|Nonlinear oscillators are ubiquitous in physical, biological, and
engineered systems. Simple external forcing, for example periodic
pulse-like forcing, can dramatically modify the behavior of
oscillators and networks of oscillators, inducing a wide range of
responses which includes entrainment (phase-locking) and chaos.
The first part of this talk concerns a systematic computational strategy for analyzing the dynamical behavior of pulse-driven oscillators. This work builds on recent theoretical advances by Q. Wang and L.-S. Young, who discovered and elucidated a general geometric mechanism underlying these phenomena. Their theory predicts some general dynamical features shared by a large class of pulse-driven oscillators; the computational strategy proposed here provides model-specific information and complements the theory. Throughout, I will illustrate the main ideas via the Hodgkin-Huxley model, a prototypical neuron model.
The second part of the talk concerns on-going work (joint with E. Shea-Brown and L.-S. Young) on the response of small oscillator networks to stochastic stimuli. This work is motivated by questions regarding the ability of neural networks to respond reliably to repeated presentations of complex signals. I will discuss preliminary numerical results and what they tell us about geometric mechanisms which can cause a network to behave unreliably.
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