This conference aims at gathering international researchers on the topic of piecewise deterministic Markov processes.
These random processes, also known as hybrid systems, are related to PDEs (growth-fragmentation, nonlocal equations, etc) and biological models (chemotaxis, neuronal activity, population dynamics etc).
|LAST NAME||First name||Affiliation|
|Malrieu||Florent||Université François-Rabelais de Tours|
|Zitt||Pierre-André||Université Paris Est - Marne la Vallée|
|Vlad||Bally||Université Paris Est Marne la Vallée|
|Löcherbach||Eva||Université de Cergy-Pontoise|
|JOUBAUD||Maud||université de Montpellier|
|Fetique||Ninon||Université de François Rabelais - Tours|
|HERBACH||Ulysse||Université Claude Bernard Lyon 1|
|Goreac||Dan||Université Paris-Est Marne-la-Vallée|
|Costa||Manon||Institut de Mathématiques de Toulouse|
|GABRIEL||Pierre||Université de Versailles|
|MARTIN||Hugo||Université Paris 6|
|Bardet||Jean-Baptiste||LMRS - Université de Rouen|
|Strickler||Edouard||Université de Neuchâtel|
|Tyran-Kaminska||Marta||University of Silesia, Katowice|
|agroum||rahma||Faculté des Sciences de Tunid|
|Bierkens||Joris||Delft University of Technology|
|Lagasquie||Gabriel||Université François-Rabelais de Tours|
|Guérin||Hélène||Université de Rennes 1|
|Radulescu||Ovidiu||Université de Monpellier|
|CONT||Rama||CNRS & Imperial College|
|Zygalakis||Konstantinos||University of Edinburgh|
In many situations one is interested to replace the small jumps in a PDMP by a diffuse component (driven by a Bronwian motion). The motivation may come from numerical problems as well as from modeling reasons. Our aim is to obtain estimates for the error which is done by such a procedure. This estimates are more or less straightforward if one deals with a “usual” stochastic equation with jumps, but becomes more tricky if one deals with PDMP's. This is because the amplitude and the intensity of the jumps depend on the position of the solution. And then it is not trivial to prove that the semigroup associated to such equations has the property of propagation of regularity. And this property is crucial in our problem. In order to illustrate our general results we treat the example of the 2 dimensional Boltzmann equation.
In this talk we consider prey-predator communities in which predators evolve much faster than prey. We introduce and study different models for the demographic and evolutionnary dynamics for these populations as piecewise deterministic processes. We will focus on different time scales where the fast dynamic reaches an equilibrium instantaneously and consider the limiting community.
The nonlinear leaky integrate-and-fire equation models the dynamics of a connected neural network. It is a structured population model in which the potential of each neuron decreases deterministically and undergoes stochastic positive jumps with a rate related to the total activity of the network. We study the existence, uniqueness, and stability of the steady states depending on the connectivity of the network. It is a joint work with Grégory Dumont.
Stochastic reaction networks are commonly used to understand the role of intracellular noise in Systems Biology. For these models, the probability distribution of the state-vector evolves according to the Chemical Master Equation (CME), whose solutions often need to be estimated via Monte Carlo methods such as the Stochastic Simulation Algorithm (SSA) by Gillespie. Such simulations become impractical for multiscale networks which exhibit wide variations in the reaction timescales as well as the copy-number magnitudes of the constituent species. To deal with this problem, hybrid simulation approaches have been developed that approximate the stochastic reaction dynamics with a suitably constructed Piecewise Deterministic Markov process (PDMP) which is computationally easier to simulate and analyze. In my talk, I will discuss how this PDMP construction can be automated and how we can adapt this construction during the simulation-run to improve the computational performance and the accuracy of the estimated CME solution. I will also present how these PDMP approximations can be exploited for the purpose of estimating the parameter sensitivities for multiscale stochastic reaction networks. These sensitivity values are extremely important for understanding the network properties and for identifying the key reactions for a specific output of interest.
The chemical master equation (CME) is the fundamental evolution equation of the stochastic description of biochemical reaction kinetics. Typically, it is impossible to solve the CME directly so that indirect approaches based on realizations of the underlying Markov jump process are used. These, however, become numerically inefficient when the system’s dynamics include fast reaction processes or species with high population levels. In many hybrid approaches, such fast reactions are approximated as continuous processes in either a stochastic or a deterministic context. Current hybrid approaches, however, almost exclusively rely on the computation of ensembles of stochastic realizations. We present a hybrid stochastic–deterministic approach to solve the CME directly. We partition molecular species into discrete and continuous species and use the WKB (Wentzel–Kramers–Brillouin) ansatz for the conditional probability distribution function (PDF) of the continuous species (given a discrete state) in combination with Laplace’s method of integral approximation. The resulting hybrid stochastic–deterministic evolution equations comprise a CME with averaged propensities for the PDF of the discrete species that is coupled to an evolution equation of the related expected levels of the continuous species for each discrete state. We illustrate the performance of the new hybrid stochastic–deterministic approach in an application to model systems of biological interest.
Continuous-time random walks are generalisations of random walks frequently used to account for the consistent observations that many molecules in living cells undergo anomalous diffusion, i.e. subdiffusion. Here, we describe the subdiffusive continuous-time random walk using age-structured partial differential equations with age renewal upon each walker jump, where the age of a walker is the time elapsed since its last jump. In the spatially-homogeneous (zero-dimensional) case, we follow the evolution in time of the age distribution. An approach inspired by relative entropy techniques allows us to obtain quantitative explicit rates for the convergence of the age distribution to a self-similar profile, which corresponds to convergence to a stationnary profile for the rescaled variables. An important difficulty arises from the fact that the equation in self-similar variables is not autonomous and we do not have a specific analytical solution. Therefore, in order to quantify the latter convergence, we estimate attraction to a time-dependent “pseudo-equilibrium”, which in turn converges to the stationnary profile.
