Slow passage through a Hopf bifurcation: New insights into the memory effect with application to neuronal bursting
Mathematics Department, Arizona State University
(May 11, 2007 3:30 PM - 4:30 PM)
In many biological, chemical and physical systems modeled mathematically as bifurcation problems, the bifurcation parameter may vary naturally and slowly with time or the parameter may be slowly varied by the experimenter. Mathematically, these are called slow passage or ramp problems . Of particular interest is when a parameter passes slowly through a Hopf bifurcation and the system's response changes from a slowly varying steady state to slowly varying oscillations. The interesting phenomena is that the transition may not occur until the parameter is considerably beyond the value predicted from a static bifurcation analysis, no matter how slow the parameter is varied, and the delay in onset is dependent on the initial state of the system (memory effect). Previous studies have focussed on linear or constant speed ramps. In this talk I will introduce the problem of slow accelerating and de-accelerating ramps, obtain new results using numerical and asymptotic methods, and apply the results to problems in nerve membrane accommodation and neuronal bursting.