Analysis of pattern dynamics for a nonlinear model of the human cortex via bifurcation theories

This thesis examines the bifurcations, i.e., the emergent behaviours, for the Waikato cortical model under the influence of the gap-junction inhibitory diffusion D₂ (identified as the Turing bifurcation parameter) and the time-to-peak for hyperpolarising GABA response γi (i.e., inhibitory rate-constant, identified as the Hopf bifurcation parameter). The cortical model simplifies the entire cortex to a cylindrical macrocolumn (∼ 1 mm³) containing ∼ 10⁵ neurons (85% excitatory, 15% inhibitory) communicating via both chemical and electrical (gap-junction) synapses. The linear stability analysis of the model equations predict the emergence of a Turing instability (in which separated areas of the cortex become activated) when gap-junction diffusivity is increased above a critical level. In addition, a Hopf bifurcation (oscillation) occurs when the inhibitory rate-constant is sufficiently small. Nonlinear interaction between these instabilities leads to spontaneous cortical patterns of neuronal activities evolving in space and time. Such model dynamics of delicately balanced interplay between Turing and Hopf instabilities may be of direct relevance to clinically observed brain dynamics such as epileptic seizure EEG spikes, deep-sleep slow-wave oscillations and cognitive gamma-waves. The relationship between the modelled brain patterns and model equations can normally be inferred from the eigenvalue dispersion curve, i.e., linear stability analysis. Sometimes we experienced mismatches between the linear stability analysis and the formed cortical patterns, which hampers us in identifying the type of instability corresponding to the emergent patterns. In this thesis, I investigate the pattern-forming mechanism of the Waikato cortical model to better understand the model nonlinearities. I first study the pattern dynamics via analysis of a simple pattern-forming system, the Brusselator model, which has a similar model structure and bifurcation phenomena as the cortical model. I apply both linear and nonlinear perturbation methods to analyse the near-bifurcation behaviour of the Brusselator in order to precisely capture the dominant mode that contributes the most to the final formed-patterns. My nonlinear analysis of the Brusselator model yields Ginzburg-Landau type amplitude equations that describe the dynamics of the most unstable mode, i.e., the dominant mode, in the vicinity of a bifurcation point. The amplitude equations at a Turing point unfold three characteristic spatial structures: honeycomb Hπ, stripes, and reentrant honeycomb H₀. A codimension-2 Turing–Hopf point (CTHP) predicts three mixed instabilities: stable Turing–Hopf (TH), chaotic TH, and bistable TH. The amplitude equations precisely determine the bifurcation conditions for these instabilities and explain the pattern-competition mechanism once the bifurcation parameters cross the thresholds, whilst driving the system into a nonlinear region where the linear stability analysis may not be applicable. Then, I apply the bifurcation theories to the cortical model for its pattern predictions. Analogous to the Brusselator model, I find cortical Turing pattens in Hπ, stripes and H₀ spatial structures. Moreover, I develop the amplitude equations for the cortical model, with which I derive the envelope frequency for the beating-waves of a stable TH mode; and propose ideas regarding emergence of the cortical chaotic mode. Apart from these pattern dynamics that the cortical model shares with the Brusselator system, the cortical model also exhibits “eye-blinking” TH patterns latticed in hexagons with localised oscillations. Although we have not found biological significance of these model pattens, the developed bifurcation theories and investigated pattern-forming mechanism may enrich our modelling strategies and help us to further improve model performance. In the last chapter of this thesis, I introduce a Turing–Hopf mechanism for the anaesthetic slow-waves, and predict a coherence drop of such slow-waves with the induction of propofol anaesthesia. To test this hypothesis, I developed an EEG coherence analysing algorithm, EEG coherence, to automatically examine the clinical EEG recordings across multiple subjects. The result shows significantly decreased coherence along the fronto-occipital axis, and increased coherence along the left- and right-temporal axis. As the Waikato cortical model is spatially homogenous, i.e., there are no explicit front-to-back or right-to-left directions, it is unable to produce different coherence changes for different regions. It appears that the Waikato cortical model best represents the cortical dynamics in the frontal region. The theory of pattern dynamics suggests that a mode transition from wave–Turing–wave to Turing–wave–Turing introduces pattern coherence changes in both positive and negative directions. Thus, a further modelling improvement may be the introduction of a cortical bistable mode where Turing and wave coexist

