arXiv subjects

Fabien Campillo

Publications and source records attributed to Fabien Campillo.

16 recordsLinked to original sources

arXiv2026

A Guided Tour of the Equations of Nonlinear Filtering for Diffusion Processes

These pedagogical notes provide an introduction to nonlinear filtering for diffusion processes. The objective of nonlinear filtering is to estimate an unobserved stochastic state from partial and noisy observations, and thereby characterize the conditional distribution of the state given the observation history. After introducing the state-observation model, we develop the main tools of the theory, including Markov semigroups, infinitesimal generators, and change-of-measure methods. This leads naturally to the Kallianpur-Striebel formula, the Zakai equation for the unnormalized filter, and the Kushner-Stratonovich equation for the normalized filter. The presentation follows, to a large extent, the classical exposition of Bain and Crisan's "Fundamentals of Stochastic Filtering", while placing particular emphasis on intuition, motivation, and the interpretation of the main concepts and equations. Many technical arguments are revisited, with additional details and comments intended to facilitate a first reading of the subject. These notes are not intended to be an exhaustive account of stochastic filtering. Their aim is instead to provide a guided tour of some of its central ideas, methods, and equations.

math.PR↗

arXiv2025

Depolarization block induction via slow NaV1.1 inactivation in Dravet syndrome

Dravet syndrome is a developmental and epileptic encephalopathy, characterized by the early onset of drug-resistant seizures and various comorbidities. Most cases of this severe and complex pathology are due to mutations of NaV1.1, a sodium channel mainly expressed in fast-spiking inhibitory neurons. Layer et al. (Front. Cell. Neurosci. 15, 2021) showed that one of these mutations alters the voltage dependence of channel activation, as well as the voltage dependence and kinetics of slow inactivation. Implementing the three effects into a computational model, they predict that altered activation has the largest impact on channel function, as it causes the most severe firing rate reduction. Using a conductance-based model tailored to the dynamics of fast-spiking inhibitory neurons, we look deeper into slow inactivation. We exploit the timescale difference between this very slow process and the rest of the system to conduct a multiple-timescale analysis. We find that, upon prolonged stimulation, the onset of slow inactivation at lower voltage in mutant channels promotes depolarization block, another possible firing deficit aside from frequency reduction. The accelerated kinetics of slow inactivation in mutant channels hastens this transition. This suggests that slow inactivation alterations might for some Dravet variant contribute to the pathological mechanism.

q-bio.NC↗

arXiv2023

The one step fixed-lag particle smoother as a strategy to improve the prediction step of particle filtering

Sequential Monte Carlo methods have been a major breakthrough in the field of numerical signal processing for stochastic dynamical state-space systems with partial and noisy observations. However, these methods still present certain weaknesses. One of the most fundamental is the degeneracy of the filter due to the impoverishment of the particles: the prediction step allows the particles to explore the state-space and can lead to the impoverishment of the particles if this exploration is poorly conducted or when it conflicts with the following observation that will be used in the evaluation of the likelihood of each particle. In this article, in order to improve this last step within the framework of the classic bootstrap particle filter, we propose a simple approximation of the one step fixed-lag smoother. At each time iteration, we propose to perform additional simulations during the prediction step in order to improve the likelihood of the selected particles.

stat.AP↗

arXiv2019

Mean-field limit of interacting 2D nonlinear stochastic spiking neurons

In this work, we propose a nonlinear stochastic model of a network of stochastic spiking neurons. We heuristically derive the mean-field limit of this system. We then design a Monte Carlo method for the simulation of the microscopic system, and a finite volume method (based on an upwind implicit scheme) for the mean-field model. The finite volume method respects numerical versions of the two main properties of the mean-field model, conservation and positivity, leading to existence and uniqueness of a numerical solution. As the size of the network tends to infinity, we numerically observe propagation of chaos and convergence from an individual description to a mean-field description. Numerical evidences for the existence of a Hopf bifurcation (synonym of synchronised activity) for a sufficiently high value of connectivity, are provided.

