The study of experimental data is a relevant task in several physical, chemical and biological applications. In particular, the analysis of chaotic dynamics in cardiac systems is crucial as it can be related to certain pathological arrhythmias. When working with short and noisy experimental time series, some standard techniques for chaos detection cannot provide reliable results because of such data characteristics. Moreover, when small datasets are available, some Deep Learning techniques cannot be applied directly (that is, using part of the data to train the network, and using the trained network to analyze the remaining dataset). To overcome all these limitations, we propose an automatic algorithm that combines Deep Learning and some selection strategies based on a mathematical model of the same nature of the experimental data. To demonstrate its performance, we test it with experimental data obtained from ex-vivo frog heart experiments, achieving highly accurate results.

This study examines the training process of Artificial Neural Networks, specifically focusing on Reservoir Computing and on binary classification tasks. Training involves optimizing the network’s parameters to minimize the error in a loss function, which quantifies the discrepancy between network outputs and target data. Several numerical optimizers are used in the literature. Here, we formulate the supervised learning problem using the recently introduced Bregman Proximal Gradient with extrapolation (BPGe) for non-convex optimization problems. We compare this new method with the classical Stochastic Gradient Descent (SGD), the Root Mean Square Propagation (RMSProp), the Adaptive Moment estimation (Adam), and two variants with Nesterov momentum (NAG and NAdam). The new approach leads to an accurate and significantly faster numerical algorithm for solving supervised learning problems on binary classification tasks in Reservoir Computing. Three test examples are presented, one based on chaotic data classification in the classical Lorenz system, the other on the Human Activity Recognition (HAR) using smartphones dataset for movement-rest classification, and the last one on the MNIST dataset for even-odd classification. We show that our approach is highly competitive with existing methods in the tests performed.

The use of hierarchical networks in motor pattern generation in hexapod robots and in insect movement provides ways to understand and control both systems. In this paper we consider the Central Pattern Generator (CPG) of insect movement consisting of six coupled neurons developed by Ghigliazza and Holmes (2004) that produces the global leg coordination pattern. We provide a detailed study of the possible gaits generated by the CPG through different numerical techniques recently developed for the study of small networks, which allow us to consider the complete model without any simplification. We combine the analysis of isolated neurons (using several three dimensional parameter spaces) to give a roadmap of the dynamics of the neurons involved, lateral phase lag plots to show the convergence and transitions towards particular patterns and a quasi-Monte Carlo sweeping method to describe different pattern routes in the parametric phase space. In all of our studies we have found the same final result: most of the observed patterns follow routes that lead to the stable tripod gait. We obtain several routes made of symmetrical patterns, but despite considering mainly a symmetric leg CPG we detect non-symmetrical patterns that provide (minor) routes present in the model. The bifurcations of the main pattern routes detected in the parameter space are studied in detail using continuation techniques. This study reveals the bifurcations that create and destroy the different routes and how for large values of some parameters only the tripod gait is present. Based on the symmetric case, a preliminary study of asymmetric configurations is done revealing the robustness of the already located patterns and routes. Due to the relevance of the tripod gait, and since more parameters are included in this study, we introduce an algorithm to locate the limit of the tripod gait in the parameter space, which shows that there is a large three parametric region where the tripod pattern is ubiquitous and highly dominant in rapidly moving insect regimes.

In image processing, total variation (TV) regularization models are commonly used to recover the blurred images. One of the most efficient and popular methods to solve the convex TV problem is the alternating direction method of multipliers (ADMM) algorithm, recently extended using the inertial proximal point method. Although all the classical studies focus on only a convex formulation, recent articles are paying increasing attention to the nonconvex methodology due to its good numerical performance and properties. In this paper, we propose to extend the classical formulation with a novel nonconvex alternating direction method of multipliers with the inertial technique (IADMM). Under certain assumptions on the parameters, we prove the convergence of the algorithm with the help of the Kurdyka-Łojasiewicz property. We also present numerical simulations on the classical TV image reconstruction problems to illustrate the efficiency of the new algorithm and its behavior compared with the well-established ADMM method.

This paper is concerned with the fast and accurate evaluation of elementary symmetric functions. We present a new compensated algorithm by applying error-free transformations to improve the accuracy of the so-called Summation Algorithm, which is used, by example, in the MATLAB's poly function. We derive a forward round off error bound and running error bound for our new algorithm. The round off error bound implies that the computed result is as accurate as if computed with twice the working precision and then rounded to the current working precision. The running error analysis provides a shaper bound along with the result, without increasing significantly the computational cost. Numerical experiments illustrate that our algorithm runs much faster than the algorithm using the classic double-double library while sharing similar error estimates. Such an algorithm can be widely applicable for example to compute characteristic polynomials from eigen values. It can also be used into the Rasch model in psychological measurement.

This article introduces the software package TIDES and revisits the use of the Taylor series method for the numerical integration of ODEs. The package TIDES provides an easy-to-use interface for standard double precision integrations, but also for quadruple precision and multiple precision integrations. The motivation for the development of this package is that more and more scientific disciplines need very high precision solution of ODEs, and a standard ODE method is not able to reach these precision levels. The TIDES package combines a preprocessor step in M athematica that generates Fortran or C programs with a library in C. Another capability of TIDES is the direct solution of sensitivities of the solution of ODE systems, which means that we can compute the solution of variational equations up to any order without formulating them explicitly. Different options of the software are discussed, and finally it is compared with other well-known available methods, as well as with different options of TIDES. From the numerical tests, TIDES is competitive, both in speed and accuracy, with standard methods, but it also provides new capabilities.