Seminars 2019

En este curso consideraremos ecuaciones diferenciales de primer orden lineales $y'(t)=H(t)y(t)$, $y(t_0)=y_0$, donde $y(t)$ es un vector (o función) y $H(t)$ una matriz (u operador en un espacio de Hilbert). Esta ecuación puede transformarse en otra equivalente: $$U'(t,t_0)=H(t)U(t,t_0)\,,$$ donde $U(t,t_0)$ es una matriz (u operador) tal que $y(t)=U(t,t_0)y(t_0)$. El método de factorización de Wei-Norman consiste en que, si $H(t)$ se puede expresar como combinación lineal de los elementos de un álgebra de Lie (en una cierta representación), entonces $U(t,t_0)$ se puede poner como producto de exponenciales de los elementos del álgebra multiplicados por funciones de $t$ que verifican un conjunto de ecuaciones diferenciales no lineales de primer orden (de tipo Ricatti). Estas ecuaciones no dependen de la representación del álgebra de Lie concreta, mientras sea fiel), tan solo de las constantes de estructura.

En la primera sesión derivaremos las expresiones generales del método. En la segunda sesión, particularizaremos a algunos ejemplos concretos de álgebras de Lie de dimensión 3.

Driven diffusive systems may undergo phase transitions to sustain atypical values of the current. This leads in some cases to symmetry-broken space-time trajectories which enhance the probability of such fluctuations. Here, we shed light on both the macroscopic large deviation properties and the microscopic origin of such spontaneous symmetry breaking in the open weakly asymmetric exclusion process. By studying the joint fluctuations of the current and a collective order parameter, we uncover the full dynamical phase diagram for arbitrary boundary driving, which is reminiscent of a $Z_2$ symmetry-breaking transition. The associated joint large deviation function becomes nonconvex below the critical point, where a Maxwell-like violation of the additivity principle is observed. At the microscopic level, the dynamical phase transition is linked to an emerging degeneracy of the ground state of the microscopic generator, from which the optimal trajectories in the symmetry-broken phase follow. In addition, we observe this symmetry-breaking phenomenon in extensive rare-event simulations, confirming our macroscopic and microscopic results.

Cortical areas in the brain are often classified as sensory areas --those receiving input from the external world --or cognitive areas --related with association, memory and more abstract mental processing. It is thought that interactions between these areas in the feedforward (sensory to cognitive) or feedback (cognitive to sensory) direction hold the key to understand attention, expectations and other brain functions. However, the underlying circuit mechanism remains poorly understood and represents a major challenge in neuroscience. We approached this problem using a large-scale computational model of the macaque cortex constrained by novel brain connectivity data. In our model, the interplay between feedforward and feedback signals depends on the fine laminar structure of the cortex, and involves complex dynamics across multiple scales. The model was tested by reproducing a wide range of experimental findings about frequency-dependent interactions between visual cortical areas. Furthermore, the model replicates the existence of a functional hierarchy as reported in recent monkey and human experiments, and suggests a mechanism for the observed dynamics of such hierarchy. Together, this work highlights the necessity of multiscale approaches and provides a modeling platform for studies of large-scale brain circuit dynamics and functions.