This thesis presents an abstract model of the mammalian neocortex. The model was constructed by taking a top-down view on the cortex, where it is assumed that cortex to a first approximation works as a system with attractor dynamics. The model deals with the processing of static inputs from the perspectives of biological mapping, algorithmic, and physical implementation, but it does not consider the temporal aspects of these inputs. The purpose of the model is twofold: Firstly, it is an abstract model of the cortex and as such it can be used to evaluate hypotheses about cortical function and structure. Secondly, it forms the basis of a general information processing system that may be implemented in computers. The characteristics of this model are studied both analytically and by simulation experiments, and we also discuss its parallel implementation on cluster computers as well as in digital hardware. The basic design of the model is based on a thorough literature study of the mammalian cortex’s anatomy and physiology. We review both the layered and columnar structure of cortex and also the long- and short-range connectivity between neurons. Characteristics of cortex that defines its computational complexity such as the time-scales of cellular processes that transport ions in and out of neurons and give rise to electric signals are also investigated. In particular we study the size of cortex in terms of neuron and synapse numbers in five mammals; mouse, rat, cat, macaque, and human. The cortical model is implemented with a connectionist type of network where the functional units correspond to cortical minicolumns and these are in turn grouped into hypercolumn modules. The learning-rules used in the model are local in space and time, which make them biologically plausible and also allows for efficient parallel implementation. We study the implemented model both as a single- and multi-layered network. Instances of the model with sizes up to that of a rat-cortex equivalent are implemented and run on cluster computers in 23% of real time. We demonstrate on tasks involving image-data that the cortical model can be used for meaningful computations such as noise reduction, pattern completion, prototype extraction, hierarchical clustering, classification, and content addressable memory, and we show that also the largest cortex equivalent instances of the model can perform these types of computations. Important characteristics of the model are that it is insensitive to limited errors in the computational hardware and noise in the input data. Furthermore, it can learn from examples and is self-organizing to some extent. The proposed model contributes to the quest of understanding the cortex and it is also a first step towards a brain-inspired computing system that can be implemented in the molecular scale computers of tomorrow. The main contributions of this thesis are: (i) A review of the size, modularization, and computational structure of the mammalian neocortex. (ii) An abstract generic connectionist network model of the mammalian cortex. (iii) A framework for a brain-inspired self-organizing information processing system. (iv) Theoretical work on the properties of the model when used as an autoassociative memory. (v) Theoretical insights on the anatomy and physiology of the cortex. (vi) Efficient implementation techniques and simulations of cortical sized instances. (vii) A fixed-point arithmetic implementation of the model that can be used in digital hardware. / QC 20100903

In this article we investigate a pitchfork bifurcation of the local attractor of a simple capsizing model proposed by Thompson. Although this is a very simple system it has a very complicate dynamic. We try to reveal some properties of this dynamic with modern numerical methods. For this reason we approximate stable and unstable manifolds which connect the steady states to obtain a complete understanding of the topology in the phase space. We also consider approximations of the Lyapunov Exponents (resp. Floquet Exponents) which indicates the pitchfork bifurcation.

