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1. Hopf bifurcation in a Mean-Field model of spiking neurons

Author

Etienne Tanré, Romain Veltz, Quentin Cormier, Université Côte d'Azur (UCA), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Mathématiques pour les Neurosciences (MATHNEURO), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), European Project: 945539,H2020,H2020-SGA-FETFLAG-HBP-2019,HBP SGA3(2020), Institut National de Recherche en Informatique et en Automatique (Inria), and European Project: 945539,HBP SGA3(2020)

Subjects
Statistics and Probability, McKean-Vlasov SDE, Holomorphic function, Mean-field interaction, Piecewise deterministic Markov process, 01 natural sciences, Measure (mathematics), 010104 statistics & probability, symbols.namesake, Mathematics - Analysis of PDEs, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Piecewise-deterministic Markov process, Hopf bifurcation, 0101 mathematics, Long time behavior, Mathematics, Toy model, Markov chain, 010102 general mathematics, Probability (math.PR), 60K35 (Primary) 35B10, 35B32, 60H10 (Secondary), Volterra integral equation, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Discrete time and continuous time, symbols, Invariant measure, Statistics, Probability and Uncertainty, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract

International audience; We study a family of non-linear McKean-Vlasov SDEs driven by a Poisson measure, modelling the mean-field asymptotic of a network of generalized Integrate-and-Fire neurons.We give sufficient conditions to have periodic solutions through a Hopf bifurcation. Our spectral conditions involve the location of the roots of an explicit holomorphic function. The proof relies on two main ingredients. First, we introduce a discrete time Markov Chain modeling the phases of the successive spikes of a neuron. The invariant measure of this Markov Chain is related to the shape of the periodic solutions. Secondly, we use the Lyapunov-Schmidt method to obtain self-consistent oscillations. We illustrate the result with a toy model for which all the spectral conditions can be analytically checked.

Published
2020

2. Long time behavior of a mean-field model of interacting neurons

Author

Romain Veltz, Quentin Cormier, Etienne Tanré, TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Mathématiques pour les Neurosciences (MATHNEURO), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and European Project: 785907,H2020,HBP SGA2(2018)

Subjects
Statistics and Probability, Current (mathematics), Applied Mathematics, 010102 general mathematics, Probability (math.PR), 60B10 (Primary) 60G55, 60K35, 45D05, 35Q92 (Secondary), Type (model theory), 01 natural sciences, Integral equation, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Stochastic differential equation, MSC Primary 60B10, secondary 60G55, 60K35, 45D05, 35Q92, Mean field theory, Modeling and Simulation, Convergence (routing), FOS: Mathematics, Constant current, Applied mathematics, Invariant measure, 0101 mathematics, Mathematics - Probability, Mathematics
Abstract

International audience; We study the long time behavior of the solution to some McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic external quantity (we call it the external current). For constant current, we obtain the convergence to the invariant measure. Using a perturbation method, we extend this result to more general external currents. Finally, we prove the result for the non-linear McKean-Vlasov equation.

Published
2020

3. On a toy network of neurons interacting through their dendrites

Author

Nicolas Fournier, Etienne Tanré, Romain Veltz, Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Mathématiques pour les Neurosciences (MATHNEURO), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Biological neural networks, Ulam’s problem, Dendrite, Longest increasing subsequence, 01 natural sciences, Combinatorics, 010104 statistics & probability, FOS: Mathematics, medicine, nonlinear stochastic differential equations, Limit (mathematics), Uniqueness, Ulam's problem, 0101 mathematics, 60K35, 60J75, 92C20, Mathematics, Propagation of chaos, Quantitative Biology::Neurons and Cognition, Probability (math.PR), 010102 general mathematics, Ode, Front (oceanography), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], medicine.anatomical_structure, 60K35, FOS: Biological sciences, Quantitative Biology - Neurons and Cognition, 92C20, Mean-field limit, Neurons and Cognition (q-bio.NC), Soma, Electric potential, 60J75, Statistics, Probability and Uncertainty, Mathematics - Probability
Abstract

Consider a large number $n$ of neurons, each being connected to approximately $N$ other ones, chosen at random. When a neuron spikes, which occurs randomly at some rate depending on its electric potential, its potential is set to a minimum value $v_{\mathrm{min}}$, and this initiates, after a small delay, two fronts on the (linear) dendrites of all the neurons to which it is connected. Fronts move at constant speed. When two fronts (on the dendrite of the same neuron) collide, they annihilate. When a front hits the soma of a neuron, its potential is increased by a small value $w_{n}$. Between jumps, the potentials of the neurons are assumed to drift in $[v_{\min },\infty )$, according to some well-posed ODE. We prove the existence and uniqueness of a heuristically derived mean-field limit of the system when $n,N\to \infty $ with $w_{n}\simeq N^{-1/2}$. We make use of some recent versions of the results of Deuschel and Zeitouni (Ann. Probab. 23 (1995) 852–878) concerning the size of the longest increasing subsequence of an i.i.d. collection of points in the plan. We also study, in a very particular case, a slightly different model where the neurons spike when their potential reach some maximum value $v_{\mathrm{max}}$, and find an explicit formula for the (heuristic) mean-field limit.

Published
2020

4. Penalisation techniques for one-dimensional reflected rough differential equations

Author

Etienne Tanré, Soledad Torres, Alexandre Richard, Fédération de Mathématiques de l'Ecole Centrale Paris (FR3487), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Mathématiques et Informatique pour la Complexité et les Systèmes (MICS), CentraleSupélec-Université Paris-Saclay, TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Centro de Investigación y Modelamiento de Fenómenos Aleatorios – Valparaíso (CIMFAV), Universidad de Valparaiso [Chile], ECOS-Sud Program Chili-France C15E05Math-AmSud project SARCS.T. is partially supported by the Project Fondecyt N. 1171335, Fédération de Mathématiques de CentraleSupélec (FR3487), CentraleSupélec, and Ecole Centrale Paris-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Reflected rough differential equation, Differential equation, Gaussian, Boundary (topology), 01 natural sciences, Gaussian noise, 010104 statistics & probability, symbols.namesake, Convergence (routing), FOS: Mathematics, Limit of a sequence, 0101 mathematics, Mathematics, Sequence, Lebesgue measure, Skorokhod problem, 34F05, 60G15, 60H10, Probability (math.PR), 010102 general mathematics, Mathematical analysis, Existence theorem, 16. Peace & justice, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Penalisation, symbols, Mathematics - Probability
Abstract