math.NA↗

arXiv2017

A numerical approach to determine mutant invasion fitness and evolutionary singular strategies

We propose a numerical approach to study the invasion fitness of a mutant and to determine evolutionary singular strategies in evolutionary structured models in which the competitive exclusion principle holds. Our approach is based on a dual representation, which consists of the modelling of the small size mutant population by a stochastic model and the computation of its corresponding deterministic model. The use of the deterministic model greatly facilitates the numerical determination of the feasibility of invasion as well as the convergence-stability of the evolutionary singular strategy. Our approach combines standard adaptive dynamics with the link between the mutant survival criterion in the stochastic model and the sign of the eigenvalue in the corresponding deterministic model. We present our method in the context of a mass-structured individual-based chemostat model. We exploit a previously derived mathematical relationship between stochastic and deterministic representations of the mutant population in the chemostat model to derive a general numerical method for analyzing the invasion fitness in the stochastic models. Our method can be applied to the broad class of evolutionary models for which a link between the stochastic and deterministic invasion fitnesses can be established.

q-bio.PE↗

arXiv2017

On the variations of the principal eigenvalue with respect to a parameter in growth-fragmentation models

We study the variations of the principal eigenvalue associated to a growth-fragmentation-death equation with respect to a parameter acting on growth and fragmentation. To this aim, we use the probabilistic individual-based interpretation of the model. We study the variations of the survival probability of the stochastic model, using a generation by generation approach. Then, making use of the link between the survival probability and the principal eigenvalue established in a previous work, we deduce the variations of the eigenvalue with respect to the parameter of the model.

math.AP↗

arXiv2015

Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models

We present two approaches to study invasion in growth-fragmentation-death mod- els. The first one is based on a stochastic individual based model, which is a piecewise deterministic branching process with a continuum of types, and the second one is based on an integro-differential model. The invasion of the population is described by the survival probability for the former model and by an eigenproblem for the latter one. We study these two notions of invasion fitness, giving different characterizations of the growth of the population, and we make links between these two complementary points of view. In particular we prove that the two approaches lead to the same crite- rion of possible invasion. Based on Krein-Rutman theory, we also give a proof of the existence of a solution to the eigenproblem, which satisfies the conditions needed for our study of the stochastic model, hence providing a set of assumptions under which both approaches can be carried out. Finally, we motivate our work in the context of adaptive dynamics in a chemostat model.

math.PR↗

arXiv2014

A modeling approach of the chemostat

Population dynamics and in particular microbial population dynamics, though they are complex but also intrinsically discrete and random, are conventionally represented as deterministic differential equations systems. We propose to revisit this approach by complementing these classic formalisms by stochastic formalisms and to explain the links between these representations in terms of mathematical analysis but also in terms of modeling and numerical simulations. We illustrate this approach on the model of chemostat.

q-bio.PE↗

arXiv2013

A mass-structured individual-based model of the chemostat: convergence and simulation

We propose a model of chemostat where the bacterial population is individually-based, each bacterium is explicitly represented and has a mass evolving continuously over time. The substrate concentration is represented as a conventional ordinary differential equation. These two components are coupled with the bacterial consumption. Mechanisms acting on the bacteria are explicitly described (growth, division and up-take). Bacteria interact via consumption. We set the exact Monte Carlo simulation algorithm of this model and its mathematical representation as a stochastic process. We prove the convergence of this process to the solution of an integro-differential equation when the population size tends to infinity. Finally, we propose several numerical simulations.

math.PR↗

arXiv2013

Estimation of the parameters of a stochastic logistic growth model

We consider a stochastic logistic growth model involving both birth and death rates in the drift and diffusion coefficients for which extinction eventually occurs almost surely. The associated complete Fokker-Planck equation describing the law of the process is established and studied. We then use its solution to build a likelihood function for the unknown model parameters, when discretely sampled data is available. The existing estimation methods need adaptation in order to deal with the extinction problem. We propose such adaptations, based on the particular form of the Fokker-Planck equation, and we evaluate their performances with numerical simulations. In the same time, we explore the identifiability of the parameters which is a crucial problem for the corresponding deterministic (noise free) model.

