Boundedness theorems and other mathematical studies of a Hodgkin-Huxley-type system of differential equations: Numerical treatment of thresholds and stationary values: Parts I, II and III

Abstract

We present here a study of the Hodgkin-Huxley system of four first-order nonlinear ordinary differential equations (with initial values) as reformulated by H.M. Lieberstein. The original Hodgkin-Huxley system modeled the development and propagation of voltage impulses on nerve fibers. It was altered by Dr. Lieberstein so as to include the effects of line inductance and line capacitance without introducing any new parameters. It included the application of a sustained constant membrane current density I 0. The inclusion of I 0 enables the system to model the electrical behavior of smooth muscle cells, pacemaker cells, receptor cells, and cells with a plateau-type behavior, in addition to nerve cells. In Part I, after a section that provides background material and preliminary facts concerning the system, local existence, uniqueness, and analyticity of solutions to the system are proved. Boundedness of the solutions is proved in Part II. It isproved there that three of the four solution components ( n, m, and h) have lower bounds of zero and upper bounds of one. The lower boundedness of the fourth component, the membrane voltage v, is then proved, followed by a proof for upper boundedness. A second proof for lower boundedness is also given. Boundedness of the solutions leads to the existence and uniqueness of solutions for all nonnegative time. The continuous dependence of the solutions on initial values and parameters follows. Besides proving that the sensitivity to parameters inherent in the original Hodgkin-Huxley model is removed, continuous dependence also limits the mathematical possibilities that could be involved in the threshold phenomenon that appears in the equations. New bounds for n, m, and h are also obtained from the fact that v is bounded. In Part III, the mathematical possibilities that might govern the threshold phenomena that appear in the model are examined and new numerical work is presented, which indicates that the model yields solutions that are intermediate to the “on” and “off” solutions expected with such phenomena. Then theorems and numerical work are presented indicating that stationary points are not involved in the threshold phenomenon, although they do determine the behavior of the system at large positive times.