International audience; In this paper we solve real-valued rough differential equations (RDEs) reflected on an irregular boundary. The solution $Y$ is constructed as the limit of a sequence $(Y^n)_{n\in\mathbb{N}}$ of solutions to RDEs with unbounded drifts $(\psi_n)_{n\in\mathbb{N}}$. The penalisation $\psi_n$ increases with $n$. Along the way, we thus also provide an existence theorem and a Doss-Sussmann representation for RDEs with a drift growing at most linearly. In addition, a speed of convergence of the sequence of penalised paths to the reflected solution is obtained. We finally use the penalisation method to prove that the law at time $t>0$ of some reflected Gaussian RDE is absolutely contiuous with respect to the Lebesgue measure.

Published
2019

5. Network of interacting neurons with random synaptic weights

Author

Etienne Tanré, Cyrille Mascart, Julien Chevallier, François Delarue, Paolo Grazieschi, Marta Leocata, Department of Mathematics, University of Warwick, Warwick Mathematics Institute (WMI), University of Warwick [Coventry]-University of Warwick [Coventry], University of Pisa - Università di Pisa, Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe BIOINFO, Modèles Discrets pour les Systèmes Complexes (Laboratoire I3S - MDSC), Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Statistique pour le Vivant et l’Homme (SVH), Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Simuler et calibrer des modèles stochastiques (TOSCA-POST), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-15-IDEX-0001,UCA JEDI,Idex UCA JEDI(2015), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), and Laboratoire Jean Alexandre Dieudonné (LJAD)

Subjects
Collective behavior, T57-57.97, Applied mathematics. Quantitative methods, Quantitative Biology::Neurons and Cognition, Computer science, 010102 general mathematics, Degenerate energy levels, Type (model theory), Topology, 01 natural sciences, Connection (mathematics), Hodgkin–Huxley model, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, QA1-939, Graph (abstract data type), Limit (mathematics), 0101 mathematics, Realization (systems), Mathematics
Abstract

Since the pioneering works of Lapicque [17] and of Hodgkin and Huxley [16], several types of models have been addressed to describe the evolution in time of the potential of the membrane of a neuron. In this note, we investigate a connected version of N neurons obeying the leaky integrate and fire model, previously introduced in [1–3,6,7,15,18,19,22]. As a main feature, neurons interact with one another in a mean field instantaneous way. Due to the instantaneity of the interactions, singularities may emerge in a finite time. For instance, the solution of the corresponding Fokker-Planck equation describing the collective behavior of the potentials of the neurons in the limit N ⟶ ∞ may degenerate and cease to exist in any standard sense after a finite time. Here we focus out on a variant of this model when the interactions between the neurons are also subjected to random synaptic weights. As a typical instance, we address the case when the connection graph is the realization of an Erdös-Renyi graph. After a brief introduction of the model, we collect several theoretical results on the behavior of the solution. In a last step, we provide an algorithm for simulating a network of this type with a possibly large value of N.

Published
2019

6. An integrate-and-fire model to generate spike trains with long-range dependence

Author

Patricio Orio, Alexandre Richard, Etienne Tanré, Mathématiques et Informatique pour la Complexité et les Systèmes (MICS), CentraleSupélec, Fédération de Mathématiques de l'Ecole Centrale Paris (FR3487), Ecole Centrale Paris-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Centro Interdisciplinario de Neurociencia de Valparaiso (CINV), Universidad de Valparaiso [Chile], TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ECOS-Sud Program Chili-France C15E05, Millenium scientific initiative of the Chilean Ministry of Economy, Development, and Tourism (P09-022-F), Advanced center for electrical and electronic engineering, Conicyt (FB0008), European Project: 720270,H2020 Pilier Excellent Science,H2020-Adhoc-2014-20,HBP SGA1(2016), and Centre National de la Recherche Scientifique (CNRS)-Ecole Centrale Paris-CentraleSupélec

Subjects
FOS: Computer and information sciences, Stationarity, Computer science, Cognitive Neuroscience, Models, Neurological, Action Potentials, Markov process, Statistics - Applications, 01 natural sciences, Measure (mathematics), Computer Science::Digital Libraries, Stochastic Integrate-and-Fire model, 03 medical and health sciences, Cellular and Molecular Neuroscience, symbols.namesake, 0302 clinical medicine, Long-range dependence, 0103 physical sciences, Range (statistics), Humans, Applications (stat.AP), Statistical physics, 010306 general physics, Neurons, Stochastic Processes, Interspike interval statistics, Quantitative Biology::Neurons and Cognition, Sensory Systems, Term (time), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Noise, Quantitative Biology - Neurons and Cognition, FOS: Biological sciences, Theory of computation, symbols, Neurons and Cognition (q-bio.NC), Train, Spike (software development), 030217 neurology & neurosurgery
Abstract

International audience; Long-range dependence (LRD) has been observed in a variety of phenomena in nature, and for several years also in the spiking activity of neurons. Often, this is interpreted as originating from a non-Markovian system. Here we show that a purely Markovian integrate-and-fire (IF) model, with a noisy slow adaptation term, can generate interspike intervals (ISIs) that appear as having LRD. However a proper analysis shows that this is not the case asymptotically. For comparison, we also consider a new model of individual IF neuron with fractional (non-Markovian) noise. The correlations of its spike trains are studied and proven to have LRD, unlike classical IF models. On the other hand, to correctly measure long-range dependence, it is usually necessary to know if the data are stationary. Thus, a methodology to evaluate stationarity of the ISIs is presented and applied to the various IF models. We explain that Markovian IF models may seem to have LRD because of non-stationarities.

