Interplay of martensitic phase transformation and plastic slip in polycrystals

Inspired by key experimental and analytical results regarding Shape Memory Alloys (SMAs), we propose a modelling framework to explore the interplay between martensitic phase transformations and plastic slip in polycrystalline materials, with an eye towards computational efficiency. The resulting framework uses a convexified potential for the internal energy density to capture the stored energy associated with transformation at the meso-scale, and introduces kinetic potentials to govern the evolution of transformation and plastic slip. The framework is novel in the way it treats plasticity on par with transformation.

We implement the framework in the setting of anti-plane shear, using a staggered implicit/explict update: we first use a Fast-Fourier Transform (FFT) solver based on an Augmented Lagrangian formulation to implicitly solve for the full-field displacements of a simulated polycrystal, then explicitly update the volume fraction of martensite and plastic slip using their respective stick-slip type kinetic laws. We observe that, even in this simple setting with an idealized material comprising four martensitic variants and four slip systems, the model recovers a rich variety of SMA type behaviors. We use this model to gain insight into the isothermal behavior of stress-stabilized martensite, looking at the effects of the relative plastic yield strength, the memory of deformation history under non-proportional loading, and several others.

We extend the framework to the generalized 3-D setting, for which the convexified potential is a lower bound on the actual internal energy, and show that the fully implicit discrete time formulation of the framework is governed by a variational principle for mechanical equilibrium. We further propose an extension of the method to finite deformations via an exponential mapping. We implement the generalized framework using an existing Optimal Transport Mesh-free (OTM) solver. We then model the $\alpha$--$\gamma$ and $\alpha$--$\varepsilon$ transformations in pure iron, with an initial attempt in the latter to account for twinning in the parent phase. We demonstrate the scalability of the framework to large scale computing by simulating Taylor impact experiments, observing nearly linear (ideal) speed-up through 256 MPI tasks. Finally, we present preliminary results of a simulated Split-Hopkinson Pressure Bar (SHPB) experiment using the $\alpha$--$\varepsilon$ model.

Producción y gestión de la industria lítica de Atxoste (Álava): Una aproximación a las sociedades Epipaleolíticas y Mesolíticas del alto Ebro

1000 p. (Anexos: 929-965 p.; bibliografía 965-1000 p.). Capítulos de discusión y conclusiones en castellano y francés.

Flight dynamics in Drosophila through a dynamically-scaled robotic approach

Flies are particularly adept at balancing the competing demands of delay tolerance, performance, and robustness during flight, which invites thoughtful examination of their multimodal feedback architecture. This dissertation examines stabilization requirements for inner-loop feedback strategies in the flapping flight of Drosophila, the fruit fly, against the backdrop of sensorimotor transformations present in the animal. Flies have evolved multiple specializations to reduce sensorimotor latency, but sensory delay during flight is still significant on the timescale of body dynamics. I explored the effect of sensor delay on flight stability and performance for yaw turns using a dynamically-scaled robot equipped with a real-time feedback system that performed active turns in response to measured yaw torque. The results show a fundamental tradeoff between sensor delay and permissible feedback gain, and suggest that fast mechanosensory feedback provides a source of active damping that compliments that contributed by passive effects. Presented in the context of these findings, a control architecture whereby a haltere-mediated inner-loop proportional controller provides damping for slower visually-mediated feedback is consistent with tethered-flight measurements, free-flight observations, and engineering design principles. Additionally, I investigated how flies adjust stroke features to regulate and stabilize level forward flight. The results suggest that few changes to hovering kinematics are actually required to meet steady-state lift and thrust requirements at different flight speeds, and the primary driver of equilibrium velocity is the aerodynamic pitch moment. This finding is consistent with prior hypotheses and observations regarding the relationship between body pitch and flight speed in fruit flies. The results also show that the dynamics may be stabilized with additional pitch damping, but the magnitude of required damping increases with flight speed. I posit that differences in stroke deviation between the upstroke and downstroke might play a critical role in this stabilization. Fast mechanosensory feedback of the pitch rate could enable active damping, which would inherently exhibit gain scheduling with flight speed if pitch torque is regulated by adjusting stroke deviation. Such a control scheme would provide an elegant solution for flight stabilization across a wide range of flight speeds.

