Complex Behavior in a Nerve Conduction Model
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The phenomenon of bursting oscillations, a behavior found in many biological and chemical systems, consists of complex periodic oscillations (active phases) separated by quasi-steady states (silent phases). Many of the mathematical systems that exhibit bursting oscillations consist of differential equations of the form: dX/dt= F(X,Y) dY/dt = E G(X,Y) E<<l where X = col(X1, X2,…, Xn) > 2 Systems of this particular form that satisfy certain conditions can be reduced and analyzed. The mathematical model studied here looks at the behavior of impulse transmission in nerve cells. Specifically, it is shown numerically that a simple variation of the Fitzhugh-Nagumo equations (a system of three, 1st order, non-linear, ordinary differential equations), known to exhibit oscillatory responses, also exhibits bursting behavior. The bursting will be explained as resulting from the slow passage through a Hopf bifurcation.