Osilatörler için faz modelleri ve hesaplamaları

Oscillators as key components of many natural and engineered systems have been a research focus for decades in many disciplines such as electronics and biology. The time keeping capability of autonomous oscillators and the synchronization of coupled oscillators are best described in terms of a scalar quantity, so-called the phase of an oscillator. Phase computations for perturbed and coupled oscillators and equations that describe phase dynamics have been quite useful in both electronics and biology in forming a rigorous understanding of oscillatory system behavior and designing oscillators that are least affected by undesired disturbances such as noise. We first review the notion of isochrons, which forms the basis for the generalized phase notion for an oscillator that we cover in a rigorous manner. The notion of isochrons for oscillators has been first introduced by Arthur Winfree in 1974 and heavily utilized in mathematical biology in studying biological oscillators. Isochrons were instrumental in introducing a notion of generalized phase for an oscillator and form the basis for oscillator perturbation analysis formulations. Calculating the isochrons of an oscillator is a very difficult task. Except for some very simple planar oscillators, isochrons can not be calculated analytically and one has to resort to numerical techniques. Previously proposed numerical methods for computing isochrons can be regarded as brute-force, which become totally impractical for non-planar oscillators with dimension more than two. In this thesis, we present a precise and carefully developed theory and numerical techniques for computing local but quadratic approximations for isochrons. Previous work offers the theory and the numerical methods needed for computing only local linear approximations. Our treatment is general and applicable to oscillators with large dimension. We present examples for isochron computations, verify our results against exact calculations in a simple analytically calculable case, test our methods on complex oscillators. We next present a unified theory of phase equations for autonomous oscillators through an assimilation of the work that has been done on oscillator analysis in both electronics and biology during the past seventy years. Based on the generalized oscillator phase notion that is founded on the theory of isochrons, we present a general framework for phase equations and derive in a unified manner a phase equation for perturbed oscillators that is exact but practically unusable, and practically useful ones that are based on linear (previously known in the literature) and quadratic (new, more accurate) isochron approximations. We discuss the utility of these phase equations in performing (semi) analytical phase computations and also describe simpler and more accurate schemes for numerical phase computations. Carefully run numerical experiments on several examples are presented which compare the accuracy of the various phase computation schemes and the phase equations described. Biochemical oscillators perform crucial functions in cells, e.g., they set up circadian clocks. Phase computation techniques for continuous oscillators that are based on isochrons have been used for characterizing the behavior of various types of oscillators under the influence of perturbations such as noise. In this thesis, we also extend the applicability of these phase equations and phase computation schemes to biochemical oscillators as discrete, molecular systems. In particular, we describe techniques for computing the instantaneous phase of discrete, molecular oscillators for SSA (Stochastic Simulation Algorithm) generated sample paths. The impact of noise that arises from the discrete and random nature of the mechanisms that make up molecular oscillators can be characterized based on these phase computations. Modeling and analysis studies of oscillators in electronics and biology seem to have progressed independently, without any cross-fertilization in between. Even though work on oscillator analysis in electronics did not directly make use of isochrons, similar concepts, models and perturbation analysis techniques, though using completely different terminology and formulations, have been developed in both disciplines. In this thesis, we reveal the connection between oscillator analysis work in these two seemingly disparate disciplines.