Computational nonlinear dynamics: Monostable stochastic resonance and a bursting neuron model

In this thesis, the power of computational nonlinear dynamics was brought to bear on two significant problems. One was generalizing the phenomenon of monostable stochastic resonance (SR) to arrays; another involved finding a better numerical algorithm for modeling a bursting neuron. Our investigation of monostable SR clarified the phenomenon of SR in monostable potentials and introduced a new measure of the system's response. We showed that SR results precisely when the natural frequency of the oscillator is ‘tuned’ using noise to match the frequency of the driving signal. By extending this research to arrays of monostable oscillators, we found multiple stochastic resonances. Bursting neurons are modeled by coupled nonlinear ordinary differential equations with intrinsic time scales that can differ by up to four orders of magnitude. Using the bifurcation diagram of a parameterized version of a bursting model to rigorously describe relationships between fast and slow variables, we designed a reduced model that has only one time scale. It successfully reproduced the dynamic behavior and temporal characteristics of the full model while offering significant savings in CPU time by obviating the need for computationally intensive variable time step algorithms.