Published
2018

7. Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size

Author

Etienne Tanré, Pierre Guiraud, Centro de Investigación y Modelamiento de Fenómenos Aleatorios – Valparaíso (CIMFAV), Universidad de Valparaiso [Chile], TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and Mathamsud Project 13MATH-04 - SIN DIUV 45/2013 (University of Valparaiso)Inria Equipe Associée Anestoc-Tosca

Subjects
0301 basic medicine, Gaussian, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], 92B25, 92B20, 60F10, 60J75, Synchronization, Stability (probability), Upper and lower bounds, Network of LIF neurons, Stochastic stability, 03 medical and health sciences, symbols.namesake, 0302 clinical medicine, Control theory, Robustness (computer science), Synchronization (computer science), Interacting Ornstein Uhlenbeck processes, FOS: Mathematics, Discrete Mathematics and Combinatorics, AMS: 92B25, 92B20, 60F10, 60J75, Stochastic neural network, Mathematics, Artificial neural network, Applied Mathematics, Probability (math.PR), Excitatory synapses, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 030104 developmental biology, FOS: Biological sciences, Quantitative Biology - Neurons and Cognition, symbols, Neurons and Cognition (q-bio.NC), Rate function, 030217 neurology & neurosurgery, Mathematics - Probability
Abstract

International audience; In the present paper, we study the synchronization in a model of neural network which can be considered as a noisy version of the model of \citet{mirollo1990synchronization}, namely, fully-connected and totally excitatory integrate and fire neural network with Gaussian white noises. Using a large deviation principle, we prove the stability of the synchronized state under stochastic perturbations. Then, we give a lower bound on the probability of synchronization for networks which are not initially synchronized. This bound shows the robustness of the emergence of synchronization in presence of small stochastic perturbations.

Published
2016

8. The first-passage time of the Brownian motion to a curved boundary: an algorithmic approach

Author

Samuel Herrmann, Etienne Tanré, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ANR-09-BLAN-0008,MANDy(2009), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), TO Simulate and CAlibrate stochastic models ( TOSCA ), Inria Sophia Antipolis - Méditerranée ( CRISAM ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut Élie Cartan de Lorraine ( IECL ), Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-09-BLAN-0008-01,MANDy,Mathematical Analysis of Neuronal Dynamics ( 2009 ), and Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Boundary (topology), 01 natural sciences, 010104 statistics & probability, Stopping time, FOS: Mathematics, Almost surely, 0101 mathematics, 65C05, 65N75, 60G40, Brownian motion, Mathematics, Randomized algorithm, Applied Mathematics, Probability (math.PR), Mathematical analysis, Hitting time, 010101 applied mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Computational Mathematics, Discrete time and continuous time, Rate of convergence, Potential theory, First-hitting-time model, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics - Probability, First-passage time, MSC: 65C05, 65N75, 60G40
Abstract

International audience; Under some weak conditions, the first-passage time of the Brownian motion to a continuous curved boundary is an almost surely finite stopping time. Its probability density function (pdf) is explicitly known only in few particular cases. Several mathematical studies proposed to approximate the pdf in a quite general framework or even to simulate this hitting time using a discrete time approximation of the Brownian motion. The authors study a new algorithm which permits to simulate the first-passage time using an iterating procedure. The convergence rate presented in this paper suggests that the method is very efficient.

Published
2016

9. Liquidity costs: a new numerical methodology and an empirical study

Author

Etienne Tanré, Christophe Michel, Denis Talay, Victor Reutenauer, Service Interest Rates and Hybrid Quantitative Research (FIM), CALYON, Fotonower, TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Stochastic control, Mathematical optimization, Stochastic Algorithms, Interest rates derivatives, Computer science, Applied Mathematics, Floating interest rate, 010102 general mathematics, Computational Finance (q-fin.CP), Function (mathematics), [QFIN.RM]Quantitative Finance [q-fin]/Risk Management [q-fin.RM], 01 natural sciences, Market liquidity, Numerical methodology, FOS: Economics and business, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Empirical research, Quantitative Finance - Computational Finance, Simple (abstract algebra), Stochastic Algorithms,Optimization optimisation,Interest rates derivatives, Optimization optimisation, 0101 mathematics, Fixed interest rate loan, Finance
Abstract

International audience; We consider rate swaps which pay a fixed rate against a floating rate in presence of bid-ask spread costs. Even for simple models of bid-ask spread costs, there is no explicit optimal strategy minimizing a risk measure of the hedging error. We here propose an efficient algorithm, based on the stochas-tic gradient method, to obtain an approximate optimal strategy without solving a stochastic control problem. We validate our algorithm by numer-ical experiments. We also develop several variants of the algorithm and discuss their performances in terms of the numerical parameters and the liquidity cost.

Published
2015

10. Global solvability of a networked integrate-and-fire model of McKean-Vlasov type

Author

Sylvain Rubenthaler, James Inglis, Etienne Tanré, François Delarue, Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ANR-09-BLAN-0008,MANDy(2009), Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)

Subjects
Statistics and Probability, Population, Proportionality (mathematics), 01 natural sciences, integrate-and-fire network, McKean nonlinear diffusion process, neuroscience, 010104 statistics & probability, FOS: Mathematics, Applied mathematics, Uniqueness, Renewal theory, 0101 mathematics, nonhomogeneous diffusion process, education, Mathematics, education.field_of_study, Quantitative Biology::Neurons and Cognition, renewal process, [SCCO.NEUR]Cognitive science/Neuroscience, 010102 general mathematics, Probability (math.PR), first hitting time density estimates, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60K35, 92C20, 60H10, Statistics, Probability and Uncertainty, 60J75, Mathematics - Probability
Abstract

We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by $\alpha$, is of great importance as the resulting system is known to blow-up for large values of $\alpha$. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when $\alpha$ is small enough., Comment: Published at http://dx.doi.org/10.1214/14-AAP1044 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2015

11. Range of Brownian Motion with Drift

Author

Etienne Tanré, Pierre Vallois, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Statistics and Probability, 010104 statistics & probability, Range (mathematics), Mathematics::Probability, General Mathematics, 010102 general mathematics, Mathematical analysis, 0101 mathematics, Statistics, Probability and Uncertainty, 01 natural sciences, Brownian motion, Mathematics
Abstract

Let (B (t); t 0) be a Brownian motion with drift > 0, starting at 0. Let us define by induction S1 = inf t 0 B (t), 1 the last time such that

Published
2006

12. On long time behavior of some coagulation processes

Author

Bernard Roynette, Etienne Tanré, Nicolas Fournier, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Particle system, Coalescence (physics), Stochastic process, Differential equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Coalescence, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Modelling and Simulation, Modeling and Simulation, Almost surely, 0101 mathematics, Non-linear stochastic differential equations with jumps, Jump process, Brownian motion, Excitation, Mathematics
Abstract

We consider an infinite system of particles characterized by their position and mass, in which coalescence occurs. Each particle endures Brownian excitation, and is subjected to the attraction of a potential. We define a stochastic process ( X t , M t ) t ⩾0 describing the evolution of the position and mass of a typical particle. We show that under some conditions, the mass process M t tends almost surely to infinity, while the position process X t tends almost surely to 0, as time tends to infinity.

