Workshop on Mathematical Methods in Data Analysis and Differential Equations

TITLES AND ABSTRACTS

Speaker: David Arroyo (Instituto de Tecnologías Físicas y de la Información, CSIC, Madrid),
Saturday 20, 10:00-10:45

Title: Trust in the trustworthy: on the design of a Misinformation Widget.

Abstract: Data production and exploitation are the very core of our economic and social reality.

Our daily activity is more determined by the way we access cloud services through our
smartphones than through our computer. The value chain associated to all these data
interfaces and computation modalities is very dependent on the quality of information sources and the trustworthiness of the mechanisms for data coding, decision making and information security.
In this seminar we will explore the design implications for the construction of a misinformation widget guiding users in assessing the trustworthiness of various sources of information. A critical aspect in the design of the widget is the identification of the best news classification tools and methodologies. To achieve this objective, on option is to rely on fact checking platforms and human experts to obtain feedback, which can be extended by leveraging the so- called wisdom of crowds and perform news curation as result of a collaborative effort among users and experts. Expert-based systems are accurate but costly and not scalable, while crowds-based systems can be biased by herding behavior. To overcome these limitations, we can ponder the developing of automatic detection techniques by means of Natural Language Processing (NLP) and more advanced Machine Learning (ML) techniques. Nonetheless, the selection of adequate models and datasets for their tuning and training is itself a challenge. Thus, we explore the option of adopting a so-called “human on the loop” approach, which integrates expert knowledge on fact checking and automatic detection of fake news and misinformation. Specifically, we propose a methodology that leverages fact-checking platforms to perform datasets labeling and the validation of the performance of NLP and ML tools for the automatic classification of information.

Speaker: Emili Balaguer-Ballester (Bournemouth University, UK), Friday 19, 18:05-18:50

Title: Stability and predictability code in neural correlations: A machine learning perspective.

Abstract: Metastable states mapping concrete perceptual or cognitive entities are well-defined in realistic models of neuronal ensembles. Recently, machine learning approaches were devised to identify them also empirically, based on weak assumptions on the mechanistic substrate of such cognitively specific attracting states. However, this theory is not always empirically conclusive. In this talk, I will first discuss scenarios and models in which this classic description is useful. Then I will focus on a recent case study in which attracting states of rat orbitofrontal cortex ensembles were not associated with specific observable variables such as rewards or choices, as often considered. Instead, metastable states mapped the foreseeability of the decision outcomes. Shared variability across trials suggested the existence of cohesive representations of this predictability, supported by higher-order neuronal correlations.

Speaker: Tomás Caraballo (Universidad de Sevilla), Saturday 20, 12:45-13:30

Title: Wong-Zakai approximations of stochastic nonlocal partial differential equations

Abstract: This talk is devoted to investigating the well-posedness and asymptotic behavior of a class of stochastic nonlocal partial differential equations driven by nonlinear noise. First, the existence of a weak martingale solution is established by using the Faedo-Galerkin approximation and an idea analogous to Da Prato and Zabczyk. Second, we show the uniqueness and continuous dependence on initial values of solutions to the above stochastic nonlocal problem when there exist some variational solutions. Third, the asymptotic local stability of steady-state solutions is analyzed either when the steady-state solutions of the deterministic problem is also solution of the stochastic one, or when this does not happen. Next, to study the global asymptotic behavior, namely, the existence of attracting sets of solutions, we consider an approximation of the noise given by Wong-Zakai’s technique using the so called colored noise. For this model, we can use the power of the theory of random dynamical systems and prove the existence of random attractors. Eventually, particularizing in the cases of additive and multiplicative noise, it is proved that the Wong-Zakai approximation models possess random attractors which converge upper-semicontinuously to the respective random attractors of the stochastic equations driven by standard Brownian motions. This fact justifies the use of this colored noise technique to approximate the asymptotic behavior of the models with general nonlinear noises, although the convergence of attractors and solutions is still an open problem.

Speaker: Alexandre Carvalho (Universidade de Sao Paulo, Sao Carlos, Brasil), Saturday 20,
11:55-12:40

Title: Inertial Manifolds, Saddle Point Property and Exponential Dichotomy

Abstract: Inertial manifold theory, saddle point property and exponential dichotomy have been treated as different topics in the literature with different proofs. As a common feature, they all have the purpose of splitting the space to understand the dynamics. We present a unified proof for the inertial manifold theorem, which as a local consequence yields the saddle-point property with a fine structure of invariant manifolds and the roughness of exponential dichotomy.

