Colored noise in oscillators. Phase-amplitude analysis and a method to avoid the Ito-Stratonovich dilemma

We investigate the effect of time-correlated noise on the phase fluctuations of nonlinear oscillators. The analysis is based on a methodology that transforms a system subject to colored noise, modeled as an Ornstein-Uhlenbeck process, into an equivalent system subject to white Gaussian noise. A description in terms of phase and amplitude deviation is given for the transformed system. Using stochastic averaging technique, the equations are reduced to a phase model that can be analyzed to characterize phase noise. We find that phase noise is a drift-diffusion process, with a noise-induced frequency shift related to the variance and to the correlation time of colored noise. The proposed approach improves the accuracy of previous phase reduced models

Tensor Computation: A New Framework for High-Dimensional Problems in EDA

Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit simulation), nonlinearity of devices and circuits, large number of design or optimization parameters (e.g. full-chip routing/placement and circuit sizing), or extensive process variations (e.g. variability/reliability analysis and design for manufacturability). The computational challenges generated by such high dimensional problems are generally hard to handle efficiently with traditional EDA core algorithms that are based on matrix and vector computation. This paper presents "tensor computation" as an alternative general framework for the development of efficient EDA algorithms and tools. A tensor is a high-dimensional generalization of a matrix and a vector, and is a natural choice for both storing and solving efficiently high-dimensional EDA problems. This paper gives a basic tutorial on tensors, demonstrates some recent examples of EDA applications (e.g., nonlinear circuit modeling and high-dimensional uncertainty quantification), and suggests further open EDA problems where the use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and System

Multi-Scale Mathematical Modelling of Brain Networks in Alzheimer's Disease

Perturbations to brain network dynamics on a range of spatial and temporal scales are believed to underpin neurological disorders such as Alzheimer’s disease (AD). This thesis combines quantitative data analysis with tools such as dynamical systems and graph theory to understand how the network dynamics of the brain are altered in AD and experimental models of related pathologies. Firstly, we use a biophysical neuron model to elucidate ionic mechanisms underpinning alterations to the dynamics of principal neurons in the brain’s spatial navigation systems in an animal model of tauopathy. To uncover how synaptic deficits result in alterations to brain dynamics, we subsequently study an animal model featuring local and long-range synaptic degeneration. Synchronous activity (functional connectivity; FC) between neurons within a region of the cortex is analysed using two-photon calcium imaging data. Long-range FC between regions of the brain is analysed using EEG data. Furthermore, a computational model is used to study relationships between networks on these different spatial scales. The latter half of this thesis studies EEG to characterize alterations to macro-scale brain dynamics in clinical AD. Spectral and FC measures are correlated with cognitive test scores to study the hypothesis that impaired integration of the brain’s processing systems underpin cognitive impairment in AD. Whole brain computational modelling is used to gain insight into the role of spectral slowing on FC, and elucidate potential synaptic mechanisms of FC differences in AD. On a finer temporal scale, microstate analyses are used to identify changes to the rapid transitioning behaviour of the brain’s resting state in AD. Finally, the electrophysiological signatures of AD identified throughout the thesis are combined into a predictive model which can accurately separate people with AD and healthy controls based on their EEG, results which are validated on an independent patient cohort. Furthermore, we demonstrate in a small preliminary cohort that this model is a promising tool for predicting future conversion to AD in patients with mild cognitive impairment