Generation of heterogeneous cellular structures by sonication

Many materials require functionally graded cellular microstructures whose porosity (i.e. ratio of the void volume to the total volume of a material) is engineered to meet specific requirements and for an optimal performance in diverse applications. Numerous applications have demonstrated the potential of porous materials in areas ranging from biomaterial science through to structural engineering. Polymeric foams are an example of a cellular material whose microstructure can be considered as a blend of material and nonmaterial zones. While a huge variety of foams can be manufactured with homogenous porosity, for heterogeneous foams there are no generic processes for controlling the distribution of porosity throughout the resulting matrix. Motivated by the desire to create a flexible process for engineering heterogeneous foams, this work has investigated how ultrasound, applied during some of the foaming stages of a polyurethane melt, affects both the cellular structure and distribution of the pore size. After reviewing the literature concerning foam chemistry, ultrasound and sonochemistry, series of experiments were performed that used an ultrasonic field created by a sonotrode irradiating in a water bath containing a strategically placed vessel filled with foaming reactants. Prior to this, the acoustic field in the bath had been accurately mapped so that the acoustic pressure conditions within the foam container were known. During the foam polymerisation reaction, the acoustic pressure in the water bath varied causing the bubbles to pulsate in a state of ‘stable cavitation’ (i.e. rectified diffusion). This pulsation of the bubbles pumped gas from the liquid to the gas phase inducing them to increase in volume. The eventual solidification resulted in a porous material with a cellular structure that reflected the acoustic field imposed upon it. The experimental results revealed how the parameters of ultrasound exposure (i.e. frequency and acoustic pressure) influenced the volume and distribution of pores within the final polyurethane matrix: it was found that porosity varies in direct proportion to both the acoustic pressure and the frequency of the ultrasound signal. The effects of ultrasound on porosity demonstrated by this work offer the prospect of a manufacturing process that can control and adjust the cellular geometry of foam and hence ensure that the resulting characteristics of the heterogeneous material match the functional requirements.Engineering and Physical Sciences Research Council (EPSRC)Neilson Endowment Fund, in the Department of Mechanical Engineerin

Semi-Parametric Drift and Diffusion Estimation for Multiscale Diffusions

We consider the problem of statistical inference for the effective dynamics of multiscale diffusion processes with (at least) two widely separated characteristic time scales. More precisely, we seek to determine parameters in the effective equation describing the dynamics on the longer diffusive time scale, i.e. in a homogenization framework. We examine the case where both the drift and the diffusion coefficients in the effective dynamics are space-dependent and depend on multiple unknown parameters. It is known that classical estimators, such as Maximum Likelihood and Quadratic Variation of the Path Estimators, fail to obtain reasonable estimates for parameters in the effective dynamics when based on observations of the underlying multiscale diffusion. We propose a novel algorithm for estimating both the drift and diffusion coefficients in the effective dynamics based on a semi-parametric framework. We demonstrate by means of extensive numerical simulations of a number of selected examples that the algorithm performs well when applied to data from a multiscale diffusion. These examples also illustrate that the algorithm can be used effectively to obtain accurate and unbiased estimates.Comment: 32 pages, 10 figure

Modeliranje turbulentnega zgorevanja vodika v eksperimentalni napravi zadrževalnega hrama

