482 research outputs found

    Tutorial of numerical continuation and bifurcation theory for systems and synthetic biology

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    Mathematical modelling allows us to concisely describe fundamental principles in biology. Analysis of models can help to both explain known phenomena, and predict the existence of new, unseen behaviours. Model analysis is often a complex task, such that we have little choice but to approach the problem with computational methods. Numerical continuation is a computational method for analysing the dynamics of nonlinear models by algorithmically detecting bifurcations. Here we aim to promote the use of numerical continuation tools by providing an introduction to nonlinear dynamics and numerical bifurcation analysis. Many numerical continuation packages are available, covering a wide range of system classes; a review of these packages is provided, to help both new and experienced practitioners in choosing the appropriate software tools for their needs.Comment: 14 pages, 2 figures, 2 table

    Advances in numerical bifurcation software : MatCont

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    The mathematical background of MatCont, a freely available toolbox, is bifurcation theory which is a field of hard analysis. Bifurcation theory treats dynamical systems from a high-level point of view. In the case of continuous dynamical systems this means that it considers nonlinear differential equations without any special form and without restrictions except for differentiability up to a sufficiently high order (in the present state of MatCont never higher than five.) The number of equations is not fixed in advance and neither is the number of variables or the number of parameters, some of which can be active and others not. The aim of bifurcation theory is to understand and classify the qualitative changes of the solutions to the differential equations under variation of the parameters. This knowledge cannot be applied to practical situations without numerical software, except in some artificially constructed situations. Matcont is a toolbox that computes bifurcation diagrams through numerical methods, namely continuation. This dissertation describes the advances and innovations that were made including the detection and continuation of new bifurcations in discrete-time systems

    A versatile software package for inter-subject correlation based analyses of fMRI

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    In the inter-subject correlation (ISC) based analysis of the functional magnetic resonance imaging (fMRI) data, the extent of shared processing across subjects during the experiment is determined by calculating correlation coefficients between the fMRI time series of the subjects in the corresponding brain locations. This implies that ISC can be used to analyze fMRI data without explicitly modeling the stimulus and thus ISC is a potential method to analyze fMRI data acquired under complex naturalistic stimuli. Despite of the suitability of ISC based approach to analyze complex fMRI data, no generic software tools have been made available for this purpose, limiting a widespread use of ISC based analysis techniques among neuroimaging community. In this paper, we present a graphical user interface (GUI) based software package, ISC Toolbox, implemented in Matlab for computing various ISC based analyses. Many advanced computations such as comparison of ISCs between different stimuli, time window ISC, and inter-subject phase synchronization are supported by the toolbox. The analyses are coupled with resampling based statistical inference. The ISC based analyses are data and computation intensive and the ISC toolbox is equipped with mechanisms to execute the parallel computations in a cluster environment automatically and with an automatic detection of the cluster environment in use. Currently, SGE-based (Oracle Grid Engine, Son of a Grid Engine, or Open Grid Scheduler) and Slurm environments are supported. In this paper, we present a detailed account on the methods behind the ISC Toolbox, the implementation of the toolbox and demonstrate the possible use of the toolbox by summarizing selected example applications. We also report the computation time experiments both using a single desktop computer and two grid environments demonstrating that parallelization effectively reduces the computing time.Peer reviewe

    PnP Maxtools: Autonomous Parameter Control in MaxMSP Utilizing MIR Algorithms

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    This research presents a new approach to computer automation through the implementation of novel real-time music information retrieval algorithms developed for this project. It documents the development of the PnP.Maxtools package, a set of open source objects designed within the popular programming environment MaxMSP. The package is a set of pre/post processing filters, objective and subjective timbral descriptors, audio effects, and other objects that are designed to be used together to compose music or improvise without the use of external controllers or hardware. The PnP.Maxtools package objects are designed to be used quickly and easily using a `plug and play\u27 style with as few initial arguments needed as possible. The PnP.Maxtools package is designed to take incoming audio from a microphone, analyze it, and use the analysis to control an audio effect on the incoming signal in real-time. In this way, the audio content has a real musical and analogous relationship with the resulting musical transformations while the control parameters become more multifaceted and better able to serve the needs of artists. The term Reflexive Automation is presented that describes this unsupervised relationship between the content of the sound being analyzed and the analogous and automatic control over a specific musical parameter. A set of compositions are also presented that demonstrate ideal usage of the object categories for creating reflexive systems and achieving fully autonomous control over musical parameters

    Engineering Education and Research Using MATLAB

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    MATLAB is a software package used primarily in the field of engineering for signal processing, numerical data analysis, modeling, programming, simulation, and computer graphic visualization. In the last few years, it has become widely accepted as an efficient tool, and, therefore, its use has significantly increased in scientific communities and academic institutions. This book consists of 20 chapters presenting research works using MATLAB tools. Chapters include techniques for programming and developing Graphical User Interfaces (GUIs), dynamic systems, electric machines, signal and image processing, power electronics, mixed signal circuits, genetic programming, digital watermarking, control systems, time-series regression modeling, and artificial neural networks

    Modification, identification and control of chaotic bubbling with electrostatic potential

