Define the Historical Example of Neuroscience?
In the 1950's Hodgkin and Huxley set out to recognize the propagation of action potentials in the squid giant axon. They perfected the voltage clamp method that is now routinely utilized to dissect the voltage-dependent dynamics of ion channels. From these data, they constructed a four-dimensional differential equation for the evolution of the action potential in this axon. Additionally, they developed a partial differential equation to study the propagation of action potentials down the axon. By transforming to travelling coordinates, they decreased the partial differential equation to a five-dimensional ordinary differential equation which they solved numerically by a shooting technique. Their physiological and theoretical work garnered them the Nobel Prize. The fallout from this beautiful formalism continues today. Approximately all biophysically based models of individual neurons are based on the formalism that they developed. A good deal of extremely nice mathematics has been developed to rigorously study this class of equations ranging from geometric singular perturbation (reviewed in Rubin and Terman, 2002) to the study of coupling among oscillatory neurons (reviewed in Kopell and Ermentrout, 2002).