Computational Nonlinear Dynamics: Monostable Stochastic Resonance and a Bursting Neuron Model

Barbara J. Breen

Most nonlinear systems cannot be analytically solved, and computational methods are invaluable in understanding them. In this thesis, the power of computational nonlinear dynamics is brought to bear on two significant problems. One is generalizing the phenomenon of monostable stochastic resonance (SR) to arrays, another involves finding a better numerical algorithm for modeling a bursting neuron.

Investigating monostable SR requires the numerical integration of stochastic differential equations to explore an eight-dimensional parameter space. Our results clarify the phenomenon of SR in monostable potentials and introduce a new measure of the system's response. We show 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 short, one-dimensional arrays of monostable oscillators, where coupling lifts the degeneracy of the natural frequencies of the oscillator, we find 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. We use the bifurcation diagram of a reduced, parameterized version of a bursting model to empirically describe relationships between fast and slow variables. Implementing these relationships in the framework of a computationally simple integrate-and-fire mechanism produces a reduced model that has only one time scale. It successfully reproduces the dynamic behavior and temporal characteristics of the full model over a wide range of activity and parameter space while offering a significant savings in CPU time by obviating the need for computationally intensive variable time step algorithms.