Published
2004

13. [Untitled]

Author

Nicolas Fournier, Madalina Deaconu, and Etienne Tanré

Subjects
Statistics and Probability, Particle system, Smoluchowski coagulation equation, General Mathematics, Open problem, 010102 general mathematics, Mathematical analysis, Monte Carlo method, Probabilistic logic, 01 natural sciences, 010104 statistics & probability, symbols.namesake, Rate of convergence, Convergence (routing), symbols, Applied mathematics, 0101 mathematics, Mathematics, Central limit theorem
Abstract

By continuing the probabilistic approach of Deaconu et al. (2001), we derive a stochastic particle approximation for the Smoluchowski coagulation equations. A convergence result for this model is obtained. Under quite stringent hypothesis we obtain a central limit theorem associated with our convergence. In spite of these restrictive technical assumptions, the rate of convergence result is interesting because it is the first obtained in this direction and seems to hold numerically under weaker hypothesis. This result answers a question closely connected to the Open Problem 16 formulated by Aldous (1999).

Published
2003

14. Particle systems with a singular mean-field self-excitation. Application to neuronal networks

Author

François Delarue, James Inglis, Etienne Tanré, Sylvain Rubenthaler, Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Mathematical and Computational Neuroscience (NEUROMATHCOMP), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS)

Subjects
Statistics and Probability, propagation of chaos, Population, Topology (electrical circuits), Type (model theory), 01 natural sciences, McKean nonlinear diffusion process, integrate-and-fire network, neuroscience, 010104 statistics & probability, Control theory, FOS: Mathematics, 0101 mathematics, education, Mathematics, Particle system, education.field_of_study, Quantitative Biology::Neurons and Cognition, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Probability (math.PR), counting process, Skorohod M1 topology, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Nonlinear system, Mean field theory, Modeling and Simulation, Spike (software development), Mathematics - Probability, Excitation
Abstract

International audience; We discuss the construction and approximation of solutions to a nonlinear McKean-Vlasov equation driven by a singular self-excitatory interaction of the mean-field type. Such an equation is intended to describe an infinite population of neurons which interact with one another. Each time a proportion of neurons 'spike', the whole network instantaneously receives an excitatory kick. The instantaneous nature of the excitation makes the system singular and prevents the application of standard results from the literature. Making use of the Skorohod M1 topology, we prove that, for the right notion of a 'physical' solution, the nonlinear equation can be approximated either by a finite particle system or by a delayed equation. As a by-product, we obtain the existence of 'synchronized' solutions, for which a macroscopic proportion of neurons may spike at the same time.

Published
2014
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15. Monte Carlo approximations of the Neumann problem

Author

Etienne Tanré, Sylvain Maire, Laboratoire des Sciences de l'Information et des Systèmes (LSIS), Centre National de la Recherche Scientifique (CNRS)-Arts et Métiers Paristech ENSAM Aix-en-Provence-Université de Toulon (UTLN)-Aix Marseille Université (AMU), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Arts et Métiers Paristech ENSAM Aix-en-Provence-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Pointwise, Stochastic process, Applied Mathematics, Quantum Monte Carlo, Mathematical analysis, Monte Carlo method, Probability (math.PR), Boundary (topology), 010103 numerical & computational mathematics, Mixed boundary condition, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Neumann Problem, Monte Carlo Methods, Neumann boundary condition, FOS: Mathematics, Spectral Methods, Boundary value problem, 0101 mathematics, Local Time Approximation, Mathematics - Probability, Mathematics
Abstract

Accepted for publication; International audience; We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes especially devised for local time approximation on the boundary of the domain. Global approximations are computed thanks to a stochastic spectral formulation taking into account the property of zero mean value of the solution. This stochastic formulation is asymptotically perfect in terms of conditioning. Numerical examples are given on the Laplace operator on a square domain with both pure Neumann and mixed Dirichlet-Neumann boundary conditions. A more general convection-diffusion equation is also numerically studied.

Published
2012
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16. Optimal stopping problems for some Markov processes

Author

Etienne Tanré, Pierre Patie, Mamadou Cissé, Ecole Nationale de la Statistique et de l'Analyse Economique (ENSAE), Ecole Nationale de la Statistique et de l'Analyse Economique, Université libre de Bruxelles (ULB), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and Finrisk

Subjects
Statistics and Probability, Class (set theory), Feller processes, excessive functions, Stochastic calculus, Markov process, 01 natural sciences, Potential theory, Linear diffusion, Negative type, 010104 statistics & probability, symbols.namesake, Optimal stopping problems, FOS: Mathematics, Applied mathematics, Optimal stopping, 0101 mathematics, 60G40, Mathematics, 60J60, Discounting, 010102 general mathematics, Probability (math.PR), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Doob’s h-transform, symbols, Statistics, Probability and Uncertainty, 60J75, 60G40 (Primary) 60J60, 60J75 (Secondary), Mathematics - Probability, Doob's $h$-transform
Abstract

In this paper, we solve explicitly the optimal stopping problem with random discounting and an additive functional as cost of observations for a regular linear diffusion. We also extend the results to the class of one-sided regular Feller processes. This generalizes the result of Beibel and Lerche [Statist. Sinica 7 (1997) 93-108] and [Teor. Veroyatn. Primen. 45 (2000) 657-669] and Irles and Paulsen [Sequential Anal. 23 (2004) 297-316]. Our approach relies on a combination of techniques borrowed from potential theory and stochastic calculus. We illustrate our results by detailing some new examples ranging from linear diffusions to Markov processes of the spectrally negative type., Published in at http://dx.doi.org/10.1214/11-AAP795 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2012

17. Viscosity solutions to optimal portfolio allocation problems in models with random time changes and transaction costs

