The Denjoy-Wolff theorem, extensions and applications

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Núria Fagella Rabionet[en] The aim of this project is to prove the Denjoy-Wolff Theorem, which deals with iteration of holomorphic self-maps of the unit disk D. It claims that either the map is conjugate to a rotation about the origin or all the points converge to a unique point in D under iteration. We will also prove that there always exists a fundamental set, an invariant subset reached by all the compact sets in a finite number of iterations and where the map is one-to-one. Fundamental sets can be classified in four different types, up to conformal conjugation. Finally, we will use this results to classify the periodic Fatou components of entire maps. For each of them, we can find a fundamental set. In the case of attracting or parabolic components or Siegel disks, the dynamics in the fundamental set is determined up to conformal conjugation. However, in the case of Baker domains three different types can occur and we will present some examples of them

Dynamics on the boundary of Fatou components

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2021, Director: Núria Fagella Rabionet[en] The aim of this project is to compile the known results about the dynamics on the boundary of invariant simply-connected Fatou components, as well as the questions which are still open concerning the topic. We focus on ergodicity and recurrence. One of the main tools to deal with this kind of questions is to study the boundary behaviour of the associate inner functions. Therefore, the project is divided in two parts. Firstly, ergodicity and recurrence are studied for inner functions. Secondly, these results are applied to study the dynamics on the boundary of invariant simply-connected Fatou components. Moreover, we study the concrete example $f(z)=z+e^{-z}$, which presents infinitely many invariant doubly-parabolic Baker domains $U_{k}$. Making use of the associate inner function, which can be computed explicitly, we give a complete characterization of the periodic points in $\partial U_{k}$ and prove the existence of uncountably many curves of non-accessible escaping points

A model for boundary dynamics of Baker domains

We consider the transcendental entire function $f(z)=z+e^{-z}$, which has a doubly parabolic Baker domain $U$ of degree two, i.e. an invariant stable component for which all iterates converge locally uniformly to infity, and for which the hyperbolic distance between successive iterates converges to zero. It is known from general results that the dynamics on the boundary is ergodic and recurrent and that the set of points in $\partial U$ whose orbit escapes to infity has zero harmonic measure. For this model we show that stronger results hold, namely that this escaping set is non-empty, it is organized in curves encoded by some symbolic dynamics, whose closure is precisely $\partial U$. We also prove that nevertheless, all escaping points in $\partial U$ are non-accessible from $U$, as opposed to points in $\partial U$ having a bounded orbit, which are all accessible. Moreover, repelling periodic points are shown to be dense in @U, answering a question posted Baranski, Fagella, Jarque and Karpinska. None of these features are known to occur for a general doubly parabolic Baker domain

Structural and functional magnetic resonance imaging in isolated REM sleep behavior disorder: A systematic review of studies using neuroimaging software.

Isolated rapid eye movement sleep behavior disorder (iRBD) is a harbinger for developing clinical synucleinopathies. Magnetic resonance imaging (MRI) has been suggested as a tool for understanding the brain bases of iRBD and its evolution. This review systematically analyzed original full text articles on structural and functional MRI in patients with video-polysomnography-confirmed iRBD according to systematic procedures suggested by Reviews and Meta-analyses (PRISMA). The literature search was conducted via the PubMed database for articles related to structural and functional MRI in iRBD from 2000 to 2020. Investigations to date have been diverse in terms of methodology, but most agree that patients with iRBD have structural changes in deep gray matter nuclei, cortical gray matter atrophy, and disrupted functional connectivity within the basal ganglia, the cortico-striatal and cortico-cortical networks. Furthermore, there is evidence that MRI detects structural and functional brain changes associated with the motor and non-motor symptoms of iRBD. The current review highlights the need for larger multicenter and longitudinal studies, using complex approaches based on data-driven and unsupervised machine learning that will help to identify structural and functional patterns of brain degeneration. In turn, this may even allow for the prediction of subsequent phenoconversion from iRBD to the clinically defined synucleinopathie

Inhibitory framing in hypersexual patients with Parkinson's disease. An fMRI pilot study

