arXiv subjects

Robert F. Melendy

Publications and source records attributed to Robert F. Melendy.

3 recordsLinked to original sources

arXiv2026

The Lie Group Basis of Neuronal Membrane Architecture: Why the Hodgkin-Huxley Equations Take Their Form

The Hodgkin-Huxley equations have described neuronal excitability for seventy years, yet their mathematical structure-gating exponents m3h and n4, exponential voltage dependencies, and bounded activation variables, has remained empirically justified rather than theoretically derived. Hodgkin and Huxley introduced voltage-dependent conductances controlled by gating variables. While these equations reproduce experimental observations, they were derived through curve-fitting without theoretical justification. Modern theoretical physics derives governing equations from symmetry principles through Lie group theory. We prove that the complete Hodgkin-Huxley equations necessarily follow from three fundamental symmetries: (1) compact conformational state spaces, (2) multiplicative conductance scaling, and (3) temporal translation invariance. These symmetries uniquely determine a Lie group structure isomorphic to SO(2) semidirect product with R2. From representation theory, we derive: boundedness from SO(2) compactness, exponential Boltzmann factors from scale invariance, specific integer exponents m3h and n4 from irreducible representations, and first-order kinetics from Lie algebra flows. This demonstrates that the HH equations are not empirical curve-fits but the unique mathematical structure mandated by fundamental symmetries. We reveal why gating variables must be bounded, voltage dependencies must be exponential, sodium requires three activation gates and one inactivation gate, potassium requires four activation gates, and kinetics must be first-order. This establishes that neural electrophysiology obeys the same theoretical framework as modern physics, where symmetries determine dynamics, providing a foundation for understanding channel mutations and network dynamics through group theory.

physics.bio-ph↗

arXiv2021

Bang-Bang Control Development of Permeability Changes in a Membrane Model

The application of systems and control theory to membrane physiology is presented here. Modeling efforts have focused on describing those physiologically realistic mechanisms which govern the regulation of membrane permeability in nerve. The motivation behind identifying such mechanisms lies in understanding the morphology of neural activity on a meaningful and analytically tractable level. The suggested merit of integrating control theory into the analysis lies in providing how a membrane effectively adapts to changes in permeability and through what governing mechanisms. The value in producing such an understanding lies in mirroring biological reality in a more formal manner than could be achieved solely through experimental means. A bang-bang control policy describing the permeability correction mechanisms is developed using Liapunov's Stability Criteria. Both changes in membrane potential and kinetic rates are required to implement the policy. The policy describes the inherent mechanisms of the membrane which act to drive its permeability from unstable firing to the resting potential state. It is shown that these permeability changes in state are governed by a switching function that depends on the membrane potential and a dominant controlling parameter. The control policy is discussed in the context of solutions of the Hodgkin-Huxley Equations of Ionic Hypothesis.

physics.bio-ph↗

arXiv2019

A Single Differential Equation Description of Membrane Properties Underlying the Action Potential and the Axon Electric Field

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, Vm and that of the accompanying cell membrane electric field, Em. To conclude, I present a novel description of Vm in terms of a single, nonlinear differential equation. This is an original stand-alone article: the major contribution is the latter, and how this description coincides with the cell membrane electric field. This work has necessitated unifying information from two preceding papers, each being concerned with the development of closed-form descriptions of the nerve action potential.

physics.bio-ph↗