arXiv subjects

Xavier Richard

Publications and source records attributed to Xavier Richard.

5 recordsLinked to original sources

arXiv2026

Stability and feasibility of Microbial Consumer-Resource Model

Microbial communities are ubiquitous in nature but how they grow on available resources is still poorly understood. Communities are complex systems harboring thousands of microbial species that interact through resource competition. The classical MacArthur consumer-resource model has been shown to underestimate formed community biomass. A recent new microbial consumer-resource model (MiCRM) considers the inclusion of inter-specific interactions mediated by metabolite exchange (cross-feeding) where the various bacterial growth byproducts can be reused by other species for their own growth. We study persistence, feasibility and stability for MiCRM under some simplifying assumptions using slow-fast approximation. We show e.g. the non-persistence of the microbial community when the number of resource species M is smaller than the number of consumer species S. We then study the stability of the slow steady state when the number or survivors S is smaller than M, and show that such equilibria are generically stable. We finally propose a stochastic slow-fast version of the model having fast Poisson steady state and study related extinction events.

q-bio.PE↗

arXiv2025

On the abelian structure of noncompetitive chemical reaction networks

Chemical reaction networks (CRNs) are foundational models for describing complex biochemical processes. We study noncompetitive CRNs, a class of networks whose static states, where the CRN is inactive, are rate independent, and that can implement ReLU neural networks. CRNs of interest in biochemistry and systems biology are embedded in complex networks so that CRNs have to respond to internal and environmental cues. We describe the network's response to such perturbations using a new Markov chain that we call CRN sandpile Markov chain, whose state space is the set of static states. The transition mechanism of the CRN sandpile Markov chain is defined by adding a molecule of a randomly chosen species to a static state, and then letting the CRN state evolve toward a new static state. A central contribution of the present work is the observation that one can associate a natural Abelian Network (AN) to each noncompetitive CRN, and use AN theory to get new mathematical results on noncompetitive CRNs. For noncompetitive CRNs on a finite state space, we use AN theory to get that only a fraction of the static states are recurrent for the CRN sandpile Markov chain. We obtain furthermore that the set of recurrent states is in one to one correspondence with the critical group of the AN, which plays a major role in AN theory. Overall, this work establishes a unified algebraic and probabilistic framework for analyzing the long-term behavior of noncompetitive CRNs. We focus on a special class of noncompetitive CRNs called generalized toppling networks, and obtain new mathematical results both for the CRN and AN settings.

q-bio.MN↗

arXiv2025

Introducing Image-Space Preconditioning in the Variational Formulation of MRI Reconstructions

The aim of the present article is to enrich the comprehension of iterative magnetic resonance imaging (MRI) reconstructions, including compressed sensing (CS) and iterative deep learning (DL) reconstructions, by describing them in the general framework of finite-dimensional inner-product spaces. In particular, we show that image-space preconditioning (ISP) and data-space preconditioning (DSP) can be formulated as non-conventional inner-products. The main gain of our reformulation is an embedding of ISP in the variational formulation of the MRI reconstruction problem (in an algorithm-independent way) which allows in principle to naturally and systematically propagate ISP in all iterative reconstructions, including many iterative DL and CS reconstructions where preconditioning is lacking. The way in which we apply linear algebraic tools to MRI reconstructions as presented in this article is a novelty. A secondary aim of our article is to offer a certain didactic material to scientists who are new in the field of MRI reconstruction. Since we explore here some mathematical concepts of reconstruction, we take that opportunity to recall some principles that may be understood for experts, but which may be hard to find in the literature for beginners. In fact, the description of many mathematical tools of MRI reconstruction is fragmented in the literature or sometimes missing because considered as a general knowledge. Further, some of those concepts can be found in mathematic manuals, but not in a form that is oriented toward MRI. For example, we think of the conjugate gradient descent, the notion of derivative with respect to non-conventional inner products, or simply the notion of adjoint. The authors believe therefore that it is beneficial for their field of research to dedicate some space to such a didactic material.

physics.med-ph↗

arXiv2023

Complete mathematical characterization of two simple toggle-switch biological systems

Prokaryotic gene expression is dynamic and noisy, and can lead to phenotypic variations. Previous work has shown that the selection of such variation can be modeled through basic reaction networks described by O.D.E. displaying a bistable behavior. While previous mathematical studies have shown that mono- or bistable behavior depends on the rate of the reactions in the system, no analytical solution of the curves delineating the actual parameter conditions that result in mono or bistability has so far been provided. In this work we provide the first explicit analytical solution for the boundary curve that separates the parameter space defining domains where double positive and double negative feedback loops become bistable.

q-bio.MN↗

arXiv2018

Mean field repulsive Kuramoto models: Phase locking and spatial signs

The phenomenon of self-synchronization in populations of oscillatory units appears naturally in neurosciences. However, in some situations, the formation of a coherent state is damaging. In this article we study a repulsive mean-field Kuramoto model that describes the time evolution of n points on the unit circle, which are transformed into incoherent phase-locked states. It has been recently shown that such systems can be reduced to a three-dimensional system of ordinary differential equations, whose mathematical structure is strongly related to hyperbolic geometry. The orbits of the Kuramoto dynamical system are then described by a ow of Möbius transformations. We show this underlying dynamic performs statistical inference by computing dynamically M-estimates of scatter matrices. We also describe the limiting phase-locked states for random initial conditions using Tyler's transformation matrix. Moreover, we show the repulsive Kuramoto model performs dynamically not only robust covariance matrix estimation, but also data processing: the initial configuration of the n points is transformed by the dynamic into a limiting phase-locked state that surprisingly equals the spatial signs from nonparametric statistics. That makes the sign empirical covariance matrix to equal 1 2 id2, the variance-covariance matrix of a random vector that is uniformly distributed on the unit circle.

nlin.AO↗