Hydrogen may be produced in a light water reactor nuclear power plant containment during a postulated severe accident. Ensuing hydrogen combustion, which is essentially inevitable, could inflict irreparable damage to the containment itself, resulting in a failure of the final protection barrier for the fission products release to the environment. Since the actual containment volumes surpass the volumes of even the largest available experimental facilities capable of conducting hydrogen combustion experiments for a few orders of magnitude, the computer-aided modelling presents itself as a formidable and accessible tool in this regard. Thus, use and development of reliable computational fluid dynamics modelling approaches for large-scale geometries is imperative for providing relatively cost-effective hydrogen combustion risk assessment. The present dissertation focuses on further theoretical investigation of hydrogen combustion, specifically hydrogen deflagration in large enclosures. This investigation is carried out through the development and validation of hydrogen combustion models and addresses some of the challenges on the way for the computer-aided modelling to become readily available for use in the real-scale containment dimensions. Specifically, it tackles the difficulties of available combustion models in producing the reliable predictions of the large-scale hydrogen deflagration experiments. The validation of combustion models was performed against the results obtained in two large-scale experimental facilities, i.e. THAI and HYKA-A2 experimental vessels. Firstly, a new combustion model was introduced, i.e. the extended eddy break-up (EEBU) model, which was developed from the existing less elaborate eddy break-up (EBU) model, with the additional treatment of the flame phenomenology also in the quasi-laminar combustion regime. This model retains beneficial characteristics of the EBU model, i.e. superior computational efficiency, while at the same time providing improved predictions for hydrogen deflagrations in the considered large-scale experiments. Furthermore, a novel approach was recommended undertaking a focused treatment of the laminar flame speed with the weighted laminar flame speed concept. This approach effectively balances the turbulent reaction rate with the buoyancy effects of the surrounding flow. It was applied to already existing extended turbulent flame speed closure (ETFC) model as well as to the newly introduced EEBU model. When properly executed, it proved to be extremely effective in improving the predictions of both models regarding the flame behavior in the considered large-scale hydrogen deflagration experiments.V primeru težke nesreče v lahkovodni jedrski elektrarni lahko pride do nastajanja vodika. Nadaljnje zgorevanje vodika, ki je v bistvu neizogibno, lahko povzroči nepopravljivo škodo na samem zadrževalnem sistemu, kar bi pomenilo odpoved končne zaščitne ograde za izpustitev radioaktivnih fisijskih produktov v okolje. Ker dejanske geometrije zadrževanja več sto krat presegajo geometrije celo največjih razpoložljivih eksperimentalnih objektov, v katerih se lahko izvaja poskuse zgorevanja vodika, se v tem pogledu računalniško podprto modeliranje ponuja kot izjemno uporabno in dostopno orodje. Zato je uporaba in razvoj zanesljivih modelov za modeliranje dinamike tekočin za obsežne geometrije nujna za zagotavljanje relativno stroškovno učinkovite ocene zgorevanja vodika. Disertacija se osredotoča na nadaljnje teoretične raziskave zgorevanja vodika, zlasti deflagracije vodika v velikih zaprtih prostorih jedrskih elektrarn. Ta raziskava se izvaja z razvojem in validacijo modelov zgorevanja vodika in naslavlja nekatere izzive na poti do računalniško podprtega modeliranja, ki bi bilo na voljo za uporabo v realnih dimenzijah zadrževalnih hramov. Konkretno se spopada s težavami razpoložljivih modelov zgorevanja pri kreiranju zanesljivih napovedi obsežnih eksperimentov deflagracije vodika. Validacija modelov izgorevanja je bila izvedena v primerjavi z rezultati pridobljenimi v dveh obsežnih eksperimentalnih objektih, tj. eksperimentalnih posodah THAI in HYKA-A2. Najprej je formuliran in predstavljen nov, t.i. razširjen model razpada vrtincev (ang. »extended eddy break-up model« - EEBU), ki smo ga razvili z nadgradnjo obstoječega modela razpada vrtincev (ang. »eddy break-up model« - EBU), z dodatno obravnavo fenomenologije plamena tudi v kvazi-laminarnem režimu zgorevanja. Novo formirani EEBU model ohranja koristne značilnosti modela EBU, tj. boljšo računsko učinkovitost, hkrati pa zagotavlja boljše napovedi deflagracije vodika v obravnavanih eksperimentih. Poleg tega je bil predstavljen tudi nov pristop, ki je usmerjen v obravnavo hitrosti laminarnega plamena s konceptom utežene hitrosti laminarnega širjenja plamena. Uveden pristop učinkovito uravnava turbulentno hitrost reakcije z vzgonskimi učinki okoliškega pretoka. Uporabljen je bil za že obstoječi razširjen model zgorevanja temelječ na turbulentni hitrosti plamena (ang. »extended turbulent flame speed closure model« - ETFC), kot tudi za novo predstavljen model EEBU. Ko je ta pristop ustrezno izveden, se izkaže za izjemno učinkovitega pri izboljšanju napovedi omenjenih modelov v zvezi z obnašanjem plamena v obravnavanih obsežnih eksperimentih deflagracije vodika

Aeronautical Engineering. A continuing bibliography, supplement 115

This bibliography lists 273 reports, articles, and other documents introduced into the NASA scientific and technical information system in October 1979