The effects of geometry and dynamics on biological pattern formation

This project examines the influence of geometry and dynamics on pattern formation in biological development . Since the work of Turing (1952) it has been known that patterns can form spontaneously given certain relatively simple conditions. The Turing mechanism involved a symmetry-breaking bifurcation from a stable spatially homogeneous state. However the development of patterns in developing organisms does not take place from such simple conditions, biological development causes pattern formation to occur within geometric structures which are complex and the environment is very noisy. This thesis examines the effects of such complexity and noise on pattern formation. The biological situations modelled in this thesis relate to the development of the mammalian cortex. The cortex is a very thin sheet, and there is evolutionary and developmental pressure to utilise cortical space to the maximum. This promotes the formation of spatial superstructures encompassing regions serving different functions. Also cortical development produces two types of pattern, one in the actual physical structure, this is common to much biological pattern formation, but also in terms of patterns of neural response which can be viewed as a feature mapping and is specific to cortical function. We examine the first type of pattern formation within the barrel field of the rat cortex, a geometric superstructure that has the properties of a Voronoi tessellation and apply a dynamical constraint from the observation that the patterns are sparse. We show that these constraints produce a distribution of patterns closer to what is observed than predictions derived from studies in a single domain of perfect circular shape. We also discover a novel effect of geometric alignment of patterns in neighbouring domains, without any physical communication between them, in a wide class of tessellations. This effect is confirmed by analysis of actual images of the subbarrel patterns in the developing rat cortex. The effect of geometry and dynamics of the second type of pattern formation is investigated in the patterns of orientation preference of neuronal response in the visual cortex of certain mammals. Where the domains are sufficiently small so that topological defects (pinwheels) cannot form the behaviour is similar to the reaction-diffusion equations. However, when there are many defects in the region alignment at the boundaries disappears

A Symmetry of the Visual World in the Architecture of the Visual Cortex.

Neuronen in der Sehrinde sprechen am besten auf orientierte visuelle Stimuli an, die sich an einer bestimmen Position im Gesichtsfeld befinden. Ihre Orts- und Orientierungspräferenzen variieren sytematisch über der Sehrrinde und sind in Karten angeordnet, der sogenannten Visuotopie- und Orientierungskarte. Experimentelle und theoretische Befunde deuten darauf hin, dass sich Orientierungskarten in einem dynamischen Prozess entwickeln, der von neuronaler Aktivität getrieben wird und von der visuellen Erfahrung abhängt. In dieser Arbeit formulieren wir ein makroskopisches Modell für die Entwicklung von Orientierungskarten in der primären Sehrinde mit dem Ziel die passende Symmetrieklasse der Dynamik zu bestimmen, welche die Muster in einem Datensatz von gemessenen Orientierungskarten des Spitzhörnchens erklären kann. Unsere statistische Analyse bringt eine neuartige Kopplung der Orientierungskarte an die zugrundeliegende Visuotopie zum Vorschein, welche sich mit der sogenannten Shift-Twist Darstellung der Euklidischen Gruppe vereinbaren läßt. Unser Ergebnis zeigt dass die Repräsentationen von Orientierung und Ortskoordinate in der Sehrinde nicht unabhängig voneinander erfolgen, und dass die Orientierungskarte an die Visuotopie gekoppelt ist. Wir entwickeln quantitative Verfahren, um den Grad dieser Kopplung zu bestimmen. Im weiteren vergleichen wir die räumliche paarweise Anordung der Kantendetektoren in der Sehrinde mit der Verteilung von Kantenpaaren in natürlichen Bildern. Wir finden, dass kollineare Anordungen von Orientierungspräferenzen für kleine Abstände unterrepräsentiert sind, im Gegensatz zur ausgeprägten Kollinearität in natürlichen Bildern. Daraus folgt, dass zur Kodierung häufig auftretender Reizanteile weniger kortikale Resourcen rekrutiert werden - eine Strategie, die der Vermeidung von Redundanz dient, und auch in anderen Bereichen des Gehirnes anzutreffen ist.Neurons in the visual cortex respond best to oriented visual stimuli at a particular position in the visual field. Their selectivities for position and orientation vary systematically along the cortical surface and are organized in maps, called the visuotopic map and orientation map, respectively. Experimental and theoretical evidence suggests that orientation map development is a dynamical process guided by neural activity and sensitive to visual experience. Here we formulate a macroscopic model for orientation map development in primary visual cortex with a particular emphasis on identifying the appropriate symmetry class of the dynamics which is able to explain the patterns in an existing dataset of tree shrew orientation maps. Our statistical analysis reveals a novel coupling of the orientation map to the underlying map of space, which is consistent with the so called shift-twist representation of the Euclidean group. Our findings show that the representation of orientation and space in the visual cortex is not independent, and that the orientation map is locked to the visuotopic map. We develop quantitative tools which allow to assess the degree of this locking. Furthermore, we compare the spatial pairwise distribution of orientation detectors along the cortical surface to the distribution of pairs of edges in natural scenes. We find that on short distances collinear arrangements of orientation preferences are underrepresented, in striking contrast to the abundance of collinearity in natural scenes. In conclusion, less cortical area is recruited to code for an abundant feature of the sensory input, in line with redundancy reduction, a strategy encountered in many different parts of the brain

