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Abstract
Van der Pol's equation for a relaxation oscillator is generalized by the addition
of terms to produce a pair of non-linear differential equations with either a stable
singular point or a limit cycle. The resulting "BVP model" has two variables of state,
representing excitability and refractoriness, and qualitatively resembles Bonhoeffer's
theoretical model for the iron wire model of nerve. This BVP model serves as a simple
representative of a class of excitable-oscillatory systems including the Hodgkin-Huxley
(HH) model of the squid giant axon. The BVP phase plane can be divided into regions
corresponding to the physiological states of nerve fiber (resting, active, refractory,
enhanced, depressed, etc.) to form a "physiological state diagram," with the help
of which many physiological phenomena can be summarized. A properly chosen projection
from the 4-dimensional HH phase space onto a plane produces a similar diagram which
shows the underlying relationship between the two models. Impulse trains occur in
the BVP and HH models for a range of constant applied currents which make the singular
point representing the resting state unstable.