Abstract: In this work we address the problem of detecting whether a sampled probability distribution of a random variable has infinite expectation. 

This issue is notably important when one  simulates stochastic particle systems with singular McKean-Vlasov interaction kernels which model complex biological phenomena.

As stated, the detection problem is ill-posed. We thus propose and analyze an asymptotic hypothesis test for independent copies of a given random variable which is supposed to  belong to an unknown domain of attraction of a stable law. 

The null hypothesis : `The random variable interest is in the domain of attraction of the Normal law' and the alternative hypothesis is `The random variable of interest is in thedomain of attraction of a stable law with index smaller than 2'.

Our key observation is that the random variable  cannot have a finite second moment when the null hypothesis is rejected.

Surprisingly, we find it useful to derive our test from the statistics of random processes. More precisely, our hypothesis test is based on a statistic which is inspired by methodologies to determine whether a semimartingale has jumps from the observation of one single path at discrete times.

We justify our test by proving asymptotic properties of discrete time functionals of Brownian bridges. We also discuss the choice of crucial parameters involved in the test and illustrate our theoretical results with numerical experiments.

This is a joint work with Héctor Olivero (Universidad de Valparaíso) 

Abstract:  Ageing’s sensitivity to natural selection has long been discussed because of its apparent negative effect on an individual’s fitness. Thanks to the recently described (Smurf) 2-phase model of ageing (data of Michael Rera) we propose a fresh angle for modeling the evolution of ageing. Indeed, by coupling a dramatic loss of fertility with a high-risk of impending death—amongst other multiple so-called hallmarks of ageing—the Smurf phenotype allowed us to consider ageing as a couple of sharp transitions. The birth–death model  we describe here is a simple life-history trait model where each asexual and haploid individual is described by its fertility period xb and survival period xd .We show that xb and xd converge during evolution to configurations xb − xd ≈ constant  in finite time. To do so, we built an individual-based stochastic model which describes the age and trait distribution dynamics of such a finite population.Then we rigorously derive the adaptive dynamics models, which describe the trait dynamics at the evolutionary time-scale. We extend the Trait Substitution Sequence with age structure and  study the limiting behaviour of this jump process when mutations are small. We show that the limiting behaviour is described by a differential equation for (xb,xd). 

Abstract:  We consider a measure-valued stochastic individual-based model describing the Darwinian evolution of an asexual population with constant population size. We investigate the asymptotic behavior of the system assuming frequent mutations, large population and small mutational effects. We prove the convergence of the mean trait process to an ODE known as the canonical equation of adaptive dynamics, which describes the evolution in time of the dominant trait in the population driven by a fitness gradient. This result holds for a certain range of scaling parameters, in particular mutations must be small enough, but not too small. This result is based on an slow-fast asymptotic analysis. We use an averaging method, inspired by Kurtz, based on controls of the moments of the population distribution and a compactness-uniqueness argument. We prove that the fast component converges to the centered Fleming-Viot process, that we recently characterized in a previous work.

[This is joint work with Vincent Hass (Université de France Comté)]

Abstract: We revisit the problem of characterising the thermodynamic limit of a fully connected

network of Hopfield-like neurons. We provide a complete description of the mean-field quations as a stochastic differential equations with coefficient depending on a mean m(t) and temporal covariance K(t,s) functions of the solution. 

We also perform numerical methods to approximate the relevent functions and to finally simulate the solution.

This is a common work with Olivier Faugeras (Inria)

Abstract:  In this talk we will review the quantum advantage provided by entangled states in parameter estimation. The relation with geometric properties of the manifold of quantum states will also be discussed.

More precisely, we will show that geodesics on this manifold correspond to dynamical quantum evolutions, which turn out to be optimal for estimating certain parameters. 

Abstract: Open quantum systems are inherently coupled to their environments, which in turn also obey quantum dynamical rules. Nevertheless, depending on particular conditions, the influence of the environment over the system could be represented by classical noise fields. Based on a microscopic unitary description we propose a measure that quantifies how far the action of an environment departs from this classical limit. In a second step, motivated by biophysical arrangements, we characterize the more general coupling mechanisms that allow describing any hybrid time-irreversible quantum-classical dynamics. 