Topics in randomized numerical linear algebra

This thesis studies three classes of randomized numerical linear algebra algorithms, namely: (i) randomized matrix sparsification algorithms, (ii) low-rank approximation algorithms that use randomized unitary transformations, and (iii) low-rank approximation algorithms for positive-semidefinite (PSD) matrices.

Randomized matrix sparsification algorithms set randomly chosen entries of the input matrix to zero. When the approximant is substituted for the original matrix in computations, its sparsity allows one to employ faster sparsity-exploiting algorithms. This thesis contributes bounds on the approximation error of nonuniform randomized sparsification schemes, measured in the spectral norm and two NP-hard norms that are of interest in computational graph theory and subset selection applications.

Low-rank approximations based on randomized unitary transformations have several desirable properties: they have low communication costs, are amenable to parallel implementation, and exploit the existence of fast transform algorithms. This thesis investigates the tradeoff between the accuracy and cost of generating such approximations. State-of-the-art spectral and Frobenius-norm error bounds are provided.

The last class of algorithms considered are SPSD "sketching" algorithms. Such sketches can be computed faster than approximations based on projecting onto mixtures of the columns of the matrix. The performance of several such sketching schemes is empirically evaluated using a suite of canonical matrices drawn from machine learning and data analysis applications, and a framework is developed for establishing theoretical error bounds.

In addition to studying these algorithms, this thesis extends the Matrix Laplace Transform framework to derive Chernoff and Bernstein inequalities that apply to all the eigenvalues of certain classes of random matrices. These inequalities are used to investigate the behavior of the singular values of a matrix under random sampling, and to derive convergence rates for each individual eigenvalue of a sample covariance matrix.

Catalytic conversion of nitrogen to ammonia by an iron model complex

Threefold symmetric Fe phosphine complexes have been used to model the structural and functional aspects of biological N₂ fixation by nitrogenases. Low-valent bridging Fe-S-Fe complexes in the formal oxidation states Fe(II)Fe(II), Fe(II)/Fe(I), and Fe(I)/Fe(I) have been synthesized which display rich spectroscopic and magnetic behavior. A series of cationic tris-phosphine borane (TPB) ligated Fe complexes have been synthesized and been shown to bind a variety of nitrogenous ligands including N₂H₄, NH₃, and NH₂-. These complexes are all high spin S = 3/2 and display EPR and magnetic characteristics typical of this spin state. Furthermore, a sequential protonation and reduction sequence of a terminal amide results in loss of NH₃ and uptake of N₂. These stoichiometric transformations represent the final steps in potential N₂ fixation schemes.

Treatment of an anionic FeN₂ complex with excess acid also results in the formation of some NH₃, suggesting the possibility of a catalytic cycle for the conversion of N₂ to NH₃ mediated by Fe. Indeed, use of excess acid and reductant results in the formation of seven equivalents of NH₃ per Fe center, demonstrating Fe mediated catalytic N₂ fixation with acids and protons for the first time. Numerous control experiments indicate that this catalysis is likely being mediated by a molecular species.

A number of other phosphine ligated Fe complexes have also been tested for catalysis and suggest that a hemi-labile Fe-B interaction may be critical for catalysis. Additionally, various conditions for the catalysis have been investigated. These studies further support the assignment of a molecular species and delineate some of the conditions required for catalysis.

Finally, combined spectroscopic studies have been performed on a putative intermediate for catalysis. These studies converge on an assignment of this new species as a hydrazido(2-) complex. Such species have been known on group 6 metals for some time, but this represents the first characterization of this ligand on Fe. Further spectroscopic studies suggest that this species is present in catalytic mixtures, which suggests that the first steps of a distal mechanism for N₂ fixation are feasible in this system.

Nonlinear dynamic transfer functions for certain retinal neuronal systems

The applicability of the white-noise method to the identification of a nonlinear system is investigated. Subsequently, the method is applied to certain vertebrate retinal neuronal systems and nonlinear, dynamic transfer functions are derived which describe quantitatively the information transformations starting with the light-pattern stimulus and culminating in the ganglion response which constitutes the visually-derived input to the brain. The retina of the catfish, Ictalurus punctatus, is used for the experiments.

