Membrane state diagrams make electrophysiological models simple

Ion channels are ubiquitous in living systems. Through interactions with membrane potential, ion channels both control metabolic events and mediate cell communication. Consequentially, membrane bioelectricity bears on fields ranging from cancer etiology to computational neuro-science. Conductance models have proven successful in quantitatively capturing these dynamics but are often considered difficult, with interpretation relegated to specialists. To facilitate research in membrane dynamics, especially in fields where roles for ion channels are just beginning to be quantified, we must make these models easy to understand.\n\nHere, we show that the membrane differential equation central to conductance models can be understood using simple circular geometry. The membrane state diagrams we construct are compact, faithful representations of conductance model state, designed to look like circular \"cells\" with currents flowing in and out. Every feature of a membrane state diagram corresponds to a physiological variable, so that insight taken from a diagram can be translated back to the underlying model. The construction is elementary: we convert conductances to angles subtended on the circle and potentials to radii; currents are then areas of the enclosed annular sectors.\n\nOur method clarifies a powerful but prohibitive modeling approach and has the potential for widespread use in both electrophysiological research and pedagogy. We illustrate how membrane state diagrams can augment traditional methods in the stability analysis of voltage equilibria and in depicting the Hodgkin-Huxley action potential, and we use the diagrams to infer the possibility of nontrivial fixed-voltage channel population dynamics by visual inspection rather than linear algebra.