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Stochastic phase dynamics in neuron models and spike time reliability

University of British Columbia

The present thesis is concerned with the stochastic phase dynamics of neuron models and spike time reliability. It is well known that noise exists in all natural systems, and some beneficial effects of noise, such as coherence resonance and noise-induced synchrony, have been observed. However, it is usually difficult to separate the effect of the nonlinear system itself from the effect of noise on the system's phase dynamics. In this thesis, we present a stochastic theory to investigate the stochastic phase dynamics of a nonlinear system. The method we use here, called ``the stochastic multi-scale method'', allows a stochastic phase description of a system, in which the contributions from the deterministic system itself and from the noise are clearly seen. Firstly, we use this method to study the noise-induced coherence resonance of a single quiescent ``neuron" (i.e. an oscillator) near a Hopf bifurcation. By calculating the expected values of the neuron's stochastic amplitude and phase, we derive analytically the dependence of the frequency of coherent oscillations on the noise level for different types of models. These analytical results are in good agreement with numerical results we obtained. The analysis provides an explanation for the occurrence of a peak in coherence measured at an intermediate noise level, which is a defining feature of the coherence resonance. Secondly, this work is extended to study the interaction and competition of the coupling and noise on the synchrony in two weakly coupled neurons. Through numerical simulations, we demonstrate that noise-induced mixed-mode oscillations occur due to the existence of multistability states for the deterministic oscillators with weak coupling. We also use the standard multi-scale method to approximate the multistability states of a normal form of such a weakly coupled system. Finally we focus on the spike time reliability that refers to the phenomenon: the repetitive application of a stochastic stimulus to a neuron generates spikes with remarkably reliable timing whereas repetitive injection of a constant current fails to do so. In contrast to many numerical and experimental studies in which parameter ranges corresponding to repetitive spiking, we show that the intrinsic frequency of extrinsic noise has no direct relationship with spike time reliability for parameters corresponding to quiescent states in the underlying system. We also present an ``energy" concept to explain the mechanism of spike time reliability. ``Energy" is defined as the integration of the waveform of the input preceding a spike. The comparison of ``energy" of reliable and unreliable spikes suggests that the fluctuation stimuli with higher ''energy" generate reliable spikes. The investigation of individual spike-evoking epochs demonstrates that they have a more favorable time profile capable of triggering reliably timed spike with relatively lower energy levels.

Thesis/Dissertation

application/pdf

2009-04-20

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/7383

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/