Abstract
A cable model that includes polarization-induced capacitive current is derived for modeling the solitonic conduction of electrotonic potentials in neuronal branchlets with microstructure containing endoplasmic membranes. A solution of the nonlinear cable equation modified for fissured intracellular medium with a source term representing charge ‘soakage’ is used to show how intracellular capacitive effects of bound electrical charges within mitochondrial membranes can influence electrotonic signals expressed as solitary waves. The elastic collision resulting from a head-on collision of two solitary waves results in localized and non-dispersing electrical solitons created by the nonlinearity of the source term. It has been shown that solitons in neurons with mitochondrial membrane and quasi-electrostatic interactions of charges held by the microstructure (i.e., charge ‘soakage’) have a slower velocity of propagation compared with solitons in neurons with microstructure, but without endoplasmic membranes. When the equilibrium potential is a small deviation from rest, the nonohmic conductance acts as a leaky channel and the solitons are small compared when the equilibrium potential is large and the outer mitochondrial membrane acts as an amplifier, boosting the amplitude of the endogenously generated solitons. These findings demonstrate a functional role of quasi-electrostatic interactions of bound electrical charges held by microstructure for sustaining solitons with robust self-regulation in their amplitude through changes in the mitochondrial membrane equilibrium potential. The implication of our results indicate that a phenomenological description of ionic current can be successfully modeled with displacement current in Maxwell’s equations as a conduction process involving quasi-electrostatic interactions without the inclusion of diffusive current. This is the first study in which solitonic conduction of electrotonic potentials are generated by polarization-induced capacitive current in microstructure and nonohmic mitochondrial membrane current.
Highlights
The electrophysiological applications of cable theory led Hodgkin and Huxley (H-H) [1] to quantitatively describe voltage-dependent currents obtained by using the voltage-clamp technique
The electrotonic signals are insensitive to the initial location of its position along the cable Xp as the hyperbolic secant function reaches a maximum value of unity at X = Xp
When the mitochondrial membrane equilibrium potential increases, the peak amplitude of the wave approaches that of the passive neuronal membrane case
Summary
The electrophysiological applications of cable theory led Hodgkin and Huxley (H-H) [1] to quantitatively describe voltage-dependent currents obtained by using the voltage-clamp technique. The remarkable success of the H-H model is a mathematical description that relates the microscopic dynamics of gated ion channels to the macroscopic behavior of membrane potential. The Frankenhaeuser and Huxley (F-H) model developed in 1964 [2] was an attempt to include in the H-H model electrodiffusion of ions within the plasma membrane. The F-H model includes electrodiffusion of membrane ion channel permeability based on a description for ionic concentration across membranes where the spatial distance reflects charge spread within the membrane and not within the cytoplasm. Analytical solutions to the F-H equations were obtained when voltage-dependent ionic channels are distributed at discrete positions throughout the membrane based on ionic cable theory [3]
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