Speaker: Aurelio Fernández Bariviera (Universitat Rovira i Virgili, Tarragona), Friday 19, 18:55-19:40

Title: Analysis of cryptocurrencies using alternative data

Abstract: Cryptocurrencies constitute a relatively new financial asset, characterized by high
volatility and 24/7 online-only trading. Contrary to stocks or bonds, cryptocurrencies cannot be related to any “fundamental” variable. However, cryptocurrency markets activity generates large volumes of data at high frequency. Traditionally, the studies in financial markets were carried out with data generated within the markets (price, traded volume, etc.). In the recent years, “alternative data” emerged as potential explanatory variables for market behavior. Such label comprises data that stems from sources different from the market itself: Twitter activity, Wikipedia queries, Google Trends, etc. Considering that cryptocurrencies are digitally native assets, we believe that online activity provided by the alternative data could enhance our understanding of the market.

Speaker: Toni Guillamón (Universitat Politècnica de Catalunya, Barcelona), Friday 19, 17:00- 17:45

Title: Empirical modelling of neuronal dynamics using artificial neural networks.

Abstract: Mathematical modelling of neuronal dynamics has experienced a fast growth in the last half century thanks, mainly, to the biophysical formalism introduced by Hodgkin and
Huxley in the 1950s. Other types of models (for instance, integrate and fire models or rate
models), although less realistic, have also contributed to understanding both single-cell and
population dynamics. However, there is still a vast amount of data that has not been
associated with a specific model, mainly because data is acquired more rapidly than it can be analyzed or because it is difficult to analyze. We have explored the identification of neuronal (single-cell) voltage traces with artificial neural networks (ANN). We present an optimized computational scheme that trains the ANN with biologically plausible input currents and is able to replicate the neuronal dynamics with high accuracy. The work has been done in collaboration with David Aquilué, Pau Fisco, Enric Fossas and Néstor Roqueiro.

Speaker: Adrián Hernández (Ingeniería de Sistemas para la Defensa de España, ISDEFE,
Madrid), Saturday 20, 10:50-11:35

Title: Differentiable programming to extend and characterize deep learning

Abstract: In recent years, the use of deep learning models has represented a great advance in
image recognition, language processing, time series prediction, game playing, etc. These
models are specific examples of differentiable programs. In this workshop we present
differentiable programming as a framework to extend and characterize deep learning. We
study the application of deep learning to unstructured data with Graph Neural Networks
(GNNs) and finally, we will see the current limitations of end-to-end differentiable models to
generalizing and reasoning beyond the data.

Speaker: Mavi Sanchez-Vives (Institut d’Investigacions Biomèdiques August Pi i Sunyer,
Barcelona), Friday 19, 16:10-16:55.

Title: Quantifying brain states and consciousness levels: analysis and numerical models of brain
activity

Abstract: The cerebral cortex as a structured network is able to spontaneously elicit different types of activity that changes over time according to the brain state. Brain states (slow wave sleep, REM sleep, wakefulness, anesthesia, coma, minimally conscious states, etc) underlie different informacion processing capabilities of the network and different consciousness states. The characterization and identification of different brain states is critical not only to understand the system and its dynamics, but also to identify consciousness levels, which is relevant in the clinic. For example, in an unresponsive wakefulness syndrome (vegetative state) the patient appears awake but unresponsive, and it is important to determine whether there is consciousness or not. A collection of analytical tools that can capture the state of the network and thus, determine the consciousness levels, will be presented. A critical one, the complexity of the network, shows a good correlate with brain states. In particular, we will discuss the perturbational complexity index, that results from the Lempel Ziv compression of the spatio-temporal matrix of cortical activity. Other measures like functional complexity, the entropy of wave  propagation, the energy- based hierarchical clustering of cortical slow waves or the spectral exponent to quantify brain states will also be presented. Finally, computational models offer insights into the dynamical properties of the brain under different states. Spiking and mean field models allow us the explore the parameter space across states that we cannot reach experimentally. We also use them to simulate local and global dynamics, to better understand network properties and transition periods in physiological and pathological conditions.