In a succession of articles published over 65 years ago, Sir Alan Lloyd Hodgkin and Sir Andrew Fielding Huxley established what now forms our physical understanding of excitation in nerve, and how the axon conducts the action potential. They uniquely quantified the movement of ions in the nerve cell during the action potential, and demonstrated that the action potential is the result of a depolarizing event across the cell membrane. They confirmed that a complete depolarization event is followed by an abrupt increase in voltage that propagates longitudinally along the axon, accompanied by considerable increases in membrane conductance. In an elegant theoretical framework, they rigorously described fundamental properties of the Na^{+} and K^{+} conductances intrinsic to the action potential.

Notwithstanding the elegance of Hodgkin and Huxley’s incisive and explicative series of discoveries, their model is mathematically complex, relies on no small number of stochastic factors, and has no analytical solution. Solving for the membrane action potential and the ionic currents requires integrations approximated using numerical methods. In this article I present an analytical formalism of the nerve action potential, _{m}_{m}_{m}_{m}

#### Keywords

- Action potential
- axon
- core conductor
- current modulation
- field-dependent current
- Hodgkin
- Huxley
- Langevin
- intracellular magnetization
- membrane conductance
- membrane depolarization
- membrane electric field
- myelinated nerve
- neuronal cable theory

A portion of this work is reprinted from [1] R.F. Melendy, Resolving the biophysics of axon transmembrane polarization in a single closed-form description.

There is abundant and well-grounded research quantifying the electrical behavior of myelinated and unmyelinated nerve fibers [3, 4, 5, 6]. This has come to include a precise understanding of membrane impedance properties, and the longitudinal voltage and ionic currents that propagate in axon membranes [7, 8, 9, 10] at the onset of a complete depolarization event. Of note is Hodgkin’s and Huxley’s quantification of ionic membrane currents and their relation to conductance and excitation in nerve [11].

Since this time, more than a few researchers have focused on rigorously describing membrane structure and electrical phenomena. A fundamental advancement in this direction was the development and prolific use of cable theory to describe signal transmission in the membrane of an axon [12, 13, 14, 15, 16].

In classic cable theory, axons are treated as core conducting cylinders of finite length, where the capacitive and conductance properties of the axon membrane are modeled as a distributed-parameter electrical network [17,18]. Consequently, quantitative determination of the membrane action potential and ionic currents requires solving a boundary-value problem. This approach provides a systematic means for realistically describing the action potential and the axon membrane field properties [19,20]. However, this method of modeling typically depends on the use of advanced analytical and numerical methods to solve the partial differential equations.

Comparably, the Hodgkin-Huxley equations of ionic hypothesis are a relatively complex system of differential equations that have no analytical solution: solving for the membrane action potential, membrane conductances, and the ionic currents requires integrations approximated using numerical methods.

The scope of this article is to derive an original, quantitative description of the membrane potential, _{m}_{m}_{m}_{m}

In this section, both electrodynamic and thermodynamic evidence will be presented in forming a basis on which the displacement of the membrane potential is described. In due course, the former will be presented in a unified, analytical description of membrane excitability, followed by a description of _{m}^{π}

One solution to the neuronal cable equations is a function describing the input resistance _{in}

where _{∞} is the input resistance of a semi-infinite cable and is proportional to the characteristic length _{m}^{2}), _{i}

It is elementary to rewrite the coth term of (1a) as _{in}_{∞} (1/tanh Χ), or as _{∞} (cosh Χ /sinh Χ). This is identical to writing (_{in}_{∞}/sinh Χ) = (_{∞} csch Χ). Since resistance is the reciprocal of conductance ^{–1}), the latter may be expressed as (1/_{in}

The relevance of (1b) is it describes how a rapid drop in the input resistance of a semi-infinite cable (_{∞}) balances with a significant increase in the leaky cable input conductance (_{in}

By and of itself, the hyperbolic conductance term (1b) is intrinsic to the displacement of the membrane potential, _{m}

The inverse variation (1c) is consistent with the fact that voltage varies inversely with conductance [25]. For initial computational generality,

A natural consequence of a depolarizing membrane is the generation of a changing magnetic field. This is supported by a body of established research corroborating the existence of time-varying magnetic fields in an axon during the nerve impulse [26, 27, 28, 29, 30].

