Dynamics of cell membrane passive depolarization: a phase portrait

Does a persistent blockage of the ionic pumps bring cell membrane voltage to zero? This apparently trivial question of basic cellular Biology stirred up an intriguing problem of nonlinear dynamics. A 3-ion model based on continuity and charge conservation proves that membrane voltage actually sets on a negative value, meaning that chemical equilibrium is never reached, rather an inversion of the Na+ concentration gradient occurs, usually hours after the blockage of the pumps. Experimental tests carried out with PC12 cells incubated with Oubaine for a period of 24 hours show an increase of cytosolic Na+ of about 266 mM/l with respect to a control sample. The result is compatible with an inversion of the Na+ gradient, which eventually brings the membrane voltage to a negative value. Reactivation of the Na+-K+ pumps even after a prolonged period of blockage (late repolarization) should lead to repolarization and revival of the cell. In the 3D space of the ionic concentrations, the dynamics of passive depolarization reveals an intriguing topology, all trajectories being stacked in parallel planes, each set ending to a unique fixed point via an infinitely dense set of lines. The dynamics of repolarization has a different phase portrait, especially in the case of late repolarization. Thus, a sequence of depolarization- repolarization cycles may result in a path wandering in the phase space, or in a closed loop, depending on the timing of the sequence.