Eva Löcherbach

Bio: Eva Löcherbach is an academic researcher from University of Paris. The author has contributed to research in topics: Limit (mathematics) & Stochastic differential equation. The author has an hindex of 17, co-authored 85 publications receiving 1242 citations. Previous affiliations of Eva Löcherbach include University of Mainz & University of Bonn.

Infinite Systems of Interacting Chains with Memory of Variable Length—A Stochastic Model for Biological Neural Nets

Abstract: We consider a new class of non Markovian processes with a countable number of interacting components. At each time unit, each component can take two values, indicating if it has a spike or not at this precise moment. The system evolves as follows. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component. This class of systems extends in a non trivial way both the interacting particle systems, which are Markovian (Spitzer in Adv. Math. 5:246–290, 1970) and the stochastic chains with memory of variable length which have finite state space (Rissanen in IEEE Trans. Inf. Theory 29(5):656–664, 1983). These features make it suitable to describe the time evolution of biological neural systems. We construct a stationary version of the process by using a probabilistic tool which is a Kalikow-type decomposition either in random environment or in space-time. This construction implies uniqueness of the stationary process. Finally we consider the case where the interactions between components are given by a critical directed Erdos-Renyi-type random graph with a large but finite number of components. In this framework we obtain an explicit upper-bound for the correlation between successive inter-spike intervals which is compatible with previous empirical findings.

Hydrodynamic limit for interacting neurons

Abstract: This paper studies the hydrodynamic limit of a stochastic process describing the time evolution of a system with N neurons with mean-field interactions produced both by chemical and by electrical synapses. This system can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. At its spiking time, the membrane potential of the spiking neuron is reset to the value 0 and, simultaneously, the membrane potentials of the other neurons are increased by an amount of potential \(\frac{1}{N} \). This mimics the effect of chemical synapses. Additionally, the effect of electrical synapses is represented by a deterministic drift of all the membrane potentials towards the average value of the system. We show that, as the system size N diverges, the distribution of membrane potentials becomes deterministic and is described by a limit density which obeys a non linear PDE which is a conservation law of hyperbolic type.

On a toy model of interacting neurons

Abstract: Cet article continue l’etude du systeme stochastique de neurones en interaction introduit par De Masi, Galves, Locherbach et Presutti (J. Stat. Phys. 158 (2015) 866–902). Le systeme est compose de $N$ neurones. Chaque neurone decharge un potentiel d’action a des instants aleatoires, a un taux qui depend de son potentiel de membrane. Ce potentiel est alors remis a $0$, et tous les autres neurones recoivent une charge supplementaire de $1/N$. De plus, des synapses electriques induisent une derive deterministe qui attire le systeme vers sa valeur moyenne. Nous etablissons la propriete de propagation du chaos lorsque $N\to\infty$, vers la solution d’une equation differentielle stochastique non-lineaire a sauts. Nous ameliorons les resultats obtenus dans (J. Stat. Phys. 158 (2015) 866–902) puisque (i) nous levons la condition de support compact imposee aux donnees initiales, (ii) nous obtenons une vitesse de convergence en $1/\sqrt{N}$. Enfin, nous proposons une etude de l’equation limite : nous decrivons la forme de ses lois marginales (en temps), nous demontrons l’existence d’une unique loi invariante non-triviale et montrons que la mesure invariante triviale n’est pas attractive. Enfin, nous obtenons la convergence vers l’equilibre dans un cas particulier.

Limit Theorems for Null Recurrent Markov Processes

Abstract: Introduction Harris recurrence Stable increasing processes and Mittag Leffler processes The main theorem Proofs for subsection 3.1 - sufficient condition Proofs for subsection 3.1 - necessary condition Nummelin splitting in discrete time Nummelin-like splitting for general continuous time Harris processes and proofs for subsection 3.3 Overview: assumptions (H1) - (H6) References.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

An Introduction To The Theory Of Point Processes

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Stochastic flows and stochastic differential equations

Abstract: 1. Stochastic processes and random fields 2. Continuous semimartingales and stochastic integrals 3. Semimartingales with spatial parameter and stochastic integrals 4. Stochastic flows 5. Convergence of stochastic flows 6. Stochastic partial differential equations.