Šiame darbe nagrinėjami fraktalų – iteruoųjų funkcijų sistemų (IFS) atraktorių – sintezės algoritmai. Pagrindinis dėmesys skiriamas pabėgimo laiko (PL)-algoritmui. PL-algoritmas yra pakankamai universalus. Pagrindinės šio algoritmo panaudojimo sričitys – netiesinių dinaminių sistemų, veikiančių kompleksinėje plokštumoje, analizė, kompleksinių daugianarių šaknų pritraukimo baseinų vizualizavimas ir kt. Geometrinių fraktalų (IFS atraktorių) sintezei šis algoritmas iki šiol nebuvo naudojamas, nors tokia galimybė, kaip teorinis rezultatas, yra žinoma. Pagrindinė to priežastis – IFS sudarančių afiniųjų transformacijų veikimo zonų atskyrimo kriterijaus nebuvimas. Tokio kriterijaus paieškai ir realizacijai darbe skiriamas didžiausias dėmesys. Rezultatas – nauja adaptyvi IFS sudarančių afiniųjų transformacijų veikimo zonų atskyrimo procedūra. Lygegrečiai spendžiama tolygaus spalvinio sintezuojamo (PL-algoritmo pagalba) fraktalinio vaizdo užpildymo problema. Pasiūlytas originalus sprendimas – problemiškai oriantuota iteracij�� skaičiaus (sintezės metu) korekcija. Darbe pristatomi ir preliminarūs su fraktalinių vaizdų (IFS atraktorių) sinteze susijusių eksperimentų rezultatai. / The contribution of this work is a new version of the escape time algoritm adapted for synthesizing fractal images, indentified with atractors of iterated functions systems (IFS). The proposesd synthesis algorithm is based on the use of shift dynamics, associated with one or another IFS. The strategy for the seperation of extended domains of the inverse affine transformations, specified by IFS, is developed. In the field of computerized real word image models (digital images) the fractal approach is of outmost importance, because it facilitates perception and understanding of the information content of an image. To say more, it provides us with a powerful means to catch sight of a fundamental real word image property generally known as self-similarity. Due to this property, the research and development of algorithms („fractal techniques“) to extract imortant fractal parameters from appropriate digital data has received significant attention in recent years. In this work, the basic concepts and ideas that are needed to describe, state and solve the problem of synthesizing fractal images, identified with attractors of IFS, are introduced and explored. A new original approach (idea), leading to practical implementation of the shift dynamical system, associated with a particular IFS, is proposed (part 3). Some experimental results are given (Part 4).

This thesis investigates the computer-based modelling and simulation of complex geospatial phenomena. Geospatial systems are real world processes which extend over some meaningful extent of the Earth's surface, such as cities and fisheries. There are many problems that require urgent attention in this domain (for example relating to sustainability) but despite increasing amounts of data and computational power there is a significant gap between the potential for model-based analyses and their actual impact on real world policy and planning. Analytical methods are confounded by the high dimensionality and nonlinearity of spatio-temporal systems and/or are hard to relate to meaningful policy decisions. Simulation-based approaches on the other hand are more heuristic and policy oriented in nature, but they are difficult to validate and almost always over-fit the data: although a given model can be calibrated on a given set of data, it usually performs very poorly on new unseen data sets. The central contribution of this thesis is a framework which is formally grounded and able to be rigourously validated, yet at the same time is interpretable in terms of real world phenomena and thus has a strong connection to domain knowledge. The scope of the thesis spans both theory and practice, and three specific contributions range along this span. Starting at the theoretical end, the first contribution concerns the conceptual and theoretical basis of the framework, which is a technique known as delay reconstruction. The underlying theory is rooted in the rather technical field of dynamical systems (itself largely based on differential topology), which has hindered its wider application and the formation of strong links with other areas. Therefore, the first contribution is an exposition of delay reconstruction in non-technical language, with a focus on explaining how some recent extensions to this theory make the concept far more widely applicable than is often assumed. The second contribution uses this theoretical foundation to develop a practical, unified framework for pattern discovery and hypothesis exploration in geo-spatial data. The central aspect of this framework is the linking of delay reconstruction with domain knowledge. This is done via the notion that determinism is not an on-off quantity, but rather that a given data set may be ascribed a particular 'degree' of determinism, and that that degree may be increased through manipulation of the data set using domain knowledge. This leads to a framework which can handle spatiotemporally complex (including multi-scale) data sets, is sensitive to the amount of data that is available, and is naturally geared to be used interactively with qualitative feedback conveyed to the user via geometry. The framework is complementary to other techniques in that it forms a scaffold within which almost all modelling approaches - including agent-based modelling - can be cast as particular kinds of 'manipulations' of the data, and as such are easily integrated. The third contribution examines the practical efficacy of the framework in a real world case study. This involves a high resolution spatio-temporal record of fishcatch data from trawlers operating in a large fishery. The study is used to test two fundamental capabilities of the framework: (i) whether real world spatio-temporal phenomena can be identified in the degree-of-determinism signature of the data set, (ii) whether the determinism-level can then be increased by manipulating the data in response to this phenomena. One of the main outcomes of this study is a clear identification of the influence of the lunar cycle on the behaviour of Tiger and Endeavour prawns. The framework allows for this to be 'non-destructively subtracted', increasing the detect-ability of further phenomena.

