The numerical solution of neural field models posed on realistic cortical domains

The mathematical modelling of neural activity is a hugely complex and prominent area of exploration that has been the focus of many researchers since the mid 1900s. Although many advancements and scientific breakthroughs have been made, there is still a great deal that is not yet understood about the brain. There have been a considerable amount of studies in mathematical neuroscience that consider the brain as a simple one-dimensional or two-dimensional domain; however, this is not biologically realistic and is primarily selected as the domain of choice to aid analytical progress. The primary aim of this thesis is to develop and provide a novel suite of codes to facilitate the computationally efficient numerical solution of large-scale delay differential equations, and utilise this to explore both neural mass and neural field models with space-dependent delays. Through this, we seek to widen the scope of models of neural activity by posing them on realistic cortical domains and incorporating real brain data to describe non-local cortical connections. The suite is validated using a selection of examples that compare numerical and analytical results, along with recreating existing results from the literature. The relationship between structural connectivity and functional connectivity is then analysed as we use an eigenmode fitting approach to inform the desired stability regimes of a selection of neural mass models with delays. Here, we explore a next-generation neural mass model developed by Coombes and Byrne [36], and compare results to the more traditional Wilson-Cowan formulation [180, 181]. Finally, we examine a variety of solutions to three different neural field models that incorporate real structural connectivity, path length, and geometric surface data, using our NFESOLVE library to efficiently compute the numerical solutions. We demonstrate how the field version of the next-generation model can yield intricate and detailed solutions which push us closer to recreating observed brain dynamics

The numerical solution of neural field models posed on realistic cortical domains

The mathematical modelling of neural activity is a hugely complex and prominent area of exploration that has been the focus of many researchers since the mid 1900s. Although many advancements and scientific breakthroughs have been made, there is still a great deal that is not yet understood about the brain. There have been a considerable amount of studies in mathematical neuroscience that consider the brain as a simple one-dimensional or two-dimensional domain; however, this is not biologically realistic and is primarily selected as the domain of choice to aid analytical progress. The primary aim of this thesis is to develop and provide a novel suite of codes to facilitate the computationally efficient numerical solution of large-scale delay differential equations, and utilise this to explore both neural mass and neural field models with space-dependent delays. Through this, we seek to widen the scope of models of neural activity by posing them on realistic cortical domains and incorporating real brain data to describe non-local cortical connections. The suite is validated using a selection of examples that compare numerical and analytical results, along with recreating existing results from the literature. The relationship between structural connectivity and functional connectivity is then analysed as we use an eigenmode fitting approach to inform the desired stability regimes of a selection of neural mass models with delays. Here, we explore a next-generation neural mass model developed by Coombes and Byrne [36], and compare results to the more traditional Wilson-Cowan formulation [180, 181]. Finally, we examine a variety of solutions to three different neural field models that incorporate real structural connectivity, path length, and geometric surface data, using our NFESOLVE library to efficiently compute the numerical solutions. We demonstrate how the field version of the next-generation model can yield intricate and detailed solutions which push us closer to recreating observed brain dynamics

The Prospect of Responsive Spacecraft Using Aeroassisted, Trans-Atmospheric Maneuvers

Comprised of exo- and trans-atmospheric trajectory segments, atmospheric re-entry represents a complex dynamical event which traditionally signals the mission end-of-life for low-Earth orbit spacecraft. Transcending this paradigm, atmospheric re-entry can be employed as a means of operational maneuver whereby aerodynamic forces can be exploited to create an aeroassisted maneuver. Utilizing a notional trans-atmospheric, lifting re-entry vehicle with L/D =6, the first phase of research demonstrates the terrestrial reachability potential for skip entry aeroassisted maneuvers. By overflying a geographically diverse set of ground targets, comparative analysis indicates a significant savings in delta V expenditure for skip entry compared with exo-atmospheric maneuvers. In the second phase, the Design of Experiments method of orthogonal arrays provides optimal vehicle and skip entry trajectory designs by employing main effects and Pareto front analysis. Depending on re-circularization altitude, the coupled optimal design can achieve an inclination change of 19.91° with 50-85% less delta V than a simple plane change. Finally, the third phase introduces the descent-boost aeroassisted maneuver as an alternative to combined Hohmann and bi-elliptic transfers in order to perform LEO injection. Compared with bi-elliptic transfers, simulations demonstrate that a lifting re-entry vehicle performing a descent-boost maneuver requires 6-12% less for injection into orbits lower than 650 km. In addition, the third phase also introduces the Maneuver Performance Number as a dimensionless means of comparative maneuver effectiveness analysis