The Wiener formulation of the white-noise theory is shown to be impractical and difficult to apply to a physical system. A different formulation based on crosscorrelation techniques is shown to be applicable to a wide range of physical systems provided certain considerations are taken into account. These considerations include the time-invariancy of the system, an optimum choice of the white-noise input bandwidth, nonlinearities that allow a representation in terms of a small number of characterizing kernels, the memory of the system and the temporal length of the characterizing experiment. Error analysis of the kernel estimates is made taking into account various sources of error such as noise at the input and output, bandwidth of white-noise input and the truncation of the gaussian by the apparatus.

Nonlinear transfer functions are obtained, as sets of kernels, for several neuronal systems: Light → Receptors, Light → Horizontal, Horizontal → Ganglion, Light → Ganglion and Light → ERG. The derived models can predict, with reasonable accuracy, the system response to any input. Comparison of model and physical system performance showed close agreement for a great number of tests, the most stringent of which is comparison of their responses to a white-noise input. Other tests include step and sine responses and power spectra.

Many functional traits are revealed by these models. Some are: (a) the receptor and horizontal cell systems are nearly linear (small signal) with certain "small" nonlinearities, and become faster (latency-wise and frequency-response-wise) at higher intensity levels, (b) all ganglion systems are nonlinear (half-wave rectification), (c) the receptive field center to ganglion system is slower (latency-wise and frequency-response-wise) than the periphery to ganglion system, (d) the lateral (eccentric) ganglion systems are just as fast (latency and frequency response) as the concentric ones, (e) (bipolar response) = (input from receptors) - (input from horizontal cell), (f) receptive field center and periphery exert an antagonistic influence on the ganglion response, (g) implications about the origin of ERG, and many others.

An analytical solution is obtained for the spatial distribution of potential in the S-space, which fits very well experimental data. Different synaptic mechanisms of excitation for the external and internal horizontal cells are implied.

Projections in a normed linear space and a generalization of the pseudo-inverse

The concept of a "projection function" in a finite-dimensional real or complex normed linear space H (the function P_M which carries every element into the closest element of a given subspace M) is set forth and examined.

If dim M = dim H - 1, then P_M is linear. If P_N is linear for all k-dimensional subspaces N, where 1 ≤ k < dim M, then P_M is linear.

The projective bound Q, defined to be the supremum of the operator norm of P_M for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, P_M is always linear, and a characterization of those norms is given.

If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when P_M is linear its adjoint P_M^H is the projection on (kernel P_M)^⊥ by the dual norm. The projective bounds of a norm and its dual are equal.

The notion of a pseudo-inverse F⁺ of a linear transformation F is extended to non-Euclidean norms. The distance from F to the set of linear transformations G of lower rank (in the sense of the operator norm ∥F - G∥) is c/∥F⁺∥, where c = 1 if the range of F fills its space, and 1 ≤ c < Q otherwise. The norms on both domain and range spaces have Q = 1 if and only if (F⁺)⁺ = F for every F. This condition is also sufficient to prove that we have (F⁺)^H = (F^H)⁺, where the latter pseudo-inverse is taken using dual norms.

In all results, the real and complex cases are handled in a completely parallel fashion.

A unified strategy to ent-kauranoid natural products : total syntheses of (–)-maoecrystal Z, (–)-trichorabdal A, and (–)-longikaurin E

The diterpenoid constituents of the Isodon plants have attracted reasearchers interested in both their chemical structures and biological properties for more than a half-century. In recent years, the isolations of new members displaying previously unprecedented ring systems and highly selective biological properties have piqued interest from the synthetic community in this class of natural products.

Reported herein is the first total synthesis of such a recently isolated diterpenoid, (–)-maoecrystal Z. The principal transformations implemented in this synthesis include two highly diastereoselective radical cyclization reactions: a Sm^(II)-mediated reductive cascade cyclization, which forms two rings and establishes four new stereocenters in a single step, and a Ti^(III)-mediated reductive epoxide-acrylate coupling that yields a functionalized spirolactone product, which forms a core bicycle of maoecrystal Z.