In this thesis we will discuss the well-posedness and long-time dynamics of curved beam and thermoelastic plates. First, we considered the Bresse system with nonlinear damping and forcing terms. For this model we show the Timoshenko system as a singular limit of the Bresse system as the arch curvature l goes to 0 and under suitable assumptions on the nonlinearity we prove the existence of a smooth global attractor with finite fractal dimension and exponential attractors as well. We also compare the Bresse system with the Timoshenko system, in the sense of upper-semicontinuity of their attractors as l → 0. Second, we study a full von Karman system, this model accounts for vertical and in plane displacements. For this system we add a nonlinear thermal coupling and free boundary conditions. It is shown that the system, without any mechanical dissipation imposed on vertical displacements, admits a global attractor which is also smooth and of finite fractal dimension. / Neste trabalho iremos discutir a existência, unicidade, dependência contínua e a dinâmica a longo prazo das soluções de um sistema de equações que modela a vibração de vigas curvas e um modelo de placas termoelásticas. Primeiro consideramos o modelo de Bresse com dissipação não linear e forças externas. Provamos que o sistema de Timoshenko pode ser obtido como limite do sistema de Bresse quando o arco de curvatura l tende para zero e sob algumas hipóteses, mostramos a existência de um atrator global com dimensão fractal finita. Também comparamos o sistema de Bresse com o sistema de Timoshenko no sentido da semicontinuidade de seus atratores quando o parâmetro l → 0. Na segunda parte estudamos o sistema de full Von Karmam. Neste modelo adicionamos efeitos térmicos e condições de fronteira do tipo livre. Mostramos que esse problema, sem dissipação mecânica no deslocamento vertical, também possui um atrator global regular com dimensão infinita.

Le système nerveux central est capable de mémoriser des percepts sur de longues échelles de temps (mémoire à long terme), ainsi que de maintenir activement ces percepts en mémoire pour quelques secondes en vue d’effectuer des tâches comportementales (mémoire de travail). Ces deux phénomènes peuvent être étudiés conjointement dans le cadre de la théorie des réseaux de neurones à attracteurs. Dans ce cadre, un percept, représenté par un patron d’activité neuronale, est stocké en mémoire à long terme et peut être chargé en mémoire de travail à condition que le réseau soit capable de maintenir de manière stable et autonome ce patron d’activité. Une telle dynamique est rendue possible par la forme spécifique de la connectivité du réseau. Ici on examine des modèles de connectivité corticale à différentes échelles, dans le but d’étudier quels circuits corticaux peuvent soutenir efficacement des dynamiques de type réseau à attracteurs. Ceci est fait en montrant comment les performances de modèles théoriques, quantifiées par la capacité de stockage des réseaux (nombre de percepts qu’il est possible de stocker, puis réutiliser), dépendent des caractéristiques de la connectivité. Une première partie est dédiée à l’étude de réseaux complètement connectés où un neurone peut potentiellement être connecté à chacun des autres neurones du réseau. Cette situation modélise des colonnes corticales dont le rayon est de l’ordre de quelques centaines de microns. On s’intéresse d’abord à la capacité de stockage de réseaux où les synapses entre neurones sont décrites par des variables binaires, modifiées de manière stochastique lorsque des patrons d’activité sont imposés sur le réseau. On étend cette étude à des cas où les synapses peuvent être dans K états discrets, ce qui, par exemple, permet de modéliser le fait que les connections entre deux cellules pyramidales voisines du cortex sont connectées par l’intermédiaire de plusieurs contacts synaptiques. Dans un second temps, on étudie des réseaux modulaires où chaque module est un réseau complètement connecté et où la connectivité entre modules est diluée. On montre comment la capacité de stockage dépend de la connectivité entre modules et de l’organisation des patrons d’activité à stocker. La comparaison avec les mesures expérimentales sur la connectivité à grande échelle du cortex permet de montrer que ces connections peuvent implémenter un réseau à attracteur à l’échelle de plusieurs aires cérébrales. Enfin on étudie un réseau dont les unités sont connectées par des poids dont l’amplitude a un coût qui dépend de la distance entre unités. On utilise une approche à la Gardner pour calculer la distribution des poids qui optimise le stockage de patrons d’activité dans ce réseau. On interprète chaque unité de ce réseau comme une aire cérébrale et on compare la distribution des poids obtenue théoriquement avec des mesures expérimentales de connectivité entre aires cérébrales. / The central nervous system is able to memorize percepts on long time scales (long-term memory), as well as actively maintain these percepts in memory for a few seconds in order to perform behavioral tasks (working memory). These two phenomena can be studied together in the framework of the attractor neural network theory. In this framework, a percept, represented by a pattern of neural activity, is stored as a long-term memory and can be loaded in working memory if the network is able to maintain, in a stable and autonomous manner, this pattern of activity. Such a dynamics is made possible by the specific form of the connectivity of the network. Here we examine models of cortical connectivity at different scales, in order to study which cortical circuits can efficiently sustain attractor neural network dynamics. This is done by showing how the performance of theoretical models, quantified by the networks storage capacity (number of percepts it is possible to store), depends on the characteristics of the connectivity. In the first part we study fully-connected networks, where potentially each neuron connects to all the other neurons in the network. This situation models cortical columns whose radius is of the order of a few hundred microns. We first compute the storage capacity of networks whose synapses are described by binary variables that are modified in a stochastic manner when patterns of activity are imposed on the network. We generalize this study to the case in which synapses can be in K discrete states, which, for instance, allows to model the fact that two neighboring pyramidal cells in cortex touches each others at multiple contact points. In the second part, we study modular networks where each module is a fully-connected network and connections between modules are diluted. We show how the storage capacity depends on the connectivity between modules and on the organization of the patterns of activity to store. The comparison with experimental measurements of large-scale connectivity suggests that these connections can implement an attractor neural network at the scale of multiple cortical areas. Finally, we study a network in which units are connected by weights whose amplitude has a cost that depends on the distance between the units. We use a Gardner's approach to compute the distribution of weights that optimizes storage in this network. We interpret each unit of this network as a cortical area and compare the obtained theoretical weights distribution with measures of connectivity between cortical areas.