Doctor of Philosophy

dissertationThe interplay of dynamics and structure is a common theme in both mathematics and biology. In this thesis, the author develops and analyzes mathematical models that give insight into the dynamics and structure of a variety of biological applications. The author presents a variety of contributions in applications of mathematics to explore biological systems across several scales. First, she analyzes pattern formation in a partial differential equation model based on two interacting proteins that are undergoing passive and active transport, respectively. This work is inspired by a longstanding problem in identifying a biophysical mechanism for the control of synaptic density in C. elegans and leads to a novel mathematical formulation of Turing-type patterns in intracellular transport. The author also demonstrates the persistence of these patterns on growing domains, and discusses extensions for a two-dimensional model. She then presents two models that explore how stochastic processes affect intracellular dynamics. First, the author and her collaborators derive effective stochastic differential equations that describe intermittent virus trafficking. Next, she shows how ion channel fluctuations lead to subthreshold oscillations in neuron models. In the final chapter, she discusses two projects for ongoing and future work: one on modeling parasite infection on dynamic social networks, and another on the bifurcation structure of localized patterns on lattices. All of these projects, presented together, chronicle the journey of the author through her mathematical development and attempts to identify, discover, create, and communicate mathematics that inspires and excites

Spatial and color hallucinations in a mathematical model of primary visual cortex

We study a simplified model of the representation of colors in the primate primary cortical visual area V1. The model is described by an initial value problem related to a Hammerstein equation. The solutions to this problem represent the variation of the activity of populations of neurons in V1 as a function of space and color. The two space variables describe the spatial extent of the cortex while the two color variables describe the hue and the saturation represented at every location in the cortex. We prove the well-posedness of the initial value problem. We focus on its stationary, i.e. independent of time, and periodic in space solutions. We show that the model equation is equivariant with respect to the direct product $\mathcal{G}$ of the group of the Euclidean transformations of the planar lattice determined by the spatial periodicity and the group of color transformations, isomorphic to $O(2)$, and study the equivariant bifurcations of its stationary solutions when some parameters in the model vary. Their variations may be caused by the consumption of drugs and the bifurcated solutions may represent visual hallucinations in space and color. Some of the bifurcated solutions can be determined by applying the Equivariant Branching Lemma (EBL) by determining the axial subgroups of $\mathcal{G}$. These define bifurcated solutions which are invariant under the action of the corresponding axial subgroup. We compute analytically these solutions and illustrate them as color images. Using advanced methods of numerical bifurcation analysis we then explore the persistence and stability of these solutions when varying some parameters in the model. We conjecture that we can rely on the EBL to predict the existence of patterns that survive in large parameter domains but not to predict their stability. On our way we discover the existence of spatially localized stable patterns through the phenomenon of “snaking”

Spatial and color hallucinations in a mathematical model of primary visual cortex

