Table of Contents

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).

- Vlad Bally (Université Paris-Est-Marne-La-Vallée)
- Joris Bierkens (TU Delft)
- Emeric Bouin (Université Paris-Dauphine)
- Manon Costa (Université de Toulouse)
- Pierre Gabriel (Université de Versailles-Saint-Quentin)
- Dan Goreac (Université Paris Est Marne La Vallée)
- Ankit Gupta (ETH Zurich)
- Wilhelm Huisinga (Potsdam)
- Thomas Lepoutre (INRIA Rhône Alpes et Université Lyon 1)
- Eva Löcherbach (Université de Cergy-Pointoise)
- Guilherme Mazanti (Université Paris-Sud)
- Diane Peurichard (Wien)
- Ovidiu Radulescu (Montpellier)
- Magali Ribot (Université d'Orléans)
- Philippe Robert (INRIA Paris)
- Édouard Strickler (Université de Neuchâtel)
- Marta Tyran-Kamińska (University of Silesia, Katowice)
- Konstantinos Zygalakis (Edinburgh)

**Monday**

- 13h30
- 15h30 Coffee break
- 16h00
**Eva Löcherbach.**Absolute continuity of the invariant measure in systems of interacting neurons - 17h00
**Wilhelm Huisinga.**Hybrid stochastic-deterministic solution of the chemical master equation

**Tuesday**

- 09h00
**Dan Goreac.**Control-Based Design Techniques for a Class of (Linear) Stochastic Networks - 10h00 Coffee break
- 10h30
**Vlad Bally.**Gaussian noise versus Poison Point measure in PDMP - 11h30
**Manon Costa.**Coevolution in slow-fast prey-predator communities - 12h30 Lunch
- 15h30 Coffee break
- 16h00
**Konstantinos Zygalakis.**Hybrid modelling of stochastic chemical kinetics - 17h00
**Émeric Bouin.**Large deviations for velocity-jump processes

**Wednesday**

- 09h30
**Marta Tyran-Kamińska.**Existence of densities for PDMPs - 10h30 Coffee break
- 11h00
**Philippe Robert.**Asymptotics of Stochastic Protein Assembly Models - 12h00 Lunch
- 14h00 Excursion to Chaumont sur Loire

**Thursday**

- 09h00
**Ankit Gupta.**Adaptive hybrid simulation and sensitivity estimation for multiscale stochastic reaction networks - 10h00 Coffee break
- 10h30
**Thomas Lepoutre.**Quantitative convergence towards a self similar profile in an age-structured renewal equation for subdiffusion - 11h30
**Joris Bierkens.**Piecewise deterministic Markov processes and efficient subsampling for MCMC - 12h30 Lunch
- 14h30
**Pierre Gabriel.**Analysis of the steady states in the leaky integrate-and-fire model - 15h30 Coffee break
- 16h00
**Magali Ribot.**A (mainly numerical) study of a hyperbolic model for chemotaxis - 17h00
**Édouard Strickler.**Random Switching between vector fields having a common equilibrium

**Friday**

- 09h00
**Diane Peurichard.**Modelling of cross-linked fiber networks: from micro to macro models - 10h00 Coffee break
- 10h30
**Guilherme Mazanti.**Deterministic and probabilistic Lyapunov exponents for linear switched systems

LAST NAME | First name | Affiliation |
---|---|---|

Agroum | Rahma | Faculté des Sciences de Tunis |

Bally | Vlad | Université Paris Est Marne la Vallée |

Bardet | Jean-Baptiste | LMRS - Université de Rouen |

Bierkens | Joris | Delft University of Technology |

Ciolek | Gabriela | Telecom ParisTech |

Cloez | Bertrand | INRA Montpellier |

Constant | Camille | LMA Université de Poitiers |

Cont | Rama | CNRS & Imperial College |

Costa | Manon | Institut de Mathématiques de Toulouse |

Fetique | Ninon | Université de François Rabelais - Tours |

Gabriel | Pierre | Université de Versailles |

Goreac | Dan | Université Paris-Est Marne-la-Vallée |

Guérin | Hélène | Université de Rennes 1 |

Gupta | Ankit | ETH Zurich |

Herbach | Ulysse | Université Claude Bernard Lyon 1 |

Huisinga | Wilhelm | Universität Potsdam |

Joubaud | Maud | université de Montpellier |

Lagasquie | Gabriel | Université François-Rabelais de Tours |

Lepoutre | Thomas | Inria |

Löcherbach | Eva | Université de Cergy-Pontoise |

Malrieu | Florent | Université François-Rabelais de Tours |

Martin | Hugo | Université Paris 6 |

Mazanti | Guilherme | Université Paris-Sud |

Monmarché | Pierre | CERMICS |

Panloup | Fabien | Université d'Angers |

Ribot | Magali | Université d'Orléans |

Robert | Philippe | INRIA Paris |

Simsekli | Umut | Télécom ParisTech |

Strickler | Edouard | Université de Neuchâtel |

Sun | Wen | INRIA PARIS |

Tyran-Kaminska | Marta | University of Silesia, Katowice |

Yvinec | Romain | INRA Nouzilly |

Zitt | Pierre-André | Université Paris Est - Marne la Vallée |

Zygalakis | Konstantinos | University of Edinburgh |

**Vlad Bally (Université Paris-Est-Marne-La-Vallée). Gaussian noise versus Poison Point measure in PDMP**

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.

