Slow Acceleration and De-acceleration through a Hopf Bifurcation with Application to Neuronal and Chemical Systems

From the periodicity of regional climate change to sustained oscillations in living cells, the transition between stationary and oscillatory behavior is often through a Hopf bifurcation. When a parameter slowly ramps through a Hopf bifurcation, stability loss is delayed considerably when compared to classical static theory. Inherent to biological, chemical, and physical systems, but often overlooked or misunderstood in the literature are nonlinear ramp problems where a parameter slowly accelerates or deaccelerates through the bifurcation point. In this talk I will briefly review, from a neuroscience perspective, the importance of the dynamic bifurcation problem. I will then present recent results that show, numerically and analytically, how slow nonlinear ramps can significantly increase or decrease the onset threshold, changing profoundly our understanding of stability loss delay in dynamic bifurcation problems. I will apply the results to membrane accommodation in nerves, neuronal elliptic bursting, and the formation of pacemakers in the Belousov-Zhabotinsky reaction. At the end of the talk I will discuss ongoing research and several open problems.