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Contributions to the mathematical biology of excitation with particular emphasis on changes in membrane permeability and on threshold phenomena

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Abstract

Simple chemical reactions of Ca++ and K+ ions with a P− ion in a membrane are assumed to be causes of changes in permeability of that membrane to K+ ions. On the basis of such a mechanism a quantative concept of membrane permeability to K+ ions is defined. The diffusion of K+ ions through such a membrane is studied mathematically in a simplified version. An applied electrical field as well as the diffusion potential of the K+ ions are considered to effect the chemical equilibrium constants of the proposed reactions. It is shown that such a system, which can be described by a set of nonlinear differential equations, may have two stable states of equilibrium which are separated by an unstable equilibrium state. As a consequence, such a system may possess a threshold. Estimations of resting potential, threshold—electrical as well as chemical—and of permeability increase, together with that of the corresponding electrical field strength are shown to have the correct order of magnitude. A possible way to derive the one-factor theory from a physical mechanism as considered here is outlined. It is pointed out that the dependence of the thresholds and the permeability changes on several parameters might be calculated on the basis of such a mechanism. As an example it is shown how in principle some of these relations are derived. Furthermore, the time course of excitatory disturbance for different intensities of the initial disturbance are derived theoretically for the case of chemical stimulation. The curves so obtained show a striking similarity in all the characteristic features with the corresponding ones which are obtained experimentally for the case of electrical stimulation of nerve. These results suggest that response to electrical and chemical stimulation is based on a common threshold phenomenon such as considered here. Finally, a more detailed mathematical description, which takes into account explicitly the diffusion of K+ ions through the membrane for a finite thickness of the membrane is outlined. The equations obtained, which seem to be infeasible of solution at the present time, suggest that it is plausible that relaxation oscillations with a threshold can be derived on the basis of such a mechanism as proposed here. Qualitative agreement with experimental evidence is indicated.

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  1. Committee on Mathematical Biology, The University of Chicago, Chicago, USA

    George Karreman

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  1. George Karreman
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Karreman, G. Contributions to the mathematical biology of excitation with particular emphasis on changes in membrane permeability and on threshold phenomena. Bulletin of Mathematical Biophysics 13, 189–243 (1951). https://doi.org/10.1007/BF02478227

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