Yue-Xian Li

We formulate and investigate a relatively new modeling paradigm by which spatially segregated dynamically active units communicate with each other through a signaling molecule that diffuses in the bulk medium between active units. The modeling studies start with a simplified setting in a one-dimensional space, where two dynamically active compartments are located at boundaries of the domain and coupled through the feedback term to the local dynamics together with flux boundary conditions at the two ends. For the symmetric steady state solution, in-phase and anti-phase synchronizations are found and Hopf bifurcation boundaries are studied using a winding number approach as well as parameter continuation methods of bifurcation theory in the case of linear coupling. Numerical studies show the existence of double Hopf points in the parameter space where center manifold and normal form theory are used to reduce the dynamics into a system of amplitude equations, which predicts the configurations of the Hopf bifurcation and stability of the two modes near the double Hopf point. The system with a periodic chain of cells is studied using Floquet theory. For the case of a single active membrane bound component, rigorous spectral results for the onset of oscillatory dynamics are obtained and in the finite domain case, a weakly nonlinear theory is developed to predict the local branching behavior near the Hopf bifurcation point. A previously developed model by Gomez et al.\cite{Gomez-Marin2007} is analyzed in detail, where the phase diagrams and the Hopf frequencies at onset are provided analytically with slow-fast type of local kinetics. A coupled cell-bulk system, with small signaling compartments, is also studied in the case of a two-dimensional bounded domain using the method of asymptotic expansions. In the very large diffusion limit we reduce the PDE cell-bulk system to a finite dimensional dynamical system, which is studied both analytically and numerically. When the diffusion rate is not very large, we show the effect of spatial distribution of cells and find the dependence of the quorum sensing threshold on influx rate.

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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.

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