Abstract: The retina, a sophisticated and highly efficient neural network, plays a pivotal role in visual processing, transforming light into neural signals that shape our perception of the world. The exploration of retinal function through molecular biology and computational neuroscience offers unprecedented opportunities to decipher these complex neural codes yet presents formidable challenges. This presentation aims to provide an overview of the current landscape of computational neuroscience in the retina, highlighting the dual nature of challenges and opportunities that define this exciting field.

Abstract: TBA

Abstract: TBA

Abstract: In this talk, we will explore the dispersion of biological polymers in nanofluidic devices, driven by pressure-induced fluid dynamics across a series of entropic barriers. The presentation is divided into two main sections. In the first section, we will examine the alignment of coarse-grained molecular dynamics simulations with empirical experiments. These reveal transitions in polymer behavior with increasing flow strength and provide physical insights through the simulations’ explicit representation of the system’s microfluidics. The second section, which is the focal point of the talk, involves the application of probabilistic tools to assess the system’s asymptotic behavior. We introduce a continuous-time Markov process to model the polymer movement across entropic barriers. The main finding is a functional central limit theorem for the polymer’s position, which includes an explicit expression for the effective diffusion coefficient derived from the model parameters. Additionally, we derive results for the asymptotic velocity of the polymer through a law of large numbers and large deviation estimates. This work received partial support from the ANID FONDECYT Regular grant No. 1221220. 

Abstract: In the modeling of a variety of models of large-scale systems of interacting agents, mathematical models assume a scaling of the interaction term that prevents a divergence of the coupling terms, and the emerging dynamics strongly depend on the choice of a scaling and connectivity coefficients. These scalings however remain mostly technical. We consider here the question of the dynamics of unscaled networks, and show that, under appropriate assumptions, they converge to a dynamics constrained on a manifold where interactions vanish. In a second part, we illustrate this result on the dynamics of a neural FitzHigh-Nagumo network composed of excitatory and inhibitory populations, communicating through chemical synapses. 

Abstract: Combining the significance of multiple experiments regarding the same scientific hypothesis is a crucial method for global hypothesis testing, with applications in meta-analysis, signal detection, and other data-integrative studies.  Such procedures consider a group of p-values to form a summary statistic to determine the overall evidence against a global null hypothesis. Mathematical studies often assume that the underlying statistics are continuous and independent, due to their homogeneous and straightforward mathematical structure. In reality, however, data and its corresponding statistics are often discrete. Discrete tests present an array of extra challenges that make the continuous framework unsuitable, and calculating the exact distribution of the summary statistic is often computationally challenging. Using tools from optimal transport, we propose an omnibus modification of a discrete statistic towards a continuous probability integral transform and show that, under mild hypothesis, the sum of discrete modifications produces an asymptotically correct test for any type I error control. Furthermore, by expressing this transformation as a likelihood ratio, we delve into the optimal choice of combination statistic for some common discrete tests.

Abstract: Under antibiotics causing DNA damage, bacteria can trigger a stress response known as the SOS response. While the expression of this stress response can make individual cells transiently able to overcome antibiotic treatment, it can also delay cell division, thus impacting in different ways the population's ability to grow and survive. In order to study the trade-offs that emerge from this phenomenon, we propose a bi-type age-structured stochastic population model that captures the phenotypic plasticity observed in the stress response. Individuals can belong to two types: either a fast-dividing but prone to death "vulnerable" type, or a slow-dividing but "tolerant" type. We study the survival probability of the population issued from a single cell as well as the population growth rate in constant and periodic environments. We show that the sensitivity of these two different notions of fitness with respect to the parameters describing the phenotypic plasticity surprisingly differs between the stochastic approach (survival probability) and the deterministic approach (population growth rate). Moreover, under a more realistic configuration of periodic stress, our results indicate that optimal population growth can only be achieved through fine-tuning simultaneously both the induction of the stress response and the repair efficiency of the damage caused by the antibiotic.    

Abstract: In this talk, we introduce a stochastic model that aims to understand the complex feedback mechanisms that affect the abundance and persistence of natural resources, with a special focus on illegal fishing. We use a case study of non-compliance with harvesting regulations in Chilean intertidal kelp forests to explain and motivate the consideration of social, ecological, biological, economic, and political factors. All of them shape fisher's decision-making processes regarding illegal activities. We will discuss the mathematical analysis of the well-posedness of the system and the conditions for non-absorption of process boundaries that define the proportion of fishermen who fail to comply with regulatory standards. We propose an approximation scheme that preserves the positivity of the trajectory, laying the groundwork for simulating potential scenarios. With this presentation, we hope to initiate a dialogue on integrating stochastic methodologies to address complex socio-ecological challenges. By shedding light on the interconnectedness of the various drivers of illegal fishing, we aim to inspire collaborative efforts toward more effective management and conservation strategies.

