A Closed-Form Adaptation of the Hodgkin-Huxley Membrane Potential and its Synthesis to a Novel Electric Circuit
Authors: Melendy Robert F. , Nguyen Loan 
Published: July 2022

AbstractIn a succession of journal papers published over 65 years ago, Sir Alan Lloyd Hodgkin and Sir Andrew Fielding Huxley discovered what now forms our contemporary understanding of excitation in nerve, and how axons conduct the action potential. Hodgkin and Huxley demonstrated that the nerve action potential is the result of a depolarizing event across a cell membrane. In an elegant theoretical framework, they established that when this depolarization event is complete, an abrupt increase in voltage gets produced that propagates longitudinally along the axon, accompanied by changes in axial conductance. Notwithstanding the elegance of Hodgkin and Huxley’s incisive and explicative series of discoveries, their model is relatively 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 paper, we present a closed-form adaptation of the Hodgkin-Huxley membrane voltage potential. The basis of our model is rooted in core conductor theory and the cable properties of neurons, with fitting parameters adapted to the classical Hodgkin-Huxley model of excitation in nerve. From this model we synthesize a novel analog circuit that simulates the dynamics of a single action potential bioelectrically equivalent to the classical Hodgkin-Huxley membrane potential. The primary novelty of our model is that it offers a bioconductive, thermodynamic, and electromagnetic explanation of how an action potential propagates in nerve in a single mathematical construct. This is in contrast to the traditional Hodgkin-Huxley equations of ionic hypothesis, which are not analytically compliant. Computational results of our model are supported by well-established quantitative descriptions of Hodgkin-Huxley’s voltage response in the membrane of an axon. Our findings provide a mechanistic understanding of how intracellular conductance, the thermodynamics of magnetization, and current modulation function together to generate excitation in nerve in a unified closed-form description. In the same manner with Hodgkin-Huxley’s findings, the model presented here corroborates (1) that the action potential is the result of a depolarizing event across a cell membrane; (2) that a complete depolarization event is followed by an abrupt increase in voltage that propagates longitudinally along the axon; (3) that the latter is accompanied by a considerable increase in membrane conductance. The work presented in this paper provides compelling evidence that three basic factors contribute to the propagated signaling in the membrane of an axon in a single, closed-form model. From our model, we synthesize a novel analog conductance-level circuit that simulates the dynamics of a single action potential bioelectrically equivalent to the classical Hodgkin-Huxley membrane potential. It’s anticipated this work will compel those in biophysics, physical biology, and in the computational neurosciences to probe deeper into the classical and quantum features of membrane magnetization and signaling. Furthermore, it’s hoped that subsequent investigations of this sort will be advanced by the computational features of this model without having to resort to numerical methods of analysis.AttributionA portion of this work is reprinted from R.F. Melendy,Resolving the biophysics of axon transmembrane polarization in a single closed-form description [1].Journal of Applied Physics,118(24), Copyright © (2015); and from R.F. Melendy,A subsequent closed-form description of propagated signaling phenomena in the membrane of an axon [2].AIP Advances,6(5), Copyright © (2016), with the permission ofAIP 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.SummaryThis work provides evidence that three basic factors contribute to propagated signaling in the membrane of an axon. The contributing factors are unified in a closed-form description. From this closed-form model we synthesize a novel analog circuit that simulates the dynamics of a single action potential that is bioelectrically equivalent to the classical Hodgkin-Huxley membrane potential.