On the outer synchronization of complex dynamical networks

Complex network models have become a major tool in the modeling and analysis of many physical, biological and social phenomena. A complex network exhibits behaviors which emerge as a consequence of interactions between its constituent elements, that is, remarkably, not the same as individual components. One particular topic that has attracted the researchers' attention is the analysis of how synchronization occurs in this class of models, with the expectation of gaining new insights of the interactions taking place in real-world complex systems. Most of the work in the literature so far has been focused on the synchronization of a collection of interconnected nodes (forming one single network), where each node is a dynamical system governed by a set of nonlinear differential equations, possibly displaying chaotic dynamics. In this thesis, we study an extended version of this problem. In particular, we consider a setup consisting of two complex networks which are coupled unidirectionally, in such a way that a set of signals from the master network are injected into the response network, and then investigate how synchronization is attained. Our analysis is fairly general. We impose few conditions on the network structure and do not assume that the nodes in a single network are synchronized. This work can be divided into two main parts; outer synchronization in fractional-order networks, and outer synchronization in ordinary networks. In both cases the system parameters are perturbed by bounded, time varying and unknown perturbations. The synchronizer feedback matrix is possibly perturbed with the same type of perturbations as well. In both cases, of fractional-order and ordinary networks, we build up several theorems that ensure the attainment of synchronization in various scenarios, including, e.g., cases in which the coupling matrix of the networks is non-diffusive (hence we can avoid this assumption, which is almost invariably made in the literature). In all the cases of interest, we show that the scheme for coupling the networks is very simple, as it reduces to the computation of a single gain matrix whose dimension is independent of the number of network nodes. The structure of the designed synchronizer is also very simple, making it convenient for real-world applications. Although all of the proposed schemes are assessed analytically, numerical results (obtained by computer simulations) are also provided to illustrate the proposed methods. ---------------------------------------Las redes complejas se han convertido en una herramienta fundamental en el análisis de muchos sistemas físicos, biológicos y sociales. Una red compleja presenta comportamientos que "emergen" como consecuencia de las interacciones entre sus elementos constituyentes pero que no se observan de forma individual en estos elementos. Un aspecto en concreto que ha atrapado la atención de muchos investigadores es el análisis de cómo se producen fenómenos de sincronización en esta clase de modelos, con la esperanza de alcanzar una mayor comprensión de las interacciones que tienen lugar en sistemas complejos del mundo real. La mayor parte del trabajo publicado hasta ahora ha estado centrado en la sincronización de una colección de nodos interconectados (que forman una única red con entidad propia), donde cada nodo es un sistema dinámico gobernado por un conjunto de ecuaciones diferenciales no lineales, posiblemente caóticas. En esta tesis estudiamos una versión extendida de este problema. En concreto, consideramos un sistema formado por dos redes complejas acopladas unidireccionalmente, de manera que un conjunto de señales de la red principal se inyectan en la red secundaria, e investigamos cómo se alcanza un estado de sincronización. Este fenómeno se conoce como "sincronización externa". Nuestro análisis es muy general. Se imponen pocas condiciones a las estructura de las redes y no es necesario suponer que los nodos de cada red estén sincronizados entre sí previamente. Esta memoria se puede dividir en dos bloques: la sincronización externa de redes descritas por ecuaciones diferenciales de orden fraccionario y la sincronización externa de redes ordinarias (descritas por ecuaciones diferenciales de orden entero). En ambos casos, se admite que los parámetros del sistema puedan estar sujetos a perturbaciones desconocidas, posiblemente variables con el tiempo, pero acotadas. La matriz de realimentación del esquema de sincronización puede sufrir el mismo tipo de perturbación. En ambos casos, con ecuaciones de orden fraccionario o entero, construimos varios teoremas que aseguran que se alcance la sincronización en escenarios diversos, incluyendo, por ejemplo, casos en los que la matriz de acoplamiento de las redes es no difusiva (por lo tanto, podemos evitar esta hipótesis, que es ubicua en la literatura). En todos los casos de interés, mostramos que el esquema necesario para interconectar las redes es muy simple, puesto que se reduce al cálculo de una única matriz de ganancia cuya dimensión es independiente de la dimensión total (número de nodos) de las redes. La estructura del sincronizados es también muy sencilla, lo que la hace potencialmente adecuada para aplicaciones del mundo real. Aunque todos los esquemas que se proponen se analizan de manera rigurosa, también se muestran resultados numéricos (obtenidos mediante simulación) para ilustrar los métodos propuestos.Programa de Doctorado en Multimedia y Comunicaciones por la Universidad Carlos III de Madrid y la Universidad Rey Juan CarlosPresidente: Ángel María Bravo Santos.- Secretario: David Luengo García.- Vocal: Irene Sendiña Nada