ADAPTIVE GRID BASED FINITE DIFFERENCE METHODS FOR SOLUTION OF HYPERBOLIC PDES: APPLICATION TO COMPUTATIONAL MECHANICS AND UNCERTAINTY QUANTIFICATION

Novel finite-difference based numerical methods for solution of linear and nonlinear hyperbolic partial differential equations (PDEs) using adaptive grids are proposed in this dissertation. The overall goal of this research is to improve the accuracy and/or computational efficiency of numerical solutions via the use of adaptive grids and suitable modifications of a given low-order order finite-difference scheme. These methods can be grouped in two broad categories. The first category of adaptive FD methods proposed in the dissertation attempt to reduce the truncation error and/or enhance the accuracy of the underlying numerical schemes via grid distribution alone. Some approaches for grid distribution considered include those based on (i) a moving uniform mesh/domain, (ii) adaptive gradient based refinement (AGBR) and (iii) unit local Courant-Freidrich-Lewy (CFL) number. The improvement in the accuracy which is obtained using these adaptive methods is limited by the underlying scheme formal order of accuracy. In the second category, the CFL based approach proposed in the first category was extended further using defect correction in order to improve the formal order of accuracy and computational efficiency significantly (i.e. by at least one order or higher). The proposed methods in this category are constructed based upon the analysis of the leading order error terms in the modified differential equation associated with the underlying partial differential equation and finite difference scheme. The error terms corresponding to regular and irregular perturbations are identified and the leading order error terms associated with regular perturbations are eliminated using a non-iterative defect correction approach while the error terms associated with irregular perturbations are eliminated using grid adaptation. In the second category of methods involving defect correction (or reduction of leading order terms of truncation error), we explored two different approaches for selection of adaptive grids. These are based on (i) optimal grid dis- tribution and (ii) remapping with monotonicity preserving interpolation. While the first category of methods may be preferred in view of ease of implementation and lower computational complexity, the second category of methods may be preferred in view of greater accuracy and computational efficiency. The two broad categories of methods, which have been applied to problems involving both bounded and unbounded domains, were also extended to multidimensional cases using a dimensional splitting approaches. The performance of these methods was demonstrated using several example problems in computational uncertainty quantification (CUQ) and computational mechanics. The results of the application of the proposed approaches all indicate improvement in both the accuracy and computational efficiency (by about three orders of magnitude in some selected cases) of underlying schemes. In the context of CUQ, all three proposed adaptive finite different solvers are combined with the Gauss-quadrature sampling technique in excitation space to obtain statistical quantities of interest for dynamical systems with parametric uncertainties from the solution of Liouville equation, which is a linear hyperbolic PDE. The numerical results for four canonical UQ problems show both enhanced computational efficiency and improved accuracy of the proposed adaptive FD solution of the Liouville equation compared to its standard/fixed domain FD solutions. Moreover, the results for canonical test problems in computational mechanics indicate that the proposed approach for increasing the formal order of the underlying FD scheme can be easily implemented in multidimensional spaces and gives an oscillation-free numerical solution with a desired order of accuracy in a reasonable computational time. This approach is shown to provide a better computational time compared to both the underlying scheme (by about three orders of magnitude) and standard FD methods of the same order of accuracy

On the use of Neural Networks to solve Differential Equations

[EN]Artificial neural networks are parametric models, generally adjusted to solve regression and classification problem. For a long time, a question has laid around regarding the possibility of using these types of models to approximate the solutions of initial and boundary value problems, as a means for numerical integration. Recent improvements in deep-learning have made this approach much attainable, and integration methods based on training (fitting) artificial neural networks have begin to spring, motivated mostly by their mesh-free nature and scalability to high dimensions. In this work, we go all the way from the most basic elements, such as the definition of artificial neural networks and well-posedness of the problems, to solving several linear and quasi-linear PDEs using this approach. Throughout this work we explain general theory concerning artificial neural networks, including topics such as vanishing gradients, non-convex optimization or regularization, and we adapt them to better suite the initial and boundary value problems nature. Some of the original contributions in this work include: an analysis of the vanishing gradient problem with respect to the input derivatives, a custom regularization technique based on the network’s parameters derivatives, and a method to rescale the subgradients of the multi-objective of the loss function used to optimize the network.[ES]Las redes neuronales son modelos paramétricos generalmente usados para resolver problemas de regresiones y clasificación. Durante bastante tiempo ha rondado la pregunta de si es posible usar este tipo de modelos para aproximar soluciones de problemas de valores iniciales y de contorno, como un medio de integración numérica. Los cambios recientes en deep-learning han hecho este enfoque más viable, y métodos basados en entrenar (ajustar) redes neuronales han empezado a surgir motivados por su no necesidad de un mallado y su buena escalabilidad a altas dimensiones. En este trabajo, vamos desde los elementos más básicos, como la definición de una red neuronal o la buena definición de los problemas, hasta ser capaces de resolver diversas EDPs lineales y casi-lineales. A lo largo del trabajo explicamos la teoría general relacionada con redes neuronales, que incluyen tópicos como los problemas de desvanecimiento de gradientes (vanishing gradient), optimización no-convexa y técnicas de regularización, y los adaptamos a la naturaleza de los problemas de valores iniciales y de contorno. Algunas de las contribuciones originales de este trabajo incluyen: un análisis del desvanecimiento de gradientes con respecto a las variables de entrada, una técnica de regularización customizada basada en las derivadas de los parámetros de la red neuronal, y un método para rescalar los subgradientes de la función de coste multi-objectivo usada para optimizar la red neuronal