Abstract: TBA

Abstract:  In this talk, we introduce a stochastic model that aims to understand the complex feedback mechanisms that affect the abundance and persistence of natural resources, with a special focus on illegal fishing. We use a case study of non-compliance with harvesting regulations in Chilean intertidal kelp forests to explain and motivate the consideration of social, ecological, biological, economic, and political factors. All of them shape fisher's decision-making processes regarding illegal activities. We will discuss the mathematical analysis of the well-posedness of the system and the conditions for non-absorption of process boundaries that define the proportion of fishermen who fail to comply with regulatory standards. We propose an approximation scheme that preserves the positivity of the trajectory, laying the groundwork for simulating potential scenarios. With this presentation, we hope to initiate a dialogue on integrating stochastic methodologies to address complex socio-ecological challenges. By shedding light on the interconnectedness of the various drivers of illegal fishing, we aim to inspire collaborative efforts toward more effective management and conservation strategies.

Abstract: We propose a family of Markov kernels on the unit interval which, after rescaling, are proved to converge in law to a large family of diffusion processes of the Wright-Fisher type.  Discrete-time line-counting processes (the ancestral lineage processes) are obtained, and we prove that moment duality relations hold for a large class of variants of the basic process. This unified, parsimonious derivation of some well-known population genetic processes allows one to easily extend the dynamics to the case where mutations and/or selection rates might depend on the instantaneous distribution of the process. In this connection we prove both, propagation of chaos for the associated particle system in mean-field regime, and the convergence of the rescaled discrete-time non-linear Wright-Fisher conglomerate to diffusions of the McKean-Vlasov type.  As an important intermediate step, it is proved that the sequence of (laws of) some non-linear Markov chains converge to the unique solution of a non-linear martingale problem. As a running motivating model, we consider an application to the evolution of an idealised infectious pathogen affecting hosts that are naturally segregated into many colonies. 

Abstract: Turbulence at multiple scales shapes the distribution and magnitude of primary production in the ocean. This talk will survey recent observations of ocean ecosystems and the ways that they are influenced by ocean currents to identify ways in which an open systems approach may advance understanding and prediction of ocean ecosystems.

Abstract: We study the large-time behavior of an ensemble of particles obeying replicator-like stochastic dynamics with mean-field interactions. We prove the propagation of chaos property and establish conditions for the strong persistence of the N-replicator system and the existence of invariant distributions for the associated McKean-Vlasov dynamics. We illustrate our findings with some simple case studies, providing numerical results.

Abstract: In distributed networks, agents are placed in nodes of a graph, and the state of such agents evolves due to interaction with neighbouring agents. A standard communication model is the so-called population protocol, where at random times, a random agent observes the state of a neighbouring agent and changes its state by incorporating the new information. From a Computer Science perspective, the main challenge is to design transition rules between states given the new information.

Within this limited framework of communication, a natural problem arises: the so-called average consensus. Here, the goal is for all agents to eventually converge to the same state, with the shared state representing the average of the initial states across all agents.

In this talk, I will present a simple dynamics for distributed averaging consensus in the setting of population protocols. We demonstrate that this dynamic asymptotically reaches consensus to a common random state F, and we provide convergence rates to this limit, as well as exact expressions for the mean and variance of F.

Abstract: The classic susceptible-infected-recovered (SIR) model, introduced by Kermack and McKendrick in 1927, is a simple system of ODEs governing the spread of an infectious disease in a large population. This model operates under the assumption that the disease is transmitted via pairwise interactions between infected and susceptible individuals. In this talk, we introduce an extension to this model, whose underlying infection mechanism consists of individuals attending randomly generated social gatherings, potentially resulting in many new infections. This gives rise to an explicit system of ODEs, wherein the force of infection term depends non-linearly on the proportion of infected individuals. Some specific instances yield models previously studied in the literature, to which our work provides a probabilistic foundation. We rigorously justify our model by showing that the system of ODEs is the mean-field limit of the jump Markov process corresponding to the evolution of the disease in a finite population. 

Abstract: TBA