Applications of Results from Linear Kinetics to the Hodgkin-Huxley Equations

Abstract

On the basis of its role in the analysis of mammillary compartmental systems, a matrix with non-zero elements in the first row, first column, and along the main diagonal and with zero elements elsewhere is called a mammillary matrix. It is pointed out that such matrices occur in a variety of biological problems including the linearized Hodgkin-Huxley equations (H-H). In considering whether such a linear system exhibits stability (all roots of the matrix with negative real parts) it is of interest to seek conditions, expressible in a simple manner in terms of the matrix elements, which lead to stability or instability. For the case when the diagonal elements, with the possible exception of the first, are negative (a condition physically guaranteed for the space-clamped axon) simple criteria for instability and stability are formulated in terms of the matrix elements. These criteria are derived by extending previous results from linear kinetics through appeal to a classical matrix theorem without recourse to the characteristic polynomial. The relation of these mathematical results to the work of Chandler, FitzHugh, and Cole on the space-clamped axon is discussed. The results are in no way restricted by the order of the matrix (which is four for the H-H equations) and other possible applications are noted.