Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels

Individual ion channels that mediate sodium and potassium currents during action-potential generation in nerve membranes open and close stochastically with transition rates that are voltage dependent. The transmembrane voltage globally couples the otherwise stochastically independent ion channels through their voltage-dependent transition rates. As the membrane channel density increases, a transition to regular, collective behavior ensues. The mathematical basis for this transition is explained in terms of a hierarchy of description contractions. Stochastic and deterministic modifications of the Hodgkin-Huxley equations are obtained.