Mathematical modelling of cancer invasion and metastatic spread

Metastatic spread—the dissemination of cancer cells from a primary tumour with subsequent re-colonisation at secondary sites in the body—causes around 90% of cancer-related deaths. Mathematical modelling may provide a complementary approach to help understand the complex mechanisms underlying metastasis. In particular, the spatiotemporal evolution of individual cancer cells during the so-called invasion-metastasis cascade—i.e. during cancer cell invasion, intravasation, vascular travel, extravasation and metastatic growth—is an aspect not yet explored through existing mathematical models. In this thesis, such a spatially explicit hybrid multi-organ metastasis modelling framework is developed. It describes the invasive growth dynamics of individual cancer cells both at a primary site and at potential secondary metastatic sites in the body, as well as their transport from the primary to the secondary sites. Throughout, the interactions between the cancer cells, matrix-degrading enzymes (MDEs) and the extracellular matrix (ECM) are accounted for. Furthermore, the individual-based framework models phenotypic variation by distinguishing between cancer cells of an epithelial-like, a mesenchymal-like and a mixed phenotype. It also describes permanent and transient mutations between these cell phenotypes in the form of epithelial-mesenchymal transition (EMT) and its reverse process mesenchymal-epithelial transition (MET). Both of these mechanisms are implemented at the biologically appropriate locations of the invasion-metastasis cascade. Finally, cancer cell dormancy and death at the metastatic sites are considered to model the frequently observed maladaptation of metastasised cancer cells to their new microenvironments. To investigate the EMT-process further, an additional three-dimensional discrete-continuum model of EMT- and MET-dependent cancer cell invasion is developed. It consists of a hybrid system of partial and stochastic differential equations that describe the evolution of epithelial-like and mesenchymal-like cancer cells, again under the consideration of MDE concentrations and the ECM density. Using inverse parameter estimation and sensitivity analysis, this model is calibrated to an in vitro organotypic assay experiment that examines the invasion of HSC-3 cancer cells

