Some Mathematical Problems from Neurobiology

Physiological background. We begin by describing, briefly, the physical setting of the model, and a little of the experimental evidence, in order to motivate the mathematical questions to be asked later. For a fuller explanation of the basic neurophysiology, see [29]. The function of the nervous system is, of course, to transmit information, and for this purpose it uses electrical impulses initiated and transmitted by individual nerve cells, or neurons. Hodgkin and Huxley developed a set of differential equations to describe the ionic and electrical events occurring during the transmission of an impulse along an axon, which is usually the filament carrying signals from the nerve cell body to other parts of the organism. These equations are based on the assumption that an axon behaves like a cylindrical electrical cable, with a conducting core and partially insulating sheath, or membrane. Both the core and the surrounding medium are fluids containing significant concentrations of charged metallic ions, mostly sodium and potassium. The currents which flow are due either to the capacitance of the membrane or to the movement of ions either through the membrane or longitudinally along the axon. Perhaps the most important feature of the theory is its reliance on changes in the permeability of the membrane to certain of these ions during the course of an impulse. (The molecular structure of the membrane which allows it to play an active role in signal transmission is unknown. The membrane is the black box referred to earlier.) The resulting nonlinear equations can be written in the form