math.ST↗

arXiv2011

Effect of population size in a Prey-Predator model

We consider a stochastic version of the basic predator-prey differential equation model. The model, which contains a parameter ωwhich represents the number of individuals for one unit of prey -- If x denotes the quantity of prey in the differential equation model x = 1 means that there are ωindividuals in the discontinuous one -- is derived from the classical birth and death process. It is shown by the mean of simulations and explained by a mathematical analysis based on results in singular perturbation theory (the so called theory of Canards) that qualitative properties of the model like persistence or extinction are dramatically sensitive to ω. For instance, in our example, if ω= 107 we have extinction and if ω= 108 we have persistence. This means that we must be very cautious when we use continuous variables in place of jump processes in dynamic population modeling even when we use stochastic differential equations in place of deterministic ones.

math.DS↗

arXiv2011

Approximation of the Fokker-Planck equation of the stochastic chemostat

We consider a stochastic model of the two-dimensional chemostat as a diffusion process for the concentration of substrate and the concentration of biomass. The model allows for the washout phenomenon: the disappearance of the biomass inside the chemostat. We establish the Fokker-Planck associated with this diffusion process, in particular we describe the boundary conditions that modelize the washout. We propose an adapted finite difference scheme for the approximation of the solution of the Fokker-Planck equation.

math.PR↗

arXiv2011

Stochastic models of the chemostat

We consider the modeling of the dynamics of the chemostat at its very source. The chemostat is classically represented as a system of ordinary differential equations. Our goal is to establish a stochastic model that is valid at the scale immediately preceding the one corresponding to the deterministic model. At a microscopic scale we present a pure jump stochastic model that gives rise, at the macroscopic scale, to the ordinary differential equation model. At an intermediate scale, an approximation diffusion allows us to propose a model in the form of a system of stochastic differential equations. We expound the mechanism to switch from one model to another, together with the associated simulation procedures. We also describe the domain of validity of the different models.

q-bio.QM↗

arXiv2011

A Markov model of land use dynamics

The application of the Markov chain to modeling agricultural succession is well known. In most cases, the main problem is the inference of the model, i.e. the estimation of the transition matrix. In this work we present methods to estimate the transition matrix from historical observations. In addition to the estimator of maximum likelihood (MLE), we also consider the Bayes estimator associated with the Jeffreys prior. This Bayes estimator will be approximated by a Markov chain Monte Carlo (MCMC) method. We also propose a method based on the sojourn time to test the adequation of Markov chain model to the dataset.

math.PR↗

arXiv2010

A spatially explicit Markovian individual-based model for terrestrial plant dynamics

An individual-based model (IBM) of a spatiotemporal terrestrial ecological population is proposed. This model is spatially explicit and features the position of each individual together with another characteristic, such as the size of the individual, which evolves according to a given stochastic model. The population is locally regulated through an explicit competition kernel. The IBM is represented as a measure-valued branching/diffusing stochastic process. The approach allows (i) to describe the associated Monte Carlo simulation and (ii) to analyze the limit process under large initial population size asymptotic. The limit macroscopic model is a deterministic integro-differential equation.

math.PR↗

arXiv2006

Parallel and interacting Markov chains Monte Carlo method

In many situations it is important to be able to propose $N$ independent realizations of a given distribution law. We propose a strategy for making $N$ parallel Monte Carlo Markov Chains (MCMC) interact in order to get an approximation of an independent $N$-sample of a given target law. In this method each individual chain proposes candidates for all other chains. We prove that the set of interacting chains is itself a MCMC method for the product of $N$ target measures. Compared to independent parallel chains this method is more time consuming, but we show through concrete examples that it possesses many advantages: it can speed up convergence toward the target law as well as handle the multi-modal case.

math.PR↗