Author

Denis Talay, Christophette Blanchet-Scalliet, Rajna Gibson Brandon, Benoîte de Saporta, Etienne Tanré, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Swiss Finance Institute [Geneva], Swiss Finance Institute, Groupe de Recherche en Economie Théorique et Appliquée (GREThA), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Quality control and dynamic reliability (CQFD), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), TOSCA, Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria), Albrecher Hansjörg, Runggaldier Wolfgang J. and Schachermayer Walter, Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Stochastic control, Mathematical optimization, viscosity solutions, 010102 general mathematics, Mathematics::Optimization and Control, Hamilton–Jacobi–Bellman equation, portfolio allocation, 01 natural sciences, [QFIN.CP]Quantitative Finance [q-fin]/Computational Finance [q-fin.CP], dynamic programming principle, transaction costs, Dynamic programming, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, HJB inequalities, Viscosity (programming), Bellman equation, Economics, ComputingMilieux_COMPUTERSANDSOCIETY, stochastic control, Asset (economics), 0101 mathematics, Portfolio optimization, Viscosity solution
Abstract

We consider a risky asset whose instantaneous rate of return takes two different values and changes from one to the other one at random times which are neither known, nor directly observable. We study the optimal allocation strategy of traders who, in the presence of cost of transactions, invest in this risky asset or in a non risky asset according to their belief on the current state of the instantaneous rate of return. We prove that the related trader’s value function is the unique viscosity solution of a system of HJB inequalities. We carefully prove the Dynamic Programming Principle and show results of numerical experiments.

Published
2009

18. Approximations of a Continuous Time Filter. Application to Optimal Allocation Problems in Finance

Author

Etienne Tanré, Sylvain Rubenthaler, Miguel Martinez, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), TOSCA, INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), and Université Nice Sophia Antipolis (1965 - 2019) (UNS)

Subjects
Statistics and Probability, Mathematical optimization, Stochastic process, Approximations of π, Applied Mathematics, Portfolios, 010102 general mathematics, Context (language use), Filter (signal processing), [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], 01 natural sciences, Applications in optimization, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Rate of convergence, Discrete time and continuous time, 60G35, 60J25, 62P05, 91G10, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Convergence (routing), Filtering problem, 0101 mathematics, Statistics, Probability and Uncertainty, Filtering, Mathematics
Abstract

International audience; In this article, we study a continuous time optimal filter and its various numerical approximations. This filter arises in an optimal allocation problem in the particular context of a non-stationary economy. We analyse the rates of convergence of the approximations of the filter when the model is misspecified and when the observations can only be made at discrete times. We give bounds that are uniform in time. Numerical results are presented.

Published
2009

19. Stochastic Spectral Formulations for Elliptic Problems

Author

Sylvain Maire, Etienne Tanré, TOSCA, Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria), Modélisation Numérique et Couplages (MNC), Université de Toulon (UTLN), L' Ecuyer, Pierre (ed.) and Owen, Art B. (ed.), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Approximations of π, Anisotropic diffusion, Mathematical analysis, Monte Carlo method, Basis function, 010103 numerical & computational mathematics, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Polynomial basis, Tensor product, Point (geometry), Numerical tests, 0101 mathematics, Mathematics
Abstract

We describe new stochastic spectral formulations with very good properties in terms of conditioning. These formulations are built by combining Monte Carlo approximations of the Feynman-Kac formula and standard deterministic approximations on basis functions. We give error bounds on the solutions obtained using these formulations in the case of linear approximations. Some numerical tests are made on an anisotropic diffusion equation using a tensor product Tchebychef polynomial basis and one random point schemes quantified or not.

Published
2009

20. Some new simulations schemes for the evaluation of Feynman–Kac representations

Author

Sylvain Maire, Etienne Tanré, TOSCA, Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Applied Mathematics, Quantum Monte Carlo, Monte Carlo method, Mathematical analysis, 010103 numerical & computational mathematics, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Hybrid Monte Carlo, 010104 statistics & probability, Dynamic Monte Carlo method, Applied mathematics, Monte Carlo integration, Monte Carlo method in statistical physics, Quasi-Monte Carlo method, 0101 mathematics, Mathematics, Monte Carlo molecular modeling
Abstract

International audience; We describe new variants of the Euler scheme and of the walk on spheres method for the Monte Carlo computation of Feynman-Kac representations. We optimize these variants using quantization for both source and boundary terms. Numerical tests are given on basic examples and on Monte Carlo versions of spectral methods for the Poisson equation. We especially introduce a new stochastic spectral formulation with very good properties in terms of conditioning.

Published
2008

21. Technical Analysis Compared to Mathematical Models Based Methods Under Parameters Mis-specification

Author

Christophette Blanchet-Scalliet, Awa Diop, Etienne Tanré, Denis Talay, Rajna Gibson, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), TOSCA, INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Sophia Antipolis - Méditerranée (CRISAM), Swiss Finance Institute [Geneva], Swiss Finance Institute, Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Economics and Econometrics, 050208 finance, Actuarial science, Mis specification, Logarithm, Mathematical model, 05 social sciences, Monte Carlo method, Stock return, Portfolio allocation, Stochastic models, Technical analysis, 0502 economics and business, Economics, Econometrics, Model specification, Trading strategy, 050207 economics, Finance, Stock (geology), Chartist
Abstract

International audience; In this study, we compare the performance of trading strategies based on possibly mis-specified mathematical models with a trading strategy based on a technical trading rule. In both cases, the trader attempts to predict a change in the drift of the stock return occurring at an unknown time.Weexplicitly compute the trader's expected logarithmic utility of wealth for the various trading strategies. We next rely on Monte Carlo numerical experiments to compare their performance. The simulations show that under parameter mis-specification, the technical analysis technique out-performs the optimal allocation strategy but not the Model and Detect strategies. The latter strategies dominance is confirmed under parameter mis-specification as long as the two stock returns' drifts are high in absolute terms.