**Joris Bierkens (TU Delft). Piecewise deterministic Markov processes and efficient subsampling for MCMC**

Markov chain Monte Carlo methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis- Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the Zig-Zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime (Bierkens, Fearnhead and Roberts (2016)). In this talk we will present a broad overview of these methods along with some theoretical analysis, in particular a Central Limit Theorem for the one-dimensional Zig-Zag process.

**Emeric Bouin (Université Paris-Dauphine). Large deviations for velocity-jump processes**

In this talk, I will present some results concerning the study of large deviations for velocity jump processes from a PDE point of view. The Chapman-Kolmogorov equation of the process being a kinetic equation, I will show how to perform an Evans-Souganidis/Freidlin type of approach directly at the kinetic level. The talk will also underline the differences between the results we obtain, and the classical results obtained for the macroscopic limit of the process (the heat equation).

**Manon Costa (Université de Toulouse). Coevolution in slow-fast prey-predator communities**

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.

**Pierre Gabriel (Université de Versailles-Saint-Quentin). Analysis of the steady states in the leaky integrate-and-fire model**

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.

**Dan Goreac (Université Paris Est Marne La Vallée. Control-Based Design Techniques for a Class of (Linear) Stochastic Networks**

We present some (controllability-type) results on a class of continuous and discrete-time models of gene networks. These models are inspired by multi-stable complex systems in which a particular stable regime is characterized by the attainability of predefined targets. The mathematical description relies on controlled piecewise linear dynamics and multiplicative noise. Should the time allow it, we will hint on alternative approaches (particularly useful for nonlinear dynamics).

**Ankit Gupta (ETH Zurich). Adaptive hybrid simulation and sensitivity estimation for multiscale stochastic reaction networks**

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.

**Wihelm Huisinga (Potsdam). Hybrid stochastic-deterministic solution of the chemical master equation**

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.

**Thomas Lepoutre (INRIA Rhône Alpes et Université Lyon 1). Quantitative convergence towards a self similar profile in an age-structured renewal equation for subdiffusion**

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.

**Eva Löcherbach (Université de Cergy-Pointoise). Absolute continuity of the invariant measure in systems of interacting neurons**

We consider the following model of a finite system of interacting neurons. Each neuron is represented by its membrane potential. After a random time (which has an intensity depending on the potential of the neuron), a neuron “spikes”, i.e. emits an action potential. At this spiking time, the potential of the spiking neuron is reset to zero, while all other neurons receive an additional amount of potential. In between successive spikes, the value of the potential of a neuron follows a deterministic evolution (basically, some leak effects imply that the potential values are all attracted to some equilibrium potential). As a consequence, the system follows an evolution which is described by a Piecewise Deterministic Markov Process (PDMP) with degenerate transitions.

**Guilherme Mazanti (Université Paris-Sud). Deterministic and probabilistic Lyapunov exponents for linear switched systems**

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.

**Diane Peurichard (Wien). Modelling of cross-linked fiber networks: from micro to macro models**

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.

**Magali Ribot (Université d'Orléans). A (mainly numerical) study of a hyperbolic model for chemotaxis**

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.

**Philippe Robert (INRIA Paris). Asymptotics of Stochastic Protein Assembly Models**

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.

**Édouard Strickler (Université de Neuchâtel). Random Switching between vector fields having a common equilibrium**

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.

**Marta Tyran-Kamińska (University of Silesia, Katowice): Existence of densities for PDMPs**

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.

**Konstantinos Zygalakis (Edingburg): Hybrid modelling of stochastic chemical kinetics**

It is well known that stochasticity can play a fundamental role in various biochemical processes, such as cell regulatory networks and enzyme cascades. Isothermal, well-mixed systems can be adequately modelled by Markov processes and, for such systems, methods such as Gillespie's algorithm are typically employed. While such schemes are easy to implement and are exact, the computational cost of simulating such systems can become prohibitive as the frequency of the reaction events increase. This has motivated numerous coarse grained schemes, where the 'fast' reactions are approximated either using Langevin dynamics or deterministically. While such approaches provide a good approximation for systems where all reactants are present in large concentrations, the approximation breaks down when the fast chemical species exist in small concentrations, giving rise to significant errors in the simulation. This is particularly problematic when using such methods to compute statistics of extinction times for chemical species, as well as computing observables of cell cycle models. In this talk, we present a hybrid scheme for simulating well-mixed stochastic kinetics, using Gillespie-type dynamics to simulate the network in regions of low reactant concentration, and chemical Langevin dynamics when the concentrations of all species is large. These two regimes are coupled via an intermediate region in which a 'blended' jump-diffusion model is introduced. Examples of gene regulatory networks involving reactions occuring at multiple scales, as well as a cell-cycle model are simulated, using the exact and hybrid scheme, and compared, both in terms weak error, as well as computational cost. Additionally we will discuss how this algorithm can use to speed up parameter inference for stochastic chemical kinetics.

The participants are welcome from Monday morning to Friday afternoon in Seillac. The train station is “Onzain-Chaumont sur Loire”.