We study a simplified model of the representation of colors in the primate primary cortical visual area V1. The model is described by an initial value problem related to a Hammerstein equation. The solutions to this problem represent the variation of the activity of populations of neurons in V1 as a function of space and color. The two space variables describe the spatial extent of the cortex while the two color variables describe the hue and the saturation represented at every location in the cortex. We prove the well-posedness of the initial value problem. We focus on its stationary, i.e. independent of time, and periodic in space solutions. We show that the model equation is equivariant with respect to the direct product G of the group of the Euclidean transformations of the planar lattice determined by the spatial periodicity and the group of color transformations, isomorphic to O(2), and study the equivariant bifurcations of its stationary solutions when some parameters in the model vary. Their variations may be caused by the consumption of drugs and the bifurcated solutions may represent visual hallucinations in space and color. Some of the bifurcated solutions can be determined by applying the Equivariant Branching Lemma (EBL) by determining the axial subgroups of G . These define bifurcated solutions which are invariant under the action of the corresponding axial subgroup. We compute analytically these solutions and illustrate them as color images. Using advanced methods of numerical bifurcation analysis we then explore the persistence and stability of these solutions when varying some parameters in the model. We conjecture that we can rely on the EBL to predict the existence of patterns that survive in large parameter domains but not to predict their stability. On our way we discover the existence of spatially localized stable patterns through the phenomenon of "snaking".Comment: 30 pages, 12 figure

Classes de dynamiques neuronales et correlations structurées par l'experience dans le cortex visuel.

Neuronal activity is often considered in cognitive neuroscience by the evoked response but most the energy used by the brain is devoted to the sustaining of ongoing dynamics in cortical networks. A combination of classification algorithms (K means, Hierarchical tree, SOM) is used on intracellular recordings of the primary visual cortex of the cat to define classes of neuronal dynamics and to compare it with the activity evoked by a visual stimulus. Those dynamics can be studied with simplified models (FitzHugh Nagumo, hybrid dynamical systems, Wilson Cowan) for which an analysis is presented. Finally, with simulations of networks composed of columns of spiking neurons, we study the ongoing dynamics in a model of the primary visual cortex and their effect on the response evoked by a stimulus. After a learning period during which visual stimuli are presented, waves of depolarization propagate through the network. The study of correlations in this network shows that the ongoing dynamics reflect the functional properties acquired during the learning period.L'activité neuronale est souvent considérée en neuroscience cognitive par la réponse évoquée mais l'essentiel de l'énergie consommée par le cerveau permet d'entretenir les dynamiques spontanées des réseaux corticaux. L'utilisation combinée d'algorithmes de classification (K means, arbre hirarchique, SOM) sur des enregistrements intracellulaires du cortex visuel primaire du chat nous permet de définir des classes de dynamiques neuronales et de les comparer l'activité évoquée par un stimulus visuel. Ces dynamiques peuvent être étudiées sur des systèmes simplifiés (FitzHugh-Nagumo, systèmes dynamiques hybrides, Wilson-Cowan) dont nous présentons l'analyse. Enfin, par des simulations de réseaux composés de colonnes de neurones, un modèle du cortex visuel primaire nous permet d'étudier les dynamiques spontanées et leur effet sur la réponse à un stimulus. Après une période d'apprentissage pendant laquelle des stimuli visuels sont presentés, des vagues de dépolarisation se propagent dans le réseau. L'étude des correlations dans ce réseau montre que les dynamiques spontanées reflètent les propriétés fonctionnelles acquises au cours de l'apprentissage