The preparation of two additional ent-kauranoid natural products, (–)-trichorabdal A and (–)-longikaurin E, is also described from a derivative of this key spirolactone. These syntheses are additionally enabled by the palladium-mediated oxidative cyclization reaction of a silyl ketene acetal precursor that is used to install the bridgehead all-carbon quaternary stereocenter and bicyclo[3.2.1]octane present in each natural product. These studies have established a synthetic relationship among three architecturally distinct ent-kaurane diterpenoids and have forged a path for the preparation of interesting unnatural ent-kauranoid structural analogs for more thorough biological study.

A multi-scale approach to shaping carbon nanotube structures for hollow microneedles

The concept of a carbon nanotube microneedle array is explored in this thesis from multiple perspectives including microneedle fabrication, physical aspects of transdermal delivery, and in vivo transdermal drug delivery experiments. Starting with standard techniques in carbon nanotube (CNT) fabrication, including catalyst patterning and chemical vapor deposition, vertically-aligned carbon nanotubes are utilized as a scaffold to define the shape of the hollow microneedle. Passive, scalable techniques based on capillary action and unique photolithographic methods are utilized to produce a CNT-polymer composite microneedle. Specific examples of CNT-polyimide and CNT-epoxy microneedles are investigated. Further analysis of the transport properties of polymer resins reveals general requirements for applying arbitrary polymers to the fabrication process.

The bottom-up fabrication approach embodied by vertically-aligned carbon nanotubes allows for more direct construction of complex high-aspect ratio features than standard top-down fabrication approaches, making microneedles an ideal application for CNTs. However, current vertically-aligned CNT fabrication techniques only allow for the production of extruded geometries with a constant cross-sectional area, such as cylinders. To rectify this limitation, isotropic oxygen etching is introduced as a novel fabrication technique to create true 3D CNT geometry. Oxygen etching is utilized to create a conical geometry from a cylindrical CNT structure as well as create complex shape transformations in other CNT geometries.

CNT-polymer composite microneedles are anchored onto a common polymer base less than 50 µm thick, which allows for the microneedles to be incorporated into multiple drug delivery platforms, including modified hypodermic syringes and silicone skin patches. Cylindrical microneedles are fabricated with 100 µm outer diameter and height of 200-250 µm with a central cavity, or lumen, diameter of 30 µm to facilitate liquid drug flow. In vitro delivery experiments in swine skin demonstrate the ability of the microneedles to successfully penetrate the skin and deliver aqueous solutions.

An in vivo study was performed to assess the ability of the CNT-polymer microneedles to deliver drugs transdermally. CNT-polymer microneedles are attached to a hand actuated silicone skin patch that holds a liquid reservoir of drugs. Fentanyl, a potent analgesic, was administered to New Zealand White Rabbits through 3 routes of delivery: topical patch, CNT-polymer microneedles, and subcutaneous hypodermic injection. Results demonstrate that the CNT-polymer microneedles have a similar onset of action as the topical patch. CNT-polymer microneedles were also vetted as a painless delivery approach compared to hypodermic injection. Comparative analysis with contemporary microneedle designs demonstrates that the delivery achieved through CNT-polymer microneedles is akin to current hollow microneedle architectures. The inherent advantage of applying a bottom-up fabrication approach alongside similar delivery performance to contemporary microneedle designs demonstrates that the CNT-polymer composite microneedle is a viable architecture in the emerging field of painless transdermal delivery.

Transformações no espaço terciário de Petrópolis: do empreendedorismo urbano às centralidades, analisando os distritos Sede e Itaipava.