Published
2007

22. Electricity Prices in a Game Theory Context

Author

Mireille Bossy, Nadia Maïzi, Geert Jan Olsder, Odile Pourtallier, Etienne Tanré, TOSCA, INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques Appliquées (CMA), MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Delft Institute of Applied Mathematics (TWA), Faculty of Electrical Engineering, Mathematics and Computer Science [Delft] (EEMCS)-Delft University of Technology (TU Delft), Constraints solving, optimization and robust interval analysis (COPRIN), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC), Alain Haurie, Georges Zaccour, Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria), Delft University of Technology (TU Delft)-Faculty of Electrical Engineering, Mathematics and Computer Science [Delft] (EEMCS), and Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Factor market, TheoryofComputation_MISCELLANEOUS, [INFO.INFO-GT]Computer Science [cs]/Computer Science and Game Theory [cs.GT], business.industry, Unit price, 020209 energy, 05 social sciences, [INFO.INFO-CE]Computer Science [cs]/Computational Engineering, Finance, and Science [cs.CE], Game, ComputerApplications_COMPUTERSINOTHERSYSTEMS, Context (language use), 02 engineering and technology, Microeconomics, Electricity prices, 0502 economics and business, Fixed price, 0202 electrical engineering, electronic engineering, information engineering, Economics, Electricity market, Electricity, 050207 economics, Market share, business, Game theory
Abstract

We consider a model of an electricity market in which S suppliers offer electricity: each supplier Si offers a maximum quantity qi at a fixed price pi. The response of the market to these offers is the quantities bought from the suppliers. The objective of the market is to satisfy its demand at minimal price. We investigate two cases. In the first case, each of the suppliers strives to maximize its market share on the market; in the second case each supplier strives to maximize its profit. We show that in both cases some Nash equilibrium exists. Nevertheless a close analysis of the equilibrium for profit maximization shows that it is not realistic. This raises the difficulty to predict the behavior of a market where the suppliers are known to be mainly interested by profit maximization.

Published
2005

23. Technical Analysis Techniques versus Mathematical Models: Boundaries of Their Validity Domains

Author

Christophette Blanchet-Scalliet, Awa Diop, Denis Talay, Etienne Tanré, Rajna Gibson, Tanré, Etienne, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Universität Zürich [Zürich] = University of Zurich (UZH), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Mathematical model, 05 social sciences, Monte Carlo method, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Moving average, Technical analysis, 0502 economics and business, Econometrics, Asset (economics), 0101 mathematics, Construct (philosophy), Instantaneous rate, Brownian motion, 050205 econometrics, Mathematics
Abstract

International audience; We aim to compare financial technical analysis techniques to strategies which depend on a mathematical model. In this paper, we consider the moving average indicator and an investor using a risky asset whose instantaneous rate of return changes at an unknown random time. We construct mathematical strategies. We compare their performances to technical analysis techniques when the model is misspecified. The comparisons are based on Monte Carlo simulations.

Published
2004

24. A pure jump Markov process associated with Smoluchowski's coagulation equation

Author

Etienne Tanré, Nicolas Fournier, Madalina Deaconu, TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Partial differential equation, Smoluchowski coagulation equation, Differential equation, Stochastic process, 010102 general mathematics, Mathematical analysis, Poisson measures, 01 natural sciences, Smoluchowski's coagulation equations, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Stochastic differential equation, symbols.namesake, Uniqueness theorem for Poisson's equation, 60K35, symbols, nonlinear stochastic differential equations, Fokker–Planck equation, 0101 mathematics, Statistics, Probability and Uncertainty, 60J75, 60H30, Jump process, Mathematics
Abstract

International audience; The aim of the present paper is to construct a stochastic process, whose law is the solution of the Smoluchowski's coagulation equation. We introduce first a modified equation, dealing with the evolution of the distribu-tion Q t (dx) of the mass in the system. The advantage we take on this is that we can perform an unified study for both continuous and discrete models. The integro-partial-differential equation satisfied by {Q t } t ≥0 can be interpreted as the evolution equation of the time marginals of a Markov pure jump process. At this end we introduce a nonlinear Poisson driven stochastic differential equation related to the Smoluchowski equation in the following way: if X t satisfies this stochastic equation, then the law of X t satisfies the modified Smoluchowski equation. The nonlinear process is richer than the Smoluchowski equation, since it provides historical information on the particles. Existence, uniqueness and pathwise behavior for the solution of this SDE are studied. Finally, we prove that the nonlinear process X can be obtained as the limit of a Marcus–Lushnikov procedure. 1. Introduction. The coagulation model governs various phenomena as for example: polymerization, aggregation of colloidal particles, formation of stars and planets, behavior of fuel mixtures in engines, etc. Smoluchowski's coagulation equation models the dynamic of such phenomena and describes the evolution of a system of clusters which coalesce in order to form bigger clusters. Each cluster is identified by its size. The only mechanism taken into account is the coalescence of two clusters, other effects as multiple coagulation are neglected. We assume also that the rate of these reactions depends on the sizes of clusters involved in the coagulation. Denoting by n(k, t) the (nonnegative) concentration of clusters of size k at time t, the discrete

Published
2002

25. A generalization of the Connection between the Additive and Multiplicative Solutions for the Smoluchowski's Coagulation Equation

Author

Madalina Deaconu, Etienne Tanré, INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
010302 applied physics, Statistics and Probability, Smoluchowski coagulation equation, Generalization, Applied Mathematics, Multiplicative function, Mathematical analysis, 01 natural sciences, Connection (mathematics), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, symbols.namesake, 0103 physical sciences, symbols, Coagulation (water treatment), Applied mathematics, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

International audience

26. Étude de la plasticité pour des neurones à décharge en interaction

Author

Pascal HELSON, Université Côte d'Azur (UCA), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Université Côte d'Azur, Etienne Tanré, Romain Veltz, Simuler et calibrer des modèles stochastiques (TOSCA-POST), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Plasticité Synaptique, Temps de Mémoire, [SCCO.NEUR]Cognitive science/Neuroscience, Recurrent Neural Network, Memory Lifetime, Champ Moyen, STDP, Mean Field, Processus de Markov Déterministes par Morceaux, Piecewise Deterministic Markov Process, Multi-échelle, [SDV.NEU]Life Sciences [q-bio]/Neurons and Cognition [q-bio.NC], Réseaux de Neurones Récurrents, [MATH]Mathematics [math], Spike Timing Dependent Plasticity, Multi-scale, Synaptic Plasticity
Abstract