Représentations de groupes de Lie et fonctionnement géométrique du cerveau

Two independent problems are considered in this thesis ; both feature non-commutative harmonic analysis−in other words, Lie group representation theory.The first problem was suggested by recent results from Neuroscience to Daniel Bennequin, who suggested it to me in turn. Our starting points are the receptive profiles of neurons in the primary visual cortex, and the geometrical properties of the maps describing how the neurons’ specializations are distributed on the surface of that part of the cortex. We recall that some remarkable properties of the "orientation maps" to be found in that cortical area are strikingly well reproduced by typical samples from Gaussian random fields on the Euclidean plane whose probability distribution is invariant under the Eucli- dean group, and whose samples probe an irreducible Plancherel factor of the quasi-regular representation of the Euclidean group on the space of functions on the Euclidean plane. We indicate that this interpretation makes it possible to build geometrical structures which call to mind, both qualitatively and qualitatively, the orientation maps of the visual cortex. Because the intervention of other groups is natural in the study of sensory information and motion planning, the construction of analogous structures on more general homogeneous spaces has an independent interest; we thus proceed to a study of invariant Gaussian random fields on riemannian homogeneous spaces, outline explicit constructions based on group representation theory (following Akiva N. Yaglom), and prove that when expressed an appropriate volume unit, the geometric measure of the zero-set of an invariant field depends only on the dimensions of the source and target spaces.The second problem was suggested by an old question from George W. Mackey and recent work by Nigel Higson ; the question was asked in 1975, and is internal to the representation theory of real reductive Lie groups. We describe a bijection between the tempered dual of a linear connected reductive Lie group and the unitary dual of its Cartan motion group. The second group is a contraction of the first, in the terminology of Inonü and Wigner ; in order to understand our bijection in terms of contractions of Lie groups, we consider an arbitrary irreducible tempered representation of the first group and introduce a family of contraction operators which make it possible to follow the individual smooth vectors (in an appropriate geometric realization for the representation) during the contraction ; we observe their convergence to the members of a carrier space for the representation of the Cartan motion group indicated by our bijection. We then use our results to give a new proof of the Connes-Kasparov conjecture in the case of linear connected reductive Lie groups, extending relatively recent work by Nigel Higson.This manuscript also contains a report on an attempt to use some matrix elements of unitary representations of the Galilei group in the study of electrophysiological recordings of the activity of Purkinje cells in the vestibulocerebellum of live (and alert) rats.Cette thèse étudie deux problèmes indépendants où l’analyse harmonique non-commutative, théorie des représentations de groupes de Lie, joue un rôle.Le premier problème a été suggéré par les neurosciences à Daniel Bennequin, qui me l’a proposé. Nous partons des profils récepteurs des neurones du cortex visuel primaire, et de la géométrie de la répartition des spécialités des neurones à la surface de cette aire corticale. Nous rappelons comment de remarquables propriétés des cartes d’orientation qu’on trouve dans cette aire corticale peuvent être reproduites par les tirages de champs aléatoires gaussiens invariants en loi dont les tirages explorent un facteur de Plancherel de la représentation quasi-régulière du groupe euclidien sur l’espace des fonctions sur le plan euclidien. Nous signalons que cette interprétation permet de construire, sur la sphère et sur le plan hyperbolique, des structures géométriques qui rappellent (qualitativement et quantitativement) les cartes d’orientation du cortex visuel. L’intervention naturelle d’autres groupes dans le traitement des informations sensorielles et la préparation du mouvement nous invite à envisager la construction d’objets analogues pour d’autres espaces homo- gènes comme une question mathématique d’intérêt indépendant. Nous étudions alors les champs aléatoires gaussiens invariants sur les espaces homogènes riemanniens, en donnant des constructions explicites issues de la théorie des représentations (d’après Akiva N. Yaglom) et en démontrant que dans une unité de volume adaptée, la mesure géométrique moyenne de l’ensemble des zéros ne dépend que des dimensions de la source et du but.Le second problème a été suggéré par George W. Mackey en 1975 ; il est interne à la théorie des représentations de groupes de Lie réductifs réels. Nous décrivons une bijection entre le dual tempéré d’un groupe de Lie linéaire connexe réductif et le dual unitaire de son groupe de déplacements de Cartan. Le second groupe est une contraction du premier, au sens d’Inonü et Wigner ; afin de comprendre la bijection précédente à l’aide de la notion de contraction, nous utilisons, pour toute représentation tempérée irréductible du premier groupe, une famille d’opérateurs de contraction qui permet de suivre le comportement des vecteurs lisses (dans une réalisation géométrique adaptée) au cours de la contraction et d’observer leur convergence vers la représentation du second groupe indiquée par notre bijection. Nous utilisons ensuite nos résultats pour donner une nouvelle preuve de la conjecture de Connes-Kasparov pour les groupes de Lie linéaires connexes réductifs, en suivant la méthode utilisée par Nigel Higson en 2008.Le manuscrit contient par ailleurs le récit d’une étude sur des données expérimentales issues d’enregistrements électrophysiologiques de l’activité de cellules de Purkinje dans le cervelet vestibulaire de rats vigiles, à l’aide d’éléments matriciels de représentations unitaires du groupe de Galilée

Mathematics & Statistics 2017 APR Self-Study & Documents

UNM Mathematics & Statistics APR self-study report, review team report, response report, and initial action plan for Spring 2017, fulfilling requirements of the Higher Learning Commission