Esta dissertação de mestrado tem por objetivo entender as transformações espaciais ocorridas no município de Petrópolis/RJ, a partir do surgimento de novas centralidades que começam a despontar principalmente a partir da década de 1980. A proposta do presente trabalho também se baseia no estudo da formação de uma economia de serviços no município, como um estágio avançado das economias capitalistas, a partir do forte declínio de sua anterior base industrial e muito incipiente base agrícola, apesar de se tratar de uma cidade média. Desta maneira, Petrópolis começa a estimular determinados ramos do setor terciário que são extremamente importantes e que ainda apresentam a possibilidade de movimentar outros ramos e setores, como por exemplo, turismo e lazer, moda (roupas e confecções), gastronomia, decoração/design e alta tecnologia. No entanto, o estímulo ao desenvolvimento de uma economia de serviços passa necessariamente por ações implementadas pelo poder público, visando remover obstáculos à atração de novos atores que investirão em cidades que atendam às suas necessidades. Por isso, além de políticas públicas, o município investe em infraestrutura urbana para alinhar-se ao empresariamento ou empreendedorismo de cidades, esperando, assim, adentrar ao circuito das city marketing. Mas este modelo de desenvolvimento terciário criou diferenças intra-urbanas no município de Petrópolis quando estimulou o surgimento de novas centralidades que, para nós, se materializa principalmente na imagem do terceiro distrito Itaipava. Esta centralidade apresenta especificidades que a torna lócus importante para uma abastada classe social que pretende não depender de deslocamentos mais longos até o distrito-Sede/Centro Histórico para consumir bens e serviços. Logo, a nova centralidade Itaipava surgirá como um espaço especializado em turismo e lazer, gastronomia, decoração/design e shopping centers, atendendo às exigências mais específicas de uma classe abastada. Porém, mesmo existindo uma nova centralidade, em nenhum momento haverá neste trabalho a construção da ideia de um centro tradicional e histórico esfacelado e agonizante. Muito pelo contrário, o que se percebeu em Petrópolis foi o inverso: o distrito-Sede/Centro Histórico continua sendo o mais dinâmico dentro do território municipal e, ao contrário daquilo observado em grandes cidades, não há uma perda de sua importância, mas sim hierarquias diferentes que se complementam. Prova disso é a sequência de dados que será apresentada no último capítulo. Enfim, um município de porte médio se qualifica enquanto uma economia de serviços, espraiando-se para as áreas afastadas do centro onde, mesmo assim, o centro tradicional não perde sua hegemonia, apesar do surgimento de novas centralidades.

A (re)valorização do patrimônio e as transformações da paisagem: o exemplo do Estado do Rio de Janeiro

Este trabalho apresenta as principais perspectivas estabelecidas a partir da utilização do patrimônio herdado da atividade canavieira no desenvolvimento da atividade turística, na região norte do estado do Rio de Janeiro. Para isso, discutiremos os principais conceitos que permitem entender como áreas consideradas rurais ou marcadas por uma história vinculada a alguma atividade agrícola passam por um processo de transformação e são inseridas em uma nova fase. Essas transformações são percebidas nos espaços chamados de rurais e aparecem hoje com uma nova perspectiva de utilização, antes eram destinados exclusivamente às atividades agrícolas. Outro aspecto verificado é que paisagens singulares acabam se constituindo em patrimônios culturais, na medida em que se tornam representações sociais. A paisagem então acaba sendo relacionada à memória e à identidade local. Nos municípios estudados ela é marcada pelas tradicionais plantações de cana-de-açúcar, que ao longo de mais de um século influenciou a vida e o cotidiano daquela porção do espaço fluminense, permitindo uma arquitetura e uma paisagem peculiar. Desta forma, nosso objetivo é entender como a paisagem é utilizada no desenvolvimento da atividade turística como recurso a ser explorado, além de tentarmos visualizar os principais agentes que promovem as iniciativas de (re)valorização patrimonial e como o espaço é modificado com tal atividade.