Hypersexuality in medicated patients with PD is caused by an increased influence of motivational drive areas and a decreased influence of inhibitory control areas due to dopaminergic medication. In this pilot study, we test a newly developed paradigm investigating the influence of dopaminergic medication on brain activation elicited by sexual pictures with and without inhibitory contextual framing. Twenty PD patients with and without hypersexuality were examined with fMRI either OFF or ON standardized dopaminergic medication. The paradigm consisted of a priming phase where either a neutral context or an inhibitory context was presented. This priming phase was either followed by a sexual or a neutral target. Sexual, compared to neutral pictures resulted in a BOLD activation of various brain regions implicated in sexual processing. Hypersexual PD patients showed increased activity compared to PD controls in these regions. There was no relevant effect of medication between the two groups. The inhibitory context elicited less activation in inhibition-related areas in hypersexual PD, but had no influence on the perception of sexual cues. The paradigm partially worked: reactivity of motivational brain areas to sexual cues was increased in hypersexual PD and inhibitory contextual framing lead to decreased activation of inhibitory control areas in PD. We could not find a medication effect and the length of the inhibitory stimulus was not optimal to suppress reactivity to sexual cues. Our data provide new insights into the mechanisms of hypersexuality and warrant a replication with a greater cohort and an optimized stimulus length in the future

Statistical inference in brain graphs using threshold-free network-based statistics

The description of brain networks as graphs where nodes represent different brain regions and edges represent a measure of connectivity between a pair of nodes is an increasingly used approach in neuroimaging research. The development of powerful methods for edge-wise grouplevel statistical inference in brain graphs while controlling for multiple-testing associated falsepositive rates, however, remains a difficult task. In this study, we use simulated data to assess the properties of threshold-free network-based statistics (TFNBS). The TFNBS combines thresholdfree cluster enhancement, a method commonly used in voxel-wise statistical inference, and network-based statistic (NBS), which is frequently used for statistical analysis of brain graphs. Unlike the NBS, TFNBS generates edge-wise significance values and does not require the a priori definition of a hard cluster-defining threshold. Other test parameters, nonetheless, need to be set. We show that it is possible to find parameters that make TFNBS sensitive to strong and topologically clustered effects, while appropriately controlling false-positive rates. Our results show that the TFNBS is an adequate technique for the statistical assessment of brain graphs

Sex differences in brain and cognition in de novo Parkinson's disease

Background and objective: Brain atrophy and cognitive impairment in neurodegenerative diseases are influenced by sex. We aimed to investigate sex differences in brain atrophy and cognition in de novo Parkinson's disease (PD) patients. Methods: Clinical, neuropsychological and T1-weighted MRI data from 205 PD patients (127 males:78 females) and 69 healthy controls (40 males:29 females) were obtained from the PPMI dataset. Results: PD males had a greater motor and rapid eye movement sleep behavior disorder symptomatology than PD females. They also showed cortical thinning in postcentral and precentral regions, greater global cortical and subcortical atrophy and smaller volumes in thalamus, caudate, putamen, pallidum, hippocampus, and brainstem, compared with PD females. Healthy controls only showed reduced hippocampal volume in males compared to females. PD males performed worse than PD females in global cognition, immediate verbal recall, and mental processing speed. In both groups males performed worse than females in semantic verbal fluency and delayed verbal recall; as well as females performed worse than males in visuospatial function. Conclusions: Sex effect in brain and cognition is already evident in de novo PD not explained by age per se, being a relevant factor to consider in clinical and translational research in PD

Cortical gray matter progression in idiopathic REM sleep behavior disorder and its relation to cognitive decline

