Abstracts

Multiscale and nonlinear properties of cardiac signals: a wavelet-based study of their variability

Françoise Argoul

Physiological systems are regulated spatially and temporally by integrated and intermingled networks of neurons. This multiscale and multifrequency regulation leads to seemingly random signals which are now classified as Network Physiology. The cardiac system in one of the most fascinating physiological system, because it is modulated by both local and distant internal factors (central and autonomic nervous systems, heart tissues and vessel mechanics) and external environmental factors (exercise and rest, wake and sleep, circadian rhythms). Despite many experimental and theoretical efforts grounded on nonlinear system and fractals concepts in the past decades, the exact role nerve tissues rhythm alterations on cardiac rhythm variability remains an open question.

The heart rate variability (HRV) was defined by clinicians by beat-to-beat interval signals. HRV exhibits temporal self-similar properties across a broad range of scales, long-range correlations and nonlinear properties, indicating fractal or multifractal structures. These fractal and nonlinear scaling features change not only with the electrical and mechanical properties of the heart excitable and muscular tissues but also with modifications of the sympathetic-parasympathetic stimuli. The complexity of the heart rate fluctuations cannot be adequately described by a single scaling parameter, a large set of scaling parameters is needed. Even more puzzling is the fact that these signals are non-stationary, which make their study a complicated task. To decipher the phase interaction of different frequency components of a cardiac signal, complex wavelet transforms offer the possibility to follow continuously the temporal fluctuations of the instantaneous frequency and its phase. Here, we compare discrete beat-to-beat interval signals with these continuous frequency and phase signals, their scale invariances and their complexity by the computation of their D(h) spectra, with the wavelet transform modulus method (WTMM).

Wavelet-based techniques have been proposed for the identification, classification and analysis of ar- rhythmic electrocardiogram signals, to distinguish normal sinus rhythm from atrial flutter, paroxysmal atrial fibrillation (AF), chronic AF and ventricular fibrillation (VT). Atrial fibrillation (AF) is an arrhythmia associated with the asynchronous contraction of the atrial muscle fibres. It is the most prevalent cardiac arrhythmia in the western world, and is associated with significant morbidity. Best available therapeutic strategies combine pharmacological and surgical means. But when successful, they do not always prevent long-term relapses. Initial surgical success becomes all the more tricky to extend as the arrhythmia maintains itself and the pathology evolves into sustained or chronic AF. This raises the open crucial issue of deciphering the mechanisms that govern the onset of AF as well as its perpetuation. Here, we propose a wavelet-based multi-scale strategy to analyze the electrical activity of human hearts recorded by catheter electrodes, positioned in the coronary sinus (CS), during episodes of chronic AF.

Because the clinical outcome of the “purely” myocardial approach remains suboptimal despite of significant technological improvement in ablation procedures with better mapping and energy delivery systems, there has been increasing evidence that dysfunction of the autonomic nervous system that encompasses the sympathetic, parasympathetic and intrinsic neural network could also be involved in the patho- genesis of AF. With a complex wavelet decomposition of AF signals, to capture their local phase dynamics, we propose to quantify the nature and strength of nonlinearities involved in these signals and propose a new marker to follow the evolution of this disease.

Multiscale Deep Learning

Richard Baraniuk

Deep (neural) networks have been applied productively in a wide range of machine learning and signal processing problems.  But a fundamental question remains:  Why do they work?  Intuitions abound, but a coherent framework for understanding, analyzing, and synthesizing deep learning architectures has remained elusive.  This talk overviews some recent progress on two fronts.  First, we build a bridge between deep networks and approximation theory via spline functions. Our key result is that a large class of deep networks can be written as a composition of max-affine spline operators (MASOs), which provide a powerful portal for analyzing the geometry of learning and prediction.  Second, we demonstrate that deep networks can be endowed with a natural multiscale structure that enables a multiresolution analysis of their prediction, for example, the decision boundary for classification tasks.

Sicut in caelo and in terra -- Data representations for seismic and gravitational waves

Eric Chassande-Mottin

In order to improve the resolution in the reconstruction of geological layers from seismic data, J Morlet, A Grossmann and colleagues proposed, in the late 70s, an alternative to the short-term Fourier representation, later called wavelet transform. Similar representations and ideas are used today in a completely different context, that of gravitational wave detection by the advanced LIGO and advanced Virgo interferometric detectors. This presentation will review data analysis issues in the field of gravitational astronomy, highlighting similarities with that encountered in seismology.

Coherent states: analysis, geometry and physics. In the footsteps of Alex Grossmann and Yves Meyer (in the age of machine learning)

Ronald Coifman

We will relate the quest for effective analytic  functional representations , to current trends in empirical machine learning and deep neural networks, to work by Grossmann and Meyer who were seeking informative waveforms for physics. Their  goal was to provide a language for expressing efficiently acoustic fields and oscillating states. 

Our current abilities to build purely empirical models directly from observations  ( by machine learning optimizations) is a continuation of some of  old work by Alex and his collaborators .

One of our goals is to describe the integration of classical methods with more recent developments in scientific machine learning.

Lovely bones: A mathematical and biological dialog.

Ingrid Daubechies

In collaboration with Biological morphologists, we have defined new distances between pairs of two-dimensional surfaces (embedded in three-dimensional space) that use both local structures and global information in the surfaces. These are motivated by the need of biological morphologists to compare different phenotypical structures, and to study relationships of living or extinct animals with their surroundings and each other. Many existing algorithms for morphological correspondences make use of carefully defined anatomical correspondence points (landmarks) on bones, for example. Our approach does not require any such preliminary marking of special features or landmarks by the user.
We then show how to use and "clean up" these distances to study collections of teeth and bones, refining the results further.

De l’algèbre linéaire pour la comparaison des séquences biologiques à l’étude de l’évolution du génome du pommier.

Claudine Landés

Cet exposé se déroulera en deux temps, une première section présentera les contributions d’Alex Grossmann pour les comparaisons de séquences biologiques en particulier dans l’étude des virus HIV1-HIV2 et de l’évolution des ADN-topoisomérases (les topoisomérases assurent le maintien de la molécule d’ADN dans une topologie correcte au cours des processus biologiques du cycle cellulaire). Dans un second temps je présenterai mes travaux de recherche en cours au sein de l’IRHS (Institut de Recherche en Horticulture et Semences) sur le génome du pommier et son histoire évolutive.