Os novos movimentos de classe: reflexões sobre a organização política da classe trabalhadora brasileira

O trabalho de pesquisa apresentado tem como objetivo analisar e discutir centralmente o processo de reconfiguração da classe trabalhadora brasileira, suas transformações e impactos ocasionados no contexto da reestruturação produtiva e das políticas neoliberais. Serão abordados aspectos referentes às características de sua organização política e sindical debatendo as potencialidades dos novos movimentos de classe, que se articulam enquanto ferramentas de transformação societária. Este novo processo de organização dos trabalhadores é tratado como fruto da realidade objetiva que os integra: precarizados e desempregados da cidade, sem-terras e sem-emprego do campo, indicando uma realidade onde a classe trabalhadora define-se de forma diferente daqueles trabalhadores que, em fins de 1970, iniciavam as greves em São Bernardo do Campo. Os novos movimentos de classe, aqui considerados, articulam a luta por uma contra-hegemonia no campo das ações políticas, de organização da classe e no campo da gestão econômica. São novos, por expressarem o resgate e a inovação de experiências políticas da classe trabalhadora em um momento onde esta sofre grandes transformações.

Gestão em saúde, uma perspectiva ergológica: com quantos gestos se faz uma gestão

Baseado na convicção de que trabalhar é gerir fruto das pesquisas com a perspectiva da Ergologia , procura-se nesta tese pensar gestão como um conceito ampliado, algo que todos os humanos operam ao trabalhar, e não somente como uma função exclusiva de administradores, no sentido restrito do termo (referido apenas aos chefes, diretores, etc.). Tendo como campo empírico o Sistema Único de Saúde (SUS) e investigando-se as proposições de alguns dos principais autores sobre o tema da Gestão e Planejamento em Saúde, verificou-se que uma exaustiva busca vem sendo empreendida por diversos agentes do campo da Saúde e por pesquisadores para aproximarem-se, compreenderem e desenvolverem melhor as habilidades, os conhecimentos, as competências e os dispositivos que permitiriam uma gestão mais eficiente do SUS e, mais especificamente, no âmbito de uma Unidade de Saúde Pública no Brasil. Estiveram em análise as práticas de gestão desenvolvidas em um Centro Municipal de Saúde do estado do Rio de Janeiro (Brasil), no qual o autor da tese, além das atividades de pesquisa, exercia a função de diretor geral. A tese teve como objetivo principal analisar, do ponto de vista da atividade, a dimensão gestionária do trabalho na Unidade de Saúde citada, a fim de discutir a viabilidade naquele local e, possivelmente em outras Unidades de Saúde do exercício de uma ergogestão, isto é, uma gestão com base nos princípios propostos pela Ergologia quando o ponto de vista da atividade tem cidadania no meio de trabalho. O referencial teórico constituiu-se de algumas abordagens clínicas do trabalho (Ergonomia da Atividade, Psicopatologia do Trabalho, Psicodinâmica do Trabalho e Clínica da Atividade, esta última em menor proporção) com elementos das contribuições do educador brasileiro Paulo Freire, do psicanalista inglês Donald Winnicott e do biólogo chileno Humberto Maturana, todas colocadas em sinergia dialética sob a orientação da perspectiva ergológica. No curso da investigação foram utilizados métodos e técnicas pertinentes a este quadro e que objetivaram possibilitar a aproximação e o diálogo com os protagonistas da atividade na Unidade de Saúde em análise. Destacam-se as influências da pesquisa-intervenção e da pesquisa etnográfica, sendo o principal dispositivo técnico utilizado aquele denominado Encontros sobre o Trabalho. A pesquisa empreendida, conjuntamente com a experiência concreta de gerenciamento (como diretor geral), permitiu concluir que o esforço de implantação da modalidade que se denomina ergogestão, privilegiando o ponto de vista da atividade, pôde colaborar para promover transformações positivas no cotidiano da Unidade posta em análise. Contudo, sua aceitação por um maior número de atores e seu desenvolvimento dependem do atendimento de algumas necessidades, apontadas pelo coletivo de trabalho como entraves a superar. Os achados aqui presentes podem contribuir para a construção de um patrimônio de informações acerca da Unidade. A partir desse patrimônio outras experiências de gerenciamento podem vir a se desenvolver, obtendo-se assim, cada vez maior êxito na gestão do processo de trabalho e na melhoria das condições do atendimento oferecido aos usuários.