Predicting one-year left ventricular mass index regression following transcatheter aortic valve replacement in patients with severe aortic stenosis: A new era is coming

Aortic stenosis (AS) is the most common valvular heart disease in the western world, particularly worrisome with an ever-aging population wherein postoperative outcome for aortic valve replacement is strongly related to the timing of surgery in the natural course of disease. Yet, guidelines for therapy planning overlook insightful, quantified measures from medical imaging to educate clinical decisions. Herein, we leverage statistical shape analysis (SSA) techniques combined with customized machine learning methods to extract latent information from segmented left ventricle (LV) shapes. This enabled us to predict left ventricular mass index (LVMI) regression a year after transcatheter aortic valve replacement (TAVR). LVMI regression is an expected phenomena in patients undergone aortic valve replacement reported to be tightly correlated with survival one and five year after the intervention. In brief, LV geometries were extracted from medical images of a cohort of AS patients using deep learning tools, and then analyzed to create a set of statistical shape models (SSMs). Then, the supervised shape features were extracted to feed a support vector regression (SVR) model to predict the LVMI regression. The average accuracy of the predictions was validated against clinical measurements calculating root mean square error and R2 score which yielded the satisfactory values of 0.28 and 0.67, respectively, on test data. Our work reveals the promising capability of advanced mathematical and bioinformatics approaches such as SSA and machine learning to improve medical output prediction and treatment planning

Robust outer synchronization between two complex networks with fractional order dynamics

Synchronization between two coupled complex networks with fractional-order dynamics, hereafter referred to as outer synchronization, is investigated in this work. In particular, we consider two systems consisting of interconnected nodes. The state variables of each node evolve with time according to a set of (possibly nonlinear and chaotic) fractional-order differential equations. One of the networks plays the role of a master system and drives the second network by way of an open-plus-closed-loop (OPCL) scheme. Starting from a simple analysis of the synchronization error and a basic lemma on the eigenvalues of matrices resulting from Kronecker products, we establish various sets of conditions for outer synchronization, i.e., for ensuring that the errors between the state variables of the master and response systems can asymptotically vanish with time. Then, we address the problem of robust outer synchronization, i.e., how to guarantee that the states of the nodes converge to common values when the parameters of the master and response networks are not identical, but present some perturbations. Assuming that these perturbations are bounded, we also find conditions for outer synchronization, this time given in terms of sets of linear matrix inequalities (LMIs). Most of the analytical results in this paper are valid both for fractional-order and integer-order dynamics. The assumptions on the inner (coupling) structure of the networks are mild, involving, at most, symmetry and diffusivity. The analytical results are complemented with numerical examples. In particular, we show examples of generalized and robust outer synchronization for networks whose nodes are governed by fractional-order Lorenz dynamics. After the seminal work in Ref. 1, complex network models have become ubiquitous in the analysis of many phenomena appearing in the physical, biological, and social sciences. In particular, the analysis of synchronization in this class of models has attracted the attention of many researchers, with the expectation of gaining new insights of the interactions taking place in real-world complex systems. Most of the work in the literature so far has been focused on the synchronization of a collection of interconnected nodes (forming one single neteork), where each node is a dynamical system governed by a set of nonlinear differential equations (possibly displaying chaotic dynamics). In this paper, we study an extended version of this problem. In particular, we consider a setup consisting of two complex networks which are coupled unidirectionally (such that a set of signals from the master network are injected into the response network) with the important peculiarity that the dynamics of the nodes, nonlinear and possibly chaotic, are governed by sets of fractional-order differential equations. Then, we study how the response network attains synchronization with the drive network. Our analysis is fairly general. We impose few conditions on the network structure, do not assume that the nodes in a single network are synchronized and provide an analytical characterization of the problem in which the master and response networks are non identical (due to the perturbation of their fixed parameters). Our analysis is based on simple definitions of the synchronization error and the stability of fractional-order systems of differential equations, but it is also valid for "ordinary" networks whose dynamics is described by integer-order differential equations. Although the main aim of the work is to provide analytical insights, some numerical illustrations (obtained by computer simulations) are also presented