Integrated batch process development based on mixed-logic dynamic optimization

Specialty chemicals industry relies on batch manufacturing, since it requires the frequent adaptation of production systems to market fluctuations. To be first in the market, batch industry requires decision-support systems for the rapid development and implementation of chemical processes. Moreover, the processes should be competitive to ensure their long-term viability. General-purpose and flexible plants and the consideration of physicochemical insights to define an efficient operation are also cornerstones for the success of specialty chemical industries. Precisely, this thesis tackles the systematic development of batch processes that are efficient, economically competitive, and environmentally friendly, to assist their agile introduction into production systems in grassroots and retrofit scenarios. Synthesis of conceptual processing schemes and plant allocation subproblems are solved simultaneously, taking into account the plant design. With this purpose, an optimization-based approach is proposed, where all structural alternatives are represented in a State-Equipment Network (SEN) superstructure, following formulated into a Mixed-Logic Dynamic Optimization (MLDO) problem which is later solved to minimize an objective function. Essentially, the strength of the proposed methodology lies in the modeling strategy which combines the different kinds of decisions of the integrated problem in a unique optimization model. Accordingly, it considers: (i) synthesis and allocation alternatives combination, (ii) dynamic process performance models and dynamic control variable profiles, (iii) discrete events associated to transitions of batch phases and operations, (iv) quantitative and qualitative information, (v) material transference synchronization to ensure batch integrity between unit procedures, and (vi) batch and semicontinuous processing elements. Different strategies can be used to solve the resulting MLDO problem. A deterministic direct-simultaneous approach is first proposed. The mixed-logic problem is reformulated into a mixed-integer one, which is fully-discretized to provide a Mixed-Integer Non-Linear Programming (MINLP) that is optimized using conventional solvers. Then, a Differential Genetic Algorithm (DGA) and a hybrid approach are presented. The purpose of these evolutionary strategies is to pose solution alternatives that keep solution goodness while seek for the improvement of computational efficiency to handle industrial-size problems. The optimization-based approach is applied in retrofit scenarios to solve the simultaneous process synthesis and plant allocation, taking into account the physical restrictions of existing plant elements. The production of specialty chemicals based on a competitive reactions system in an existing reactor network is first defined through process development and improvement according to different economic scenarios, decision criteria, and plant modifications. Additionally, a photo-Fenton process is optimized to eliminate an emergent wastewater pollutant in a given pilot plant, pursuing the minimization of processing time and cost. Batch process development in grassroots scenarios is also proven to be a problem of utmost importance to deal with uncertainty in future markets. Seeking for plant flexibility in several demand scenarios, the expected profit is maximized through a two-stage stochastic formulation that includes simultaneous plant design, process synthesis, and plant allocation decisions. A heuristic solution algorithm is used to handle the problem complexity. A grassroots plant design is defined to implement the previous competitive reaction system, where decisions like the feed-forward trajectories or operating modes allow the adaptation of master recipes to different demands. Finally, an acrylic fiber production example is presented to illustrate process development decisions like the selection of tasks, technological alternatives, chemicals, and solvent reuse.La indústria de productes químics especials es basa en la fabricació discontinua, ja que permet adaptar de forma freqüent els sistemes de producció en funció de les fluctuacions de mercat. Per ser líder al sector, són necessàries eines de suport a la decisió que ajudin a l’àgil desenvolupament i implementació de nous processos. A més, aquests han de ser competitius per garantir la seva viabilitat a llarg termini. Altres peces clau per una operació eficient són l’ús de plantes flexibles així com l’estudi dels fenòmens fisicoquímics. Aquesta tesis aborda justament el desenvolupament sistemàtic de processos químics discontinus que siguin eficients, econòmicament competitius i ecològics, per contribuir a la seva ràpida introducció en els sistemes de producció, tant en escenaris de plantes existents com des de les bases. En concret, es planteja la resolució simultània de la síntesi conceptual d’esquemes de procés i l’assignació d’equips, tenint en compte el disseny de la planta. Amb aquest objectiu, es proposa una metodologia de solució basada en optimització, on les alternatives estructurals es representen en una Xarxa d’Estats i Equips (SEN per les sigles en anglès) que es formula mitjançant un problema d’Optimització Dinàmica Mixta-Lògica (MLDO per les sigles en anglès) que es resol minimitzant una funció objectiu. La solidesa de la metodologia proposada rau en la estratègia de modelat del problema MLDO, que integra els diferents tipus de decisions en un sol model d’optimització. En concret, es consideren: (i) la combinació d’alternatives de síntesi i assignació d’equips, (ii) models de procés i trajectòries de control dinàmics, (iii) esdeveniments discrets associats al canvi de fase i operació, (iv) informació quantitativa i qualitativa, (v) sincronització de transferències de material en tasques consecutives, i (vi) elements de processat discontinus i semi-continus. Existeixen diverses estratègies per resoldre el problema MLDO resultant. En aquesta tesi es proposa en primer lloc un mètode determinístic directe-simultani, on