Neuronal activity as the behavior of a differential system

A geometric approach to the activity of membrane equations suggested examining the response of molluscan isopotential somata to K
⁺
-channel blockers (four-aminopyridine and tetraethylammonium), [Ca
²⁺
], and applied currents using conventional and phase plane displays. Multiple equilibria and subcritical and supercritical bifurcations of periodic activity from equilibria are demonstrated, analogous to those obtained for the Hodgkin-Huxley differential system. The molluscan somatic membrane is a more complicated excitation system than the Hodgkin-Huxley equations, and using K
⁺
-channel blockers, the bifurcations of periodic activity from periodic activity are also demonstrated. In both the numerical and experimental excitation systems the richness of stable behavior is greater than that seen in the corresponding partial differential systems. Thus the behavior possible in isopotential regions of neurons is more complex than that behavior which can be transmitted by propagating action potentials along long axons.