Bursting Oscillations along the Axon of a Neuron
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The body transfers electrical signals from one part to another by means of neurons. The first relatively complete mathematical model of nerve conduction was published by Hodgkin and Huxley. They. subsequently won the Nobel Prize for this work. The focus of this paper is to reveal insight into the nature of bursting oscillations produced by the FitzHugh-Nagumo-Rinzel equations - a simplified version of the Hodgkin and Huxley equations. As previously mentioned, neurons transfer electrical signals. The conveyed signals are electrical potentials arising from the potential drop across the membrane of the neuron. When observed at a fixed location, the electrical potential promulgates an action potential, a single nerve impulse. If the stimuli is large enough an oscillation reaction will arise. The oscillation reaction or bursting is r' characterized by successive action potentials (active phase), positioned between periods of quiescence (quiet phase). An analysis of the FitzHugh-Nagumo-Rinzel equations was performed using· Fourier transformations, Hopf bifurcation diagrams, and perturbation theory. Although the Fourier analysis cannot prove the oscillations are quasiperiodic (i.e. the ratio of the frequencies that comprise the oscillations is irrational), they did, however, lend support to the claim.