Multiple equilibria and exotic behaviour in excitable membranes

The excitation equation for an excitable membrane dV/dt = F(V) may have multiple equilibria where F(V) = 0, and these may be stable or unstable. We demonstrate multiple equilibria in the Hodgkin-Huxley equations when either -gK or [Ca2+]0 is lowered in the presence of a hyperpolarizing current density. Under these conditions molluscan somata exhibit exotic behaviours -- endogenous paroxysmal depolarising shifts and complex multiple spikes reminiscent of the normal complex activity of some mammalian central neurones. Complex discharge waveforms can be an expression of membrane (differential) properties, rather than electrotonic, geometric (partial differential) behaviour.