A common thread that runs through these studies is that the bioelectric activity present during the action potential produces a current in a volume conductor. For instance, the current density ^{–2}) throughout a volume conductor generates a biomagnetic field,

Biological tissue has been shown to have paramagnetic properties, particularly in the presence of Ca^{+} and Na^{+} ions [31, 32, 33]. It’s therefore relevant to consider intracellular magnetization as an intrinsic membrane property and particularly, over the action potential cycle. Langevin’s paramagnetic equation is suitable in this circumstance: (^{–1} or J⋅T^{–1}⋅m^{–3}), ^{–1})], ^{−23} J⋅K^{–1}), and

Langevin’s equation predicts that a paramagnetic material saturates asymptotically to the line (_{Na} of the axon membrane. This produces a marked increase in the current density throughout the conducting medium [36] and subsequently, an appreciable increase in the magnetization of the intracellular membrane. By Langevin’s relation, it stands to reason that this intracellular magnetization saturates as all the moments become aligned against the biomagnetic field during a complete polarization event.

Based on this hypothesis, the hyperbolic conductance term (1c) and Langevin’s thermodynamic relation are asserted to vary together, such that:

The inverse variation (2) has left-hand units of Ω (since Langevin’s relation is dimensionless). As with (1c),

An accepted and reliable method for depolarizing the excitable cells of a membrane involves variations in voltage-clamping techniques [37,38]. Regardless of method, the sensors utilized in voltage-clamping exploit the properties of the membrane potential and ionic current signals [39]. These signals are not fundamental. They’re constructed of sinusoidal harmonics of the form

One can deduce ^{B· dl = μ0I(B).}

This prompts a fundamental question: can one quantify ^{2}^{2} (A⋅T ^{–2}).

As is characteristic of the classically understood Na^{+} and K^{+} time-dependent currents, it’s reasonable to assert

To mathematically synthesize a function for _{n}_{n}^{n} x^{n}^{–1}^{n}

Neurons of membranes have been shown to have natural frequency-selective feedback properties [45,46]. It stands to reason that such properties would influence how the field-strength current ^{2}^{2}.

On the premise of the preceding discussion, it’s reasonable to expect that ^{2}^{2} would exhibit fluctuations through the membrane over the action potential cycle. This is supported by the elementary fact that a magnetic field cannot instantaneously collapse in an axon as the action potential transitions from depolarization to the hyperpolarizing afterpotential.

One plausible conjecture is that the current modulation signal behaves according to ^{2}^{2} ∝ _{n}^{–3}) is sufficient to completely depolarize the membrane.

There are chaotic nonlinearities associated with initiation of the nerve impulse by membrane depolarization [49,50]. When this is taken in conjunction with the oscillatory nature of the spherical Bessel functions _{n}^{2}^{2} will exhibit unstable oscillations for the period of the action potential cycle [51, 52, 53].

Without exception, unstable eigenvalues are almost always present in dynamic systems: in biological systems, there are intrinsic control mechanisms that operate in the presence of unstable equilibrium points to produce a stable response, often ^{n} x^{n}^{–1}^{n}^{2}. On that account, it follows that ^{0} (^{2} ≡ ^{2}^{2} = (2^{0} ) × (^{0} ) is a consequence of Ampere’s law. The axon radius is ^{~}μm) and ^{0} is the vacuum permeability of free space (4^{–7} H⋅m^{–1}).

The question posed –

By Ohm’s law, _{m}^{th} derivative of

The right-hand units of (3a) are V⋅T^{–2}. As before, _{m}

(3b) has units of V and offers an initial analytical description of the membrane action potential _{m}_{m}

The numerator of (3b) is a current term having units of amps (A). Electrical current in the axon per unit area of axon cross section is ^{–2}), where ^{2}. This current density may also be described as _{m}_{m}_{m}^{–1}) and _{m}^{2})_{m}_{m}

The electric field is considered constant along the axon longitudinal axis but is radially-dependent, such that _{m}_{m}_{m}_{m}

Consider now the introduction of a proportionality constant _{m}^{–1}) × (V⋅m^{–1}). The conjecture therefore is that _{0}_{m}_{0} is the vacuum permittivity of free space (8.854 × 10^{–12} F⋅m^{–1}). Substituting these relations into (3b) gives:

The units of volts are preserved in going from (3b) to (3d). It will be subsequently shown that (3d) gives a correct description of the classic nerve action potential and the cell membrane electric field. Computational results for _{m}_{m}

A Matlab algorithm was developed to computationally test the modeling suitability of (3d). This required a practical choice of physical membrane parameters [14,23,25,35,36]:

Axon thickness (myelinated): Δ

Axon (“cable”) length: 0 ≤

Length constant:

Resistance (unit area of membrane): _{m}^{2}

Intracellular resistivity: _{i}

Input resistance (semi ∞ cable): _{∞} =

20.3718(32) MΩ

Nonlinear magnetization (unitless): 0 ≤ (

Action potential cycle time: 0 ≤

Vacuum permittivity _{0}: 8.854(10^{–12}) F/m

The Matlab algorithm was used to first compute an appropriate number of

The magnetization factor tanh (

An asymptotic series expansion [58] of this factor was performed to reveal the sensitivities associated with each of the terms in the tanh argument. In consequence, it was established that this factor was best-fit to an exponential function having the form ^{tanh (}_{nπμB}^{/}_{kT}^{)}.

A first prediction for

For the sin argument of (3d), the Matlab algorithm returned a best-fit iteration of

This reasoning supported the notion of

A double-precision floating point Matlab algorithm was written to compute

Where the intracellular resting potential of the membrane

The Matlab algorithm was used to establish an estimate of _{0}Δ_{rE}^{2}_{m}^{–8}. If all previously discussed hypotheses are sound, this estimate will have the units V^{2}⋅F⋅m^{–2}.

The conducted research is not related to either human or animal use.

To demonstrate that (4a) gives a correct description of the classic nerve action potential and the cell membrane electric field, computational results for _{m}_{m}

_{m}

The thickness of a myelinated cell membrane is Δ^{0}ΔrE^{2}_{m}^{–12} F⋅m^{–1}) × (2 μm) × ^{2}_{m}^{–8} V^{2}⋅F⋅m^{–2}. This results in _{m}^{4} V⋅m^{–1}.

A classic axon membrane model will have a potential difference between the interior and exterior side of the membrane of Δ_{m}_{m}_{m}^{4} V⋅m^{–1} [23,60,63]. This theoretical result is highly consistent with the computation of the electric field from the analytical model (4a), having a percent error ≈ 2.3%. This is an initial confirmation that (4a) is a correct description of the classical membrane action potential cycle, _{m}

_{m}

To further establish that (4a) provides a correct description of a classical action potential cycle, a computational profile of _{m}

Compiling all preceding factors into the Matlab algorithm gives a restoration voltage of _{m}

Figure 1 is a plot of _{m}

In the classical Hodgkin-Huxley model, it’s well-known that the lipid bilayer of the axon membrane is modeled as a lumped-capacitance _{m}_{m}_{0}Δ

Also well-known to the Hodgkin-Huxley model is the quantification of the ionic current flow _{C}_{m}_{C}_{m}_{m}_{m}

The relationship between the membrane electric and magnetic fields _{m}_{m}_{m}^{2}_{m}_{m}_{m}_{m}_{m}^{2}_{m}_{m}_{C}_{m}

The field current term in _{0}Δ_{m}^{2} has units of V^{2}⋅F⋅m^{–2}: but this is also units of T^{2}. The point is that _{C}

It is hoped this makes clear the implicit manifestation of the membrane current underlying the action potential and its relationship to _{m}

From

So then _{m}_{m}

As previously mentioned, the electric field is constant along the axon’s longitudinal axis, but is radial-dependent such that _{m}_{m}_{m}

Where ^{–3})Γ^{–1}. Equation _{m}

The development of an original, quantitative description of the membrane (action) potential displacement _{m}

Evidence was given that three principal factors form a basis on which the membrane potential displacement is described. These three factors are the axon leaky cable conductance, intracellular membrane magnetization, and membrane current modulation.

These three hypothesized factors were unified in a single analytical form for quantitatively determining _{m}

Beginning with substitution of established membrane parameters, the range of phenomena to which the analytical form is relevant was demonstrated by: (_{m}_{m}

One of the novelties of this work is that it provides a mechanistic understanding of how intracellular conductance, the thermodynamics of magnetization, and current modulation function together to generate excitation in nerve.

Another novel feature of this work is the statistical mechanics description of intracellular magnetization, and how this phenomenon relates to the presence of ions in the membrane channel.

The significance of this model is that it offers an original and fundamental advancement in the understanding of the action potential in a unified analytical description. It provides a conductive, thermodynamic, and electromagnetic explanation of how an action potential propagates in nerve in a single mathematical construct.

Another significant feature of this model is that it offers a new and rigorous description of the action potential, quantified as a single, nonlinear differential equation in _{m}

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