In this thesis, we study a phenomenon that may be responsible for our memory capacity: the synaptic plasticity. It modifies the links between neurons over time. This phenomenon is stochastic: it is the result of a series of diverse and numerous chemical processes. The aim of the thesis is to propose a model of plasticity for interacting spiking neurons. The main difficulty is to find a model that satisfies the following conditions: it must be both consistent with the biological results of the field and simple enough to be studied mathematically and simulated with a large number of neurons.In a first step, from a rather simple model of plasticity, we study the learning of external signals by a neural network as well as the forgetting time of this signal when the network is subjected to other signals (noise). The mathematical analysis allows us to control the probability to misevaluate the signal. From this, we deduce explicit bounds on the time during which a given signal is kept in memory.Next, we propose a model based on stochastic rules of plasticity as a function of the occurrence time of the neural electrical discharges (Spike Timing Dependent Plasticity, STDP). This model is described by a Piecewise Deterministic Markov Process (PDMP). The long time behaviour of such a neural network is studied using a slow-fast analysis. In particular, sufficient conditions are established under which the process associated with synaptic weights is ergodic. Finally, we make the link between two levels of modelling: the microscopic and the macroscopic approaches. Starting from the dynamics presented at a microscopic level (neuron model and its interaction with other neurons), we derive an asymptotic dynamics which represents the evolution of a typical neuron and its incoming synaptic weights: this is the mean field analysis of the model. We thus condense the information on the dynamics of the weights and that of the neurons into a single equation, that of a typical neuron.; Dans cette thèse nous étudions un phénomène susceptible d’être responsable de notre capacité de mémoire: la plasticité synaptique. C’est le changement des liens entre les neurones au cours du temps. Ce phénomène est stochastique: c’est le résultat d’une suite de divers et nombreux mécanismes chimiques. Le but de la thèse est de proposer un modèle de plasticité pour des neurones à décharge en interaction. La principale difficulté consiste à trouver un modèle qui satisfait les conditions suivantes: ce modèle doit être à la fois cohérent avec les résultats biologiques dans le domaine et assez simple pour être étudié mathématiquement et simulé avec un grand nombre de neurones.Dans un premier temps, à partir d’un modèle assez simple de plasticité, on étudie l’apprentissage de signaux extérieurs par un réseau de neurones ainsi que le temps d’oubli de ce signal lorsque le réseau est soumis à d’autres signaux (bruit). L’analyse mathématique nous permet de contrôler la probabilité d’une mauvaise évaluation du signal. On en déduit un minorant du temps de mémoire du signal en fonction des paramètres.Ensuite, nous proposons un modèle basé sur des règles stochastiques de plasticité fonction du temps d’occurrence des décharges électriques neurales (STDP en anglais). Ce modèle est décrit par un Processus de Markov Déterministe par Morceaux (PDMP en anglais). On étudie le comportement en temps long d’un tel réseau de neurones grâce à une analyse lent-rapide. En particulier, on trouve des conditions suffisantes pour lesquelles le processus associé aux poids synaptiques est ergodique.Enfin, nous faisons le lien entre deux niveaux de modélisation: l’approche microscopique et celle macroscopique. À partir des dynamiques présentées d’un point de vu microscopique (modèle du neurone et son interaction avec les autres neurones), on détermine une dynamique limite qui représente l’évolution d’un neurone typique et de ses poids synaptiques entrant: c’est l’analyse champ moyen du modèle. On condense ainsi l’information sur la dynamique des poids et celle des neurones dans une seule équation, celle d’un neurone typique.

Published
2021

27. Long time behavior of a mean-field model of interacting spiking neurons

Author

Quentin Cormier, Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria), Simuler et calibrer des modèles stochastiques (TOSCA-POST), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Côte d'Azur, Etienne Tanré, and Romain Veltz

Subjects
Comportement en temps long, Equation Champ moyen, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], McKean-Vlasov stochastic processes, Processus stochastique de McKean-Vlasov, Mean-field interaction, Hopf bifurcation, Long time behavior, Bifurcation de Hopf, Volterra integral equation, Equation intégrale de Volterra
Abstract

We study the long time behavior of a McKean-Vlasov stochastic differential equation (SDE), driven by a Poisson measure. In neuroscience, this SDE models the dynamics of the membrane potential of a typical neuron in a large network. The model can be derived by considering a finite network of generalized Integrate-And-Fire neurons and by taking the limit where the number of neurons goes to infinity. Hence the McKean-Vlasov SDE is a mean-field model of spiking neurons.We study existence and uniqueness of the solution this McKean-Vlasov SDE and describe its invariant probability measures. For small enough interaction parameter J, we prove uniqueness and global stability of the invariant measure. For J arbitrary large however, the invariant measures may not be unique. We give a sufficient condition ensuring the local stability of such a given invariant probability measure. Our criterion involves the location of the zeros of an explicit holomorphic function associated to the considered stationary solution. When all the zeros have negative real part, we prove that stability holds. We then give sufficient general conditions ensuring the existence of periodic solutions through a Hopf bifurcation: at some critical interaction parameter J0, the invariant probability losses its stability and periodic solutions appear for J close to J0. To obtain these results, we combine probabilistic and deterministic methods. In particular, a key tool in this analysis is a nonlinear Volterra Integral equation satisfied by the synaptic current.Finally, we illustrate these results with examples which are tractable analytically. Additionally, we give numerical methods to approximate the solution of the mean-field equation and to predict numerically the bifurcations.; Nous étudions le comportement en temps long d'une équation différentielle stochastique (EDS) de type McKean-Vlasov, dirigée par une mesure de Poisson. En neurosciences, cette EDS modélise la dynamique du potentiel de membrane d'un neurone typique dans un grand réseau. Le modèle peut-être obtenu en considérant un réseau fini de neurones de type Intègre-Et-Tire généralisé et en prenant la limite où le nombre de neurones tend vers l'infini. Cette EDS est donc un modèle champ moyen de neurones à décharge.Nous étudions l'existence et l'unicité de la solution de cette EDS McKean-Vlasov et nous donnons ses mesures de probabilité invariantes. Si le paramètre d'interaction J est suffisamment petit, nous prouvons l'unicité et la stabilité globale de la mesure invariante. Pour un J quelconque cependant, il peut y avoir plusieurs mesures de probabilité invariantes. Nous donnons une condition suffisante assurant la stabilité locale d'une telle mesure invariante. Notre critère fait intervenir les zéros d'une fonction holomorphe associée à la solution stationnaire considérée. Lorsque tous les zéros sont de partie réelle négative, nous prouvons la stabilité. Nous donnons finalement des conditions générales suffisantes assurant l'existence de solutions périodiques par le biais d'une bifurcation de Hopf : pour un certain paramètre d'interaction critique J0, la probabilité invariante perd sa stabilité et des solutions périodiques apparaissent pour J suffisamment proche de J0. Pour obtenir ces résultats, nous combinons des méthodes probabilistes et déterministes. En particulier, dans cette analyse, un outil clé est l'équation intégrale de Volterra non linéaire satisfaite par le courant synaptique. Enfin, nous illustrons ces résultats par des exemples que l'on peut traiter de manière analytique. En outre, nous donnons des méthodes numériques pour approximer la solution de l'équation champ moyen et pour prédire numériquement les bifurcations.