el model mixt-lògic es transforma en un mixt-enter. Aquest es discretitza al seu torn de forma completa per obtenir un problema de Programació No-Lineal Mixta-Entera (MINLP per les sigles en anglès) el qual es pot resoldre utilitzant algoritmes d’optimització convencionals. A més, es presenten un Algoritme Genètic Diferencial (DGA per les sigles en anglès) i un mètode híbrid. Totes dues estratègies esdevenen alternatives de cerca amb l’objectiu de mantenir la bondat de la solució i millorar l’eficàcia de computació per tractar problemes de dimensió industrial. La metodologia de solució proposada s’aplica al desenvolupament de processos discontinus en escenaris de plantes existents, tenint en compte les restriccions físiques dels equips. Un primer exemple aborda la manufactura de productes químics basada en un sistema de reaccions competitives. Concretament, es desenvolupa i millora el procés de producció implementat en una xarxa de reactors considerant diferents escenaris econòmics, criteris de decisió, i modificacions de planta. En un segon exemple, s’optimitza el procés foto-Fenton per ser executat en una planta pilot per eliminar contaminants emergents. Buscant integrar el desenvolupament de procés i el disseny de plantes flexibles en escenaris de base, es presenta una formulació estocàstica en dues etapes per a optimitzar el benefici esperat d’acord a diversos escenaris de demanda. Per gestionar la complexitat d’aquest problema es proposa la utilització d’una heurística. Com a exemple, es planteja el disseny d’una planta de base on implementar l’anterior sistema de reaccions competitives. Decisions com les trajectòries dinàmiques de control o la configuració d’equips permeten adaptar la recepta màster en funció de la demanda. Un darrer exemple defineix el procés de producció de fibra acrílica, il·lustrant decisions com la selecció de tasques, tecnologia, reactius o reutilització de dissolvents.La industria productos químicos especiales se basa en la fabricación discontinua, la cual permite la adaptación frecuente de los sistemas de producción en función de las fluctuaciones de mercado. Para ser líder en el sector, son necesarias herramientas de soporte a la decisión que contribuyan al ágil desarrollo e implementación de nuevos procesos. Además, éstos deben ser competitivos para garantizar su viabilidad a largo plazo. Otras piezas clave para una operación eficiente son la utilización de plantas flexibles y el estudio de los fenómenos fisicoquímicos. Esta tesis aborda justamente el desarrollo sistemático de procesos químicos discontinuos que sean eficientes, económicamente competitivos y ecológicos, para contribuir a su rápida introducción en los sistemas de producción, ya sea en escenarios de plantas existentes o desde las bases. En particular, se plantea la resoluciónsimultánea de la síntesis conceptual de esquemas de proceso y la asignación de equipos, teniendo en cuenta además el diseño de planta.Con este fin, se propone una metodología de solución basada en optimización, donde todas las alternativas estructurales se representan en una Red de Estados y Equipos (SENpor sus siglas en inglés) que se formula mediante un problema de Optimización Dinámica Mixta-Lógica (MLDO por sus siglas en inglés) que se resuelve minimizando una función objetivo. La solidez de la metodología propuesta reside en la estrategia de modelado delproblema MLDO, que integra los diferentes tipos de decisiones en un solo modelo de optimización. En concreto, se consideran: (i) la combinación de alternativas de síntesis y asignación de equipos, (ii) modelos de proceso y trayectorias de control dinámicos, (iii)eventos discretos asociados al cambio de fase y operación, (iv) información cuantitativa y cualitativa, (v) sincronización de la transferencia de material en tareas consecutivas, y(vi) elementos de procesado discontinuos y semicontinuos.Existen diversas estrategias para resolver el problema MLDO resultante. En esta tesis se propone en primer lugar un método determinístico directo-simultáneo, donde el problema mixto-lógico se reformula en un mixto-entero. A su vez, éste se discretiza de formacompleta para obtener un problema de Programación No-Lineal Mixta-Entera (MINLP por sus siglas en inglés) el cual se puede resolver mediante algoritmos de optimización convencionales. Además, se presentan un Algoritmo Genético Diferencial (DGA por sussiglas en inglés) y un método híbrido. Ambas estrategias se plantean como alternativas de búsqueda con objeto de mantener la bondad de la solución y mejorar la eficacia de computación para tratar problemas de dimensión industrial.La metodología de solución propuesta se aplica al desarrollo de procesos discontinuos en escenarios con plantas existentes, teniendo en cuenta las restricciones físicas de los equipos. Un primer ejemplo aborda la fabricación de productos químicos basada en un sistema de reacciones competitivas. En concreto, se desarrolla y mejora el proceso de producción a implementar en una red de reactores considerando diferentes escenarios económicos, criterios de decisión, y modificaciones de planta. En un segundo ejemplo,se optimiza el proceso foto-Fenton a ser ejecutado en una planta piloto para eliminar contaminantes emergentes.Persiguiendo la integración del desarrollo de proceso con el diseño de plantas flexi-bles en escenarios base, se presenta asimismo una formulación estocástica en dos etapas para optimizar el beneficio esperado de acuerdo a varios escenarios de demanda. Paramanejar la complejidad de dicho problema se propone la utilización de una heurística.Como ejemplo, se plantea el diseño de una planta de base para implementar el anterior sistema de reacciones competitivas, donde decisiones como las trayectorias dinámicas de control o la configuración de equipos permiten adaptar la receta máster en función de lademandas. Por último, se presenta un ejemplo donde se define el proceso de producción de fibra acrílica, ilustrando decisiones como la selección de tareas, alternativas tecnológicas, reactivos químicos o la reutilización de disolventes.Postprint (published version