Background: Idiopathic Rapid eye movement sleep behavior disorder (IRBD) is recognized as the prodromal stage of the alpha-Synucleinopathies. Although some studies have addressed the characterization of brain structure in IRBD, little is known about its progression. Objective: The present work aims at further characterizing gray matter progression throughout IRBD relative to normal aging and investigating how these changes are associated with cognitive decline. Methods: Fourteen patients with polysomnography-confirmed IRBD and 18 age-matched healthy controls (HC) underwent neuropsychological, olfactory, motor, and T1-weighted MRI evaluation at baseline and follow-up. We compared the evolution of cortical thickness (CTh), subcortical volumes, smell, motor and cognitive performance in IRBD and HC after a mean of 1.6 years. FreeSurfer was used for CTh and volumetry preprocessing and analyses. The symmetrized percent of change (SPC) of the CTh was correlated with the SPC of motor and neuropsychological performance. Results: IRBD and HC differed significantly in the cortical thinning progression in regions encompassing bilateral superior parietal and precuneus, the right cuneus, the left occipital pole and lateral orbitofrontal gyri (FWE corrected, p < 0.05). The Visual form discrimination test showed worse progression in the IRBD relative to HC, that was associated with gray matter loss in the right superior parietal and the left precuneus. Increasing motor signs in IRBD were related to cortical thinning mainly involving frontal regions, and late-onset IRBD was associated with cortical thinning involving posterior areas (FWE corrected, p < 0.05). Despite finding olfactory identification deficits in IRBD, results did not show decline over the disease course. Conclusion: Progression in IRBD patients is characterized by parieto-occipital and orbitofrontal thinning and visuospatial loss. The cognitive decline in IRBD is associated with degeneration in parietal regions

Cortical atrophy patterns in early Parkinson's disease patients using hierarchical cluster analysis.

INTRODUCTION: Cortical brain atrophy detectable with MRI in non-demented advanced Parkinson's disease (PD) is well characterized, but its presence in early disease stages is still under debate. We aimed to investigate cortical atrophy patterns in a large sample of early untreated PD patients using a hypothesis-free data-driven approach. METHODS: Seventy-seven de novo PD patients and 50 controls from the Parkinson's Progression Marker Initiative database with T1-weighted images in a 3-tesla Siemens scanner were included in this study. Mean cortical thickness was extracted from 360 cortical areas defined by the Human Connectome Project Multi-Modal Parcellation version 1.0, and a hierarchical cluster analysis was performed using Ward's linkage method. A general linear model with cortical thickness data was then used to compare clustering groups using FreeSurfer software. RESULTS: We identified two patterns of cortical atrophy. Compared with controls, patients grouped in pattern 1 (n = 33) were characterized by cortical thinning in bilateral orbitofrontal, anterior cingulate, and lateral and medial anterior temporal gyri. Patients in pattern 2 (n = 44) showed cortical thinning in bilateral occipital gyrus, cuneus, superior parietal gyrus, and left postcentral gyrus, and they showed neuropsychological impairment in memory and other cognitive domains. CONCLUSIONS: Even in the early stages of PD, there is evidence of cortical brain atrophy. Neuroimaging clustering analysis is able to detect two subgroups of cortical thinning, one with mainly anterior atrophy, and the other with posterior predominance and worse cognitive performance

Progression of Parkinson's disease patients' subtypes based on cortical thinning: 4-year follow-up

Background. Three cortical atrophy patterns were previously identified in non-demented Parkinson's disease patients using a data-driven approach based on cortical thickness data: i) parieto-temporal pattern of atrophy with worse cognitive performance (pattern 1), ii) occipital and frontal cortical atrophy with younger disease onset (pattern 2), and iii) non-detectable cortical atrophy (pattern 3). We aimed to investigate the evolution of these three patterns over time. Methods. Magnetic resonance imaging and neuropsychological assessment were obtained at baseline and follow-up (3.8±0.4 year apart) in a group of 45 Parkinson's disease patients and 22 healthy controls. FreeSurfer was used for cortical thickness analysis and global atrophy measures. Results. Temporo-parietal cortical thinning occurred in pattern 2, 3 and controls groups, and patients showed decline in processing speed (as measured by the Stroop Word-Color test, the Symbol Digits Modalities test and the Trail Making Test Part B) and in semantic fluency (animals). Pattern 3 patients showed more progressive cortical thinning in the left prefrontal cortex than controls and more right occipital thinning than pattern 2 patients over time. Pattern 1 patients had greater compromise in activities of the daily living and suffered higher attrition rate. Conclusion. The Parkinson's disease phenotypes identified using cluster analysis of cortical thickness data showed different progression over time. The presence of prefrontal thinning and younger disease onset at baseline was associated to less cortical degeneration, while non-atrophic patients progressed showing a temporo-parietal cortical thinning