  Electrical signals in excitable cells involve spatially localized ionic
fluxes through ion channels and pumps on cellular lipid membranes. Common
approaches to understand how these localized fluxes spread assume that the
membrane and the surrounding electrolyte comprise an equivalent circuit of
capacitors and resistors, which ignores the localized nature of transmembrane
ion transport, the resulting ionic gradients and electric fields, and their
spatiotemporal relaxation. Here, we consider a model of localized ion pumping
across a lipid membrane, and use theory and simulation to investigate how the
electrochemical signal propagates spatiotemporally in- and out-of-plane along
the membrane. The localized pumping generates long-ranged electric fields with
three distinct scaling regimes along the membrane: a constant potential
near-field region, an intermediate "monopolar" region, and a far-field
"dipolar" region. Upon sustained pumping, the monopolar region expands radially
in-plane with a steady speed that is enhanced by the dielectric mismatch and
the finite thickness of the lipid membrane. For unmyelinated lipid membranes in
physiological settings, we find remarkably fast propagation speeds of $\sim\!40
\, \mathrm{m/s}$, allowing faster ionic reorganization compared to bare
diffusion. Together, our work shows that transmembrane ionic fluxes induce
transient long-ranged electric fields in electrolyte solutions, which may play
hitherto unappreciated roles in biological signaling.