Neural and computational representations of decision variables

These studies explore how, where, and when representations of variables critical to decision-making are represented in the brain. In order to produce a decision, humans must first determine the relevant stimuli, actions, and possible outcomes before applying an algorithm that will select an action from those available. When choosing amongst alternative stimuli, the framework of value-based decision-making proposes that values are assigned to the stimuli and that these values are then compared in an abstract “value space” in order to produce a decision. Despite much progress, in particular regarding the pinpointing of ventromedial prefrontal cortex (vmPFC) as a region that encodes the value, many basic questions remain. In Chapter 2, I show that distributed BOLD signaling in vmPFC represents the value of stimuli under consideration in a manner that is independent of the type of stimulus it is. Thus the open question of whether value is represented in abstraction, a key tenet of value-based decision-making, is confirmed. However, I also show that stimulus-dependent value representations are also present in the brain during decision-making and suggest a potential neural pathway for stimulus-to-value transformations that integrates these two results.

More broadly speaking, there is both neural and behavioral evidence that two distinct control systems are at work during action selection. These two systems compose the “goal-directed system”, which selects actions based on an internal model of the environment, and the “habitual” system, which generates responses based on antecedent stimuli only. Computational characterizations of these two systems imply that they have different informational requirements in terms of input stimuli, actions, and possible outcomes. Associative learning theory predicts that the habitual system should utilize stimulus and action information only, while goal-directed behavior requires that outcomes as well as stimuli and actions be processed. In Chapter 3, I test whether areas of the brain hypothesized to be involved in habitual versus goal-directed control represent the corresponding theorized variables.

The question of whether one or both of these neural systems drives Pavlovian conditioning is less well-studied. Chapter 4 describes an experiment in which subjects were scanned while engaged in a Pavlovian task with a simple non-trivial structure. After comparing a variety of model-based and model-free learning algorithms (thought to underpin goal-directed and habitual decision-making, respectively), it was found that subjects’ reaction times were better explained by a model-based system. In addition, neural signaling of precision, a variable based on a representation of a world model, was found in the amygdala. These data indicate that the influence of model-based representations of the environment can extend even to the most basic learning processes.

Knowledge of the state of hidden variables in an environment is required for optimal inference regarding the abstract decision structure of a given environment and therefore can be crucial to decision-making in a wide range of situations. Inferring the state of an abstract variable requires the generation and manipulation of an internal representation of beliefs over the values of the hidden variable. In Chapter 5, I describe behavioral and neural results regarding the learning strategies employed by human subjects in a hierarchical state-estimation task. In particular, a comprehensive model fit and comparison process pointed to the use of "belief thresholding". This implies that subjects tended to eliminate low-probability hypotheses regarding the state of the environment from their internal model and ceased to update the corresponding variables. Thus, in concert with incremental Bayesian learning, humans explicitly manipulate their internal model of the generative process during hierarchical inference consistent with a serial hypothesis testing strategy.

Generalizations and extensions of the Fokker-Planck-Kolmogorov equations

The problem of determining probability density functions of general transformations of random processes is considered in this thesis. A method of solution is developed in which partial differential equations satisfied by the unknown density function are derived. These partial differential equations are interpreted as generalized forms of the classical Fokker-Planck-Kolmogorov equations and are shown to imply the classical equations for certain classes of Markov processes. Extensions of the generalized equations which overcome degeneracy occurring in the steady-state case are also obtained.

The equations of Darling and Siegert are derived as special cases of the generalized equations thereby providing unity to two previously existing theories. A technique for treating non-Markov processes by studying closely related Markov processes is proposed and is seen to yield the Darling and Siegert equations directly from the classical Fokker-Planck-Kolmogorov equations.

As illustrations of their applicability, the generalized Fokker-Planck-Kolmogorov equations are presented for certain joint probability density functions associated with the linear filter. These equations are solved for the density of the output of an arbitrary linear filter excited by Markov Gaussian noise and for the density of the output of an RC filter excited by the Poisson square wave. This latter density is also found by using the extensions of the generalized equations mentioned above. Finally, some new approaches for finding the output probability density function of an RC filter-limiter-RC filter system driven by white Gaussian noise are included. The results in this case exhibit the data required for complete solution and clearly illustrate some of the mathematical difficulties inherent to the use of the generalized equations.