pi \text{X} \right)}$$</tex-math></alternatives></disp-formula>In review [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_001_w2aab3b7d147b1b6b1ab2ab1Aa">1</a>] (p. 107), <italic>Gin</italic> is the leaky cable input conductance along the longitudinal length of neuronal fiber (Ω–1) and Χ (Chi) is a normalized length (dimensionless) [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_007_w2aab3b7d147b1b6b1ab2ab7Aa">7</a>,<a ref-type="bibr" href="#j_joeb-2019-0009_ref_008_w2aab3b7d147b1b6b1ab2ab8Aa">8</a>]. The hyperbolic tangent term (p. 108) is Langevin’s thermodynamic relation [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_009_w2aab3b7d147b1b6b1ab2ab9Aa">9</a>]. The sine term (p. 108-109) is the current-modulation function and is a magnetic field-dependent current. On p. 110 [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_001_w2aab3b7d147b1b6b1ab2ab1Aa">1</a>] the myelinated thickness of the axon is given as Δ<italic>r</italic> and <italic>ε</italic>0 is the vacuum permittivity of free space (8.854 x 10–12 F/m). The relationship between <italic>Vm</italic> and Γ is given as <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_joeb-2019-0009_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>V</mi><mi>m</mi></msub><mo>=</mo><mi>Γ</mi><msup><mi>E</mi><mn>2</mn></msup><msub><mrow></mrow><mi>m</mi></msub><mo>−</mo><mn>67.9</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup><mtext>V</mtext></mrow></math><tex-math>${{V}_{m}}=\Gamma {{E}^{2}}_{m}-67.9\times {{10}^{-3}}\text{V}$</tex-math></alternatives></inline-formula>(p. 112), where <italic>Em</italic> is the membrane electric field (V/m).From classical electrodynamics [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_010_w2aab3b7d147b1b6b1ab2ac10Aa">10</a>], Melendy showed that (p. 110) <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_joeb-2019-0009_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>E</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><mi>π</mi><msup><mi>a</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mi>ρ</mi><msup><mi>m</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mrow><mi>t</mi><mi>sin</mi><mi>n</mi><mi>π</mi><mi>t</mi></mrow><mo>)</mo></mrow><mo>=</mo><msup><mi>B</mi><mn>2</mn></msup><msub><mrow></mrow><mi>m</mi></msub><mrow><mo>(</mo><mrow><mn>2</mn><mi>π</mi><mi>a</mi><mtext>/</mtext><msub><mi>μ</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></math><tex-math>${{E}_{m}}\left( \pi {{a}^{2}} \right)\rho {{m}^{-1}}\left( t\sin n\pi t \right)={{B}^{2}}_{m}\left( 2\pi a\text{/}{{\mu }_{0}} \right)$</tex-math></alternatives></inline-formula>(<italic>t</italic> sin <italic>nπt</italic>), where <italic>a</italic> is the axon radius (μm), <italic>ρm</italic> is the longitudinal membrane resistivity (Ω-m), <italic>Bm</italic> 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 <italic>Em</italic> and <italic>Bm</italic> is Melendy’s origianly-derived model (p. 109) but that includes identical numerical parameters as in (1b):<disp-formula id="j_joeb-2019-0009_eq_003_w2aab3b7d147b1b6b1ab1b2b8Aa"><label>(1c)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_joeb-2019-0009_eq_005.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msub><mi>V</mi><mi>m</mi></msub><mo>=</mo><mfrac><mrow><msubsup><mi>B</mi><mi>m</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2</mn><mi>π</mi><mi>a</mi></mrow><mrow><msub><mi>μ</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>t</mi><mrow><mn>0.5</mn><mi>e</mi></mrow></msup><mi>sin</mi><mi>π</mi><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>0.5</mn><mi>π</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi>tanh</mi><mrow><mo>(</mo><mrow><mn>4</mn><mi>π</mi><mfrac><mrow><mi>μ</mi><mi>B</mi></mrow><mrow><mi>k</mi><mi>T</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></msup><mrow><mo>(</mo><mrow><msub><mi>G</mi><mrow><mi>i</mi><mi>n</mi></mrow></msub><mi>cosh</mi><mi>π</mi><mtext>X</mtext></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mn>67.9</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup><mtext>V</mtext></mrow></math><tex-math>$${{V}_{m}}=\frac{B_{m}^{2}\left( \frac{2\pi a}{{{\mu }_{0}}} \right)\left( {{t}^{0.5e}}\sin \pi t \right)}{{{\left( 0.5\pi \right)}^{\tanh \left( 4\pi \frac{\mu B}{kT} \right)}}\left( {{G}_{in}}\cosh \pi \text{X} \right)}-67.9\times {{10}^{-3}}\text{V}$$</tex-math></alternatives></disp-formula>In other words, (1c) produces identical results to (4b) (p. 111) [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_001_w2aab3b7d147b1b6b1ab2ab1Aa">1</a>] for <italic>Bm</italic> = 2.964(74) × 10–5 Wb/m2 and <italic>a</italic> = 4.710(94) μm.The laws of electrodynamics state that <italic>Bm</italic> = Φ<italic>m</italic>/πa2, where Φ<italic>m</italic> is the membrane magnetic flux (Wb). The axon cross-sectional area [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_001_w2aab3b7d147b1b6b1ab2ab1Aa">1</a>] is <italic>π</italic>a2 = 6.972(15) × 10–11 m2, resulting in a flux of Φ<italic>m</italic> = 2.067(06) × 10–15 Wb. This is very nearly the value of the magnetic flux quantum, Φ0 = <italic>h</italic>/2<italic>e</italic> [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_011_w2aab3b7d147b1b6b1ab2ac11Aa">11</a>, <a ref-type="bibr" href="#j_joeb-2019-0009_ref_012_w2aab3b7d147b1b6b1ab2ac12Aa">12</a>, <a ref-type="bibr" href="#j_joeb-2019-0009_ref_013_w2aab3b7d147b1b6b1ab2ac13Aa">13</a>], where <italic>h</italic> = 6.626 069 934 x 10–34 J/Hz is the currently reported measured value of Planck’s constant [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_014_w2aab3b7d147b1b6b1ab2ac14Aa">14</a>] and <italic>e</italic> = 1.602 176 634 x 10–19 C is elementary charge [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_015_w2aab3b7d147b1b6b1ab2ac15Aa">15</a>].</sec><sec id="j_joeb-2019-0009_s_004_w2aab3b7d147b1b6b1ab1b3Aa"><label>III</label><div>Summary</div>The development of an original, quantitative model of the membrane (action) potential <italic>Vm</italic> was presented in [<a ref-type="bibr" href="#j_joeb-2019-0009_ref_001_w2aab3b7d147b1b6b1ab2ab1Aa">1</a>]. 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 <italic>Bm</italic> and subsequently, the membrane magnetic flux, Φ<italic>m</italic>. It was discovered that Φ<italic>m</italic> = 2.067(06) × 10–15 Wb, which is very nearly the value of the magnetic flux quantum, Φ0 = <italic>h</italic>/2<italic>e</italic>. This is evidence for quantized magnetic flux in the membrane of an axon.</sec><sec id="j_joeb-2019-0009_s_005_w2aab3b7d147b1b6b1ab1b4Aa"><label>IV</label><div>Ethical Approval</div>The conducted research reported in this article is not related to either human or animal use.</sec><sec id="j_joeb-2019-0009_s_006_w2aab3b7d147b1b6b1ab1b5Aa"><label>V</label><div>Compliance with Ethical Standards</div>Conflict of Interests: The author Robert F. Melendy, Ph.D. declares that I have no conflict of interest(s).</sec></div>

