Spectral dimension reduction of complex dynamical networks

Dynamical networks are powerful tools for modeling a broad range of complex systems, including financial markets, brains, and ecosystems. They encode how the basic elements (nodes) of these systems interact altogether (via links) and evolve (nodes' dynamics). Despite substantial progress, little is known about why some subtle changes in the network structure, at the so-called critical points, can provoke drastic shifts in its dynamics. We tackle this challenging problem by introducing a method that reduces any network to a simplified low-dimensional version. It can then be used to describe the collective dynamics of the original system. This dimension reduction method relies on spectral graph theory and, more specifically, on the dominant eigenvalues and eigenvectors of the network adjacency matrix. Contrary to previous approaches, our method is able to predict the multiple activation of modular networks as well as the critical points of random networks with arbitrary degree distributions. Our results are of both fundamental and practical interest, as they offer a novel framework to relate the structure of networks to their dynamics and to study the resilience of complex systems.Comment: 16 pages, 8 figure

Beyond Wilson-Cowan dynamics: oscillations and chaos without inhibition

Fifty years ago, Wilson and Cowan developed a mathematical model to describe the activity of neural populations. In this seminal work, they divided the cells in three groups: active, sensitive and refractory, and obtained a dynamical system to describe the evolution of the average firing rates of the populations. In the present work, we investigate the impact of the often neglected refractory state and show that taking it into account can introduce new dynamics. Starting from a continuous-time Markov chain, we perform a rigorous derivation of a mean-field model that includes the refractory fractions of populations as dynamical variables. Then, we perform bifurcation analysis to explain the occurance of periodic solutions in cases where the classical Wilson-Cowan does not predict oscillations. We also show that our mean-field model is able to predict chaotic behavior in the dynamics of networks with as little as two populations.Comment: 14 pages, 14 figure

Analyse des bifurcations dans un modèle du flutter auriculaire

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal

Counting Involutions on Multicomplex Numbers

We show that there is a bijection between real-linear automorphisms of the multicomplex numbers of order $n$ and signed permutations of length $2^{n-1}$. This allows us to deduce a number of results on the multicomplex numbers, including a formula for the number of involutions on multicomplex spaces which generalizes a recent result on the bicomplex numbers and contrasts drastically with the quaternion case. We also generalize this formula to $r$-involutions and obtain a formula for the number of involutions preserving elementary imaginary units. The proofs rely on new elementary results pertaining to multicomplex numbers that are surprisingly unknown in the literature, including a count and a representation theorem for numbers squaring to $\pm 1$

Counting hidden neural networks

We apply combinatorial tools, including P´olya’s theorem, to enumerate all possible networks for which (1) the network contains distinguishable input and output nodes as well as partially distinguishable intermediate nodes; (2) all connections are directed and for each pair of nodes, there are at most two connections, that is, at most one connection per direction; (3) input nodes send connections but don’t receive any, while output nodes receive connections but don’t send any; (4) every intermediate node receives a path from an input node and sends a path to at least one output node; and (5) input nodes don’t send direct connections to output nodes. We first obtain the generating function for the number of such networks, and then use it to obtain precise estimates for the number of networks. Finally, we develop a computer algorithm that allows us to generate these networks. This work could become useful in the field of neuroscience, in which the problem of deciphering the structure of hidden networks is of the utmost importance, since there are several instances in which the activity of input and output neurons can be directly measured, while no direct access to the intermediate network is possible. Our results can also be used to count the number of finite automata in which each cell plays a relevant role

Bounds for the counting function of the Jordan-Pólya numbers

summary:A positive integer $n$ is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number $x$

APPleSOSS: A Producer of ProfiLEs for SOSS. Application to the NIRISS SOSS Mode

The SOSS mode of the NIRISS instrument is poised to be one of the workhorse modes for exoplanet atmosphere observations with the newly launched James Webb Space Telescope. One of the challenges of the SOSS mode, however, is the physical overlap of the first two diffraction orders of the G700XD grism on the detector. Recently, the ATOCA algorithm was developed and implemented as an option in the official JWST pipeline, as a method to extract SOSS spectra by decontaminating the detector -- that is, separating the first and second orders. Here, we present APPleSOSS (A Producer of ProfiLEs for SOSS), which generates the spatial profiles for each diffraction order upon which ATOCA relies. We validate APPleSOSS using simulated SOSS time series observations of WASP-52b, and compare it to ATOCA extractions using two other spatial profiles (a best and worst case scenario on-sky), as well as a simple box extraction performed without taking into account the order contamination. We demonstrate that APPleSOSS traces retain a high degree of fidelity to the true underlying spatial profiles, and therefore yield accurate extracted spectra. We further confirm that the effects of the order contamination for relative measurements (e.g., exoplanet transmission or emission observations) is small -- the transmission spectrum obtained from each of our four tests, including the contaminated box extraction, deviates by $\lesssim$0.1$\sigma$ from the atmosphere model input into our noiseless simulations. We further confirm via a retrieval analysis that the atmosphere parameters (metallicity and C/O) obtained from each transmission spectrum are consistent at the 1$\sigma$ level with the true underlying values.Comment: 12 pages, 9 figures. Submitted to PAS

Transiting Exoplanet Studies and Community Targets for JWST's Early Release Science Program

The James Webb Space Telescope will revolutionize transiting exoplanet atmospheric science due to its capability for continuous, long-duration observations and its larger collecting area, spectral coverage, and spectral resolution compared to existing space-based facilities. However, it is unclear precisely how well JWST will perform and which of its myriad instruments and observing modes will be best suited for transiting exoplanet studies. In this article, we describe a prefatory JWST Early Release Science (ERS) program that focuses on testing specific observing modes to quickly give the community the data and experience it needs to plan more efficient and successful future transiting exoplanet characterization programs. We propose a multi-pronged approach wherein one aspect of the program focuses on observing transits of a single target with all of the recommended observing modes to identify and understand potential systematics, compare transmission spectra at overlapping and neighboring wavelength regions, confirm throughputs, and determine overall performances. In our search for transiting exoplanets that are well suited to achieving these goals, we identify 12 objects (dubbed "community targets") that meet our defined criteria. Currently, the most favorable target is WASP-62b because of its large predicted signal size, relatively bright host star, and location in JWST's continuous viewing zone. Since most of the community targets do not have well-characterized atmospheres, we recommend initiating preparatory observing programs to determine the presence of obscuring clouds/hazes within their atmospheres. Measurable spectroscopic features are needed to establish the optimal resolution and wavelength regions for exoplanet characterization. Other initiatives from our proposed ERS program include testing the instrument brightness limits and performing phase-curve observations.(Abridged)Comment: This is a white paper that originated from an open discussion at the Enabling Transiting Exoplanet Science with JWST workshop held November 16 - 18, 2015 at STScI (http://www.stsci.edu/jwst/science/exoplanets). Accepted for publication in PAS