Flushing waste in the central nervous system
The central nervous system has a tiny extracellular space easily filled by flows from nerve and glia. Potassium ions in that space can block signaling in nerve fibers and thus become a toxic waste. Sleep is said to flush toxic wastes from the brain, in the glymphatic hypothesis. Qualitative hypotheses like this are difficult to test and can lead to more discussion than knowledge. Numbers are needed because flows in complex structures are complex. We construct models that are field theories built on conservation laws written as partial differential equations in three dimensions and time. Models can be molded to fit complex structures in the brain. Models can include channels/transporters as they are discovered. The differential equations fit experimental data from the optic nerve in some detail, as do equations of the lens of the eye. Simplified structures are assumed by biological and clinical communities because they are needed to deal with disease and design experiments. Here, simplified structures are derived, not assumed. Simplified models are studied using perturbation expansions, with errors determined from numerical solutions. Computation shows that extracellular potassium is maintained by bulk flow, mostly in the glia. The glia acts as a pipe that moves potassium by convection away from the nerve membrane, presumably into blood vessels, as proposed by the glymphatic hypothesis.