Published
2021

28. Long time behavior of a mean-field model of interacting spiking neurons

Author

Cormier, Quentin, Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria), Simuler et calibrer des modèles stochastiques (TOSCA-POST), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Côte d'Azur, Etienne Tanré, and Romain Veltz

Subjects
Comportement en temps long, Equation Champ moyen, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], McKean-Vlasov stochastic processes, Processus stochastique de McKean-Vlasov, Mean-field interaction, Hopf bifurcation, Long time behavior, Bifurcation de Hopf, Volterra integral equation, Equation intégrale de Volterra
Abstract

We study the long time behavior of a McKean-Vlasov stochastic differential equation (SDE), driven by a Poisson measure. In neuroscience, this SDE models the dynamics of the membrane potential of a typical neuron in a large network. The model can be derived by considering a finite network of generalized Integrate-And-Fire neurons and by taking the limit where the number of neurons goes to infinity. Hence the McKean-Vlasov SDE is a mean-field model of spiking neurons.We study existence and uniqueness of the solution this McKean-Vlasov SDE and describe its invariant probability measures. For small enough interaction parameter J, we prove uniqueness and global stability of the invariant measure. For J arbitrary large however, the invariant measures may not be unique. We give a sufficient condition ensuring the local stability of such a given invariant probability measure. Our criterion involves the location of the zeros of an explicit holomorphic function associated to the considered stationary solution. When all the zeros have negative real part, we prove that stability holds. We then give sufficient general conditions ensuring the existence of periodic solutions through a Hopf bifurcation: at some critical interaction parameter J0, the invariant probability losses its stability and periodic solutions appear for J close to J0. To obtain these results, we combine probabilistic and deterministic methods. In particular, a key tool in this analysis is a nonlinear Volterra Integral equation satisfied by the synaptic current.Finally, we illustrate these results with examples which are tractable analytically. Additionally, we give numerical methods to approximate the solution of the mean-field equation and to predict numerically the bifurcations.; Nous étudions le comportement en temps long d'une équation différentielle stochastique (EDS) de type McKean-Vlasov, dirigée par une mesure de Poisson. En neurosciences, cette EDS modélise la dynamique du potentiel de membrane d'un neurone typique dans un grand réseau. Le modèle peut-être obtenu en considérant un réseau fini de neurones de type Intègre-Et-Tire généralisé et en prenant la limite où le nombre de neurones tend vers l'infini. Cette EDS est donc un modèle champ moyen de neurones à décharge.Nous étudions l'existence et l'unicité de la solution de cette EDS McKean-Vlasov et nous donnons ses mesures de probabilité invariantes. Si le paramètre d'interaction J est suffisamment petit, nous prouvons l'unicité et la stabilité globale de la mesure invariante. Pour un J quelconque cependant, il peut y avoir plusieurs mesures de probabilité invariantes. Nous donnons une condition suffisante assurant la stabilité locale d'une telle mesure invariante. Notre critère fait intervenir les zéros d'une fonction holomorphe associée à la solution stationnaire considérée. Lorsque tous les zéros sont de partie réelle négative, nous prouvons la stabilité. Nous donnons finalement des conditions générales suffisantes assurant l'existence de solutions périodiques par le biais d'une bifurcation de Hopf : pour un certain paramètre d'interaction critique J0, la probabilité invariante perd sa stabilité et des solutions périodiques apparaissent pour J suffisamment proche de J0. Pour obtenir ces résultats, nous combinons des méthodes probabilistes et déterministes. En particulier, dans cette analyse, un outil clé est l'équation intégrale de Volterra non linéaire satisfaite par le courant synaptique. Enfin, nous illustrons ces résultats par des exemples que l'on peut traiter de manière analytique. En outre, nous donnons des méthodes numériques pour approximer la solution de l'équation champ moyen et pour prédire numériquement les bifurcations.

Published
2021

29. Étude de l'équation limite d'un modèle de neurones en interactions

Author

Cormier, Quentin, École normale supérieure de Lyon (ENS de Lyon), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Inria - Sophia Antipolis, Etienne Tanré, Romain Veltz, and École normale supérieure - Lyon (ENS Lyon)

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2017

30. States Reduction on Markov Processes

Author

Papic, Alexis, TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ConicytEquipe associée Inria Anestoc, Inria Sophia Antipolis, Université Pierre et Marie Curie, and Etienne Tanré

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Abstract

International audience; A Markov process is a stochastic process that satisfies the Markov property, in which the future is independent of the past given the present. We first consider a Markov process over the real line with values on a finite set, where the law is defined by exponentially distributed jumps and a transition measure according to which the location of the process at the jump time is chosen; or indistinctly by the generator matrix. We also study Piecewise deterministic Markov processes, a more complex process that consists on two sub-processes: one on a continuous-space and the other on a discrete-space, and together are a Markov process involving a deterministic motion punctuated by random jumps. In the case when there are multiple weakly irreducible classes and the generator matrix can be rewritten as a double scales generator for a small parameter ε, we present a method to approximate the process to a two-scales process: a slow-process on a reduced state space and fast-process inside each new class, and we prove an approximation error of the laws of order ε. We present simulation examples and an application to the sodium channel in the Hodgkin and Huxley model, where we separate the voltage-gates of type h and m into two different time scales.

Published
2016

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