Tato diplomová práce se zabývá rekonstrukcí dynamického modelu produkce polyhydroxyalkanoátů (PHA) termofilní bakterií Schlegelella thermodepolymerans. První kapitola poskytuje čtenářům krátký úvod do systémové biologie a matematické teorie grafů. Na ni navazuje druhá kapitola zabývající se různými přístupy v dynamickém modelování, včetně běžně používaných nástrojů pro dynamickou analýzu komplexních systémů. Třetí kapitola pak sleduje další pojmy a možnosti týkající se analýzy modelu. Následující kapitola se zaměřuje na metabolomiku a často používané laboratorní techniky a pátá kapitola je pak věnována polyhydroxyalkanoátům, zejména jejich chemické struktuře a vlastnostem. V kapitole šesté je navržen obecný booleovský model pro produkci PHA termofilními bakteriemi. Kapitola sedmá se poté zaměřuje na zdokonalení modelu se zaměřením na S. thermodepolymerans. Výsledný dynamický model je podroben analýze a výsledky jsou diskutovány.

It is an important question in machine learning to understand how neural networks learn. This thesis sheds further light onto this by studying autoencoder neural networks which can memorize data by storing it as attractors.What this means is that an autoencoder can learn a training set and later produce parts or all of this training set even when using other inputs not belonging to this set. We seek out to illuminate the effect on how ReLU networks handle memorization when trained with different setups: with and without bias, for different widths and depths, and using two different types of training images -- from the CIFAR10 dataset and randomly generated. For this, we created controlled experiments in which we train autoencoders and compute the eigenvalues of their Jacobian matrices to discern the number of data points stored as attractors.We also manually verify and analyze these results for patterns and behavior. With this thesis we broaden the understanding of ReLU autoencoders: We find that the structure of the data has an impact on the number of attractors. For instance, we produced autoencoders where every training image became an attractor when we trained with random pictures but not with CIFAR10. Changes to depth and width on these two types of data also show different behaviour.Moreover, we observe that loss has less of an impact than expected on attractors of trained autoencoders.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21412号 / 理博第4432号 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 堤 誉志雄, 教授 泉 正己, 教授 上 正明 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the `generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student. The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the `exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012. In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'. In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions. In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation. Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from `affine' to `bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.