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    The effect of an applied electric potential on the dynamics of bubble formation from a single nozzle was studied experimentally. Bubbling of dry nitrogen into glycerol through a nozzle having an electrified tip was examined by pressure measurements upstream of the nozzle. As the applied electric potential is increased from zero, the bubble size is reduced and the bubbling frequency is increased. It has been found that, at constant flow rate, bubble-formation dynamics exhibit the classic signs of a period-doubling bifurcation to chaos with increasing applied potential. The behavior is similar to that for increasing flow rate of the well-studied dripping faucet and of bubble formation in liquids. A three-dimensional bifurcation map was determined as a function of gas flow rate and electrostatic potential. The data indicate that although bifurcation route is similar for applied voltage and flow, the effect on bubbling for applied voltage is significantly smaller than that for flow as measured by dimensionless variables. Electrostatic potential was successfully harnessed as the manipulated variable to track a set-point of a constant average bubbling frequency with disturbances dues to flow-rate It was found that an increase/decrease in electrostatic potential can be used to compensate a 50% change in the flow rate. A control scheme suggested by Ott, Grebogi and Yorke was implemented to tame the chaos of bubbles with partial success Software tools namely the Bubble Toolbox for Chaotic Analysis and the Automation Workbench for LabView were developed in conjunction with this experiment

    Computational dynamical systems analysis : Bogdanov-Takens points and an economic model

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    The subject of this thesis is the bifurcation analysis of dynamical systems (ordinary differential equations and iterated maps). A primary aim is to study the branch of homoclinic solutions that emerges from a Bogdanov-Takens point. The problem of approximating such branch has been studied intensively but neither an exact solution was ever found nor a higher-order approximation has been obtained. We use the classical ``blow-up'' technique to reduce an appropriate normal form near a Bogdanov-Takens bifurcation in a generic smooth autonomous ordinary differential equations to a perturbed Hamiltonian system. With a regular perturbation method and a generalization of the Lindstedt-Poincare' perturbation method, we derive two explicit third-order corrections of the unperturbed homoclinic orbit and parameter value. We prove that both methods lead to the same homoclinic parameter value as the classical Melnikov technique and the branching method. We show that the regular perturbation method leads to a ``parasitic turn'' near the saddle point while the Lindstedt-Poincare' solution does not have this turn, making it more suitable for numerical implementation. To obtain the normal form on the center manifold, we apply the standard parameter dependent center manifold reduction combined with the normalization, using the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we correct the parameter transformation existing in the literature. The generic homoclinic predictors are applied to explicitly compute the homoclinic solutions in the Gray-Scott kinetic model. The actual implementation of both predictors in the MATLAB continuation package MatCont and five numerical examples illustrating its efficiency are discussed. Besides, the thesis discusses the possibility to use the derived homoclinic predictor of generic ordinary differential equations to continue the branches of homoclinic tangencies in the Bogdanov-Takens map. The second part of this thesis is devoted to the application of bifurcation theory to analyze the dynamic and chaotic behaviors of a nonlinear economic model. The thesis studies the monopoly model with cubic price and quadratic marginal cost functions. We present fundamental corrections to the earlier studies of the model and a complete discussion of the existence of cycles of period 4. A numerical continuation method is used to compute branches of solutions of period 5, 10, 13 and 17 and to determine the stability regions of these solutions. General formulas for solutions of period 4 are derived analytically. We show that the solutions of period 4 are never linearly asymptotically stable. A nonlinear stability criterion is combined with basin of attraction analysis and simulation to determine the stability region of the 4-cycles. This corrects the erroneous linear stability analysis in previous studies of the model. The chaotic and periodic behavior of the monopoly model are further analyzed by computing the largest Lyapunov exponents, and this confirms the above mentioned results

    Detection, Prediction and Control of Epileptic Seizures

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    abstract: From time immemorial, epilepsy has persisted to be one of the greatest impediments to human life for those stricken by it. As the fourth most common neurological disorder, epilepsy causes paroxysmal electrical discharges in the brain that manifest as seizures. Seizures have the effect of debilitating patients on a physical and psychological level. Although not lethal by themselves, they can bring about total disruption in consciousness which can, in hazardous conditions, lead to fatality. Roughly 1\% of the world population suffer from epilepsy and another 30 to 50 new cases per 100,000 increase the number of affected annually. Controlling seizures in epileptic patients has therefore become a great medical and, in recent years, engineering challenge. In this study, the conditions of human seizures are recreated in an animal model of temporal lobe epilepsy. The rodents used in this study are chemically induced to become chronically epileptic. Their Electroencephalogram (EEG) data is then recorded and analyzed to detect and predict seizures; with the ultimate goal being the control and complete suppression of seizures. Two methods, the maximum Lyapunov exponent and the Generalized Partial Directed Coherence (GPDC), are applied on EEG data to extract meaningful information. Their effectiveness have been reported in the literature for the purpose of prediction of seizures and seizure focus localization. This study integrates these measures, through some modifications, to robustly detect seizures and separately find precursors to them and in consequence provide stimulation to the epileptic brain of rats in order to suppress seizures. Additionally open-loop stimulation with biphasic currents of various pairs of sites in differing lengths of time have helped us create control efficacy maps. While GPDC tells us about the possible location of the focus, control efficacy maps tells us how effective stimulating a certain pair of sites will be. The results from computations performed on the data are presented and the feasibility of the control problem is discussed. The results show a new reliable means of seizure detection even in the presence of artifacts in the data. The seizure precursors provide a means of prediction, in the order of tens of minutes, prior to seizures. Closed loop stimulation experiments based on these precursors and control efficacy maps on the epileptic animals show a maximum reduction of seizure frequency by 24.26\% in one animal and reduction of length of seizures by 51.77\% in another. Thus, through this study it was shown that the implementation of the methods can ameliorate seizures in an epileptic patient. It is expected that the new knowledge and experimental techniques will provide a guide for future research in an effort to ultimately eliminate seizures in epileptic patients.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201
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