When considering the exponential behavior of a linear switched system, one may take a deterministic or a probabilistic point of view. The deterministic case consists in characterizing the maximal Lyapunov exponent of the system with respect to all possible switching signals belonging to a given class, whereas, in the probabilistic approach, one provides a random model for the switching signals and considers the almost sure Lyapunov exponent. The latter is always less than or equal to the former: the “worst” possible exponential behavior with random switching is at most as bad as the “worst” possible deterministic exponential behavior. In this talk, we address the question of how much smaller the probabilistic Lyapunov exponent can be. We will prove that, for a certain class of deterministic switched systems, the maximal Lyapunov exponent is lower bounded, but, for its probabilistic counterpart, the almost sure Lyapunov exponent can be made arbitrarily small by a suitable choice of parameters appearing in the system, thus proving that the gap between deterministic and probabilistic Lyapunov exponents can be arbitrarily large. This talk is based on joint works with Yacine Chitour, Fritz Colonius, and Mario Sigalotti.
In this talk, we will study a complex network composed of fibers having the ability to cross-link or unlink each other via random processes in time (Poisson processes), and to align with each other at the cross links. This model aims to describe networks of collagen fibers in a fibrous tissue. We will first present a microscopic model which features the following basic rules: We assume the existence of a fiber unit element (or monomer) modeled as a line segment of fixed length. We suppose that two fiber elements that cross each other may form a link, thereby creating a longer fiber. The fibers have the ability to branch off and to achieve complex network topologies. We include fiber resistance to bending by assuming the existence of a torque which, in the absence of any other force, makes the two linked fiber elements align with each other. Fibers are also subject to random positional and orientational noise and to external positional and orientational potential forces. Finally, cross-links may also be removed to model possible fiber breakage or depolymerization.
We then formally derive a kinetic model for the fiber and cross-links distribution functions, and consider the fast linking/unlinking regime in which the model can be reduced to the fiber distribution function only. Then, we investigate its diffusion limit. The resulting macroscopic model consists of a system of nonlinear diffusion equations for the fiber density and mean orientation. In the case of a homogeneous fiber density, we show that the model is elliptic. We finally will present simulation results which show the good correspondence between the microscopic and the macroscopic models.
Le chimiotactisme est un phénomène présent dans de nombreux comportements biologiques, où les organismes se déplacent sous l'effet d'une substance chimique. Cet exposé commencera par une introduction à la modélisation du chimiotactisme et aux liens entre les différents modèles déterministes pour le chimiotactisme. Nous donnerons ensuite quelques renseignements sur un modèle hyperbolique de chimiotactisme : calcul de solutions stationnaires, simulations numériques, schéma numérique adapté de type well-balanced, comparaison avec les autres modèles.
Self-assembly of proteins is a biological phenomenon which gives rise to spontaneous formation of amyloid fibrils or polymers. A striking feature of this process is that the starting point of the chemical reaction, called nucleation exhibits an important variability among replicated experiments. In the case of neuro-degenerative diseases, such as Alzheimer or Prion diseases, this variation can be of the order of several years. Two populations of chemical components are involved: monomers (proteins) and polymers. Initially there are only monomers. There are two reactions for the polymerization of a monomer: either two monomers collide to combine into a polymer or a monomer is polymerised after the encounter of a polymer. After a general introduction, the talk will present several stochastic models of increasing complexity from a basic two-dimensional Markovian model to more sophisticated multi-dimensional processes. The problem of fitting parameters will be presented with data coming from experiments in biochemistry. Mathematically, the techniques use classical stochastic calculus methods, stochastic averaging principles and a set of classical ODE's, the Becker-Doring equations. It is shown that a set of different time scales play an important role for these models.
The talk is based on joint works with Marie Doumic, Sarah Eugène, Wen Sun and Wei-Feng Xue.
When one consider a non-linear system x'=F(x), the local behavior at an equilibrium point (say 0), is given by the sign of the real parts of the eigenvalues of the Jacobian matrix of F at 0. The aim of this talk is to give similar results when X is now a PDMP that randomly switches between several vector fields having 0 as an equilibrium. Briefly put, our main result is that the long term behavior of the process is determined by the behavior of the process obtained by linearization at the origin and, under suitable irreducibility and hypoellipticity conditions, by the sign of the top Lyapunov exponent. We provide several paradoxical examples coming from epidemiological models where all the deterministic systems converge to 0 but when switching between them, the PDMP go away from 0…and vice versa. This talk is based on a joint work with Michel Benaïm.
The evolution of densities of the given piecewise deterministic Markov process can be studied with the help of substochastic semigroups on the space of integrable functions. We show how one can associate the semigroup to the given process using the Kato perturbation theorem. Nontrivial fixed points of the semigroup give rise to absolutely continuous invariant measures for the process. We also give relationships between the existence of invariant densities for the semigroup and for the discrete-time Markov process obtained by taking the continuous time process at the consecutive jump times.
The participants are welcome from Monday morning to Friday afternoon in Seillac. The train station is “Onzain-Chaumont sur Loire”.