Ion flux through a single transmembrane channel is the fundamental unit of bioelectricity, yet classical equivalent-circuit models do not explain the emergence of macroscopic responses from the microscopic currents. Toward bridging this gap, we develop a boundary layer theory that resolves nanoscale diffuse charge dynamics generated by a single ion channel current. Our analysis shows that the spatiotemporal dynamics are shaped by the coupling between charge-containing boundary layers at surfaces and the electroneutral bulk: ions translocated through the channel accumulate in the diffuse layers, thereby dynamically reshaping the bulk electric field; the altered field in turn drives further ionic reorganization. We identify distinct regimes of membrane charging, derive closed-form expressions for the membrane potential in each regime, and validate these results with large-scale simulations. We apply this framework to two biologically inspired membrane geometries: a cylindrical, axon-like membrane and a planar membrane stimulated by electrodes, reminiscent of electrophysiological experiments. At short times, ionic reorganization is dominated by dipolar electric field from newly displaced charges at the channel. The electric field is shaped by the geometry and electrical properties of surfaces. At long times, charge transport appears diffusive, but the underlying transport is shaped by electric fields rather than thermal fluctuations—the effective diffusivity scales linearly with system size. In cylindrical membranes, this scaling correlates with the signal conduction velocity, which typically increases with axon diameter. In electrode-bound systems, the dynamics asymptotically reduce to an equivalent-circuit description consistent with cable theory, clarifying its regime of validity. More broadly, our framework captures charge reorganization near membranes with microscopic detail at greatly reduced computational cost relative to full simulations of the governing equations. It thus provides a foundation for modeling the collective dynamics of multiple ion channels.
Row et al. (Sun,) studied this question.