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].
Method
Melendy [1] demonstrated that the differential equation:
$${{\left[ \frac{\partial {{V}_{m}}}{\partial \left( \Delta r \right)} \right]}^{2}}-\frac{1}{\Gamma }{{V}_{m}}-u=0$$
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)Γ –1, where:
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 ${{V}_{m}}=\Gamma {{E}^{2}}_{m}-67.9\times {{10}^{-3}}\text{V}$(p. 112), where Em is the membrane electric field (V/m).
From classical electrodynamics [10], Melendy showed that (p. 110) ${{E}_{m}}\left( \pi {{a}^{2}} \right)\rho {{m}^{-1}}\left( t\sin n\pi t \right)={{B}^{2}}_{m}\left( 2\pi a\text{/}{{\mu }_{0}} \right)$(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):
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].
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.
Ethical Approval
The conducted research reported in this article is not related to either human or animal use.
Compliance with Ethical Standards
Conflict of Interests: The author Robert F. Melendy, Ph.D. declares that I have no conflict of interest(s).