Evidence for quantized magnetic flux in an axon

Journal of Electrical Bioimpedance

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Evidence for quantized magnetic flux in an axon"}

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...

Evidence for quantized magnetic flux in an axon

Published Online: Nov 06, 2019

Page range: 63 - 64

Received: Jun 27, 2019

DOI: https://doi.org/10.2478/joeb-2019-0009

Keywords
Action potential, axon, cable theory, field-dependent current modulation, fluxoid, Hodgkin, Huxley, Langevin, intracellular magnetization, magnetic flux quantum, membrane depolarization, membrane electric field, membrane magnetic field, myelinated nerve, quantized magnetic flux

© 2019 Melendy, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Evidence for quantized magnetic flux in an axon

Published Online: Nov 06, 2019

Page range: 63 - 64

Received: Jun 27, 2019

DOI: https://doi.org/10.2478/joeb-2019-0009

KeywordsAction potential, axon, cable theory, field-dependent current modulation, fluxoid, Hodgkin, Huxley, Langevin, intracellular magnetization, magnetic flux quantum, membrane depolarization, membrane electric field, membrane magnetic field, myelinated nerve, quantized magnetic flux

© 2019 Melendy, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Action potential, axon, cable theory, field-dependent current modulation, fluxoid, Hodgkin, Huxley, Langevin, intracellular magnetization, magnetic flux quantum, membrane depolarization, membrane electric field, membrane magnetic field, myelinated nerve, quantized magnetic flux